highertopoi

April 9, 2017

Annals of Mathematics Studies
Number 170

highertopoi

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highertopoi

April 9, 2017

Higher Topos Theory

Jacob Lurie

PRINCETON UNIVERSITY PRESSPRINCETON AND OXFORD
2009

highertopoi

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Copyright c 2009 by Princeton University Press
Published by Princeton University Press
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In the United Kingdom: Princeton University Press
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All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Lurie, Jacob, 1977Higher topos theory / Jacob Lurie.
p. cm. (Annals of mathematics studies ; no. 170)
Includes bibliographical references and index.
ISBN 978-0-691-14048-3 (hardcover : alk. paper) – ISBN 978-0-691-14049-0(pbk. : alk. paper) 1. Toposes. 2. Categories (Mathematics) I. Title.
QA169.L87 2009
512’.62–dc22
2008038170
British Library Cataloging-in-Publication Data is available
This book has been composed in Times in LATEXThe publisher would like to acknowledge the authors of this volume forproviding the camera-ready copy from which this book was printed.
Printed on acid-free paper. ∞press.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1highertopoi

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Contents

Prefacevii

Chapter 1. An Overview of Higher Category Theory
1.1
1.2

Foundations for Higher Category Theory
The Language of Higher Category Theory

Chapter 2. Fibrations of Simplicial Sets
2.1
2.2
2.3
2.4

Left Fibrations
Simplicial Categories and ∞-Categories
Inner Fibrations
Cartesian Fibrations

Chapter 3. The ∞-Category of ∞-Categories
3.1
3.2
3.3

Marked Simplicial Sets
Straightening and Unstraightening
Applications

Chapter 4. Limits and Colimits
4.1
4.2
4.3
4.4

Cofinality
Techniques for Computing Colimits
Kan Extensions
Examples of Colimits

1
1
26
53
55
72
95
114
144
146
168
204
222
222
239
260
292

Chapter 5. Presentable and Accessible ∞-Categories

314∞-Categories of Presheaves
Adjoint Functors∞-Categories of Inductive Limits
Accessible ∞-Categories
Presentable ∞-Categories

315
334
379
417
456

5.1
5.2
5.3
5.4
5.5

Chapter 6. ∞-Topoi
6.1
6.2
6.3
6.4
6.5

∞-Topoi: Definitions and Characterizations
Constructions of ∞-Topoi
The ∞-Category of ∞-Topoi
n-Topoi
Homotopy Theory in an ∞-Topos

Chapter 7. Higher Topos Theory in Topology

527
528
570
600
638
657
688

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CONTENTS

7.1
7.2
7.3

Paracompact Spaces
Dimension Theory
The Proper Base Change Theorem

Appendix.
A.1 Category Theory
A.2 Model Categories
A.3 Simplicial Categories

689
717
749
787
787
809
850

Bibliography

915

General Index

921

Index of Notation

929

highertopoi

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Preface

Let X be a nice topological space (for example, a CW complex). One goalof algebraic topology is to study the topology of X by means of algebraic
invariants, such as the singular cohomology groups Hn (X; G) of X with coefficients in an abelian group G. These cohomology groups have proven to be
an extremely useful tool largely because they enjoy excellent formal properties (which have been axiomatized by Eilenberg and Steenrod, see [26]) andbecause they tend to be very computable. However, the usual definition ofHn (X; G) in terms of singular G-valued cochains on X is perhaps somewhat
unenlightening. This raises the following question: can we understand thecohomology group Hn (X; G) in more conceptual terms?
As a first step toward answering this question, we observe that Hn (X; G)
is a representable functor of X. That is, there exists an Eilenberg-MacLane
space K(G, n) and a universal cohomology class η ∈ Hn (K(G, n); G) suchthat, for any nice topological space X, pullback of η determines a bijection[X, K(G, n)] → Hn (X; G).
Here [X, K(G, n)] denotes the set of homotopy classes of maps from X toK(G, n). The space K(G, n) can be characterized up to homotopy equivalence by the above property or by the formula(∗ if k 6= nπk K(G, n) 'G if k = n.
In the case n = 1, we can be more concrete. An Eilenberg-MacLane space
K(G, 1) is called a classifying space for G and is typically denoted by BG.
The universal cover of BG is a contractible space EG, which carries a free
action of the group G by covering transformations. We have a quotient map
π : EG → BG. Each fiber of π is a discrete topological space on which thegroup G acts simply transitively. We can summarize the situation by sayingthat EG is a G-torsor over the classifying space BG. For every continuous
e : EG ×BG X has the structure of
map X → BG, the fiber product X
a G-torsor on X: that is, it is a space endowed with a free action of Ge
and a homeomorphism X/G' X. This construction determines a map
from [X, BG] to the set of isomorphism classes of G-torsors on X. If X is asufficiently well-behaved space (such as a CW complex), then this map is abijection. We therefore have (at least) three different ways of thinking abouta cohomology class η ∈ H1 (X; G):highertopoi

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PREFACE(1) As a G-valued singular cocycle on X, which is well-defined up to
coboundaries.
(2) As a continuous map X → BG, which is well-defined up to homotopy.
(3) As a G-torsor on X, which is well-defined up to isomorphism.
These three points of view are equivalent if the space X is sufficiently nice.
However, they are generally quite different from one another. The singularcohomology of a space X is constructed using continuous maps from simplices ∆k into X. If there are not many maps into X (for example, if every
path in X is constant), then we cannot expect singular cohomology to tell
us very much about X. The second definition uses maps from X into theclassifying space BG, which (ultimately) relies on the existence of continuous
real-valued functions on X. If X does not admit many real-valued functions,then the set of homotopy classes [X, BG] is also not a very useful invariant.
For such spaces, the third approach is the most powerful: there is a goodtheory of G-torsors on an arbitrary topological space X.
There is another reason for thinking about H1 (X; G) in the language ofG-torsors: it continues to make sense in situations where the traditional ideas
e is a G-torsor on a topological space X, then theof topology break down. If X
e → X is a local homeomorphism; we may therefore identify
projection map X
e with a sheaf of sets F on X. The action of G on X
e determines an action ofXe (with its G-action)
G on F. The sheaf F (with its G-action) and the space X
determine each other up to canonical isomorphism. Consequently, we canformulate the definition of a G-torsor in terms of the category ShvSet (X) of
sheaves of sets on X without ever mentioning the topological space X itself.
The same definition makes sense in any category which bears a sufficiently
strong resemblance to the category of sheaves on a topological space: forexample, in any Grothendieck topos. This observation allows us to construct
a theory of torsors in a variety of nonstandard contexts, such as the étale
topology of algebraic varieties (see [2]).
Describing the cohomology of X in terms of the sheaf theory of X has still
another advantage, which comes into play even when the space X is assumed
to be a CW complex. For a general space X, isomorphism classes of G-torsors
on X are classified not by the singular cohomology H1sing (X; G) but by thesheaf cohomology H1sheaf (X; G) of X with coefficients in the constant sheaf
G associated to G. This sheaf cohomology is defined more generally for any
sheaf of groups G on X. Moreover, we have a conceptual interpretation of
H1sheaf (X; G) in general: it classifies G-torsors on X (that is, sheaves F on X
which carry an action of G and locally admit a G-equivariant isomorphism
F ' G) up to isomorphism. The general formalism of sheaf cohomology isextremely useful, even if we are interested only in the case where X is a nicetopological space: it includes, for example, the theory of cohomology withcoefficients in a local system on X.
Let us now attempt to obtain a similar interpretation for cohomology
classes η ∈ H2 (X; G). What should play the role of a G-torsor in this case?

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To answer this question, we return to the situation where X is a CW complex, so that η can be identified with a continuous map X → K(G, 2).
We can think of K(G, 2) as the classifying space of a group: not the discrete group G but instead the classifying space BG (which, if built in a
sufficiently careful way, comes equipped with the structure of a topologicalabelian group). Namely, we can identify K(G, 2) with the quotient E/BG,
where E is a contractible space with a free action of BG. Any cohomology
class η ∈ H2 (X; G) determines a map X → K(G, 2) (which is well-defined
e = E ×BG X. We now
up to homotopy), and we can form the pullback X
e as a torsor over X: not for the discrete group G but instead forthink of X
its classifying space BG.
To complete the analogy with our analysis in the case n = 1, we woulde → X as defining some kind of sheaf F on
like to interpret the fibration X
the space X. This sheaf F should have the property that for each x ∈ X, theex ' BG. Since the space BG is notstalk Fx can be identified with the fiber X
discrete (or homotopy equivalent to a discrete space), the situation cannot beadequately described in the usual language of set-valued sheaves. However,the classifying space BG is almost discrete: since the homotopy groups πi BGvanish for i > 1, we can recover BG (up to homotopy equivalence) from its
fundamental groupoid. This suggests that we might try to think about F asa “groupoid-valued sheaf” on X, or a stack (in groupoids) on X.
Remark. The condition that each stalk Fx be equivalent to a classifying
space BG can be summarized by saying that F is a gerbe on X: more precisely, it is a gerbe banded by the constant sheaf G associated to G. We refer
the reader to [31] for an explanation of this terminology and a proof thatsuch gerbes are indeed classified by the sheaf cohomology group H2sheaf (X; G).
For larger values of n, even the language of stacks is not sufficient to dee → X. To adscribe the nature of the sheaf F associated to the fibration X
dress the situation, Grothendieck proposed (in his infamous letter to Quillen;see [35]) that there should be a theory of n-stacks on X for every integern ≥ 0. Moreover, for every sheaf of abelian groups G on X, the cohomology group Hn+1
sheaf (X; G) should have an interpretation as classifying a special
type of n-stack: namely, the class of n-gerbes banded by G (for a discussion in the case n = 2, we refer the reader to [13]; we will treat the generalcase in §7.2.2). In the special case where the space X is a point (and where
we restrict our attention to n-stacks in groupoids), the theory of n-stacks
on X should recover the classical homotopy theory of n-types: that is, CW
complexes Z such that the homotopy groups πi (Z, z) vanish for i > n (andevery base point z ∈ Z). More generally, we should think of an n-stack (ingroupoids) on a general space X as a “sheaf of n-types” on X.
When n = 0, an n-stack on a topological space X is simply a sheaf of sets
on X. The collection of all such sheaves can be organized into a category
ShvSet (X), and this category is a prototypical example of a Grothendiecktopos. The main goal of this book is to obtain an analogous understanding

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PREFACEof the situation for n > 0. More precisely, we would like answers to thefollowing questions:(Q1) Given a topological space X, what should we mean by a “sheaf ofn-types” on X?
(Q2) Let Shv≤n (X) denote the collection of all sheaves of n-types on X.
What sort of a mathematical object is Shv≤n (X)?
(Q3) What special features (if any) does Shv≤n (X) possess?
Our answers to questions (Q2) and (Q3) may be summarized as follows(our answer to (Q1) is more elaborate, and we will avoid discussing it forthe moment):(A2) The collection Shv≤n (X) has the structure of an ∞-category.
(A3) The ∞-category Shv≤n (X) is an example of an (n + 1)-topos: that is,
an ∞-category which satisfies higher-categorical analogues of Giraud’s
axioms for Grothendieck topoi (see Theorem 6.4.1.5).
Remark. Grothendieck’s vision has been realized in various ways thanks to
the work of a number of mathematicians (most notably Brown, Joyal, andJardine; see for example [41]), and their work can also be used to provide
answers to questions (Q1) and (Q2) (for more details, we refer the reader to§6.5.2). Question (Q3) has also been addressed (at least in the limiting casen = ∞) by Toën and Vezzosi (see [79]) and in unpublished work of Rezk.
To provide more complete versions of the answers (A2) and (A3), we
will need to develop the language of higher category theory. This is generally regarded as a technical and forbidding subject, but fortunately we will
only need a small fragment of it. More precisely, we will need a theory of(∞, 1)-categories: higher categories C for which the k-morphisms of C are
required to be invertible for k > 1. In Chapter 1, we will present such a theory: namely, one can define an ∞-category to be a simplicial set satisfying
a weakened version of the Kan extension condition (see Definition 1.1.2.4;
simplicial sets satisfying this condition are also called weak Kan complexesor quasi-categories in the literature). Our intention is that Chapter 1 canbe used as a short “user’s guide” to ∞-categories: it contains many of thebasic definitions and explains how many ideas from classical category theory
can be extended to the ∞-categorical context. To simplify the exposition,
we have deferred many proofs until later chapters, which contain a morethorough account of the theory. The hope is that Chapter 1 will be useful to readers who want to get the flavor of the subject without becoming
overwhelmed by technical details.
In Chapter 2 we will shift our focus slightly: rather than study individual
examples of ∞-categories, we consider families of ∞-categories {CD }D∈D
parametrized by the objects of another ∞-category D. We might expecthighertopoi

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such a family to be given by a map of ∞-categories p : C → D: given sucha map, we can then define each CD to be the fiber product C ×D {D}. Thisdefinition behaves poorly in general (for example, the fibers CD need not
be ∞-categories), but it behaves well if we make suitable assumptions on
the map p. Our goal in Chapter 2 is to study some of these assumptions
in detail and to show that they lead to a good relative version of higher
category theory.
One motivation for the theory of ∞-categories is that it arises naturally
in addressing questions like (Q2) above. More precisely, given a collection ofmathematical objects {Fα } whose definition has a homotopy-theoretic flavor(like n-stacks on a topological space X), one can often organize the collection{Fα } into an ∞-category (in other words, there exists an ∞-category C
whose vertices correspond to the objects Fα ). Another important example isprovided by higher category theory itself: the collection of all ∞-categories
can itself be organized into a (very large) ∞-category, which we will denoteby Cat∞ . Our goal in Chapter 3 is to study Cat∞ and to show that it
can be characterized by a universal property: namely, functors χ : D →Cat∞ are classified (up to equivalence) by certain kinds of fibrations C → D(see Theorem 3.2.0.1 for a more precise statement). Roughly speaking, thiscorrespondence assigns to a fibration C → D the functor χ given by theformula χ(D) = C ×D {D}.
Classically, category theory is a useful tool not so much because of the lightit sheds on any particular mathematical discipline but instead because categories are so ubiquitous: mathematical objects in many different settings(sets, groups, smooth manifolds, and so on) can be organized into categories. Moreover, many elementary mathematical concepts can be describedin purely categorical terms and therefore make sense in each of these settings.
For example, we can form products of sets, groups, and smooth manifolds:
each of these notions can simply be described as a Cartesian product in therelevant category. Cartesian products are a special case of the more general
notion of limit, which plays a central role in classical category theory. In
Chapter 4, we will make a systematic study of limits (and the dual theory of
colimits) in the ∞-categorical setting. We will also introduce the more general theory of Kan extensions, in both absolute and relative versions; thistheory plays a key technical role throughout the later parts of the book.
In some sense, the material of Chapters 1 through 4 of this book should be
regarded as purely formal. Our main results can be summarized as follows:
there exists a reasonable theory of ∞-categories, and it behaves in more orless the same way as the theory of ordinary categories. Many of the ideasthat we introduce are straightforward generalizations of their ordinary counterparts (though proofs in the ∞-categorical setting often require a bit ofdexterity in manipulating simplicial sets), which will be familiar to mathematicians who are acquainted with ordinary category theory (as presented,
for example, in [53]). In Chapter 5, we introduce ∞-categorical analoguesof more sophisticated concepts from classical category theory: presheaves,
Pro-categories and Ind-categories, accessible and presentable categories, andhighertopoi

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PREFACE

localizations. The main theme is that most of the ∞-categories which appear“in nature” are large but are nevertheless determined by small subcategories.
Taking careful advantage of this fact will allow us to deduce a number ofpleasant results, such as an ∞-categorical version of the adjoint functor theorem (Corollary 5.5.2.9).
In Chapter 6 we come to the heart of the book: the study of ∞-topoi, the∞-categorical analogues of Grothendieck topoi. The theory of ∞-topoi is ouranswer to the question (Q3) in the limiting case n = ∞ (we will also study
the analogous notion for finite values of n). Our main result is an analogue ofGiraud’s theorem, which asserts the equivalence of “extrinsic” and “intrinsic”
approaches to the subject (Theorem 6.1.0.6). Roughly speaking, an ∞-topos
is an ∞-category which “looks like” the ∞-category of all homotopy types.
We will show that this intuition is justified in the sense that it is possible toreconstruct a large portion of classical homotopy theory inside an arbitrary∞-topos. In other words, an ∞-topos is a world in which one can “do”
homotopy theory (much as an ordinary topos can be regarded as a world inwhich one can “do” other types of mathematics).
In Chapter 7 we will discuss some relationships between our theory of∞-topoi and ideas from classical topology. We will show that, if X is a paracompact space, then the ∞-topos of “sheaves of homotopy types” on X canbe interpreted in terms of the classical homotopy theory of spaces over X.
Another main theme is that various ideas from geometric topology (suchas dimension theory and shape theory) can be described naturally using thelanguage of ∞-topoi. We will also formulate and prove “nonabelian” generalizations of classical cohomological results, such as Grothendieck’s vanishingtheorem for the cohomology of Noetherian topological spaces and the properbase change theorem.
Prerequisites and Suggested Reading
We have made an effort to keep this book as self-contained as possible.
The main prerequisite is familiarity with the classical homotopy theory ofsimplicial sets (good references include [57] and [32]; we have also provided
a very brief review in §A.2.7). The remaining material that we need is either
described in the appendix or developed in the body of the text. However,
our exposition of this background material is often somewhat terse, and thereader might benefit from consulting other treatments of the same ideas.
Some suggestions for further reading are listed below.
Warning. The list of references below is woefully incomplete. We have notattempted, either here or in the body of the text, to give a comprehensive
survey of the literature on higher category theory. We have also not attempted to trace all of the ideas presented to their origins or to present a
detailed history of the subject. Many of the topics presented in this bookhave appeared elsewhere or belong to the mathematical folklore; it should
not be assumed that uncredited results are due to the author.

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• Classical Category Theory: Large portions of this book are devoted to providing ∞-categorical generalizations of the basic notionsof category theory. A good reference for many of the concepts we use
is MacLane’s book [53] (see also [1] and [55] for some of the moreadvanced material of Chapter 5).
• Classical Topos Theory: Our main goal in this book is to describe an∞-categorical version of topos theory. Familiarity with classical topos
theory is not strictly necessary (we will define all of the relevant concepts as we need them) but will certainly be helpful. Good references
include [2] and [54].
• Model Categories: Quillen’s theory of model categories provides auseful tool for studying specific examples of ∞-categories, including thetheory of ∞-categories itself. We will summarize the theory of model
categories in §A.2; more complete references include [40], [38], and [32].
• Higher Category Theory: There are many approaches to the theoryof higher categories, some of which look quite different from the approach presented in this book. Several other possibilities are presented
in the survey article [48]. More detailed accounts can be found in [49],[72], and [76].
In this book, we consider only (∞, 1)-categories: that is, higher categories in which all k-morphisms are assumed to be invertible for k > 1.
There are a number of convenient ways to formalize this idea: via
simplicial categories (see, for example, [21] and [7]), via Segal categories ([72]), via complete Segal spaces ([65]), or via the theory of ∞categories presented in this book (other references include [43], [44],[61], and [10]). The relationship between these various approaches is
described in [8], and an axiomatic framework which encompasses all ofthem is described in [77].
• Higher Topos Theory: The idea of studying a topological space X
via the theory of sheaves of n-types (or n-stacks) on X goes back at
least to Grothendieck ([35]) and has been taken up a number of times
in recent years. For small values of n, we refer the reader to [31], [75],[13], [45], and [62]. For the case n = ∞, we refer the reader to [14],[41], [39], and [78]. A very readable introduction to some of these ideas
can be found in [4].
Higher topos theory itself can be considered an abstraction of thisidea: rather than studying sheaves of n-types on a particular topological space X, we instead study general n-categories with the sameformal properties. This idea has been implemented in the work of Toënand Vezzosi (see [79] and [80]), resulting in a theory which is essentially equivalent to the one presented in this book. (A rather different
variation on this idea in the case n = 2 can also be found in [11].) The

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PREFACEsubject has also been greatly influenced by the unpublished ideas ofCharles Rezk.

TERMINOLOGY
Here are a few comments on some of the terminology which appears in thisbook:
• The word topos will always mean Grothendieck topos.
• We let Set∆ denote the category of simplicial sets. If J is a linearlyordered set, we let ∆J denote the simplicial set given by the nerve ofJ, so that the collection of n-simplices of ∆J can be identified with thecollection of all nondecreasing maps {0, . . . , n} → J. We will frequently
apply this notation when J is a subset of {0, . . . , n}; in this case, we
can identify ∆J with a subsimplex of the standard n-simplex ∆n (atleast if J 6= ∅; if J = ∅, then ∆J is empty).
• We will refer to a category C as presentable if it is locally presentablein the terminology of [55].
• Unless otherwise specified, the term ∞-category will be used to indicate
a higher category in which all n-morphisms are invertible for n > 1.
• We will study higher categories using Joyal’s theory of quasi-categories.
However, we do not always follow Joyal’s terminology. In particular,
we will use the term ∞-category to refer to what Joyal calls a quasicategory (which are, in turn, the same as the weak Kan complex ofBoardman and Vogt); we will use the terms inner fibration and inner
anodyne map where Joyal uses mid-fibration and mid-anodyne map.
• Let n ≥ 0. We will say that a homotopy type X (described by either
a topological space or a Kan complex) is n-truncated if the homotopy groups πi (X, x) vanish for every point x ∈ X and every i > n.
By convention, we say that X is (−1)-truncated if it is either emptyor (weakly) contractible, and (−2)-truncated if X is (weakly) contractible.
• Let n ≥ 0. We will say that a homotopy type X (described either by atopological space or a Kan complex) is n-connective if X is nonempty
and the homotopy groups πi (X, x) vanish for every point x ∈ X andevery integer i < n. By convention, we will agree that every homotopy
type X is (−1)-connective.
• More generally, we will say that a map of homotopy types f : X → Y isn-truncated (n-connective) if the homotopy fibers of f are n-truncated(n-connective).

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Remark. For n ≥ 1, a homotopy type X is n-connective if and only if it is(n−1)-connected (in the usual terminology). In particular, X is 1-connective
if and only if it is path-connected.
Warning. In this book, we will often be concerned with sheaves on a topological space X (or some Grothendieck site) taking values in an ∞-category
C. The most “universal” case is that in which C is the ∞-category S of spaces.
Consequently, the term “sheaf on X” without any other qualifiers will typically refer to a sheaf of spaces on X rather than a sheaf of sets on X. We
will see that the collection of all S-valued sheaves on X can be organized
into an ∞-category, which we denote by Shv(X). In particular, Shv(X) will
not denote the ordinary category of set-valued sheaves on X; if we need toconsider this latter object, we will denote it by ShvSet (X).

ACKNOWLEDGEMENTS
This book would never have come into existence without the advice and encouragement of many people. In particular, I would like to thank Vigleik Angeltveit, Ben Antieau, Clark Barwick, Lukas Brantner, Kestutis Cesnavicius,
Rex Cheung, Vladimir Drinfeld, Dustin Clausen, Pierre Deligne, Gal Dor,
Matt Emerton, John Francis, Daniel Gerigk, Harry Gindi, Moritz Groth,Andre Henriques, Marc Hoyois, Viktor Kleen, Edoardo Lanari, Avi Levy,
Sergey Lysenko, Akhil Mathew, Nori Minami, Yogesh More, James Parson,
Luis Alexandre Pereira, Anatoly Preygel, Steven Sam, Elena Sendroiu, David
Spivak, Herman Stel, James Wallbridge, and Samuel Wüthrich for many suggestions and corrections which have improved the readability of this book;
Andre Joyal, who was kind enough to share with me a preliminary versionof his work on the theory of quasi-categories; Charles Rezk, for explaining
to me a very conceptual reformulation of the axioms for ∞-topoi (which
we will describe in §6.1.3); Bertrand Toën and Gabriele Vezzosi, for many
stimulating conversations about their work (which has considerable overlap
with the material treated here); Mike Hopkins, for his advice and support
throughout my time as a graduate student; Max Lieblich, for offering encouragement during early stages of this project; Josh Nichols-Barrer, for sharing
with me some of his ideas about the foundations of higher category theory. I
would also like to thank my copyeditor Carol Dean and my editors Kathleen
Cioffi, Vickie Kearn, and Anna Pierrehumbert at Princeton University Pressfor helping to make this the best book that it can be. Finally, I would like tothank the American Institute of Mathematics for supporting me throughout
the (seemingly endless) process of writing and revising this work.

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Higher Topos Theory

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highertopoi

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Chapter One
An Overview of Higher Category Theory
This chapter is intended as a general introduction to higher category theory.
We begin with what we feel is the most intuitive approach to the subject
using topological categories. This approach is easy to understand but difficult
to work with when one wishes to perform even simple categorical constructions. As a remedy, we will introduce the more suitable formalism of ∞categories (called weak Kan complexes in [10] and quasi-categories in [43]),
which provides a more convenient setting for adaptations of sophisticatedcategory-theoretic ideas. Our goal in §1.1.1 is to introduce both approaches
and to explain why they are equivalent to one another. The proof of thisequivalence will rely on a crucial result (Theorem 1.1.5.13) which we will
prove in §2.2.
Our second objective in this chapter is to give the reader an idea of how towork with the formalism of ∞-categories. In §1.2, we will establish a vocabulary which includes ∞-categorical analogues (often direct generalizations)of most of the important concepts from ordinary category theory. To keep
the exposition brisk, we will postpone the more difficult proofs until later
chapters of this book. Our hope is that, after reading this chapter, a reader
who does not wish to be burdened with the details will be able to understand(at least in outline) some of the more conceptual ideas described in Chapter
5 and beyond.

1.1 FOUNDATIONS FOR HIGHER CATEGORY THEORY
1.1.1 Goals and Obstacles
Recall that a category C consists of the following data:(1) A collection {X, Y, Z, . . .} whose members are the objects of C. We
typically write X ∈ C to indicate that X is an object of C.
(2) For every pair of objects X, Y ∈ C, a set HomC (X, Y ) of morphisms
from X to Y . We will typically write f : X → Y to indicate thatf ∈ HomC (X, Y ) and say that f is a morphism from X to Y .
(3) For every object X ∈ C, an identity morphism idX ∈ HomC (X, X).
(4) For every triple of objects X, Y, Z ∈ C, a composition map
HomC (X, Y ) × HomC (Y, Z) → HomC (X, Z).

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Given morphisms f : X → Y and g : Y → Z, we will usually denotethe image of the pair (f, g) under the composition map by gf or g ◦ f .
These data are furthermore required to satisfy the following conditions,which guarantee that composition is unital and associative:(5) For every morphism f : X → Y , we have idY ◦f = f = f ◦ idX in
HomC (X, Y ).
(6) For every triple of composable morphismsfgh

W → X → Y → Z,
we have an equality h ◦ (g ◦ f ) = (h ◦ g) ◦ f in HomC (W, Z).
The theory of categories has proven to be a valuable organization tool inmany areas of mathematics. Mathematical structures of virtually any typecan be viewed as the objects of a suitable category C, where the morphisms
in C are given by structure-preserving maps. There is a veritable legion of
examples of categories which fit this paradigm:• The category Set whose objects are sets and whose morphisms aremaps of sets.
• The category Grp whose objects are groups and whose morphisms aregroup homomorphisms.
• The category Top whose objects are topological spaces and whose morphisms are continuous maps.
• The category Cat whose objects are (small) categories and whose morphisms are functors. (Recall that a functor F from C to D is a map
which assigns to each object C ∈ C another object F C ∈ D, and toeach morphism f : C → C 0 in C a morphism F (f ) : F C → F C 0 in D,
so that F (idC ) = idF C and F (g ◦ f ) = F (g) ◦ F (f ).)• ···In general, the existence of a morphism f : X → Y in a category C
reflects some relationship that exists between the objects X, Y ∈ C. In somecontexts, these relationships themselves become basic objects of study andcan be fruitfully organized into categories:
Example 1.1.1.1. Let Grp be the category whose objects are groups andwhose morphisms are group homomorphisms. In the theory of groups, one
is often concerned only with group homomorphisms up to conjugacy. The
relation of conjugacy can be encoded as follows: for every pair of groups
G, H ∈ Grp, there is a category Map(G, H) whose objects are group homomorphisms from G to H (that is, elements of HomGrp (G, H)), where amorphism from f : G → H to f 0 : G → H is an element h ∈ H suchthat hf (g)h−1 = f 0 (g) for all g ∈ G. Note that two group homomorphisms
f, f 0 : G → H are conjugate if and only if they are isomorphic when viewed
as objects of Map(G, H).

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Example 1.1.1.2. Let X and Y be topological spaces and let f0 , f1 : X →Y be continuous maps. Recall that a homotopy from f0 to f1 is a continuous
map f : X × [0, 1] → Y such that f |X × {0} coincides with f0 and f |X × {1}coincides with f1 . In algebraic topology, one is often concerned not withthe category Top of topological spaces but with its homotopy category: thatis, the category obtained by identifying those pairs of morphisms f0 , f1 :X → Y which are homotopic to one another. For many purposes, it is betterto do something a little bit more sophisticated: namely, one can form acategory Map(X, Y ) whose objects are continuous maps f : X → Y andwhose morphisms are given by (homotopy classes of) homotopies.
Example 1.1.1.3. Given a pair of categories C and D, the collection of all
functors from C to D is itself naturally organized into a category Fun(C, D),
where the morphisms are given by natural transformations. (Recall that,given a pair of functors F, G : C → D, a natural transformation α : F → G
is a collection of morphisms {αC : F (C) → G(C)}C∈C which satisfy thefollowing condition: for every morphism f : C → C 0 in C, the diagram
F (C)
αC


G(C)

F (f )/ F (C 0 )
αC 0


G(f )
/ G(C 0 )commutes in D.)
In each of these examples, the objects of interest can naturally be organized into what is called a 2-category (or bicategory): we have not only acollection of objects and a notion of morphisms between objects but also
a notion of morphisms between morphisms, which are called 2-morphisms.
The vision of higher category theory is that there should exist a good notionof n-category for all n ≥ 0 in which we have not only objects, morphisms,
and 2-morphisms but also k-morphisms for all k ≤ n. Finally, in some sortof limit we might hope to obtain a theory of ∞-categories, where there aremorphisms of all orders.
Example 1.1.1.4. Let X be a topological space and 0 ≤ n ≤ ∞. We canextract an n-category π≤n X (roughly) as follows. The objects of π≤n X are
the points of X. If x, y ∈ X, then the morphisms from x to y in π≤n X
are given by continuous paths [0, 1] → X starting at x and ending at y.
The 2-morphisms are given by homotopies of paths, the 3-morphisms byhomotopies between homotopies, and so forth. Finally, if n < ∞, then two
n-morphisms of π≤n X are considered to be the same if and only if they arehomotopic to one another.
If n = 0, then π≤n X can be identified with the set π0 X of path componentsof X. If n = 1, then our definition of π≤n X agrees with the usual definition
for the fundamental groupoid of X. For this reason, π≤n X is often called thefundamental n-groupoid of X. It is called an n-groupoid (rather than a merehighertopoi

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n-category) because every k-morphism of π≤n X has an inverse (at least up
to homotopy).
There are many approaches to realizing the theory of higher categories. We
might begin by defining a 2-category to be a “category enriched over Cat.”In other words, we consider a collection of objects together with a categoryof morphisms Hom(A, B) for any two objects A and B and composition
functors cABC : Hom(A, B) × Hom(B, C) → Hom(A, C) (to simplify the discussion, we will ignore identity morphisms for a moment). These functors are
required to satisfy an associative law, which asserts that for any quadruple(A, B, C, D) of objects, the diagram
Hom(A, B) × Hom(B, C) × Hom(C, D)/ Hom(A, C) × Hom(C, D)
Hom(A, B) × Hom(B, D)/ Hom(A, D)

commutes; in other words, one has an equality of functors
cACD ◦ (cABC × 1) = cABD ◦ (1 × cBCD )from Hom(A, B) × Hom(B, C) × Hom(C, D) to Hom(A, D). This leads tothe definition of a strict 2-category.
At this point, we should object that the definition of a strict 2-category
violates one of the basic philosophical principles of category theory: one
should never demand that two functors F and F 0 be equal to one another.
Instead one should postulate the existence of a natural isomorphism between F and F 0 . This means that the associative law should not take theform of an equation but of additional structure: a collection of isomorphisms
γABCD : cACD ◦ (cABC × 1) ' cABD ◦ (1 × cBCD ). We should further demandthat the isomorphisms γABCD be functorial in the quadruple (A, B, C, D)
and satisfy certain higher associativity conditions, which generalize the “Pentagon axiom” described in §A.1.3. After formulating the appropriate conditions, we arrive at the definition of a weak 2-category.
Let us contrast the notions of strict 2-category and weak 2-category. The
former is easier to define because we do not have to worry about the higher
associativity conditions satisfied by the transformations γABCD . On theother hand, the latter notion seems more natural if we take the philosophy of category theory seriously. In this case, we happen to be lucky: thenotions of strict 2-category and weak 2-category turn out to be equivalent.
More precisely, any weak 2-category is equivalent (in the relevant sense) toa strict 2-category. The choice of definition can therefore be regarded as aquestion of aesthetics.
We now plunge onward to 3-categories. Following the above program, wemight define a strict 3-category to consist of a collection of objects together
with strict 2-categories Hom(A, B) for any pair of objects A and B, together with a strictly associative composition law. Alternatively, we could
seek a definition of weak 3-category by allowing Hom(A, B) to be a weak

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52-category, requiring associativity only up to natural 2-isomorphisms, which
satisfy higher associativity laws up to natural 3-isomorphisms, which in turnsatisfy still higher associativity laws of their own. Unfortunately, it turns outthat these notions are not equivalent.
Both of these approaches have serious drawbacks. The obvious problem
with weak 3-categories is that an explicit definition is extremely complicated(see [33], where a definition is given along these lines), to the point where
it is essentially unusable. On the other hand, strict 3-categories have theproblem of not being the correct notion: most of the weak 3-categories which
occur in nature are not equivalent to strict 3-categories. For example, thefundamental 3-groupoid of the 2-sphere S 2 cannot be described using thelanguage of strict 3-categories. The situation only gets worse (from either
point of view) as we pass to 4-categories and beyond.
Fortunately, it turns out that major simplifications can be introduced if
we are willing to restrict our attention to ∞-categories in which most of
the higher morphisms are invertible. From this point forward, we will use
the term (∞, n)-category to refer to ∞-categories in which all k-morphisms
are invertible for k > n. The ∞-categories described in Example 1.1.1.4(when n = ∞) are all (∞, 0)-categories. The converse, which asserts thatevery (∞, 0)-category has the form π≤∞ X for some topological space X,
is a generally accepted principle of higher category theory. Moreover, the∞-groupoid π≤∞ X encodes the entire homotopy type of X. In other words,(∞, 0)-categories (that is, ∞-categories in which all morphisms are invertible) have been extensively studied from another point of view: they are
essentially the same thing as “spaces” in the sense of homotopy theory, andthere are many equivalent ways to describe them (for example, we can use
CW complexes or simplicial sets).
Convention 1.1.1.5. We will sometimes refer to (∞, 0)-categories as ∞groupoids and (∞, 2)-categories as ∞-bicategories. Unless we specify otherwise, the generic term “∞-category” will refer to an (∞, 1)-category.
In this book, we will restrict our attention almost entirely to the theoryof ∞-categories (in which we have only invertible n-morphisms for n ≥ 2).
Our reasons are threefold:(1) Allowing noninvertible n-morphisms for n > 1 introduces a numberof additional complications to the theory at both technical and conceptual levels. As we will see throughout this book, many ideas fromcategory theory generalize to the ∞-categorical setting in a naturalway. However, these generalizations are not so straightforward if weallow noninvertible 2-morphisms. For example, one must distinguishbetween strict and lax fiber products, even in the setting of “classical”
2-categories.
(2) For the applications studied in this book, we will not need to consider(∞, n)-categories for n > 2. The case n = 2 is of some relevancehighertopoi

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because the collection of (small) ∞-categories can naturally be viewed
as a (large) ∞-bicategory. However, we will generally be able to exploit
this structure in an ad hoc manner without developing any general
theory of ∞-bicategories.
(3) For n > 1, the theory of (∞, n)-categories is most naturally viewed as a
special case of enriched (higher) category theory. Roughly speaking, ann-category can be viewed as a category enriched over (n−1)-categories.
As we explained above, this point of view is inadequate because it
requires that composition satisfies an associative law up to equality,
while in practice the associativity holds only up to isomorphism orsome weaker notion of equivalence. In other words, to obtain the correct definition we need to view the collection of (n−1)-categories as ann-category, not as an ordinary category. Consequently, the naive approach is circular: though it does lead to a good theory of n-categories,
we can make sense of it only if the theory of n-categories is already inplace.
Thinking along similar lines, we can view an (∞, n)-category as an∞-category which is enriched over (∞, n − 1)-categories. The collection of (∞, n − 1)-categories is itself organized into an (∞, n)-category
Cat(∞,n−1) , so at a first glance this definition suffers from the sameproblem of circularity. However, because the associativity propertiesof composition are required to hold up to equivalence, rather than up
to arbitrary natural transformation, the noninvertible k-morphisms inCat(∞,n−1) are irrelevant for k > 1. One can define an (∞, n)-category
to be a category enriched over Cat(∞,n−1) , where the latter is regarded as an ∞-category by discarding noninvertible k-morphisms for2 ≤ k ≤ n. In other words, the naive inductive definition of higher category theory is reasonable provided that we work in the ∞-categoricalsetting from the outset. We refer the reader to [76] for a definition ofn-categories which follows this line of thought.
The theory of enriched ∞-categories is a useful and important one but
will not be treated in this book. Instead we refer the reader to [51]for an introduction using the same language and formalism we employ
here.
Though we will not need a theory of (∞, n)-categories for n > 1, thecase n = 1 is the main subject matter of this book. Fortunately, the above
discussion suggests a definition. Namely, an ∞-category C should consist of acollection of objects and an ∞-groupoid MapC (X, Y ) for every pair of objects
X, Y ∈ C. These ∞-groupoids can be identified with topological spaces,and should be equipped with an associative composition law. As before, we
are faced with two choices as to how to make this precise: do we requireassociativity on the nose or only up to (coherent) homotopy? Fortunately,
the answer turns out to be irrelevant: as in the theory of 2-categories, any∞-category with a coherently associative multiplication can be replaced byhighertopoi

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AN OVERVIEW OF HIGHER CATEGORY THEORY

7

an equivalent ∞-category with a strictly associative multiplication. We are
led to the following:
Definition 1.1.1.6. A topological category is a category which is enrichedover CG, the category of compactly generated (and weakly Hausdorff) topological spaces. The category of topological categories will be denoted by
Cattop .
More explicitly, a topological category C consists of a collection of objects
together with a (compactly generated) topological space MapC (X, Y ) for any
pair of objects X, Y ∈ C. These mapping spaces must be equipped with anassociative composition law given by continuous maps
MapC (X0 , X1 ) × MapC (X1 , X2 ) × · · · × MapC (Xn−1 , Xn ) → MapC (X0 , Xn )(defined for all n ≥ 0). Here the product is taken in the category of compactly
generated topological spaces.
Remark 1.1.1.7. The decision to work with compactly generated topological spaces, rather than arbitrary spaces, is made in order to facilitate thecomparison with more combinatorial approaches to homotopy theory. Thisis a purely technical point which the reader may safely ignore.
It is possible to use Definition 1.1.1.6 as a foundation for higher category theory: that is, to define an ∞-category to be a topological category.
However, this approach has a number of technical disadvantages. We will
describe an alternative (though equivalent) formalism in the next section.
1.1.2 ∞-Categories
Of the numerous formalizations of higher category theory, Definition 1.1.1.6
is the quickest and most transparent. However, it is one of the most difficult
to actually work with: many of the basic constructions of higher category
theory give rise most naturally to (∞, 1)-categories for which the composition of morphisms is associative only up to (coherent) homotopy (for several
examples of this phenomenon, we refer the reader to §1.2). In order to remain in the world of topological categories, it is necessary to combine these
constructions with a “straightening” procedure which produces a strictly
associative composition law. Although it is always possible to do this (seeTheorem 2.2.5.1), it is much more technically convenient to work from theoutset within a more flexible theory of (∞, 1)-categories. Fortunately, there
are many candidates for such a theory, including the theory of Segal categories ([72]), the theory of complete Segal spaces ([65]), and the theory ofmodel categories ([40], [38]). To review all of these notions and their interrelationships would involve too great a digression from the main purpose ofthis book. However, the frequency with which we will encounter sophisticatedcategorical constructions necessitates the use of one of these more efficientapproaches. We will employ the theory of weak Kan complexes, which goes

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back to Boardman-Vogt ([10]). These objects have subsequently been studied more extensively by Joyal ([43], [44]), who calls them quasi-categories.
We will simply call them ∞-categories.
To get a feeling for what an ∞-category C should be, it is useful to consider
two extreme cases. If every morphism in C is invertible, then C is equivalent
to the fundamental ∞-groupoid of a topological space X. In this case, higher
category theory reduces to classical homotopy theory. On the other hand, if C
has no nontrivial n-morphisms for n > 1, then C is equivalent to an ordinary
category. A general formalism must capture the features of both of theseexamples. In other words, we need a class of mathematical objects which
can behave both like categories and like topological spaces. In §1.1.1, we
achieved this by “brute force”: namely, we directly amalgamated the theoryof topological spaces and the theory of categories by considering topologicalcategories. However, it is possible to approach the problem more directly
using the theory of simplicial sets. We will assume that the reader has somefamiliarity with the theory of simplicial sets; a brief review of this theory is
included in §A.2.7, and a more extensive introduction can be found in [32].
The theory of simplicial sets originated as a combinatorial approach tohomotopy theory. Given any topological space X, one can associate a simplicial set Sing X, whose n-simplices are precisely the continuous maps |∆n | →X, where |∆n | = {(x0 , . . . , xn ) ∈ [0, 1]n+1 |x0 + . . . + xn = 1} is the standard n-simplex. Moreover, the topological space X is determined, up to weakhomotopy equivalence, by Sing X. More precisely, the singular complex functor X 7→ Sing X admits a left adjoint, which carries every simplicial set K to
its geometric realization |K|. For every topological space X, the counit map| Sing X| → X is a weak homotopy equivalence. Consequently, if one is onlyinterested in studying topological spaces up to weak homotopy equivalence,
one might as well work with simplicial sets instead.
If X is a topological space, then the simplicial set Sing X has an important
property, which is captured by the following definition:
Definition 1.1.2.1. Let K be a simplicial set. We say that K is a Kancomplex if, for any 0 ≤ i ≤ n and any diagram of solid arrows
/K
Λni
>
_

∆n ,
there exists a dotted arrow as indicated rendering the diagram commutative.
Here Λni ⊆ ∆n denotes the ith horn, obtained from the simplex ∆n bydeleting the interior and the face opposite the ith vertex.
The singular complex of any topological space X is a Kan complex: thisfollows from the fact that the horn |Λni | is a retract of the simplex |∆n |in the category of topological spaces. Conversely, any Kan complex K “behaves like” a space: for example, there are simple combinatorial recipes for
extracting homotopy groups from K (which turn out be isomorphic to thehighertopoi

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AN OVERVIEW OF HIGHER CATEGORY THEORY

homotopy groups of the topological space |K|). According to a theorem of
Quillen (see [32] for a proof), the singular complex and geometric realization provide mutually inverse equivalences between the homotopy categoryof CW complexes and the homotopy category of Kan complexes.
The formalism of simplicial sets is also closely related to category theory.
To any category C, we can associate a simplicial set N(C) called the nerveof C. For each n ≥ 0, we let N(C)n = MapSet∆ (∆n , N(C)) denote the set of
all functors [n] → C. Here [n] denotes the linearly ordered set {0, . . . , n},regarded as a category in the obvious way. More concretely, N(C)n is the setof all composable sequences of morphisms
f1

f2

fn

C0 → C1 → · · · → Cn
having length n. In this description, the face map di carries the above sequence tofi+1 ◦fi

fi−1

f1

fi+2

fn

fi+2

fn

C0 → · · · → Ci−1 −→ Ci+1 → · · · → Cn
while the degeneracy si carries it to
f1

fi

idC

fi+1

C0 → · · · → Ci →i Ci → Ci+1 → · · · → Cn .
It is more or less clear from this description that the simplicial set N(C) is
just a fancy way of encoding the structure of C as a category. More precisely,
we note that the category C can be recovered (up to isomorphism) fromits nerve N(C). The objects of C are simply the vertices of N(C): that is,
the elements of N(C)0 . A morphism from C0 to C1 is given by an edgeφ ∈ N(C)1 with d1 (φ) = C0 and d0 (φ) = C1 . The identity morphism froman object C to itself is given by the degenerate simplex s0 (C). Finally, givenφψ

a diagram C0 → C1 → C2 , the edge of N(C) corresponding to ψ ◦ φ may beuniquely characterized by the fact that there exists a 2-simplex σ ∈ N(C)2with d2 (σ) = φ, d0 (σ) = ψ, and d1 (σ) = ψ ◦ φ.
It is not difficult to characterize those simplicial sets which arise as thenerve of a category:
Proposition 1.1.2.2. Let K be a simplicial set. Then the following conditions are equivalent:(1) There exists a small category C and an isomorphism K ' N(C).
(2) For each 0 < i < n and each diagram
Λni
_

/K
>


∆n ,
there exists a unique dotted arrow rendering the diagram commutative.

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Proof. We first show that (1) ⇒ (2). Let K be the nerve of a small category
C and let f0 : Λni → K be a map of simplicial sets, where 0 < i < n. We
wish to show that f0 can be extended uniquely to a map f : ∆n → K. For
0 ≤ k ≤ n, let Xk ∈ C be the image of the vertex {k} ⊆ Λni . For 0 < k ≤ n,
let gk : Xk−1 → Xk be the morphism in C determined by the restriction
f0 |∆{k−1,k} . The composable chain of morphisms
g1

g2

gn

X0 → X1 → · · · → Xn
determines an n-simplex f : ∆n → K. We will show that f is the desired
solution to our extension problem (the uniqueness of this solution is evident:
if f 0 : ∆n → K is any other map with f 0 |Λni = f0 , then f 0 must correspondto the same chain of morphisms in C, so that f 0 = f ). It will suffice to prove
the following for every 0 ≤ j ≤ n:(∗j ) If j 6= i, thenf |∆{0,...,j−1,j+1,...,n} = f0 |∆{0,...,j−1,j+1,...,n} .
To prove (∗j ), it will suffice to show that f and f0 have the same restriction0
to ∆{k,k } , where k and k 0 are adjacent elements of the linearly ordered set{0, . . . , j − 1, j + 1, . . . , n} ⊆ [n]. If k and k 0 are adjacent in [n], then thisfollows by construction. In particular, (∗) is automatically satisfied if j = 0
or j = n. Suppose instead that k = j − 1 and k 0 = j + 1, where 0 < j < n.
If n = 2, then j = 1 = i and we obtain a contradiction. We may therefore
assume that n > 2, so that either j − 1 > 0 or j + 1 < n. Without loss ofgenerality, j − 1 > 0, so that ∆{j−1,j+1} ⊆ ∆{1,...,n} . The desired conclusion
now follows from (∗0 ).
We now prove the converse. Suppose that the simplicial set K satisfies(2); we claim that K is isomorphic to the nerve of a small category C. We
construct the category C as follows:(i) The objects of C are the vertices of K.
(ii) Given a pair of objects x, y ∈ C, we let HomC (x, y) denote the collectionof all edges e : ∆1 → K such that e|{0} = x and e|{1} = y.
(iii) Let x be an object of C. Then the identity morphism idx is the edgeof K defined by the compositione∆1 → ∆0 → K.
(iv) Let f : x → y and g : y → z be morphisms in C. Then f and g together
determine a map σ0 : Λ21 → K. In view of condition (2), the map σ0
can be extended uniquely to a 2-simplex σ : ∆2 → K. We define thecomposition g ◦ f to be the morphism from x to z in C corresponding
to the edge given by the compositionσ∆1 ' ∆{0,2} ⊆ ∆2 → K.

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We first claim that C is a category. To prove this, we must verify thefollowing axioms:(a) For every object y ∈ C, the identity idy is a unit with respect tocomposition. In other words, for every morphism f : x → y in C andevery morphism g : y → z in C, we have idy ◦f = f and g ◦ idy =g. These equations are “witnessed” by the 2-simplices s1 (f ), s0 (g) ∈HomSet∆ (∆2 , K).
(b) Composition is associative. That is, for every sequence of composable
morphismsfghw → x → y → z,
we have h ◦ (g ◦ f ) = (h ◦ g) ◦ f . To prove this, let us first choose
2-simplices σ012 and σ123 as indicated below:
?x
?y
f

g

g

h

g◦f
h◦g
/ y
/  z.
wx
Now choose a 2-simplex σ023 corresponding to a diagram
?y
g◦f

h

h◦(g◦f )/ z.
wThese three 2-simplices together define a map τ0 : Λ32 → K. Since K
satisfies condition (2), we can extend τ0 to a 3-simplex τ : ∆3 → K.
The compositionτ∆2 ' ∆{0,1,3} ⊆ ∆3 → K
corresponds to the diagram
>x
f

w

h◦g

h◦(g◦f )/ z,which witnesses the associativity axiom h ◦ (g ◦ f ) = (h ◦ g) ◦ f .
It follows that C is a well-defined category. By construction, we have acanonical map of simplicial sets φ : K → N C. To complete the proof, it
will suffice to show that φ is an isomorphism. We will prove, by induction onn ≥ 0, that φ induces a bijection HomSet∆ (∆n , K) → HomSet∆ (∆n , N C). For
n = 0 and n = 1, this is obvious from the construction. Assume therefore thatn ≥ 2 and choose an integer i such that 0 < i < n. We have a commutative
diagram/ HomSet∆ (∆n , N C)
HomSet∆ (∆n , K)
HomSet∆ (Λni , K)
/ HomSet∆ (Λn , N C).
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Since K and N C both satisfy (2) (for N C, this follows from the first part ofthe proof), the vertical maps are bijective. It will therefore suffice to showthat the lower horizontal map is bijective, which follows from the inductive
hypothesis.
We note that condition (2) of Proposition 1.1.2.2 is very similar to Definition 1.1.2.1. However, it is different in two important respects. First, it
requires the extension condition only for inner horns Λni with 0 < i < n.
Second, the asserted condition is stronger in this case: not only does any
map Λni → K extend to the simplex ∆n , but the extension is unique.
Remark 1.1.2.3. It is easy to see that it is not reasonable to expect condition (2) of Proposition 1.1.2.2 to hold for outer horns Λni where i ∈ {0, n}.
Consider, for example, the case where i = n = 2 and where K is the nerveof a category C. Giving a map Λ22 → K corresponds to supplying the solid
arrows in the diagram> C1!
/ C2 ,

C0

and the extension condition would amount to the assertion that one could
always find a dotted arrow rendering the diagram commutative. This is truein general only when the category C is a groupoid.
We now see that the notion of a simplicial set is a flexible one: a simplicial
set K can be a good model for an ∞-groupoid (if K is a Kan complex) or for
an ordinary category (if it satisfies the hypotheses of Proposition 1.1.2.2).
Based on these observations, we might expect that some more general classof simplicial sets could serve as models for ∞-categories in general.
Consider first an arbitrary simplicial set K. We can try to envision K as a
generalized category whose objects are the vertices of K (that is, the elementsof K0 ) and whose morphisms are the edges of K (that is, the elements of
K1 ). A 2-simplex σ : ∆2 → K should be thought of as a diagram
>Y
φ

X

ψ
θ



/Z

together with an identification (or homotopy) between θ and ψ ◦ φ which
witnesses the “commutativity” of the diagram. (In higher category theory,commutativity is not merely a condition: the homotopy θ ' ψ ◦ φ is an additional datum.) Simplices of larger dimension may be thought of as verifying
the commutativity of certain higher-dimensional diagrams.
Unfortunately, for a general simplicial set K, the analogy outlined above
is not very strong. The essence of the problem is that, though we may referto the 1-simplices of K as morphisms, there is in general no way to compose

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13

them. Taking our cue from the example of N(C), we might say that a morphism θ : X → Z is a composition of morphisms φ : X → Y and ψ : Y → Z
if there exists a 2-simplex σ : ∆2 → K as in the diagram indicated above.
We must now consider two potential difficulties: the desired 2-simplex σ may
not exist, and if it does exist it may not be unique, so that we have morethan one choice for the composition θ.
The existence requirement for σ can be formulated as an extension condition on the simplicial set K. We note that a composable pair of morphisms(ψ, φ) determines a map of simplicial sets Λ21 → K. Thus, the assertion
that σ can always be found may be formulated as a extension property: any
map of simplicial sets Λ21 → K can be extended to ∆2 , as indicated in thefollowing diagram:
Λ21
_

/K
>


∆2 .
The uniqueness of θ is another matter. It turns out to be unnecessary(and unnatural) to require that θ be uniquely determined. To understandthis point, let us return to the example of the fundamental groupoid of a
topological space X. This is a category whose objects are the points x ∈ X.
The morphisms between a point x ∈ X and a point y ∈ X are given bycontinuous paths p : [0, 1] → X such that p(0) = x and p(1) = y. Twosuch paths are considered to be equivalent if there is a homotopy between
them. Composition in the fundamental groupoid is given by concatenationof paths. Given paths p, q : [0, 1] → X with p(0) = x, p(1) = q(0) = y, andq(1) = z, the composite of p and q should be a path joining x to z. There
are many ways of obtaining such a path from p and q. One of the simplest
is to define(p(2t)
if 0 ≤ t ≤ 12
r(t) =q(2t − 1) if 21 ≤ t ≤ 1.
However, we could just as well use the formula(p(3t)
if 0 ≤ t ≤ 31
0
r (t) =
1
q( 3t−12 ) if 3 ≤ t ≤ 1
to define the composite path. Because the paths r and r0 are homotopic to
one another, it does not matter which one we choose.
The situation becomes more complicated if we try to think 2-categorically.
We can capture more information about the space X by considering its fundamental 2-groupoid. This is a 2-category whose objects are the points of
X, whose morphisms are paths between points, and whose 2-morphisms are
given by homotopies between paths (which are themselves considered modulo homotopy). In order to have composition of morphisms unambiguously
defined, we would have to choose some formula once and for all. Moreover,highertopoi

14

April 9, 2017

CHAPTER 1

there is no particularly compelling choice; for example, neither of the formulas written above leads to a strictly associative composition law.
The lesson to learn from this is that in higher-categorical situations, weshould not necessarily ask for a uniquely determined composition of twomorphisms. In the fundamental groupoid example, there are many choices for
a composite path, but all of them are homotopic to one another. Moreover,in keeping with the philosophy of higher category theory, any path which ishomotopic to the composite should be just as good as the composite itself.
From this point of view, it is perhaps more natural to view composition
as a relation than as a function, and this is very efficiently encoded in theformalism of simplicial sets: a 2-simplex σ : ∆2 → K should be viewed as“evidence” that d0 (σ) ◦ d2 (σ) is homotopic to d1 (σ).
Exactly what conditions on a simplicial set K will guarantee that it behaves like a higher category? Based on the above argument, it seems reasonable to require that K satisfy an extension condition with respect to certain
horn inclusions Λni , as in Definition 1.1.2.1. However, as we observed in Remark 1.1.2.3, this is reasonable only for the inner horns where 0 < i < n,which appear in the statement of Proposition 1.1.2.2.
Definition 1.1.2.4. An ∞-category is a simplicial set K which has thefollowing property: for any 0 < i < n, any map f0 : Λni → K admits anextension f : ∆n → K.
Definition 1.1.2.4 was first formulated by Boardman and Vogt ([10]). They
referred to ∞-catgories as weak Kan complexes, motivated by the obvious
analogy with Definition 1.1.2.1. Our terminology places more emphasis on
the analogy with the characterization of ordinary categories given in Proposition 1.1.2.2: we require the same extension conditions but drop the uniqueness assumption.
Example 1.1.2.5. Any Kan complex is an ∞-category. In particular, if X
is a topological space, then we may view its singular complex Sing X as an∞-category: this is one way of defining the fundamental ∞-groupoid π≤∞ X
of X introduced informally in Example 1.1.1.4.
Example 1.1.2.6. The nerve of any category is an ∞-category. We will
occasionally abuse terminology by identifying a category C with its nerve
N(C); by means of this identification, we may view ordinary category theory
as a special case of the study of ∞-categories.
The weak Kan condition of Definition 1.1.2.4 leads to a very elegant andpowerful version of higher category theory. This theory has been developed
by Joyal in [43] and [44] (where simplicial sets satisfying the condition ofDefinition 1.1.2.4 are called quasi-categories) and will be used throughout
this book.
Notation 1.1.2.7. Depending on the context, we will use two different
notations in connection with simplicial sets. When emphasizing their role ashighertopoi

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15∞-categories, we will often denote them by calligraphic letters such as C, D,
and so forth. When casting simplicial sets in their different (though related)role as representatives of homotopy types, we will employ capital Roman
letters. To avoid confusion, we will also employ the latter notation when wewish to contrast the theory of ∞-categories with some other approach tohigher category theory, such as the theory of topological categories.
1.1.3 Equivalences of Topological Categories
We have now introduced two approaches to higher category theory: one
based on topological categories and one based on simplicial sets. These two
approaches turn out to be equivalent to one another. However, the equivalence itself needs to be understood in a higher-categorical sense. We take ourcue from classical homotopy theory, in which we can take the basic objects
to be either topological spaces or simplicial sets. It is not true that every
Kan complex is isomorphic to the singular complex of a topological space orthat every CW complex is homeomorphic to the geometric realization of asimplicial set. However, both of these statements become true if we replace
the words “isomorphic to” by “homotopy equivalent to.” We would like toformulate a similar statement regarding our approaches to higher category
theory. The first step is to find a concept which replaces homotopy equivalence. If F : C → D is a functor between topological categories, under whatcircumstances should we regard F as an equivalence (so that C and D really
represent the same higher category)?
The most naive answer is that F should be regarded as an equivalence if
it is an isomorphism of topological categories. This means that F inducesa bijection between the objects of C and the objects of D, and a homeomorphism MapC (X, Y ) → MapD (F (X), F (Y )) for every pair of objects
X, Y ∈ C. However, it is immediately obvious that this condition is far
too strong; for example, in the case where C and D are ordinary categories(which we may view also as topological categories where all morphism setsare endowed with the discrete topology), we recover the notion of an isomorphism between categories. This notion does not play an important role incategory theory. One rarely asks whether or not two categories are isomorphic; instead, one asks whether or not they are equivalent. This suggests thefollowing definition:
Definition 1.1.3.1. A functor F : C → D between topological categories
is a strong equivalence if it is an equivalence in the sense of enriched category theory. In other words, F is a strong equivalence if it induces homeomorphisms MapC (X, Y ) → MapD (F (X), F (Y )) for every pair of objects
X, Y ∈ C, and every object of D is isomorphic (in D) to F (X) for some
X ∈ C.
The notion of strong equivalence between topological categories has thevirtue that, when restricted to ordinary categories, it reduces to the usual
notion of equivalence. However, it is still not the right definition: for a pairhighertopoi

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of objects X and Y of a higher category C, the morphism space MapC (X, Y )should itself be well-defined only up to homotopy equivalence.
Definition 1.1.3.2. Let C be a topological category. The homotopy category
hC is defined as follows:• The objects of hC are the objects of C.
• If X, Y ∈ C, then we define HomhC (X, Y ) = π0 MapC (X, Y ).
• Composition of morphisms in hC is induced from the composition ofmorphisms in C by applying the functor π0 .
Example 1.1.3.3. Let C be the topological category whose objects are
CW complexes, where MapC (X, Y ) is the set of continuous maps from X
to Y , equipped with the (compactly generated version of the) compact-open
topology. We will denote the homotopy category of C by H and refer to H
as the homotopy category of spaces.
There is a second construction of the homotopy category H which will play
an important role in what follows. First, we must recall a bit of terminology
from classical homotopy theory.
Definition 1.1.3.4. A map f : X → Y between topological spaces is said
to be a weak homotopy equivalence if it induces a bijection π0 X → π0 Y , andif for every point x ∈ X and every i ≥ 1, the induced map of homotopy
groupsπi (X, x) → πi (Y, f (x))
is an isomorphism.
Given a space X ∈ CG, classical homotopy theory ensures the existence ofa CW complex X 0 equipped with a weak homotopy equivalence φ : X 0 →X. Of course, X 0 is not uniquely determined; however, it is unique up to
canonical homotopy equivalence, so that the assignment
X 7→ [X] = X 0
determines a functor θ : CG → H. By construction, θ carries weak homotopy
equivalences in CG to isomorphisms in H. In fact, θ is universal with respect
to this property. In other words, we may describe H as the category obtained from CG by formally inverting all weak homotopy equivalences. Thisis one version of Whitehead’s theorem, which is usually stated as follows: every weak homotopy equivalence between CW complexes admits a homotopy
inverse.
We can now improve upon Definition 1.1.3.2 slightly. We first observe thatthe functor θ : CG → H preserves products. Consequently, we can apply theconstruction of Remark A.1.4.3 to convert any topological category C intoa category enriched over H. We will denote this H-enriched category by hC
and refer to it as the homotopy category of C. More concretely, the homotopy
category hC may be described as follows:highertopoi

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17(1) The objects of hC are the objects of C.
(2) For X, Y ∈ C, we have
MaphC (X, Y ) = [MapC (X, Y )].
(3) The composition law on hC is obtained from the composition law on
C by applying the functor θ : CG → H.
Remark 1.1.3.5. If C is a topological category, we have now defined hC
in two different ways: first as an ordinary category and later as a category
enriched over H. These two definitions are compatible with one another inthe sense that hC (regarded as an ordinary category) is the underlying category of hC (regarded as an H-enriched category). This follows immediatelyfrom the observation that for every topological space X, there is a canonicalbijection π0 X ' MapH (∗, [X]).
If C is a topological category, we may imagine that hC is the object which
is obtained by forgetting the topological morphism spaces of C and remembering only their (weak) homotopy types. The following definition codifies
the idea that these homotopy types should be “all that really matter.”Definition 1.1.3.6. Let F : C → D be a functor between topological categories. We will say that F is a weak equivalence, or simply an equivalence, if
the induced functor hC → hD is an equivalence of H-enriched categories.
More concretely, a functor F is an equivalence if and only if the following
conditions are satisfied:• For every pair of objects X, Y ∈ C, the induced map
MapC (X, Y ) → MapD (F (X), F (Y ))is a weak homotopy equivalence of topological spaces.
• Every object of D is isomorphic in hD to F (X) for some X ∈ C.
Remark 1.1.3.7. A morphism f : X → Y in D is said to be an equivalence
if the induced morphism in hD is an isomorphism. In general, this is much
weaker than the condition that f be an isomorphism in D; see Proposition
1.2.4.1.
It is Definition 1.1.3.6 which gives the correct notion of equivalence between topological categories (at least, when one is using them to describe
higher category theory). We will agree that all relevant properties of topological categories are invariant under this notion of equivalence. We say that
two topological categories are equivalent if there is an equivalence between
them, or more generally if there is a chain of equivalences joining them.
Equivalent topological categories should be regarded as interchangeable forall relevant purposes.

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Remark 1.1.3.8. According to Definition 1.1.3.6, a functor F : C → D isan equivalence if and only if the induced functor hC → hD is an equivalence.
In other words, the homotopy category hC (regarded as a category which isenriched over H) is an invariant of C which is sufficiently powerful to detect
equivalences between ∞-categories. This should be regarded as analogous tothe more classical fact that the homotopy groups πi (X, x) of a CW complex
X are homotopy invariants which detect homotopy equivalences between CW
complexes (by Whitehead’s theorem). However, it is important to rememberthat hC does not determine C up to equivalence, just as the homotopy typeof a CW complex is not determined by its homotopy groups.
1.1.4 Simplicial Categories
In the previous sections we introduced two very different approaches to thefoundations of higher category theory: one based on topological categories,
the other on simplicial sets. In order to prove that they are equivalent to
one another, we will introduce a third approach which is closely related to
the first but shares the combinatorial flavor of the second.
Definition 1.1.4.1. A simplicial category is a category which is enrichedover the category Set∆ of simplicial sets. The category of simplicial categories (where morphisms are given by simplicially enriched functors) will bedenoted by Cat∆ .
Remark 1.1.4.2. Every simplicial category can be regarded as a simplicial
object in the category Cat. Conversely, a simplicial object of Cat arises froma simplicial category if and only if the underlying simplicial set of objects isconstant.
Like topological categories, simplicial categories can be used as models ofhigher category theory. If C is a simplicial category, then we will generally
think of the simplicial sets MapC (X, Y ) as encoding homotopy types or ∞groupoids.
Remark 1.1.4.3. If C is a simplicial category with the property that each of
the simplicial sets MapC (X, Y ) is an ∞-category, then we may view C itself asa kind of ∞-bicategory. We will not use this interpretation of simplicial categories in this book. Usually we will consider only fibrant simplicial categories;that is, simplicial categories for which the mapping objects MapC (X, Y ) are
Kan complexes.
The relationship between simplicial categories and topological categories
is easy to describe. Let Set∆ denote the category of simplicial sets and CG thecategory of compactly generated (and weakly Hausdorff) spaces. We recall
that there exists a pair of adjoint functors
Set∆ o

||
Sing

/ CG

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19

which are called the geometric realization and singular complex functors,
respectively. Both of these functors commute with finite products. Consequently, if C is a simplicial category, we may define a topological category| C | in the following way:
• The objects of | C | are the objects of C.
• If X, Y ∈ C, then Map| C | (X, Y ) = | MapC (X, Y )|.
• The composition law for morphisms in | C | is obtained from the composition law on C by applying the geometric realization functor.
Similarly, if C is a topological category, we may obtain a simplicial category Sing C by applying the singular complex functor to each of the morphism
spaces individually. The singular complex and geometric realization functors
determine an adjunction between Cat∆ and Cattop . This adjunction should
be understood as determining an equivalence between the theory of simplicial categories and the theory of topological categories. This is essentially aformal consequence of the fact that the geometric realization and singularcomplex functors determine an equivalence between the homotopy theoryof topological spaces and the homotopy theory of simplicial sets. More precisely, we recall that a map f : S → T of simplicial sets is said to be a weakhomotopy equivalence if the induced map |S| → |T | of topological spaces isa weak homotopy equivalence. A theorem of Quillen (see [32] for a proof)
asserts that the unit and counit morphisms
S → Sing |S|| Sing X| → X
are weak homotopy equivalences for every (compactly generated) topologicalspace X and every simplicial set S. It follows that the category obtained fromCG by inverting weak homotopy equivalences (of spaces) is equivalent to thecategory obtained from Set∆ by inverting weak homotopy equivalences. We
use the symbol H to denote either of these (equivalent) categories.
If C is a simplicial category, we let hC denote the H-enriched category
obtained by applying the functor Set∆ → H to each of the morphism spacesof C. We will refer to hC as the homotopy category of C. We note that this is
the same notation that was introduced in §1.1.3 for the homotopy categoryof a topological category. However, there is little risk of confusion: the above
remarks imply the existence of canonical isomorphisms
hC ' h| C |hD ' hSing D
for every simplicial category C and every topological category D.
Definition 1.1.4.4. A functor C → C0 between simplicial categories is anequivalence if the induced functor hC → hC0 is an equivalence of H-enrichedcategories.

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In other words, a functor C → C0 between simplicial categories is an equivalence if and only if the geometric realization | C | → | C0 | is an equivalenceof topological categories. In fact, one can say more. It follows easily from thepreceding remarks that the unit and counit maps
C → Sing | C || Sing D | → D
induce isomorphisms between homotopy categories. Consequently, if we areworking with topological or simplicial categories up to equivalence, we arealways free to replace a simplicial category C by | C | or a topological category
D by Sing D. In this sense, the notions of topological category and simplicial
category are equivalent, and either can be used as a foundation for higher
category theory.
1.1.5 Comparing ∞-Categories with Simplicial Categories
In §1.1.4, we introduced the theory of simplicial categories and explainedwhy (for our purposes) it is equivalent to the theory of topological categories. In this section, we will show that the theory of simplicial categories
is also closely related to the theory of ∞-categories. Our discussion requires
somewhat more elaborate constructions than were needed in the previous
sections; a reader who does not wish to become bogged down in details is
urged to skip ahead to §1.2.1.
We will relate simplicial categories with simplicial sets by means of thesimplicial nerve functor
N : Cat∆ → Set∆ ,originally introduced by Cordier (see [16]). The nerve of an ordinary category
C is characterized by the formula
HomSet∆ (∆n , N(C)) = HomCat ([n], C);
here [n] denotes the linearly ordered set {0, . . . , n} regarded as a category.
This definition makes sense also when C is a simplicial category but is clearly
not very interesting: it makes no use of the simplicial structure on C. In orderto obtain a more interesting construction, we need to replace the ordinary
category [n] by a suitable “thickening,” a simplicial category which we will
denote by C[∆n ].
Definition 1.1.5.1. Let J be a finite nonempty linearly ordered set. The
simplicial category C[∆J ] is defined as follows:• The objects of C[∆J ] are the elements of J.
• If i, j ∈ J, then(∅if j < i
MapC[∆J ] (i, j) =N(Pi,j ) if i ≤ j.
Here Pi,j denotes the partially ordered set {I ⊆ J : (i, j ∈ I) ∧ (∀k ∈I)[i ≤ k ≤ j]}.

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21• If i0 ≤ i1 ≤ · · · ≤ in , then the composition
MapC[∆J ] (i0 , i1 ) × · · · × MapC[∆J ] (in−1 , in ) → MapC[∆J ] (i0 , in )is induced by the map of partially ordered sets
Pi0 ,i1 × · · · × Pin−1 ,in → Pi0 ,in(I1 , . . . , In ) 7→ I1 ∪ · · · ∪ In .
In order to help digest Definition 1.1.5.1, let us analyze the structure of
the topological category | C[∆n ]|. The objects of this category are elementsof the set [n] = {0, . . . , n}. For each 0 ≤ i ≤ j ≤ n, the topological space
Map| C[∆n ]| (i, j) is homeomorphic to a cube; it may be identified with the setof all functions p : {k ∈ [n] : i ≤ k ≤ j} → [0, 1] which satisfy p(i) = p(j) = 1.
The morphism space Map| C[∆n ]| (i, j) is empty when j < i, and compositionof morphisms is given by concatenation of functions.
Remark 1.1.5.2. Let us try to understand better the simplicial categoryC[∆n ] and its relationship to the ordinary category [n]. These categories havethe same objects: the elements of {0, . . . , n}. In the category [n], there is aunique morphism qij : i → j whenever i ≤ j. By virtue of the uniqueness,these elements satisfy qjk ◦ qij = qik for i ≤ j ≤ k.
In the simplicial category C[∆n ], there is a vertex pij ∈ MapC[∆n ] (i, j) for
each i ≤ j, given by the element {i, j} ∈ Pij . We note that pjk ◦ pij 6= pik(except in degenerate cases where i = j or j = k). Instead, the collection of
all compositions
pin−1 in ◦ pin−2 in−1 ◦ · · · ◦ pi0 i1 ,where i = i0 < i1 < · · · < in−1 < in = j constitute all of the different
vertices of the cube MapC[∆n ] (i, j). The weak contractibility of MapC[∆n ] (i, j)
expresses the idea that although these compositions do not coincide, they
are all canonically homotopic to one another. We observe that there is a
(unique) functor C[∆n ] → [n] which is the identity on objects. This functor
is an equivalence of simplicial categories. We can summarize the situation
informally as follows: the simplicial category C[∆n ] is a thickened version of[n] where we have dropped the strict associativity condition
qjk ◦ qij = qik
and instead have imposed associativity only up to (coherent) homotopy. (Wecan formulate this idea more precisely by saying that C[∆• ] is a cofibrantreplacement for [•] with respect to a suitable model structure on the categoryof cosimplicial objects of Cat∆ .)The construction J 7→ C[∆J ] is functorial in J, as we now explain.
Definition 1.1.5.3. Let f : J → J 0 be a monotone map between linearly0ordered sets. The simplicial functor C[f ] : C[∆J ] → C[∆J ] is defined asfollows:

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0• For each object i ∈ C[∆J ], C[f ](i) = f (i) ∈ C[∆J ].
• If i ≤ j in J, then the map MapC[∆J ] (i, j) → MapC[∆J 0 ] (f (i), f (j))
induced by f is the nerve of the map
Pi,j → Pf (i),f (j)I 7→ f (I).
Remark 1.1.5.4. Using the notation of Remark 1.1.5.2, we note that Definition 1.1.5.3 has been rigged so that the functor C[f ] carries the vertex
pij ∈ MapC[∆J ] (i, j) to the vertex pf (i)f (j) ∈ MapC[∆J 0 ] (f (i), f (j)).
It is not difficult to check that the construction described in Definition
1.1.5.3 is well-defined and compatible with composition in f . Consequently,
we deduce that C determines a functor∆ → Cat∆∆n 7→ C[∆n ],which we may view as a cosimplicial object of Cat∆ .
Definition 1.1.5.5. Let C be a simplicial category. The simplicial nerve
N(C) is the simplicial set described by the formula
HomSet∆ (∆n , N(C)) = HomCat∆ (C[∆n ], C).
If C is a topological category, we define the topological nerve N(C) of C to
be the simplicial nerve of Sing C.
Remark 1.1.5.6. If C is a simplicial (topological) category, we will often
abuse terminology by referring to the simplicial (topological) nerve of C
simply as the nerve of C.
Warning 1.1.5.7. Let C be a simplicial category. Then C can be regarded
as an ordinary category by ignoring all simplices of positive dimension in
the mapping spaces of C. The simplicial nerve of C does not coincide with
the nerve of this underlying ordinary category. Our notation is therefore potentially ambiguous. We will adopt the following convention: whenever C isa simplicial category, N(C) will denote the simplicial nerve of C unless we
specify otherwise. Similarly, if C is a topological category, then the topological nerve of C does not generally coincide with the nerve of the underlying
category; the notation N(C) will be used to indicate the topological nerve
unless otherwise specified.
Example 1.1.5.8. Any ordinary category C may be considered as a simplicial category by taking each of the simplicial sets HomC (X, Y ) to be constant.
In this case, the set of simplicial functors C[∆n ] → C may be identified withthe set of functors from [n] into C. Consequently, the simplicial nerve of C
agrees with the ordinary nerve of C as defined in §1.1.2. Similarly, the ordinary nerve of C can be identified with the topological nerve of C, where C isregarded as a topological category with discrete morphism spaces.

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AN OVERVIEW OF HIGHER CATEGORY THEORY

In order to get a feel for what the nerve of a topological category C looks
like, let us explicitly describe its low-dimensional simplices:• The 0-simplices of N(C) may be identified with the objects of C.
• The 1-simplices of N(C) may be identified with the morphisms of C.
• To give a map from the boundary of a 2-simplex into N(C) is to give adiagram (not necessarily commutative)
>Y
fXY

X

fY Z
fXZ

/ Z.

To give a 2-simplex of N(C) having this specified boundary is equivalent
to giving a path from fXZ to fY Z ◦ fXY in MapC (X, Z).
The category Cat∆ of simplicial categories admits (small) colimits. Consequently, by formal nonsense, the functor C : ∆ → Cat∆ extends uniquely (upto unique isomorphism) to a colimit-preserving functor Set∆ → Cat∆ , which
we will denote also by C. By construction, the functor C is left adjoint to thesimplicial nerve functor N. For each simplicial set S, we can view C[S] as
the simplicial category “freely generated” by S: every n-simplex σ : ∆n → Sdetermines a functor C[∆n ] → C[S], which we can think of as a homotopy
coherent diagram [n] → C[S].
Example 1.1.5.9. Let A be a partially ordered set. The simplicial category
C[N A] can be constructed using the following generalization of Definition
1.1.5.1:• The objects of C[N A] are the elements of A.
• Given a pair of elements a, b ∈ A, the simplicial set MapC[N A] (a, b) canbe identified with N Pa,b , where Pa,b denotes the collection of linearlyordered subsets S ⊆ A with least element a and largest element b,
partially ordered by inclusion.
• Given a sequence of elements a0 , . . . , an ∈ A, the composition map
MapC[N A] (a0 , a1 ) × · · · × MapC[N A] (an−1 , an ) → MapC[N A] (a0 , an )is induced by the map of partially ordered sets
Pa0 ,a1 × · · · × Pan−1 ,an → Pa0 ,an(S1 , . . . , Sn ) 7→ S1 ∪ · · · ∪ Sn .
Proposition 1.1.5.10. Let C be a simplicial category having the property
that, for every pair of objects X, Y ∈ C, the simplicial set MapC (X, Y ) is a
Kan complex. Then the simplicial nerve N(C) is an ∞-category.

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Proof. We must show that if 0 < i < n, then N(C) has the right extension
property with respect to the inclusion Λni ⊆ ∆n . Rephrasing this in thelanguage of simplicial categories, we must show that C has the right extension
property with respect to the simplicial functor C[Λni ] → C[∆n ]. To prove this,we make use of the following observations concerning C[Λni ], which we viewas a simplicial subcategory of C[∆n ]:• The objects of C[Λni ] are the objects of C[∆n ]: that is, elements of theset [n].
• For 0 ≤ j ≤ k ≤ n, the simplicial set MapC[Λni ] (j, k) coincides with
MapC[∆n ] (j, k) unless j = 0 and k = n (note that this condition fails
if i = 0 or i = n).
Consequently, every extension problem
Λni
_

F

/ N(C)<

∆n
is equivalent to
MapC[Λni ] (0, n)/ MapC (F (0), F (n))
5MapC[∆n ] (0, n).
Since the simplicial set on the right is a Kan complex by assumption, it
suffices to verify that the left vertical map is anodyne. This follows by inspection: the simplicial set MapC[∆n ] (0, n) can be identified with the cube(∆1 ){1,...,n−1} . Under this identification, MapC[Λni ] (0, n) corresponds to thesimplicial subset of (∆1 ){1,...,n−1} obtained by removing the interior of the
cube together with one of its faces.
Remark 1.1.5.11. The proof of Proposition 1.1.5.10 actually provides aslightly stronger result: if F : C → D is a functor between simplicial categories which induces Kan fibrations MapC (C, C 0 ) → MapD (F (C), F (C 0 ))for every pair of objects C, C 0 ∈ C, then the associated map N(C) → N(D)is an inner fibration of simplicial sets (see Definition 2.0.0.3).
Corollary 1.1.5.12. Let C be a topological category. Then the topologicalnerve N(C) is an ∞-category.
Proof. This follows immediately from Proposition 1.1.5.10 (note that the
singular complex of any topological space is a Kan complex).
We now cite the following theorem, which will be proven in §2.2.4 andrefined in §2.2.5:

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Theorem 1.1.5.13. Let C be a topological category and let X, Y ∈ C beobjects. Then the counit map| MapC[N(C)] (X, Y )| → MapC (X, Y )is a weak homotopy equivalence of topological spaces.
Using Theorem 1.1.5.13, we can explain why the theory of ∞-categories is
equivalent to the theory of topological categories (or equivalently, simplicial
categories). The adjoint functors N and | C[•]| are not mutually inverse equivalences of categories. However, they are homotopy inverse to one another.
To make this precise, we need to introduce a definition.
Definition 1.1.5.14. Let S be a simplicial set. The homotopy category hS
is defined to be the homotopy category hC[S ] of the simplicial category C[S].
We will often view hS as a category enriched over the homotopy category
H of spaces via the construction of §1.1.4: that is, for every pair of vertices
x, y ∈ S, we have MaphS (x, y) = [MapC[S] (x, y)]. A map f : S → T of
simplicial sets is a categorical equivalence if the induced map hS → hT is anequivalence of H-enriched categories.
Remark 1.1.5.15. In [44], Joyal uses the term “weak categorical equivalence” for what we have called a categorical equivalence, and reserves theterm “categorical equivalence” for a stronger notion of equivalence.
Remark 1.1.5.16. We have introduced the term “categorical equivalence,”
rather than simply “equivalence” or “weak equivalence,” in order to avoid
confusing the notion of categorical equivalence of simplicial sets with the(more classical) notion of weak homotopy equivalence of simplicial sets.
Remark 1.1.5.17. It is immediate from the definition that f : S → T isa categorical equivalence if and only if C[S] → C[T ] is an equivalence (ofsimplicial categories) if and only if | C[S]| → | C[T ]| is an equivalence (oftopological categories).
We now observe that the adjoint functors (| C[•]|, N) determine an equivalence between the theory of simplicial sets (up to categorical equivalence)and that of topological categories (up to equivalence). In other words, for
any topological category C the counit map | C[N(C)]| → C is an equivalence of
topological categories, and for any simplicial set S the unit map S → N | C[S]|is a categorical equivalence of simplicial sets. In view of Remark 1.1.5.17, thesecond assertion is a formal consequence of the first. Moreover, the first assertion is merely a reformulation of Theorem 1.1.5.13.
Remark 1.1.5.18. The reader may at this point object that we have obtained a comparison between the theory of topological categories and thetheory of simplicial sets but that not every simplicial set is an ∞-category.
However, every simplicial set is categorically equivalent to an ∞-category.
In fact, Theorem 1.1.5.13 implies that every simplicial set S is categoricallyequivalent to the nerve of the topological category | C[S]|, which is an ∞category (Corollary 1.1.5.12).

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1.2 THE LANGUAGE OF HIGHER CATEGORY THEORYOne of the main goals of this book is to demonstrate that many ideas fromclassical category theory can be adapted to the setting of higher categories.
In this section, we will survey some of the simplest examples.
1.2.1 The Opposite of an ∞-Category
If C is an ordinary category, then the opposite category Cop is defined in thefollowing way:
• The objects of Cop are the objects of C.
• For X, Y ∈ C, we have HomCop (X, Y ) = HomC (Y, X). Identity morphisms and composition are defined in the obvious way.
This definition generalizes without change to the setting of topological or
simplicial categories. Adapting this definition to the setting of ∞-categories
requires a few additional words. We may define more generally the opposite
of a simplicial set S as follows: for any finite nonempty linearly ordered set
J, we set S op (J) = S(J op ), where J op denotes the same set J endowed withthe opposite ordering. More concretely, we have Snop = Sn , but the face anddegeneracy maps on S op are given by the formulasop(di : Snop → Sn−1
) = (dn−i : Sn → Sn−1 )op(si : Snop → Sn+1
) = (sn−i : Sn → Sn+1 ).

The formation of opposite categories is fully compatible with all of the constructions we have introduced for passing back and forth between different
models of higher category theory.
It is clear from the definition that a simplicial set S is an ∞-category if andonly if its opposite S op is an ∞-category: for 0 < i < n, S has the extension
property with respect to the horn inclusion Λni ⊆ ∆n if and only if S op has
the extension property with respect to the horn inclusion Λnn−i ⊆ ∆n .
1.2.2 Mapping Spaces in Higher Category Theory
If X and Y are objects of an ordinary category C, then one has a well-definedset HomC (X, Y ) of morphisms from X to Y . In higher category theory, one
has instead a morphism space MapC (X, Y ). In the setting of topological or
simplicial categories, this morphism space (either a topological space or asimplicial set) is an inherent feature of the formalism. It is less obvious how
to define MapC (X, Y ) in the setting of ∞-categories. However, it is at least
clear what to do on the level of the homotopy category.
Definition 1.2.2.1. Let S be a simplicial set containing vertices x and y
and let H denote the homotopy category of spaces. We define MapS (x, y) =highertopoi

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MaphS (x, y) ∈ H to be the object of H representing the space of maps fromx to y in S. Here hS denotes the homotopy category of S regarded as a
H-enriched category (Definition 1.1.5.14).
Warning 1.2.2.2. Let S be a simplicial set. The notation MapS (X, Y ) has
two very different meanings. When X and Y are vertices of S, then ournotation should be interpreted in the sense of Definition 1.2.2.1, so that
MapS (X, Y ) is an object of H. If X and Y are objects of (Set∆ )/S , then we
instead let MapS (X, Y ) denote the simplicial mapping object
Y X ×S X {φ} ∈ Set∆ ,where φ denotes the structural morphism X → S. We trust that it will
be clear from the context which of these two definitions applies in a given
situation.
We now consider the following question: given a simplicial set S containing
a pair of vertices x and y, how can we compute MapS (x, y)? We have defined
MapS (x, y) as an object of the homotopy category H, but for many purposes
it is important to choose a simplicial set M which represents MapS (x, y). The
most obvious candidate for M is the simplicial set MapC[S] (x, y). The advantages of this definition are that it works in all cases (that is, S does not need
to be an ∞-category) and comes equipped with an associative composition
law. However, the construction of the simplicial set MapC[S] (x, y) is quite
complicated. Furthermore, MapC[S] (x, y) is usually not a Kan complex, so
it can be difficult to extract algebraic invariants like homotopy groups evenwhen a concrete description of its simplices is known.
In order to address these shortcomings, we will introduce another simplicial set which represents the homotopy type MapS (x, y) ∈ H, at least when
S is an ∞-category. We define a new simplicial set HomR
S (x, y), the space ofright morphisms from x to y, by letting HomSet∆ (∆n , HomR
S (x, y)) denotethe set of all z : ∆n+1 → S such that z|∆{n+1} = y and z|∆{0,...,n} is aconstant simplex at the vertex x. The face and degeneracy operations on
HomR
S (x, y)n are defined to coincide with corresponding operations on Sn+1 .
We first observe that when S is an ∞-category, HomR
S (x, y) really is a
“space”:
Proposition 1.2.2.3. Let C be an ∞-category containing a pair of objects
x and y. The simplicial set HomR
C (x, y) is a Kan complex.
Proof. It is immediate from the definition that if C is a ∞-category, thenM = HomR
C (x, y) satisfies the Kan extension condition for every horn inclusion Λni ⊆ ∆n , where 0 < i ≤ n. This implies that M is a Kan complex(Proposition 1.2.5.1).
Remark 1.2.2.4. If S is a simplicial set and x, y, z ∈ S0 , then there is no
obvious composition law
RR
HomR
S (x, y) × HomS (y, z) → HomS (x, z).

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We will later see that if S is an ∞-category, then there is a composition law
which is well-defined up to a contractible space of choices. The absence of a
canonical choice for a composition law is the main drawback of HomR
S (x, y) incomparison with MapC[S] (x, y). The main goal of §2.2 is to show that if S is
an ∞-category, then there is a (canonical) isomorphism between HomR
S (x, y)and MapC[S] (x, y) in the homotopy category H. In particular, we will conclude that HomR
S (x, y) represents MapS (x, y) whenever S is an ∞-category.
Remark 1.2.2.5. The definition of HomR
S (x, y) is not self-dual: that is,R
L
HomR
S op (x, y) 6= HomS (y, x) in general. Instead, we define HomS (x, y) =LR
op
HomS op (y, x) , so that HomS (x, y)n is the set of all z ∈ Sn+1 such thatz|∆{0} = x and z|∆{1,...,n+1} is the constant simplex at the vertex y.
Although the simplicial sets HomLS (x, y) and HomR
S (x, y) are generally not
isomorphic to one another, they are homotopy equivalent whenever S is an∞-category. To prove this, it is convenient to define a third, self-dual, space
1of morphisms: let HomS (x, y) = {x} ×S S ∆ ×S {y}. In other words, to givean n-simplex of HomS (x, y), one must give a map f : ∆n × ∆1 → S suchthat f |∆n × {0} is constant at x and f |∆n × {1} is constant at y. We observethat there exist natural inclusions
L
HomR
S (x, y) ,→ HomS (x, y) ←- HomS (x, y),which are induced by retracting the cylinder ∆n × ∆1 onto certain maximaldimensional simplices. We will later show (Corollary 4.2.1.8) that these inclusions are homotopy equivalences provided that S is an ∞-category.
1.2.3 The Homotopy Category
For every ordinary category C, the nerve N(C) is an ∞-category. Informally,
we can describe the situation as follows: the nerve functor is a fully faithful inclusion from the bicategory of categories to the ∞-bicategory of ∞categories. Moreover, this inclusion has a left adjoint:
Proposition 1.2.3.1. The nerve functor Cat → Set∆ is right adjoint to
the functor h: Set∆ → Cat, which associates to every simplicial set S its
homotopy category hS (here we ignore the H-enrichment of hS ).
Proof. Let us temporarily distinguish between the nerve functor N : Cat →Set∆ and the simplicial nerve functor N0 : Cat∆ → Set∆ . These two functors
are related by the fact that N can be written as a compositioni

N0

Cat ⊆ Cat∆ → Set∆ .
The functor π0 : Set∆ → Set is a left adjoint to the inclusion functor Set →Set∆ , so the functor
Cat∆ → Cat
C 7→ hC

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AN OVERVIEW OF HIGHER CATEGORY THEORY

29

is left adjoint to i. It follows that N = N0 ◦i has a left adjoint, given by thecomposition
C[•]hSet∆ → Cat∆ → Cat,
which coincides with the homotopy category functor h : Set∆ → Cat bydefinition.
Remark 1.2.3.2. The formation of the homotopy category is literally left
adjoint to the inclusion Cat ⊆ Cat∆ . The analogous assertion is not quite truein the setting of topological categories because the functor π0 : CG → Set isa left adjoint only when restricted to locally path-connected spaces.
Warning 1.2.3.3. If C is a simplicial category, then we do not necessarily
expect that hC ' hN(C). However, this is always the case when C is fibrantin the sense that every simplicial set MapC (X, Y ) is a Kan complex.
Remark 1.2.3.4. If S is a simplicial set, Joyal ([44]) refers to the categoryhS as the fundamental category of S. This is motivated by the observationthat if S is a Kan complex, then hS is the fundamental groupoid of S in theusual sense.
Our objective for the remainder of this section is to obtain a more explicit
understanding of the homotopy category hS of a simplicial set S. Proposition
1.2.3.1 implies that hS admits the following presentation by generators andrelations:• The objects of hS are the vertices of S.
• For every edge φ : ∆1 → S, there is a morphism φ from φ(0) to φ(1).
• For each σ : ∆2 → S, we have d0 (σ) ◦ d2 (σ) = d1 (σ).
• For each vertex x of S, the morphism s0 x is the identity idx .
If S is an ∞-category, there is a much more satisfying construction of
the category hS . We will describe this construction in detail since it nicely
illustrates the utility of the weak Kan condition of Definition 1.1.2.4.
Let C be an ∞-category. We will construct a category π(C) (which we will
eventually show to be equivalent to the homotopy category hC). The objectsof π(C) are the vertices of C. Given an edge φ : ∆1 → C, we shall say that φ
has source C = φ(0) and target C 0 = φ(1) and write φ : C → C 0 . For eachobject C of C, we let idC denote the degenerate edge s0 (C) : C → C.
Let φ : C → C 0 and φ0 : C → C 0 be a pair of edges of C having the same
source and target. We will say that φ and φ0 are homotopic if there is a
2-simplex σ : ∆2 → C, which we depict as follows:
>C
φ

0
idC 0

!
φ0
/ C 0.
C
In this case, we say that σ is a homotopy between φ and φ0 .

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Proposition 1.2.3.5. Let C be an ∞-category and let C and C 0 be objectsof C. Then the relation of homotopy is an equivalence relation on the edges
joining C to C 0 .
Proof. Let φ : ∆1 → C be an edge. Then s1 (φ) is a homotopy from φ toitself. Thus homotopy is a reflexive relation.
Suppose next that φ, φ0 , φ00 : C → C 0 are edges with the same source andtarget. Let σ be a homotopy from φ to φ0 , and σ 0 a homotopy from φ to
φ00 . Let σ 00 : ∆2 → C denote the constant map at the vertex C 0 . We have acommutative diagramΛ31
_(σ 00 ,•,σ 0 ,σ)

8/ C

τ


∆3 .
Since C is an ∞-category, there exists a 3-simplex τ : ∆3 → C as indicated by
the dotted arrow in the diagram. It is easy to see that d1 (τ ) is a homotopy
from φ0 to φ00 .
As a special case, we can take φ = φ00 ; we then deduce that the relationof homotopy is symmetric. It then follows immediately from the above thatthe relation of homotopy is also transitive.
Remark 1.2.3.6. The definition of homotopy that we have given is notevidently self-dual; in other words, it is not immediately obvious that ahomotopic pair of edges φ, φ0 : C → C 0 of an ∞-category C remain homotopic
when regarded as edges in the opposite ∞-category Cop . To prove this, let σ
be a homotopy from φ to φ0 and consider the commutative diagram
Λ32
_(σ,s1 φ,•,s0 φ)

/8 C

τ


∆3 .
The assumption that C is an ∞-category guarantees a 3-simplex τ rendering
the diagram commutative. The face d2 τ may be regarded as a homotopy
from φ0 to φ in Cop .
We can now define the morphism sets of the category π(C): given vertices
X and Y of C, we let Homπ(C) (X, Y ) denote the set of homotopy classes ofedges φ : X → Y in C. For each edge φ : ∆1 → C, we let [φ] denote thecorresponding morphism in π(C).
We define a composition law on π(C) as follows. Suppose that X, Y , andZ are vertices of C and that we are given edges φ : X → Y , ψ : Y → Z. The
pair (φ, ψ) determines a map Λ21 → C. Since C is an ∞-category, this map
extends to a 2-simplex σ : ∆2 → C. We now define [ψ] ◦ [φ] = [d1 σ].

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AN OVERVIEW OF HIGHER CATEGORY THEORY

Proposition 1.2.3.7. Let C be an ∞-category. The composition law on π(C)is well-defined. In other words, the homotopy class [ψ] ◦ [φ] does not dependon the choice of ψ representing [ψ], the choice of φ representing [φ], or the
choice of the 2-simplex σ.
Proof. We begin by verifying the independence of the choice of σ. Supposethat we are given two 2-simplices σ, σ 0 : ∆2 → C, satisfying
d0 σ = d0 σ 0 = ψ
d2 σ = d2 σ 0 = φ.
Consider the diagramΛ31
_(s1 ψ,•,σ 0 ,σ)

/8 C

τ


∆3 .
Since C is an ∞-category, there exists a 3-simplex τ as indicated by thedotted arrow. It follows that d1 τ is a homotopy from d1 σ to d1 σ 0 .
We now show that [ψ] ◦ [φ] depends on ψ and φ only up to homotopy. Inview of Remark 1.2.3.6, the assertion is symmetric with respect to ψ and φ;it will therefore suffice to show that [ψ] ◦ [φ] does not change if we replace
φ by a morphism φ0 which is homotopic to φ. Let σ be a 2-simplex withd0 σ = ψ and d2 σ = φ, and let σ 0 be a homotopy from φ to φ0 . Consider the
diagramΛ31_(s0 ψ,•,σ,σ 0 )

/8 C

τ


∆3 .
Again, the hypothesis that C is an ∞-category guarantees the existence ofa 3-simplex τ as indicated in the diagram. Let σ 00 = d1 τ . Then [ψ] ◦ [φ0 ] =[d1 σ 0 ]. But d2 σ 00 = d1 σ 0 by construction, so that [ψ] ◦ [φ] = [ψ] ◦ [φ0 ], as
desired.
Proposition 1.2.3.8. If C is an ∞-category, then π(C) is a category.
Proof. Let C be a vertex of C. We first verify that [idC ] is an identity withrespect to the composition law on π(C). For every edge φ : C 0 → C in C, the2-simplex s1 (φ) verifies the equation[idC ] ◦ [φ] = [φ].
This proves that idC is a left identity; the dual argument (Remark 1.2.3.6)
shows that [idC ] is a right identity.
The only other thing we need to check is the associative law for composition in π(C). Suppose we are given a composable sequence of edges
φ

φ0

φ00

C → C 0 → C 00 → C 000 .

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CHAPTER 1

Choose 2-simplices σ, σ 0 , σ 00 : ∆2 → C corresponding to diagrams
>C

0

φ

φ0

!
/ C 00

ψ

C

00
C
>
ψ

φ00

!
/ C 000

θ

C

0

00
C
>

φ

φ00

ψ0

C0

!
/ C 000 ,respectively. Then [φ0 ] ◦ [φ] = [ψ], [φ00 ] ◦ [ψ] = [θ], and [φ00 ] ◦ [φ0 ] = [ψ 0 ].
Consider the diagramΛ32
_(σ 00 ,σ 0 ,•,σ)

/8 C

τ


∆3 .
Since C is an ∞-category, there exists a 3-simplex τ rendering the diagram
commutative. Then d2 (τ ) verifies the equation [ψ 0 ] ◦ [φ] = [θ], so that([φ00 ] ◦ [φ0 ]) ◦ [φ] = [θ] = [φ00 ] ◦ [ψ] = [φ00 ] ◦ ([φ0 ] ◦ [φ]),as desired.
We now show that if C is an ∞-category, then π(C) is naturally equivalent(in fact, isomorphic) to hC.
Proposition 1.2.3.9. Let C be an ∞-category. There exists a unique functor
F : hC → π(C) with the following properties:(1) On objects, F is the identity map.
(2) For every edge φ of C, F (φ) = [φ].
Moreover, F is an isomorphism of categories.
Proof. The existence and uniqueness of F follows immediately from our presentation of hC by generators and relations. It is obvious that F is bijective
on objects and surjective on morphisms. To complete the proof, it will suffice
to show that F is faithful.
We first show that every morphism f : x → y in hC may be written as
φ for some φ ∈ C1 . Since the morphisms in hC are generated by morphisms

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AN OVERVIEW OF HIGHER CATEGORY THEORY

having the form φ under composition, it suffices to show that the set of suchmorphisms contains all identity morphisms and is stable under composition.
The first assertion is clear since s0 x = idx . For the second, we note that if
φ : x → y and φ0 : y → z are composable edges, then there exists a 2-simplex
σ : ∆2 → C which we may depict as follows:
φ

x

?y

φ0

ψ

/ z.

Thus φ0 ◦ φ = ψ.
Now suppose that φ, φ0 : x → y are such that [φ] = [φ0 ]; we wish to showthat φ = φ0 . By definition, there exists a homotopy σ : ∆2 → C joining φ
and φ0 . The existence of σ entails the relation
idy ◦φ = φ0in the homotopy category hC, so that φ = φ0 , as desired.
1.2.4 Objects, Morphisms, and Equivalences
As in ordinary category theory, we may speak of objects and morphisms in ahigher category C. If C is a topological (or simplicial) category, these should
be understood literally as the objects and morphisms in the underlying category of C. We may also apply this terminology to ∞-categories (or evenmore general simplicial sets): if S is a simplicial set, then the objects of S
are the vertices ∆0 → S, and the morphisms of S are the edges ∆1 → S. A
morphism φ : ∆1 → S is said to have source X = φ(0) and target Y = φ(1);we will often denote this by writing φ : X → Y . If X : ∆0 → S is an objectof S, we will write idX = s0 (X) : X → X and refer to this as the identity
morphism of X.
If f, g : X → Y are two morphisms in a higher category C, then f andg are homotopic if they determine the same morphism in the homotopy
category hC. In the setting of ∞-categories, this coincides with the notionof homotopy introduced in the previous section. In the setting of topologicalcategories, this simply means that f and g lie in the same path componentof MapC (X, Y ). In either case, we will sometimes indicate this relationshipbetween f and g by writing f ' g.
A morphism f : X → Y in an ∞-category C is said to be an equivalence
if it determines an isomorphism in the homotopy category hC. We say that
X and Y are equivalent if there is an equivalence between them (in otherwords, if they are isomorphic as objects of hC).
If C is a topological category, then the requirement that a morphism f :X → Y be an equivalence is quite a bit weaker than the requirement that f
be an isomorphism. In fact, we have the following:
Proposition 1.2.4.1. Let f : X → Y be a morphism in a topological category. The following conditions are equivalent:highertopoi

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CHAPTER 1(1) The morphism f is an equivalence.
(2) The morphism f has a homotopy inverse g : Y → X: that is, a morphism g such that f ◦ g ' idY and g ◦ f ' idX .
(3) For every object Z ∈ C, the induced map MapC (Z, X) → MapC (Z, Y )is a homotopy equivalence.
(4) For every object Z ∈ C, the induced map MapC (Z, X) → MapC (Z, Y )is a weak homotopy equivalence.
(5) For every object Z ∈ C, the induced map MapC (Y, Z) → MapC (X, Z)
is a homotopy equivalence.
(6) For every object Z ∈ C, the induced map MapC (Y, Z) → MapC (X, Z)
is a weak homotopy equivalence.
Proof. It is clear that (2) is merely a reformulation of (1). We will show that(2) ⇒ (3) ⇒ (4) ⇒ (1); the implications (2) ⇒ (5) ⇒ (6) ⇒ (1) follow using
the same argument.
To see that (2) implies (3), we note that if g is a homotopy inverse tof , then composition with g gives a map MapC (Z, Y ) → MapC (Z, X) which
is homotopy inverse to composition with f . It is clear that (3) implies (4).
Finally, if (4) holds, then we note that X and Y represent the same functor
on hC so that f induces an isomorphism between X and Y in hC.
Example 1.2.4.2. Let C be the category of CW complexes which we regard
as a topological category by endowing each of the sets HomC (X, Y ) with the(compactly generated) compact open topology. A pair of objects X, Y ∈ C
are equivalent (in the sense defined above) if and only if they are homotopy
equivalent (in the sense of classical topology).
If C is an ∞-category (topological category, simplicial category), then weshall write X ∈ C to mean that X is an object of C. We will generally
understand that all meaningful properties of objects are invariant underequivalence. Similarly, all meaningful properties of morphisms are invariant
under homotopy and under composition with equivalences.
In the setting of ∞-categories, there is a very useful characterization ofequivalences which is due to Joyal.
Proposition 1.2.4.3 (Joyal [44]). Let C be an ∞-category and φ : ∆1 → C
a morphism of C. Then φ is an equivalence if and only if, for every n ≥ 2 andevery map f0 : Λn0 → C such that f0 |∆{0,1} = φ, there exists an extension of
f0 to ∆n .
The proof requires some ideas which we have not yet introduced and will
be given in §2.1.2.

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35

1.2.5 ∞-Groupoids and Classical Homotopy Theory
Let C be an ∞-category. We will say that C is an ∞-groupoid if the homotopy category hC is a groupoid: in other words, if every morphism in C is anequivalence. In §1.1.1, we asserted that the theory of ∞-groupoids is equivalent to classical homotopy theory. We can now formulate this idea in a very
precise way:
Proposition 1.2.5.1 (Joyal [43]). Let C be a simplicial set. The following
conditions are equivalent:(1) The simplicial set C is an ∞-category, and its homotopy category hC
is a groupoid.
(2) The simplicial set C satisfies the extension condition for all horn inclusions Λni ⊆ ∆n for 0 ≤ i < n.
(3) The simplicial set C satisfies the extension condition for all horn inclusions Λni ⊆ ∆n for 0 < i ≤ n.
(4) The simplicial set C is a Kan complex; in other words, it satisfies theextension condition for all horn inclusions Λni ⊆ ∆n for 0 ≤ i ≤ n.
Proof. The equivalence (1) ⇔ (2) follows immediately from Proposition
1.2.4.3. Similarly, the equivalence (1) ⇔ (3) follows by applying Proposition 1.2.4.3 to Cop . We conclude by observing that (4) ⇔ (2) ∧ (3).
Remark 1.2.5.2. The assertion that we can identify ∞-groupoids withspaces is less obvious in other formulations of higher category theory. For
example, suppose that C is a topological category whose homotopy category
hC is a groupoid. For simplicity, we will assume furthermore that C has
a single object X. We may then identify C with the topological monoid
M = HomC (X, X). The assumption that hC is a groupoid is equivalent tothe assumption that the discrete monoid π0 M is a group. In this case, one
can show that the unit map M → ΩBM is a weak homotopy equivalence,
where BM denotes the classifying space of the topological monoid M . In
other words, up to equivalence, specifying C (together with the object X) is
equivalent to specifying the space BM (together with its base point).
Informally, we might say that the inclusion functor i from Kan complexes
to ∞-categories exhibits the ∞-category of (small) ∞-groupoids as a fullsubcategory of the ∞-bicategory of (small) ∞-categories. Conversely, every ∞-category C has an “underlying” ∞-groupoid, which is obtained bydiscarding the noninvertible morphisms of C:
Proposition 1.2.5.3 ([44]). Let C be an ∞-category. Let C0 ⊆ C be thelargest simplicial subset of C having the property that every edge of C0 is
an equivalence in C. Then C0 is a Kan complex. It may be characterized by
the following universal property: for any Kan complex K, the induced map
HomSet∆ (K, C0 ) → HomSet∆ (K, C) is a bijection.

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Proof. It is straightforward to check that C0 is an ∞-category. Moreover, if
f is a morphism in C0 , then f has a homotopy inverse g ∈ C. Since g isitself an equivalence in C, we conclude that g belongs to C0 and is therefore
a homotopy inverse to f in C0 . In other words, every morphism in C0 is anequivalence, so that C0 is a Kan complex by Proposition 1.2.5.1. To prove
the last assertion, we observe that if K is an ∞-category, then any map ofsimplicial sets φ : K → C carries equivalences in K to equivalences in C. In
particular, if K is a Kan complex, then φ factors (uniquely) through C0 .
We can describe the situation of Proposition 1.2.5.3 by saying that C0
is the largest Kan complex contained in C. The functor C 7→ C0 is rightadjoint to the inclusion functor from Kan complexes to ∞-categories. It iseasy to see that this right adjoint is an invariant notion: that is, a categoricalequivalence of ∞-categories C → D induces a homotopy equivalence C0 → D0
of Kan complexes.
Remark 1.2.5.4. It is easy to give analogous constructions in the case of
topological or simplicial categories. For example, if C is a topological category, then we can define C0 to be another topological category with the same
objects as C, where MapC0 (X, Y ) ⊆ MapC (X, Y ) is the subspace consistingof equivalences in MapC (X, Y ), equipped with the subspace topology.
Remark 1.2.5.5. We will later introduce a relative version of the construction described in Proposition 1.2.5.3, which applies to certain families of∞-categories (Corollary 2.4.2.5).
Although the inclusion functor from Kan complexes to ∞-categories does
not literally have a left adjoint, it does have such an in a higher-categoricalsense. This left adjoint is computed by any “fibrant replacement” functor(for the usual model structure) from Set∆ to itself, for example, the functor
S 7→ Sing |S|. The unit map u : S → Sing |S| is always a weak homotopy
equivalence but generally not a categorical equivalence. For example, if S is
an ∞-category, then u is a categorical equivalence if and only if S is a Kancomplex. In general, Sing |S| may be regarded as the ∞-groupoid obtained
from S by freely adjoining inverses to all the morphisms in S.
Remark 1.2.5.6. The inclusion functor i and its homotopy-theoretic left
adjoint may also be understood using the formalism of localizations of model
categories. In addition to its usual model category structure, the category
Set∆ of simplicial sets may be endowed with the Joyal model structure, which
we will define in §2.2.5. These model structures have the same cofibrations(in both cases, the cofibrations are simply the monomorphisms of simplicial sets). However, the Joyal model structure has fewer weak equivalences(categorical equivalences rather than weak homotopy equivalences) and consequently more fibrant objects (all ∞-categories rather than only Kan complexes). It follows that the usual homotopy theory of simplicial sets is a localization of the homotopy theory of ∞-categories. The identity functor fromhighertopoi

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Set∆ to itself determines a Quillen adjunction between these two homotopy
theories, which plays the role of i and its left adjoint.
1.2.6 Homotopy Commutativity versus Homotopy Coherence
Let C be an ∞-category (topological category, simplicial category). To a first
approximation, working in C is like working in its homotopy category hC:
up to equivalence, C and hC have the same objects and morphisms. The
main difference between hC and C is that in C one must not ask whether
or not morphisms are equal; instead one should ask whether or not they
are homotopic. If so, the homotopy itself is an additional datum which wewill need to consider. Consequently, the notion of a commutative diagram
in hC, which corresponds to a homotopy commutative diagram in C, is quite
unnatural and usually needs to be replaced by the more refined notion of ahomotopy coherent diagram in C.
To understand the problem, let us suppose that F : I → H is a functorfrom an ordinary category I into the homotopy category of spaces H. In
other words, F assigns to each object X ∈ I a space (say, a CW complex)
F (X), and to each morphism φ : X → Y in I a continuous map of spaces
F (φ) : F (X) → F (Y ) (well-defined up to homotopy), such that F (φ ◦ ψ) is
homotopic to F (φ) ◦ F (ψ) for any pair of composable morphisms φ, ψ in I.
In this situation, it may or may not be possible to lift F to an actual functor
Fe from I to the ordinary category of topological spaces such that Fe induces
a functor I → H which is naturally isomorphic to F . In general, there are
obstructions to both the existence and the uniqueness of the lifting Fe, evenup to homotopy. To see this, let us suppose for a moment that Fe exists, sothat there exist homotopies kφ : Fe(φ) ' F (φ). These homotopies determine
additional data on F : namely, one obtains a canonical homotopy hφ,ψ fromF (φ ◦ ψ) to F (φ) ◦ F (ψ) by composing
F (φ ◦ ψ) ' Fe(φ ◦ ψ) = Fe(φ) ◦ Fe(ψ) ' F (φ) ◦ F (ψ).
The functor F to the homotopy category H should be viewed as a first
approximation to Fe; we obtain a second approximation when we take into
account the homotopies hφ,ψ . These homotopies are not arbitrary: the associativity of composition gives a relationship between hφ,ψ , hψ,θ , hφ,ψ◦θ , andhφ◦ψ,θ , for a composable triple of morphisms (φ, ψ, θ) in I. This relationshipmay be formulated in terms of the existence of a certain higher homotopy,
which is once again canonically determined by Fe (and the homotopies kφ ).
To obtain the next approximation to Fe, we should take these higher homotopies into account and formulate the associativity properties that they
enjoy, and so on. Roughly speaking, a homotopy coherent diagram in C isa functor F : I → hC together with all of the extra data that would beavailable if we were able to lift F to a functor Fe : I → C.
The distinction between homotopy commutativity and homotopy coherence is arguably the main difficulty in working with higher categories. The
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the setting of a general topological category. However, the amount of datarequired to specify a homotopy coherent diagram is considerable, so theconcept is quite difficult to employ in practical situations.
Remark 1.2.6.1. Let I be an ordinary category and let C be a topologicalcategory. Any functor F : I → C determines a homotopy coherent diagram
in C (with all of the homotopies involved being constant). For many topological categories C, the converse fails: not every homotopy-coherent diagram
in C can be obtained in this way, even up to equivalence. In these cases, it is
the notion of homotopy coherent diagram which is fundamental; a homotopy
coherent diagram should be regarded as “just as good” as a strictly commutative diagram for ∞-categorical purposes. As evidence for this, we remarkthat given an equivalence C0 → C, a strictly commutative diagram F : I → C
cannot always be lifted to a strictly commutative diagram in C0 ; however, it
can always be lifted (up to equivalence) to a homotopy coherent diagram in
C0 .
One of the advantages of working with ∞-categories is that the definitionof a homotopy coherent diagram is easy to formulate. We can simply definea homotopy coherent diagram in an ∞-category C to be a map of simplicial
sets f : N(I) → C. The restriction of f to simplices of low dimension encodes
the induced map on homotopy categories. Specifying f on higher-dimensional
simplices gives precisely the “coherence data” that the above discussion callsfor.
Remark 1.2.6.2. Another possible approach to the problem of homotopy
coherence is to restrict our attention to simplicial (or topological) categories
C in which every homotopy coherent diagram is equivalent to a strictly commutative diagram. For example, this is always true when C arises from a
simplicial model category (Proposition 4.2.4.4). Consequently, in the framework of model categories, it is possible to ignore the theory of homotopy
coherent diagrams and work with strictly commutative diagrams instead.
This approach is quite powerful, particularly when combined with the observation that every simplicial category C admits a fully faithful embedding
into a simplicial model category (for example, one can use a simplicially enriched version of the Yoneda embedding). This idea can be used to show thatevery homotopy coherent diagram in C can be “straightened” to a commutative diagram, possibly after replacing C by an equivalent simplicial category(for a more precise version of this statement, we refer the reader to Corollary
4.2.4.7).
1.2.7 Functors Between Higher Categories
The notion of a homotopy coherent diagram in an higher category C is a
special case of the more general notion of a functor F : I → C between
higher categories (specifically, it is the special case in which I is assumed
to be an ordinary category). Just as the collection of all ordinary categories

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forms a bicategory (with functors as morphisms and natural transformations
as 2-morphisms), the collection of all ∞-categories can be organized into an∞-bicategory. In particular, for any ∞-categories C and C0 , we expect to beable to construct an ∞-category Fun(C, C0 ) of functors from C to C0 .
In the setting of topological categories, the construction of an appropriate
mapping object Fun(C, C0 ) is quite difficult. The naive guess is that Fun(C, C0 )should be a category of topological functors from C to C0 : that is, functors
which induce continuous maps between morphism spaces. However, we sawin §1.2.6 that this notion is generally too rigid, even in the special case where
C is an ordinary category.
Remark 1.2.7.1. Using the language of model categories, one might saythat the problem is that not every topological category is cofibrant. If C
is a cofibrant topological category (for example, if C = | C[S]|, where S isa simplicial set), then the collection of topological functors from C to C0
is large enough to contain representatives for every ∞-categorical functorfrom C to C0 . Most ordinary categories are not cofibrant when viewed as
topological categories. More importantly, the property of being cofibrant is
not stable under products, so that naive attempts to construct a mapping
object Fun(C, C0 ) need not give the correct answer even when C itself isassumed cofibrant (if C is cofibrant, then we are guaranteed to have “enough”
topological functors C → C0 to represent all functors between the underlying∞-categories but not necessarily enough natural transformations between
them; note that the product C ×[1] is usually not cofibrant, even in thesimplest nontrivial case where C = [1].) This is arguably the most important
technical disadvantage of the theory of topological (or simplicial) categories
as an approach to higher category theory.
The construction of functor categories is much easier to describe in theframework of ∞-categories. If C and D are ∞-categories, then we can simply
define a functor from C to D to be a map p : C → D of simplicial sets.
Notation 1.2.7.2. Let C and D be simplicial sets. We let Fun(C, D) denotethe simplicial set MapSet∆ (C, D) parametrizing maps from C to D. We will
use this notation only when D is an ∞-category (the simplicial set C will often, but not always, be an ∞-category as well). We will refer to Fun(C, D) as
the ∞-category of functors from C to D (see Proposition 1.2.7.3 below). We
will refer to morphisms in Fun(C, D) as natural transformations of functors,
and equivalences in Fun(C, D) as natural equivalences.
Proposition 1.2.7.3. Let K be an arbitrary simplicial set.
(1) For every ∞-category C, the simplicial set Fun(K, C) is an ∞-category.
(2) Let C → D be a categorical equivalence of ∞-categories. Then the
induced map Fun(K, C) → Fun(K, D) is a categorical equivalence.
(3) Let C be an ∞-category and K → K 0 a categorical equivalence of simplicial sets. Then the induced map Fun(K 0 , C) → Fun(K, C) is a categorical equivalence.

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The proof makes use of the Joyal model structure on Set∆ and will begiven in §2.2.5.
1.2.8 Joins of ∞-Categories
Let C and C0 be ordinary categories. We will define a new category C ? C0 ,called the join of C and C0 . An object of C ? C0 is either an object of C or an
object of C0 . The morphism sets are given as follows:
HomC (X, Y ) if X, Y ∈ CHom 0 (X, Y ) if X, Y ∈ C0
C
HomC ? C0 (X, Y ) =∅
if X ∈ C0 , Y ∈ C∗
if X ∈ C, Y ∈ C0 .
Composition of morphisms in C ? C0 is defined in the obvious way.
The join construction described above is often useful when discussing diagram categories, limits, and colimits. In this section, we will introduce a
generalization of this construction to the ∞-categorical setting.
Definition 1.2.8.1. If S and S 0 are simplicial sets, then the simplicial set
S ? S 0 is defined as follows: for each nonempty finite linearly ordered set J,
we seta(S ? S 0 )(J) =S(I) × S 0 (I 0 ),J=I∪I 0where the union is taken over all decompositions of J into disjoint subsets
I and I 0 , satisfying i < i0 for all i ∈ I, i0 ∈ I 0 . Here we allow the possibility
that either I or I 0 is empty, in which case we agree to the convention that
S(∅) = S 0 (∅) = ∗.
More concretely, we have[

(S ? S 0 )n = Sn ∪ Sn0 ∪Si × Sj0 .

i+j=n−1

The join operation endows Set∆ with the structure of a monoidal category(see §A.1.3). The identity for the join operation is the empty simplicial set∅ = ∆−1 . More generally, we have natural isomorphisms φij : ∆i−1 ? ∆j−1 '∆(i+j)−1 for all i, j ≥ 0.
Remark 1.2.8.2. The operation ? is essentially determined by the isomorphisms φij , together with its behavior under the formation of colimits: forany fixed simplicial set S, the functors
T 7→ T ? S
T 7→ S ? T
commute with colimits when regarded as functors from Set∆ to the undercategory (Set∆ )S/ of simplicial sets under S.

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Passage to the nerve carries joins of categories into joins of simplicial sets.
More precisely, for every pair of categories C and C0 , there is a canonicalisomorphism
N(C ? C0 ) ' N(C) ? N(C0 ).
(The existence of this isomorphism persists when we allow C and C0 to be
simplicial or topological categories and apply the appropriate generalizationof the nerve functor.) This suggests that the join operation on simplicial sets
is the appropriate ∞-categorical analogue of the join operation on categories.
We remark that the formation of joins does not commute with the functor
C[•]. However, the simplicial category C[S ?S 0 ] contains C[S] and C[S 0 ] as full
(simplicial) subcategories and contains no morphisms from objects of C[S 0 ] toobjects of C[S]. Consequently, there is unique map φ : C[S ?S 0 ] → C[S]?C[S 0 ]which reduces to the identity on C[S] and C[S 0 ]. We will later show that φis an equivalence of simplicial categories (Corollary 4.2.1.4).
We conclude by recording a pleasant property of the join operation:
Proposition 1.2.8.3 (Joyal [44]). If S and S 0 are ∞-categories, then S ? S 0
is an ∞-category.
Proof. Let p : Λni → S ? S 0 be a map, with 0 < i < n. If p carries Λni entirely
into S ⊆ S ? S 0 or into S 0 ⊆ S ? S 0 , then we deduce the existence of an
extension of p to ∆n using the assumption that S and S 0 are ∞-categories.
Otherwise, we may suppose that p carries the vertices {0, . . . , j} into S, andthe vertices {j + 1, . . . , n} into S 0 . We may now restrict p to obtain maps∆{0,...,j} → S∆{j+1,...,n} → S 0 ,which together determine a map ∆n → S ? S 0 extending p.
Notation 1.2.8.4. Let K be a simplicial set. The left cone K / is defined
to be the join ∆0 ? K. Dually, the right cone K . is defined to be the join
K ?∆0 . Either cone contains a distinguished vertex (belonging to ∆0 ), which
we will refer to as the cone point.
1.2.9 Overcategories and Undercategories
Let C be an ordinary category and X ∈ C an object. The overcategory C/X
is defined as follows: the objects of C/X are morphisms Y → X in C having
target X. Morphisms are given by commutative triangles
/Z

Y

X

~

and composition is defined in the obvious way.

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One can rephrase the definition of the overcategory as follows. Let [0]denote the category with a single object possessing only an identity morphism. Then specifying an object X ∈ C is equivalent to specifying a functor
x : [0] → C. The overcategory C/X may then be described by the following
universal property: for any category C0 , we have a bijection
Hom(C0 , C/X ) ' Homx (C0 ?[0], C),
where the subscript on the right hand side indicates that we consider only
those functors C0 ?[0] → C whose restriction to [0] coincides with x.
Our goal in this section is to generalize the construction of overcategories
to the ∞-categorical setting. Let us begin by working in the framework of
topological categories. In this case, there is a natural candidate for the relevant overcategory. Namely, if C is a topological category containing an object
X, then the overcategory C/X (defined as above) has the structure of a topological category where each morphism space MapC/X (Y, Z) is topologized asa subspace of MapC (Y, Z) (here we are identifying an object of C/X with its
image in C). This topological category is usually not a model for the correct∞-categorical slice construction. The problem is that a morphism in C/X
consists of a commutative triangle/ZY~
X
of objects over X. To obtain the correct notion, we should also allow triangles
which commute only up to homotopy.
Remark 1.2.9.1. In some cases, the naive overcategory C/X is a good
approximation to the correct construction: see Lemma 6.1.3.13.
In the setting of ∞-categories, Joyal has given a much simpler descriptionof the desired construction (see [43]). This description will play a vitally
important role throughout this book. We begin by noting the following:
Proposition 1.2.9.2 ([43]). Let S and K be simplicial sets, and p : K →S an arbitrary map. There exists a simplicial set S/p with the following
universal property:
HomSet∆ (Y, S/p ) = Homp (Y ? K, S),
where the subscript on the right hand side indicates that we consider only
those morphisms f : Y ? K → S such that f |K = p.
Proof. One defines (S/p )n to be Homp (∆n ? K, S). The universal property
holds by definition when Y is a simplex. It holds in general because bothsides are compatible with the formation of colimits in Y .
Let p : K → S be as in Proposition 1.2.9.2. If S is an ∞-category, we
will refer to S/p as an overcategory of S or as the ∞-category of objects of
S over p. The following result guarantees that the operation of passing toovercategories is well-behaved:

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Proposition 1.2.9.3. Let p : K → C be a map of simplicial sets and supposethat C is an ∞-category. Then C/p is an ∞-category. Moreover, if q : C → C0
is a categorical equivalence of ∞-categories, then the induced map C/p →C0/qp is a categorical equivalence as well.
The proof requires a number of ideas which we have not yet introduced
and will be postponed (see Corollary 2.1.2.2 for the first assertion, and §2.4.5
for the second).
Remark 1.2.9.4. Let C be an ∞-category. In the particular case wherep : ∆n → C classifies an n-simplex σ ∈ Cn , we will often write C/σ in place of
C/p . In particular, if X is an object of C, we let C/X denote the overcategory
C/p , where p : ∆0 → C has image X.
Remark 1.2.9.5. Let p : K → C be a map of simplicial sets. The preceding
discussion can be dualized, replacing Y ? K by K ? Y ; in this case we denotethe corresponding simplicial set by Cp/ , which (if C is an ∞-category) we
will refer to as an undercategory of C. In the special case where K = ∆n andp classifies a simplex σ ∈ Cn , we will also write Cσ/ for Cp/ ; in particular, we
will write CX/ when X is an object of C.
Remark 1.2.9.6. If C is an ordinary category and X ∈ C, then there is a
canonical isomorphism N(C)/X ' N(C/X ). In other words, the overcategory
construction for ∞-categories can be regarded as a generalization of therelevant construction from classical category theory.

1.2.10 Fully Faithful and Essentially Surjective Functors
Definition 1.2.10.1. Let F : C → D be a functor between topological categories (simplicial categories, simplicial sets). We will say that F is essentiallysurjective if the induced functor hF : hC → hD is essentially surjective (thatis, if every object of D is equivalent to F (X) for some X ∈ C).
We will say that F is fully faithful if hF is a fully faithful functor of
H-enriched categories. In other words, F is fully faithful if and only if,for every pair of objects X, Y ∈ C, the induced map MaphC (X, Y ) →MaphD (F (X), F (Y )) is an isomorphism in the homotopy category H.
Remark 1.2.10.2. Because Definition 1.2.10.1 makes reference only to thehomotopy categories of C and D, it is invariant under equivalence and underoperations which pass between the various models for higher category theory
that we have introduced.
Just as in ordinary category theory, a functor F is an equivalence if andonly if it is fully faithful and essentially surjective.

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1.2.11 Subcategories of ∞-Categories
Let C be an ∞-category and let (hC)0 ⊆ hC be a subcategory of its homotopy
category. We can then form a pullback diagram of simplicial sets:
C0

/C


N(hC)0/ N(hC).

We will refer to C0 as the subcategory of C spanned by (hC)0 . In general, we
will say that a simplicial subset C0 ⊆ C is a subcategory of C if it arises via
this construction.
Remark 1.2.11.1. We use the term “subcategory,” rather than “sub-∞category,” in order to avoid awkward language. The terminology is not meant
to suggest that C0 is itself a category or is isomorphic to the nerve of acategory.
In the case where (hC)0 is a full subcategory of hC, we will say that C0
is a full subcategory of C. In this case, C0 is determined by the set C00 of
those objects X ∈ C which belong to C0 . We will then say that C0 is the fullsubcategory of C spanned by C00 .
It follows from Remark 1.2.2.4 that the inclusion C0 ⊆ C is fully faithful.
In general, any fully faithful functor f : C00 → C factors as a composition
f0

f 00

C00 → C0 → C,
where f 0 is an equivalence of ∞-categories and f 00 is the inclusion of the fullsubcategory C0 ⊆ C spanned by the set of objects f (C000 ) ⊆ C0 .
1.2.12 Initial and Final Objects
If C is an ordinary category, then an object X ∈ C is said to be final if
HomC (Y, X) consists of a single element for every Y ∈ C. Dually, an object
X ∈ C is initial if it is final when viewed as an object of Cop . The goal ofthis section is to generalize these definitions to the ∞-categorical setting.
If C is a topological category, then a candidate definition immediately
presents itself: we could ignore the topology on the morphism spaces andconsider those objects of C which are final when C is regarded as an ordinary
category. This requirement is unnaturally strong. For example, the category
CG of compactly generated Hausdorff spaces has a final object: the topological space ∗, consisting of a single point. However, there are objects of CG
which are equivalent to ∗ (any contractible space) but not isomorphic to ∗(and therefore not final objects of CG, at least in the classical sense). Since
any reasonable ∞-categorical notion is stable under equivalence, we need tofind a weaker condition.
Definition 1.2.12.1. Let C be a topological category (simplicial category,
simplicial set). An object X ∈ C is final if it is final in the homotopy category

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hC, regarded as a category enriched over H. In other words, X is final if andonly if for each Y ∈ C, the mapping space MaphC (Y, X) is weakly contractible(that is, a final object of H).
Remark 1.2.12.2. Since Definition 1.2.12.1 makes reference only to thehomotopy category hC, it is invariant under equivalence and under passing
between the various models for higher category theory.
In the setting of ∞-categories, it is convenient to employ a slightly moresophisticated definition, which we borrow from [43].
Definition 1.2.12.3. Let C be a simplicial set. A vertex X of C is strongly
final if the projection C/X → C is a trivial fibration of simplicial sets.
In other words, a vertex X of C is strongly final if and only if any map
f0 : ∂ ∆n → C such that f0 (n) = X can be extended to a map f : ∆n → C.
Proposition 1.2.12.4. Let C be an ∞-category containing an object Y.
The object Y is strongly final if and only if, for every object X ∈ C, the Kancomplex HomR
C (X, Y ) is contractible.
Proof. The “only if” direction is clear: the space HomR
C (X, Y ) is the fiber ofthe projection p : C/Y → C over the vertex X. If p is a trivial fibration, thenthe fiber is a contractible Kan complex. Since p is a right fibration (Corollary
2.1.2.2), the converse holds as well (Lemma 2.1.3.4).
Corollary 1.2.12.5. Let C be a simplicial set. Every strongly final object of
C is a final object of C. The converse holds if C is an ∞-category.
Proof. Let [0] denote the category with a single object and a single morphism. Suppose that Y is a strongly final vertex of C. Then there exists a
retraction of C. onto C carrying the cone point to Y . Consequently, we obtain a retraction of (H-enriched) homotopy categories from hC ? [0] to hC
carrying the unique object of [0] to Y . This implies that Y is final in hC, so
that Y is a final object of C.
To prove the converse, we note that if C is an ∞-category, then the Kancomplex HomR
C (X, Y ) represents the homotopy type MapC (X, Y ) ∈ H; by
Proposition 1.2.12.4 this space is contractible for all X if and only if Y isstrongly final.
Remark 1.2.12.6. The above discussion dualizes in an evident way, so that
we have a notion of initial objects of an ∞-category C.
Example 1.2.12.7. Let C be an ordinary category containing an object X.
Then X is a final (initial) object of the ∞-category N(C) if and only if it isa final (initial) object of C in the usual sense.
Remark 1.2.12.8. Definition 1.2.12.3 is only natural in the case where C is
an ∞-category. For example, if C is not an ∞-category, then the collectionof strongly final vertices of C need not be stable under equivalence.

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An ordinary category C may have more than one final object, but any
two final objects are uniquely isomorphic to one another. In the setting of∞-categories, an analogous statement holds but is slightly more complicatedbecause the word “unique” needs to be interpreted in a homotopy-theoretic
sense:
Proposition 1.2.12.9 (Joyal). Let C be a ∞-category and let C0 be the fullsubcategory of C spanned by the final vertices of C. Then C0 either is empty
or is a contractible Kan complex.
Proof. We wish to prove that every map p : ∂ ∆n → C0 can be extended toan n-simplex of C0 . If n = 0, this is possible unless C0 is empty. For n > 0,
the desired extension exists because p carries the nth vertex of ∂ ∆n to a
final object of C.
1.2.13 Limits and Colimits
An important consequence of the distinction between homotopy commutativity and homotopy coherence is that the appropriate notions of limit andcolimit in a higher category C do not coincide with the notions of limit andcolimit in the homotopy category hC (where limits and colimits often do notexist). Limits and colimits in C are often referred to as homotopy limits andhomotopy colimits to avoid confusing them with ordinary limits and colimits.
Homotopy limits and colimits can be defined in a topological category, but
the definition is rather complicated. We will review a few special cases here
and discuss the general definition in the Appendix (§A.2.8).
Example 1.2.13.1. Let {Xα } be aQfamily of objects in a topological category C. A homotopy product X = α Xα is an object of C equipped withmorphisms fα : X → Xα which induce a weak homotopy equivalence
Y
MapC (Y, X) →MapC (Y, Xα )αfor every object Y ∈ C.
Passing to path components and using the fact that π0 commutes withproducts, we deduce thatY
HomhC (Y, X) 'HomhC (Y, Xα ),αso that any homotopy product in C is also a product in hC. In particular,
the object X is determined up to canonical isomorphism in hC.
In the special case where the index set is empty, we recover the notionof a final object of C: an object X for which each of the mapping spaces
MapC (Y, X) is weakly contractible.
Example 1.2.13.2. Given two morphisms π : X → Z and ψ : Y → Z
in a topological category C, let us define MapC (W, X ×hZ Y ) to be thespace consisting of points p ∈ MapC (W, X) and q ∈ MapC (W, Y ) together

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47

with a path r : [0, 1] → MapC (W, Z) joining π ◦ p to ψ ◦ q. We endow
MapC (W, X ×hZ Y ) with the obvious topology, so that X ×hZ Y can be viewed
as a presheaf of topological spaces on C. A homotopy fiber product for X
and Y over Z is an object of C which represents this presheaf up to weakhomotopy equivalence. In other words, it is an object P ∈ C equipped witha point p ∈ MapC (P, X ×hZ Y ) which induces weak homotopy equivalences
MapC (W, P ) → MapC (W, X ×hZ Y ) for every W ∈ C.
We note that if there exists a fiber product (in the ordinary sense) X ×Z Y
in the category C, then this ordinary fiber product admits a (canonically determined) map to the homotopy fiber product (if the homotopy fiber productexists). This map need not be an equivalence, but it is an equivalence in many
good cases. We also note that a homotopy fiber product P comes equippedwith a map to the fiber product X ×Z Y taken in the category hC (if thisfiber product exists); this map is usually not an isomorphism.
Remark 1.2.13.3. Homotopy limits and colimits in general may be described in relation to homotopy limits of topological spaces. The homotopy
limit X of a diagram of objects {Xα } in an arbitrary topological category
C is determined, up to equivalence, by the requirement that there exists anatural weak homotopy equivalence
MapC (Y, X) ' holim{MapC (Y, Xα )}.
Similarly, the homotopy colimit of the diagram is characterized by the existence of a natural weak homotopy equivalence
MapC (X, Y ) ' holim{MapC (Xα , Y )}.
For a more precise discussion, we refer the reader to Remark A.3.3.13.
In the setting of ∞-categories, limits and colimits are quite easy to define:
Definition 1.2.13.4 (Joyal [43]). Let C be an ∞-category and let p : K → C
be an arbitrary map of simplicial sets. A colimit for p is an initial object of
Cp/ , and a limit for p is a final object of C/p .
Remark 1.2.13.5. According to Definition 1.2.13.4, a colimit of a diagram
p : K → C is an object of Cp/ . We may identify this object with a map
p : K . → C extending p. In general, we will say that a map p : K . → C is acolimit diagram if it is a colimit of p = p|K. In this case, we will also abuseterminology by referring to p(∞) ∈ C as a colimit of p, where ∞ denotes thecone point of K . .
If p : K → C is a diagram, we will sometimes write lim(p) to denote−→a colimit of p (considered either as an object of Cp/ or of C), and lim(p)←−to denote a limit of p (as either an object of C/p or an object of C). This
notation is slightly abusive since lim(p) is not uniquely determined by p.
−→
This phenomenon is familiar in classical category theory: the colimit of adiagram is not unique but is determined up to canonical isomorphism. In
the ∞-categorical setting, we have a similar uniqueness result: Proposition
1.2.12.9 implies that the collection of candidates for lim(p), if nonempty, is−→parametrized by a contractible Kan complex.

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Remark 1.2.13.6. In §4.2.4, we will show that Definition 1.2.13.4 agrees
with the classical theory of homotopy (co)limits when we specialize to thecase where C is the nerve of a topological category.
Remark 1.2.13.7. Let C be an ∞-category, C0 ⊆ C a full subcategory, andp : K → C0 a diagram. Then C0p/ = C0 ×C Cp/ . In particular, if p has a colimit
in C and that colimit belongs to C0 , then the same object may be regarded
as a colimit for p in C0 .
Let f : C → C0 be a map between ∞-categories. Let p : K → C be adiagram in C having a colimit x ∈ Cp/ . The image f (x) ∈ C0f p/ may or may
not be a colimit for the composite map f ◦ p. If it is, we will say that f
preserves the colimit of the diagram p. Often we will apply this terminology
not to a particular diagram p but to some class of diagrams: for example,
we may speak of maps f which preserve coproducts, pushouts, or filteredcolimits (see §4.4 for a discussion of special classes of colimits). Similarly, we
may ask whether or not a map f preserves the limit of a particular diagramor various families of diagrams.
We conclude this section by giving a simple example of a colimit-preserving
functor.
Proposition 1.2.13.8. Let C be an ∞-category and let q : T → C andp : K → C/q be two diagrams. Let p0 denote the composition of p with the
projection C/q → C. Suppose that p0 has a colimit in C. Then
(1) The diagram p has a colimit in C/q , and that colimit is preserved by
the projection C/q → C.
(2) An extension pe : K . → C/q is a colimit of p if and only if the composition
K . → C/q → C
is a colimit of p0 .
Proof. We first prove the “if” direction of (2). Let pe : K . → C/q be suchthat the composite map pe0 : K . → C is a colimit of p0 . We wish to showthat pe is a colimit of p. We may identify pe with a map K ? ∆0 ? T → C. For
this, it suffices to show that for any inclusion A ⊆ B of simplicial sets, it ispossible to solve the lifting problem depicted in the following diagram:`/5 C(K ? B ? T ) K?A?T (K ? ∆0 ? A ? T )_
K ? ∆0 ? B ? T.
Because pe0 is a colimit of p0 , the projection
Cpf0 / → Cp0 /is a trivial fibration of simplicial sets and therefore has the right lifting
property with respect to the inclusion A ? T ⊆ B ? T .

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AN OVERVIEW OF HIGHER CATEGORY THEORY49

We now prove (1). Let pe0 : K . → C be a colimit of p0 . Since the projection
Cpf0 / → Cp0 / is a trivial fibration, it has the right lifting property with respect
to T : this guarantees the existence of an extension pe : K . → C lifting pe0 .
The preceding analysis proves that pe is a colimit of p.
Finally, the “only if” direction of (2) follows from (1) since any two colimitsof p are equivalent.
1.2.14 Presentations of ∞-Categories
Like many other types of mathematical structures, ∞-categories can be described by generators and relations. In particular, it makes sense to speak ofa finitely presented ∞-category C. Roughly speaking, C is finitely presented
if it has finitely many objects and its morphism spaces are determined by
specifying a finite number of generating morphisms, a finite number of relations among these generating morphisms, a finite number of relations among
the relations, and so forth (a finite number of relations in all).
Example 1.2.14.1. Let C be the free higher category generated by a single
object X and a single morphism f : X → X. Then C is a finitely presented∞-category with a single object and HomC (X, X) = {1, f, f 2 , . . .} is infinite
and discrete. In particular, we note that the finite presentation of C does notguarantee finiteness properties of the morphism spaces.
Example 1.2.14.2. If we identify ∞-groupoids with spaces, then giving apresentation for an ∞-groupoid corresponds to giving a cell decompositionof the associated space. Consequently, the finitely presented ∞-groupoids
correspond precisely to the finite cell complexes.
Example 1.2.14.3. Suppose that C is a higher category with only two
objects X and Y , that X and Y have contractible endomorphism spaces,
and that HomC (X, Y ) is empty. Then C is completely determined by themorphism space HomC (Y, X), which may be arbitrary. In this case, C isfinitely presented if and only if HomC (Y, X) is a finite cell complex (up tohomotopy equivalence).
The idea of giving a presentation for an ∞-category is very naturally
encoded in Joyal’s model structure on the category of simplicial sets, which
we will discuss in §2.2.4). This model structure can be described as follows:• The fibrant objects of Set∆ are precisely the ∞-categories.
• The weak equivalences in Set∆ are precisely those maps p : S → S 0
which induce equivalences C[S] → C[S 0 ] of simplicial categories.
If S is an arbitrary simplicial set, we can choose a “fibrant replacement”
for S: that is, a categorical equivalence S → C, where C is an ∞-category. For
example, we can take C to be the nerve of the topological category | C[S]|.
The ∞-category C is well-defined up to equivalence, and we may regard it ashighertopoi

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CHAPTER 1

an ∞-category “generated by” S. The simplicial set S itself can be thoughtof as a “blueprint” for building C. We may view S as generated from theempty (simplicial) set by adjoining nondegenerate simplices. Adjoining a 0simplex to S has the effect of adding an object to the ∞-category C, andadjoining a 1-simplex to S has the effect of adjoining a morphism to C.
Higher-dimensional simplices can be thought of as encoding relations among
the morphisms.
1.2.15 Set-Theoretic Technicalities
In ordinary category theory, one frequently encounters categories in which
the collection of objects is too large to form a set. Generally speaking, thisdoes not create any difficulties so long as we avoid doing anything which is
obviously illegal (such as considering the “category of all categories” as an
object of itself).
The same issues arise in the setting of higher category theory and arein some sense even more of a nuisance. In ordinary category theory, one
generally allows a category C to have a proper class of objects but stillrequires HomC (X, Y ) to be a set for fixed objects X, Y ∈ C. The formalismof ∞-categories treats objects and morphisms on the same footing (they areboth simplices of a simplicial set), and it is somewhat unnatural (thoughcertainly possible) to directly impose the analogous condition; see §5.4.1 fora discussion.
There are several means of handling the technical difficulties inherent inworking with large objects (in either classical or higher category theory):(1) One can employ some set-theoretic device that enables one to distinguish between “large” and “small”. Examples include:– Assuming the existence of a sufficient supply of (Grothendieck)universes.
– Working in an axiomatic framework which allows both sets andclasses (collections of sets which are possibly too large for themselves to be considered sets).
– Working in a standard set-theoretic framework (such as ZermeloFrankel) but incorporating a theory of classes through some ad
hoc device. For example, one can define a class to be a collectionof sets which is defined by some formula in the language of set
theory.
(2) One can work exclusively with small categories, and mirror the distinction between large and small by keeping careful track of relative
sizes.
(3) One can simply ignore the set-theoretic difficulties inherent in discussing large categories.

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51

Needless to say, approach (2) yields the most refined information. However, it has the disadvantage of burdening our exposition with an additional
layer of technicalities. On the other hand, approach (3) will sometimes beinadequate because we will need to make arguments which play off the distinction between a large category and a small subcategory which determines
it. Consequently, we shall officially adopt approach (1) for the remainderof this book. More specifically, we assume that for every cardinal κ0 , there
exists a strongly inaccessible cardinal κ ≥ κ0 . We then let U(κ) denote thecollection of all sets having rank < κ, so that U(κ) is a Grothendieck universe: in other words, U(κ) satisfies all of the usual axioms of set theory.
We will refer to a mathematical object as small if it belongs to U(κ) (or isisomorphic to such an object), and essentially small if it is equivalent (inwhatever relevant sense) to a small object. Whenever it is convenient to doso, we will choose another strongly inaccessible cardinal κ0 > κ to obtain alarger Grothendieck universe U(κ0 ) in which U(κ) becomes small.
For example, an ∞-category C is essentially small if and only if it satisfies
the following conditions:• The set of isomorphism classes of objects in the homotopy category hC
has cardinality < κ.
• For every morphism f : X → Y in C and every i ≥ 0, the homotopy
set πi (HomR
C (X, Y ), f ) has cardinality < κ.
For a proof and further discussion, we refer the reader to §5.4.1.
Remark 1.2.15.1. The existence of the strongly inaccessible cardinal κ
cannot be proven from the standard axioms of set theory, and the assumptionthat κ exists cannot be proven consistent with the standard axioms for set
theory. However, it should be clear that assuming the existence of κ is merely
the most convenient of the devices mentioned above; none of the results
proven in this book will depend on this assumption in an essential way.
1.2.16 The ∞-Category of Spaces
The category of sets plays a central role in classical category theory. The
main reason is that every category C is enriched over sets: given a pair ofobjects X, Y ∈ C, we may regard HomC (X, Y ) as an object of Set. In thehigher-categorical setting, the proper analogue of Set is the ∞-category S ofspaces, which we will now introduce.
Definition 1.2.16.1. Let Kan denote the full subcategory of Set∆ spannedby the collection of Kan complexes. We will regard Kan as a simplicial
category. Let S = N(Kan) denote the (simplicial) nerve of Kan. We willrefer to S as the ∞-category of spaces.
Remark 1.2.16.2. For every pair of objects X, Y ∈ Kan, the simplicial set
MapKan (X, Y ) = Y X is a Kan complex. It follows that S is an ∞-category(Proposition 1.1.5.10).

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Remark 1.2.16.3. There are many other ways to construction a suitable“∞-category of spaces.” For example, we could instead define S to be the(topological) nerve of the category of CW complexes and continuous maps.
All that really matters is that we have a ∞-category which is equivalent to
S = N(Kan). We have selected Definition 1.2.16.1 for definiteness and tosimplify our discussion of the Yoneda embedding in §5.1.3.
Remark 1.2.16.4. We will occasionally need to distinguish between large
and small spaces. In these contexts, we will let S denote the ∞-category ofsmall spaces (defined by taking the simplicial nerve of the category of small
Kan complexes), and b
S the ∞-category of large spaces (defined by taking
the simplicial nerve of the category of all Kan complexes). We observe that
S is a large ∞-category and that b
S is even bigger.

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Chapter Two
Fibrations of Simplicial Sets
Many classes of morphisms which play an important role in the homotopy
theory of simplicial sets can be defined by their lifting properties (we refer
the reader to §A.1.2 for a brief discussion and a summary of the terminology
employed below).
Example 2.0.0.1. A morphism p : X → S of simplicial sets which has theright lifting property with respect to every horn inclusion Λni ⊆ ∆n is calleda Kan fibration. A morphism i : A → B which has the left lifting property
with respect to every Kan fibration is said to be anodyne.
Example 2.0.0.2. A morphism p : X → S of simplicial sets which has theright lifting property with respect to every inclusion ∂ ∆n ⊆ ∆n is called atrivial fibration. A morphism i : A → B has the left lifting property withrespect to every trivial Kan fibration if and only if it is a cofibration: thatis, if and only if i is a monomorphism of simplicial sets.
By definition, a simplicial set S is a ∞-category if it has the extension
property with respect to all horn inclusions Λni ⊆ ∆n with 0 < i < n. As inclassical homotopy theory, it is convenient to introduce a relative version ofthis condition.
Definition 2.0.0.3 (Joyal). A morphism f : X → S of simplicial sets is• a left fibration if f has the right lifting property with respect to allhorn inclusions Λni ⊆ ∆n , 0 ≤ i < n.
• a right fibration if f has the right lifting property with respect to allhorn inclusions Λni ⊆ ∆n , 0 < i ≤ n.
• an inner fibration if f has the right lifting property with respect to allhorn inclusions Λni ⊆ ∆n , 0 < i < n.
A morphism of simplicial sets i : A → B is
• left anodyne if i has the left lifting property with respect to all left
fibrations.
• right anodyne if i has the left lifting property with respect to all rightfibrations.
• inner anodyne if i has the left lifting property with respect to all inner
fibrations.

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Remark 2.0.0.4. Joyal uses the terms “mid-fibration” and “mid-anodyne
morphism” for what we have chosen to call inner fibrations and inner anodyne morphisms.
The purpose of this chapter is to study the notions of fibration defined
above, which are basic tools in the theory of ∞-categories. In §2.1, we study
the theory of right (left) fibrations p : X → S, which can be viewed as
the ∞-categorical analogue of categories (co)fibered in groupoids over S. We
will apply these ideas in §2.2 to show that the theory of ∞-categories is
equivalent to the theory of simplicial categories.
There is also an analogue of the more general theory of (co)fibered categories whose fibers are not necessarily groupoids: this is the theory of(co)Cartesian fibrations, which we will introduce in §2.4. Cartesian and coCartesian fibrations are both examples of inner fibrations, which we will
study in §2.3.
Remark 2.0.0.5. To help orient the reader, we summarize the relationship between many of the classes of fibrations which we will study in thisbook. If f : X → S is a map of simplicial sets, then we have the following
implications:f is a trivial
fibration
f is a Kan
fibrations{
f is a left
fibration#+
f is a rightfibration
f is a coCartesianfibration
f is a Cartesianfibration#+

f is acategoricalfibrationt|
f is an inner
fibration.
In general, none of these implications is reversible.
Remark 2.0.0.6. The small object argument (Proposition A.1.2.5) showshighertopoi

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FIBRATIONS OF SIMPLICIAL SETS

that every map X → Z of simplicial sets admits a factorizationpq

X → Y → Z,
where p is anodyne (left anodyne, right anodyne, inner anodyne, a cofibration) and q is a Kan fibration (left fibration, right fibration, inner fibration,
trivial fibration).
Remark 2.0.0.7. The theory of left fibrations (left anodyne maps) is dual
to the theory of right fibrations (right anodyne maps): a map S → T is a left
fibration (left anodyne map) if and only if the induced map S op → T op is aright fibration (right anodyne map). Consequently, we will generally confine
our remarks in §2.1 to the case of left fibrations; the analogous statements
for right fibrations will follow by duality.

2.1 LEFT FIBRATIONSIn this section, we will study the class of left fibrations between simplicial
sets. We begin in §2.1.1 with a review of some classical category theory:
namely, the theory of categories cofibered in groupoids (over another category). We will see that the theory of left fibrations is a natural ∞-categoricalgeneralization of this idea. In §2.1.2, we will show that the class of left fibrations is stable under various important constructions, such as the formationof slice ∞-categories.
It follows immediately from the definition that every Kan fibration of
simplicial sets is a left fibration. The converse is false in general. However,
it is possible to give a relatively simple criterion for testing whether or nota left fibration f : X → S is a Kan fibration. We will establish this criterionin §2.1.3 and deduce some of its consequences.
The classical theory of Kan fibrations has a natural interpretation in thelanguage of model categories: a map p : X → S is a Kan fibration if andonly if X is a fibrant object of (Set∆ )/S , where the category (Set∆ )/S isequipped with its usual model structure. There is a similar characterizationof left fibrations: a map p : X → S is a left fibration if and only if X is a
fibrant object of (Set∆ )/S with respect to a certain model structure which
we will refer to as the covariant model structure. We will define the covariant
model structure in §2.1.4 and give an overview of its basic properties.
2.1.1 Left Fibrations in Classical Category Theory
Before beginning our study of left fibrations, let us recall a bit of classicalcategory theory. Let D be a small category and suppose we are given afunctorχ : D → Gpd,
where Gpd denotes the category of groupoids (where the morphisms are given
by functors). Using the functor χ, we can extract a new category Cχ via theclassical Grothendieck construction:

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CHAPTER 2• The objects of Cχ are pairs (D, η), where D ∈ D and η is an object of
the groupoid χ(D).
• Given a pair of objects (D, η), (D0 , η 0 ) ∈ Cχ , a morphism from (D, η) to(D0 , η 0 ) in Cχ is given by a pair (f, α), where f : D → D0 is a morphismin D and α : χ(f )(η) ' η 0 is an isomorphism in the groupoid χ(D0 ).
• Composition of morphisms is defined in the obvious way.
There is an evident forgetful functor F : Cχ → D, which carries an object(D, η) ∈ Cχ to the underlying object D ∈ D. Moreover, it is possible toreconstruct χ from the category Cχ (together with the forgetful functor F ) atleast up to equivalence; for example, if D is an object of D, then the groupoid
χ(D) is canonically equivalent to the fiber product Cχ ×D {D}. Consequently,
the Grothendieck construction sets up a dictionary which relates functors
χ : D → Gpd with categories Cχ admitting a functor F : Cχ → D. However,this dictionary is not perfect; not every functor F : C → D arises via theGrothendieck construction described above. To clarify the situation, we recall
the following definition:
Definition 2.1.1.1. Let F : C → D be a functor between categories. We
say that C is cofibered in groupoids over D if the following conditions are
satisfied:(1) For every object C ∈ C and every morphism η : F (C) → D in D, there
e such that F (eexists a morphism ηe : C → Dη ) = η.
(2) For every morphism η : C → C 0 in C and every object C 00 ∈ C, themap
HomC (C 0 , C 00 )

HomC (C, C 00 ) ×HomD (F (C),F (C 00 )) HomD (F (C 0 ), F (C 00 ))is bijective.
Example 2.1.1.2. Let χ : D → Gpd be a functor from a category D to thecategory of groupoids. Then the forgetful functor Cχ → D exhibits Cχ as
cofibered in groupoids over D.
Example 2.1.1.2 admits a converse: suppose we begin with a category C
cofibered in groupoids over D. Then, for every object D ∈ D, the fiber
CD = C ×D {D} is a groupoid. Moreover, for every morphism f : D → D0
in D, it is possible to construct a functor f! : CD → CD0 as follows: for eachC ∈ CD , choose a morphism f : C → C 0 covering the map D → D0 and set
f! (C) = C 0 . The map f may not be uniquely determined, but it is unique
up to isomorphism and depends functorially on C. Consequently, we obtainhighertopoi

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FIBRATIONS OF SIMPLICIAL SETS

a functor f! , which is well-defined up to isomorphism. We can then try todefine a functor χ : D → Gpd by the formulas
D 7→ CDf 7→ f! .
Unfortunately, this does not quite work: since the functor f! is determined
only up to canonical isomorphism by f , the identity (f ◦ g)! = f! ◦ g! holds
only up to canonical isomorphism rather than up to equality. This is merely
a technical inconvenience; it can be addressed in (at least) two ways:• The groupoid χ(D) = C ×D {D} can be described as the category offunctors G fitting into a commutative diagram
=C
G

{D}

F


/ D.

If we replace the one-point category {D} with the undercategory DD/
in this definition, then we obtain a groupoid equivalent to χ(D) which
depends on D in a strictly functorial fashion.
• Without modifying the definition of χ(D), we can realize χ as a functorfrom D to an appropriate bicategory of groupoids.
We may summarize the above discussion informally by saying that theGrothendieck construction establishes an equivalence between functors χ :D → Gpd and categories cofibered in groupoids over D.
The theory of left fibrations should be regarded as an ∞-categorical generalization of Definition 2.1.1.1. As a preliminary piece of evidence for thisassertion, we offer the following:
Proposition 2.1.1.3. Let F : C → D be a functor between categories.
Then C is cofibered in groupoids over D if and only if the induced map
N(F ) : N(C) → N(D) is a left fibration of simplicial sets.
Proof. Proposition 1.1.2.2 implies that N(F ) is an inner fibration. It follows
that N(F ) is a left fibration if and only if it has the right lifting property
with respect to Λn0 ⊆ ∆n for all n > 0. When n = 1, the relevant lifting
property is equivalent to (1) of Definition 2.1.1.1. When n = 2 (n = 3), therelevant lifting property is equivalent to the surjectivity (injectivity) of themap described in (2). For n > 3, the relevant lifting property is automatic(since a map Λn0 → S extends uniquely to ∆n when S is isomorphic to thenerve of a category).
Let us now consider the structure of a general left fibration p : X → S.
In the case where S consists of a single vertex, Proposition 1.2.5.1 assertsthat p is a left fibration if and only if X is a Kan complex. Since the class ofhighertopoi

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left fibrations is stable under pullback, we deduce that for any left fibration
p : X → S and any vertex s of S, the fiber Xs = X ×S {s} is a Kancomplex (which we can think of as the ∞-categorical analogue of a groupoid).
Moreover, these Kan complexes are related to one another. More precisely,
suppose that f : s → s0 is an edge of the simplicial set S and consider the
inclusion i : Xs ' Xs × {0} ⊆ Xs × ∆1 . In §2.1.2, we will prove that i is leftanodyne (Corollary 2.1.2.7). It follows that we can solve the lifting problem
/6 X{0} × Xs_p
∆1 × Xs

/ ∆1

f


/ S.

Restricting the dotted arrow to {1} × Xs , we obtain a map f! : Xs → Xs0 .
Of course, f! is not unique, but it is uniquely determined up to homotopy.
Lemma 2.1.1.4. Let q : X → S be a left fibration of simplicial sets. The
assignments ∈ S0 7→ Xs
f ∈ S1 7→ f!
determines a (covariant) functor from the homotopy category hS into thehomotopy category H of spaces.
Proof. Let f : s → s0 be an edge of S. We note the following characterizationof the morphism f! in H. Let K be any simplicial set and suppose we are
given homotopy classes of maps η ∈ HomH (K, Xs ), η 0 ∈ HomH (K, Xs0 ).
Then η 0 = f! ◦ η if and only if there exists a map p : K × ∆1 → X such thatq ◦ p is given by the compositionf

K × ∆1 → ∆1 → S,
η is the homotopy class of p|K × {0}, and η 0 is the homotopy class of p|K ×{1}.
Now consider any 2-simplex σ : ∆2 → S, which we will depict as
f

u

>v

g

h

/ w.

We note that the inclusion Xu × {0} ⊆ Xu × ∆2 is left anodyne (Corollary
2.1.2.7). Consequently there exists a map p : Xu × ∆2 → X such thatp|Xu × {0} is the inclusion Xu ⊆ X and q ◦ p is the composition Xu × ∆2 →σ
∆2 → S. Then f! ' p|Xu × {1}, h! = p|Xu × {2}, and the map p|Xu × ∆{1,2}verifies the equationh! = g! ◦ f!
in HomH (Xu , Xw ).

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We can summarize the situation informally as follows. Fix a simplicial set
S. To give a left fibration q : X → S, one must specify a Kan complex Xs
for each “object” of S, a map f! : Xs → Xs0 for each “morphism” f : s → s0
of S, and “coherence data” for these morphisms for each higher-dimensional
simplex of S. In other words, giving a left fibration ought to be more or lessequivalent to giving a functor from S to the ∞-category S of spaces. Lemma
2.1.1.4 can be regarded as a weak version of this assertion; we will provesomething considerably more precise in §2.1.4 (see Theorem 2.2.1.2).
We close this section by establishing two simple properties of left fibrations, which will be needed in the proof of Proposition 1.2.4.3:
Proposition 2.1.1.5. Let p : C → D be a left fibration of ∞-categories andlet f : X → Y be a morphism in C such that p(f ) is an equivalence in D.
Then f is an equivalence in C.
Proof. Let g be a homotopy inverse to p(f ) in D so that there exists a
2-simplex of D depicted as follows:
p(Y )
;

p(f )

g

#
/ p(X).

idp(X)

p(X)Since p is a left fibration, we can lift this to a diagram
>Y
f

X

g

/X

idX

in C. It follows that g ◦ f ' idX , so that f admits a left homotopy inverse.
Since p(g) = g is an equivalence in D, the same argument proves that g has
a left homotopy inverse. This left homotopy inverse must be homotopic to f ,since f is a right homotopy inverse to g. Thus f and g are homotopy inversein the ∞-category C, so that f is an equivalence, as desired.
Proposition 2.1.1.6. Let p : C → D be a left fibration of ∞-categories, let
Y be an object of C, and let f : X → p(Y ) be an equivalence in D. Then
there exists a morphism f : X → Y in C such that p(f ) = f (automaticallyan equivalence in view of Proposition 2.1.1.5).
Proof. Let g : p(Y ) → X be a homotopy inverse to f in C. Since p is a left
fibration, there exists a morphism g : Y → X such that g = p(g). Since f
and g are homotopy inverse to one another, there exists a 2-simplex of D
which we can depict as follows:
p(g)

p(Y )

p(X)
;
idp(Y )f#
/ p(Y ).

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Applying the assumption that p is a left fibration once more, we can lift thisto a diagram
?X
g

f

/ Y,

idY

Y
which proves the existence of f .

2.1.2 Stability Properties of Left Fibrations
The purpose of this section is to show that left fibrations of simplicial sets exist in abundance. Our main results are Proposition 2.1.2.1 (which is our basic
source of examples for left fibrations) and Corollary 2.1.2.9 (which assertsthat left fibrations are stable under the formation of functor categories).
Let C be an ∞-category and let S denote the ∞-category of spaces. One
can think of a functor from C to S as a “cosheaf of spaces” on C. By analogy
with ordinary category theory, one might expect that the basic example ofsuch a cosheaf would be the cosheaf corepresented by an object C of C;
roughly speaking this should be given by the functor
D 7→ MapC (C, D).
As we saw in §2.1.1, it is natural to guess that such a functor can be ene → C. There is a natural candidate for C:e thecoded by a left fibration C
undercategory CC/ . Note that the fiber of the map
f : CC/ → Cover the object D ∈ C is the Kan complex HomLC (C, D). The assertion thatf is a left fibration is a consequence of the following more general result:
Proposition 2.1.2.1 (Joyal). Suppose we are given a diagram of simplicial
sets
p

q

A ⊆ B → X → S,
where q is an inner fibration. Let r = q ◦ p : B → S, p0 = p|A, and r0 = r|A.
Then the induced map
Xp/ → Xp0 / ×Sr0 / Sr/
is a left fibration. If the map q is already a left fibration, then the inducedmap
X/p → X/p0 ×S/r0 S/r
is a left fibration as well.
Proposition 2.1.2.1 immediately implies the following half of Proposition
1.2.9.3, which we asserted earlier without proof:

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Corollary 2.1.2.2 (Joyal). Let C be an ∞-category and p : K → C an arbitrary diagram. Then the projection Cp/ → C is a left fibration. In particular,
Cp/ is itself an ∞-category.
Proof. Apply Proposition 2.1.2.1 in the case where X = C, S = ∆0 , andA = ∅.
We can also use Proposition 2.1.2.1 to prove Proposition 1.2.4.3, which
was stated without proof in §1.2.4.
Proposition. Let C be an ∞-category and φ : ∆1 → C a morphism of C.
Then φ is an equivalence if and only if, for every n ≥ 2 and every map
f0 : Λn0 → C such that f0 |∆{0,1} = φ, there exists an extension of f0 to ∆n .
Proof. Suppose first that φ is an equivalence and let f0 be as above. To
find the desired extension of f0 , we must produce the dotted arrow in theassociated diagram/ C/∆n−2
:

{0}
_

q


∆1

φ

0



/ C/ ∂ ∆n−2 .

The projection map p : C/ ∂ ∆n−2 → C is a right fibration (Corollary 2.1.2.2).
Since φ0 is a preimage of φ under p, Proposition 2.1.1.5 implies that φ0 is
an equivalence. Because q is a right fibration (Corollary 2.1.2.2 again), theexistence of the dotted arrow follows from Proposition 2.1.1.6.
We now prove the converse. Let φ : X → Y be a morphism in C andconsider the map Λ20 → C indicated in the following diagram:
>Y
φ

X

ψ
idX

/ X.

The assumed extension property ensures the existence of the dotted morphism ψ : Y → X and a 2-simplex σ which verifies the identity ψ ◦ φ ' idX .
We now consider the map
τ0 : Λ30(•,s0 φ,s1 φ,σ)/ C.

Once again, our assumption allows us to extend τ0 to a 3-simplex τ : ∆3 → C,
and the face d0 τ verifies the identity φ ◦ ψ ' idY . It follows that ψ is a
homotopy inverse to φ, so that φ is an equivalence in C.
We now turn to the proof of Proposition 2.1.2.1. It is an easy consequenceof the following more basic observation:

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Lemma 2.1.2.3 (Joyal [44]). Let f : A0 ⊆ A and g : B0 ⊆ B be inclusionsof simplicial sets. Suppose either that f is right anodyne or that g is leftanodyne. Then the induced inclusiona(A ? B0 ) ⊆ A ? B
h : (A0 ? B)
A0 ?B0is inner anodyne.
Proof. We will prove that h is inner anodyne whenever f is right anodyne;
the other assertion follows by a dual argument.
Consider the class of all morphisms f for which the conclusion of thelemma holds (for any inclusion g). This class of morphisms is weakly saturated; to prove that it contains all right anodyne morphisms, it suffices toshow that it contains each of the inclusions f : Λnj ⊆ ∆n for 0 < j ≤ n. We
may therefore assume that f is of this form.
Now consider the collection of all inclusions g for which h is inner anodyne(where f is now fixed). This class of morphisms is also weakly saturated;
to prove that it contains all inclusions, it suffices to show that the lemma
holds when g is of the form ∂ ∆m ⊆ ∆m . In this case, h can be identified
with the inclusion Λn+m+1⊆ ∆n+m+1 , which is inner anodyne becausej
0 < j ≤ n < n + m + 1.
The following result can be proven by exactly the same argument:
Lemma 2.1.2.4 (Joyal). Let f : A0 → A and g : B0 → B be inclusions ofsimplicial sets. Suppose that f is left anodyne. Then the induced inclusiona(A0 ? B)(A ? B0 ) ⊆ A ? B
A0 ?B0is left anodyne.
Proof of Proposition 2.1.2.1. After unwinding the definitions, the first assertion follows from Lemma 2.1.2.3 and the second from Lemma 2.1.2.4.
For future reference, we record the following counterpart to Proposition
2.1.2.1:
Proposition 2.1.2.5 (Joyal). Let π : S → T be a map of simplicial sets,p : B → S a map of simplicial sets, i : A ⊆ B an inclusion of simplicial
sets, p0 = p|A, p0 = π ◦ p, and p00 = π ◦ p0 = p0 |A. Suppose either that i isright anodyne and π is an inner fibration, or that i is anodyne and π is aleft fibration. Then the induced map
φ : Sp/ → Sp0 / ×Tp0 / Tp0 /0is a trivial Kan fibration.
Proof. Consider the class of all cofibrations i : A → B for which φ is a trivial
Kan fibration for every inner fibration (left fibration) π : S → T . It is notdifficult to see that this is a weakly saturated class of morphisms; thus, ithighertopoi

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63

FIBRATIONS OF SIMPLICIAL SETSmsuffices to consider the case where A = Λm
for 0 < j ≤ m
j and B = ∆(0 ≤ j ≤ m).
Let q : ∂ ∆n → Sp/ be a map and suppose we are given an extension ofφ ◦ q to ∆n . We wish to find a compatible extension of q. Unwinding thedefinitions, we are given a mapan(Λmr : (∆m ? ∂ ∆n )j ? ∆ ) → S,nΛmj ?∂ ∆

which we wish to extend to ∆m ? ∆n in a manner that is compatible with agiven extension ∆m ? ∆n → T of the composite map π ◦ r. The existence ofsuch an extension follows immediately from the assumption that π has theright lifting property with respect to the horn inclusion Λn+m+1⊆ ∆n+m+1 .
j
The remainder of this section is devoted to the study of the behavior ofleft fibrations under exponentiation. Our goal is to prove an assertion of thefollowing form: if p : X → S is a left fibration of simplicial sets, then sois the induced map X K → S K , for every simplicial set K (this is a special
case of Corollary 2.1.2.9 below). This is an easy consequence of the following
characterization of left anodyne maps, which is due to Joyal:
Proposition 2.1.2.6 (Joyal [44]). The following collections of morphisms
generate the same weakly saturated class of morphisms of Set∆ :(1) The collection A1 of all horn inclusions Λni ⊆ ∆n , 0 ≤ i < n.
(2) The collection A2 of all inclusionsa(∆m × {0})(∂ ∆m × ∆1 ) ⊆ ∆m × ∆1 .
∂ ∆m ×{0}(3) The collection A3 of all inclusionsa(S 0 × {0})(S × ∆1 ) ⊆ S 0 × ∆1 ,S×{0}0where S ⊆ S .
Proof. Let S ⊆ S 0 be as in (3). Working cell by cell on S 0 , we deduce thatevery morphism in A3 can be obtained as an iterated pushout of morphisms
belonging to A2 . Conversely, A2 is contained in A3 , which proves that theygenerate the same weakly saturated collection of morphisms.
To proceed with the proof, we must first introduce a bit of notation. The(n + 1)-simplices of ∆n × ∆1 are indexed by order-preserving maps
[n + 1] → [0, . . . , n] × [0, 1].
We let σk denote the map((m, 0)
if m ≤ kσk (m) =(m − 1, 1) if m > k.

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We will also denote by σk the corresponding (n + 1)-simplex of ∆n × ∆1 .
We note that {σk }0≤k≤n are precisely the nondegenerate (n + 1)-simplicesof ∆n × ∆1 .
We define a collection {X(k)}0≤k≤n+1 of simplicial subsets of ∆n × ∆1 bydescending induction on k. We begin by settingaX(n + 1) = (∆n × {0})(∂ ∆n × ∆1 ).
∂ ∆n ×{0}Assuming that X(k + 1) has been defined, we let X(k) ⊆ ∆n × ∆1 be theunion of X(k + 1) and the simplex σk (together with all the faces of σk ). We
note that this description exhibits X(k) as a pushoutaX(k + 1)
∆n+1
Λn+1
k

and also that X(0) = ∆n × ∆1 . It follows that each step in the chain ofinclusions
X(n + 1) ⊆ X(n) ⊆ · · · ⊆ X(1) ⊆ X(0)is contained in the class of morphisms generated by A1 , so that the inclusion
X(n + 1) ⊆ X(0) is generated by A1 .
To complete the proof, we show that each inclusion in A1 is a retract of
an inclusion in A3 . More specifically, the inclusion Λni ⊆ ∆n is a retract ofa(∆n × {0})(Λni × ∆1 ) ⊆ ∆n × ∆1
Λni ×{0}

so long as 0 ≤ i < n. We will define the relevant mapsjr∆n → ∆n × ∆1 → ∆nand leave it to the reader to verify that they are compatible with the relevant
subobjects. The map j is simply the inclusion ∆n ' ∆n × {1} ⊆ ∆n × ∆1 .
The map r is induced by a map of partially ordered sets, which we will also
denote by r. It may be described by the formulas(mr(m, 0) =i

if m 6= i + 1
if m = i + 1

r(m, 1) = m.

Corollary 2.1.2.7. Let i : A → A0 be left anodyne and let j : B → B 0 be acofibration. Then the induced mapa(A × B 0 )(A0 × B) → A0 × B 0
A×B

is left anodyne.

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Proof. This follows immediately from Proposition 2.1.2.6, which characterizes the class of left anodyne maps as the class generated by A3 .
Remark 2.1.2.8. A basic fact in the homotopy theory of simplicial sets
is that the analogue of Corollary 2.1.2.7 also holds for the class of anodyne
maps of simplicial sets. Since the class of anodyne maps is generated (as aweakly saturated class of morphisms) by the class of left anodyne maps andthe class of right anodyne maps, this classical fact follows from Corollary
2.1.2.7 (together with the dual assertion concerning right anodyne maps).
Corollary 2.1.2.9. Let p : X → S be a left fibration and let i : A → B be
any cofibration of simplicial sets. Then the induced map q : X B → X A ×S A
S B is a left fibration. If i is left anodyne, then q is a trivial Kan fibration.
Corollary 2.1.2.10 (Homotopy Extension Lifting Property). Let p : X →S be a map of simplicial sets. Then p is a left fibration if and only if theinduced map
1

X ∆ → X {0} ×S {0} S ∆is a trivial Kan fibration of simplicial sets.

1

For future use, we record the following criterion for establishing that amorphism is left anodyne:
Proposition 2.1.2.11. Let p : X → S be a map of simplicial sets, lets : S → X be a section of p, and let h ∈ HomS (X × ∆1 , X) be a (fiberwise)
simplicial homotopy from s ◦ p = h|X × {0} to idX = h|X × {1}. Then s isleft anodyne.
Proof. Consider a diagram
g

S
s

f

/Y
>
q



g0
/Z
X
where q is a left fibration. We must show that it is possible to find a map f
rendering the diagram commutative. DefineaF0 : (S × ∆1 )(X × {0}) → Y
S×{0}to be the composition of g with the projection onto S. Now consider the
diagram`F0
3/ Y(S × ∆1 ) S×{0} (X × {0})
F

q



g 0 ◦h/ Z.
X × ∆1Since q is a left fibration and the left vertical map is left anodyne, it ispossible to supply the dotted arrow F as indicated. Now we observe thatf = F |X × {1} has the desired properties.

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2.1.3 A Characterization of Kan Fibrations
Let p : X → S be a left fibration of simplicial sets. As we saw in §2.1.1,
p determines for each vertex s of S a Kan complex Xs , and for each edgef : s → s0 a map of Kan complexes f! : Xs → Xs0 (which is well-defined up
to homotopy). If p is a Kan fibration, then the same argument allows us toconstruct a map Xs0 → Xs , which is a homotopy inverse to f! . Our goal inthis section is to prove the following converse:
Proposition 2.1.3.1. Let p : S → T be a left fibration of simplicial sets.
The following conditions are equivalent:(1) The map p is a Kan fibration.
(2) For every edge f : t → t0 in T , the map f! : St → St0 is an isomorphismin the homotopy category H of spaces.
Lemma 2.1.3.2. Let p : S → T be a left fibration of simplicial sets. Supposethat S and T are Kan complexes and that p is a homotopy equivalence. Thenp induces a surjection from S0 to T0 .
Proof. Fix a vertex t ∈ T0 . Since p is a homotopy equivalence, there exists a
vertex s ∈ S0 and an edge e joining p(s) to t. Since p is a left fibration, thisedge lifts to an edge e0 : s → s0 in S. Then p(s0 ) = t.
Lemma 2.1.3.3. Let p : S → T be a left fibration of simplicial sets. Supposethat T is a Kan complex. Then p is a Kan fibration.
Proof. We note that the projection S → ∗, being a composition of left fibrations S → T and T → ∗, is a left fibration, so that S is also a Kan complex.
Let A ⊆ B be an anodyne inclusion of simplicial sets. We must show thatthe map p : S B → S A ×T A T B is surjective on vertices. Since S and T are
Kan complexes, the maps T B → T A and S B → S A are trivial fibrations. It
follows that p is a homotopy equivalence and a left fibration. Now we simply
apply Lemma 2.1.3.2.
Lemma 2.1.3.4. Let p : S → T be a left fibration of simplicial sets. Supposethat for every vertex t ∈ T , the fiber St is contractible. Then p is a trivial
Kan fibration.
Proof. It will suffice to prove the analogous result for right fibrations (we dothis in order to keep the notation we use below consistent with that employedin the proof of Proposition 2.1.2.6).
Since p has nonempty fibers, it has the right lifting property with respect
to the inclusion ∅ = ∂ ∆0 ⊆ ∆0 . Let n > 0, let f : ∂ ∆n → S be any map,and let g : ∆n → T be an extension of p ◦ f . We must show that there exists
an extension fe : ∆n → S with g = p ◦ fe.
Pulling back via the map G, we may suppose that T = ∆n and g is theidentity map, so that S is an ∞-category. Let t denote the initial vertex of

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T . There is a unique map g 0 : ∆n × ∆1 → T such that g 0 |∆n × {1} = g and
g 0 |∆n × {0} is constant at the vertex t.
Since the inclusion ∂ ∆n × {1} ⊆ ∂ ∆n × ∆1 is right anodyne, there exists
an extension f 0 of f to ∂ ∆n × ∆1 which covers g 0 | ∂ ∆n × ∆1 . To complete
the proof, it suffices to show that we can extend f 0 to a map fe0 : ∆n × ∆1 →S (such an extension is automatically compatible with g 0 in view of ourassumptions that T = ∆n and n > 0). Assuming this has been done, we
simply define fe = fe0 |∆n × {1}.
Recall the notation of the proof of Proposition 2.1.2.6 and filter the simplicial set ∆n × ∆1 by the simplicial subsets
X(n + 1) ⊆ · · · ⊆ X(0) = ∆n × ∆1 .
We extend the definition of f 0 to X(m) by a descending induction on m.
When m = n + 1, we note that X(n + 1) is obtained from ∂ ∆n × ∆1 byadjoining the interior of the simplex ∂ ∆n × {0}. Since the boundary of thissimplex maps entirely into the contractible Kan complex St , it is possible toextend f 0 to X(n + 1).
Now suppose the definition of f 0 has been extended to X(i + 1). We notethat X(i) is obtained from X(i + 1) by pushout along a horn inclusion
Λn+1⊆ ∆n+1 . If i > 0, then the assumption that S is an ∞-categoryiguarantees the existence of an extension of f 0 to X(i). When i = 0, we notethat f 0 carries the initial edge of σ0 into the fiber St . Since St is a Kancomplex, f 0 carries the initial edge of σ0 to an equivalence in S, and thedesired extension of f 0 exists by Proposition 1.2.4.3.
Proof of Proposition 2.1.3.1. Suppose first that (1) is satisfied and let f :t → t0 be an edge in T . Since p is a right fibration, the edge f induces a
map f ∗ : St0 → St which is well-defined up to homotopy. It is not difficult
to check that the maps f ∗ and f! are homotopy inverse to one another; inparticular, f! is a homotopy equivalence. This proves that (1) ⇒ (2).
Assume now that (2) is satisfied. A map of simplicial sets is a Kan fibration
if and only if it is both a right fibration and a left fibration; consequently,
it will suffice to prove that p is a right fibration. According to Corollary
2.1.2.10, it will suffice to show that1

q : S ∆ → S {1} ×T {1} T ∆1

is a trivial Kan fibration. Corollary 2.1.2.9 implies that q is a left fibration.
By Lemma 2.1.3.4, it suffices to show that the fibers of q are contractible.
Fix an edge f : t → t0 in T . Let X denote the simplicial set of sectionsof the projection S ×T ∆1 → ∆1 , where ∆1 maps into T via the edge f .
Consider the map q 0 : X → St0 given by evaluation at the final vertex of ∆1 .
1
Since q and q 0 have the same fibers (over points of S {1} ×T {1} T ∆ whose
second projection is the edge f ), it will suffice to show that q 0 is a trivial
fibration for every choice of f .
Consider the projection r : X → St . Since p is a left fibration, r is a trivial
fibration. Because St is a Kan complex, so is X. Lemma 2.1.3.3 implies thathighertopoi

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q 0 is a Kan fibration. We note that f! is obtained by choosing a section ofr and then composing with q 0 . Consequently, assumption (2) implies that q 0
is a homotopy equivalence and thus a trivial fibration, which completes theproof.
Remark 2.1.3.5. Lemma 2.1.3.4 is an immediate consequence of Proposition 2.1.3.1 since any map between contractible Kan complexes is a homotopy
equivalence. Lemma 2.1.3.3 also follows immediately (if T is a Kan complex,
then its homotopy category is a groupoid, so that any functor hT → H
carries edges of T to invertible morphisms in H).
2.1.4 The Covariant Model Structure
In §2.1.2, we saw that a left fibration p : X → S determines a functor χfrom hS to the homotopy category H, carrying each vertex s to the fiber
Xs = X ×S {s}. We would like to formulate a more precise relationshipbetween left fibrations over S and functors from S into spaces. For this, it isconvenient to employ Quillen’s language of model categories. In this section,
we will show that the category (Set∆ )/S can be endowed with the structureof a simplicial model category whose fibrant objects are precisely the left
fibrations X → S. In §2.2, we will describe an ∞-categorical version of theGrothendieck construction which is implemented by a right Quillen functor(Set∆ )C[S] → (Set∆ )/S ,which we will eventually prove to be a Quillen equivalence (Theorem 2.2.1.2).
Warning 2.1.4.1. We will assume throughout this section that the reader
is familiar with the theory of model categories as presented in §A.2. We will
also assume familiarity with the model structure on the category Cat∆ ofsimplicial categories (see §A.3.2).
Definition 2.1.4.2. Let f : X →`S be a map of simplicial sets. The leftcone of f is the simplicial set S X X / . We will denote the left cone off by C / (f ).
` Dually, we define the right cone of f to be the simplicial set
C . (f ) = S X X . .
Remark 2.1.4.3. Let f : X → S be a map of simplicial sets. There is a
canonical monomorphism of simplicial sets S → C / (f ). We will generally
identify S with its image under this monomorphism and thereby regard S asa simplicial subset of C / (f ). We note that there is a unique vertex of C / (f )which does not belong to S. We will refer to this vertex as the cone point of
C / (f ).
Example 2.1.4.4. Let S be a simplicial set and let idS denote the identity
map from S to itself. Then C / (idS ) and C . (idS ) can be identified with S /and S . , respectively.
Definition 2.1.4.5. Let S be a simplicial set. We will say that a map
f : X → Y in (Set∆ )/S is ahighertopoi

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(C) covariant cofibration if it is a monomorphism of simplicial sets.
(W ) covariant equivalence if the induced map
a
a
X/
S →Y/
S
X

Y

is a categorical equivalence.
(F ) covariant fibration if it has the right lifting property with respect toevery map which is both a covariant cofibration and a covariant equivalence.
Lemma 2.1.4.6. Let S be a simplicial set. Then every left anodyne map in(Set∆ )/S is a covariant equivalence.
Proof. By general nonsense, it suffices to prove the result for a generating
left anodyne inclusion of the form Λni ⊆ ∆n , where 0 ≤ i < n. In otherwords, we must show any mapaai : (Λni )/S → (∆n )/
S
Λn
i

∆n

is a categorical equivalence. We now observe that i is a pushout of the innern+1
anodyne inclusion Λn+1.
i+1 ⊆ ∆
Proposition 2.1.4.7. Let S be a simplicial set. The covariant cofibrations,covariant equivalences, and covariant fibrations determine a left proper combinatorial model structure on (Set∆ )/S .
Proof. It suffices to show that conditions (1), (2), and (3) of Proposition
A.2.6.15 are met. We consider each in turn:(1) The class (W ) of weak equivalences is perfect.
` This follows from Corollary A.2.6.14 since the functor X 7→ X / X S commutes with filteredcolimits.
(2) It is clear that the class (C) of cofibrations is generated by a set.
We must show that weak equivalences are stable under pushouts bycofibrations. In other words, suppose we are given a pushout diagram
X

j

/Y

i



j0
/ Y0
X0
in (Set∆ )/S , where i is a covariant cofibration and j is a covariant
equivalence. We must show that j 0 is a covariant equivalence. We obtaina pushout diagram in Cat∆ :`/ C[Y / ` S]
C[X /
S]
X

C[(X 0 )/`

X 0 S]Y/ C[(Y 0 )/ ` 0 S].
Yhighertopoi

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This diagram is homotopy coCartesian because Cat∆ is a left proper
model category (the construction Z 7→ C[Z] carries monomorphisms of
simplicial sets to cofibrations of simplicial categories; see the proof of
Theorem 2.2.5.1). Since the upper horizontal map is an equivalence, so
is the bottom horizontal map; thus j 0 is a covariant equivalence.
(3) We must show that a map p : X → Y in Set∆ , which has the right
lifting property with respect to every map in (C), belongs to (W ). We
note in that case that p is a trivial Kan fibration and therefore admits a
section s : Y → X. We will show
induce mutually inverse` that p and s `isomorphisms between C[X / X S] and C[Y / Y S] in the homotopy
category hCat∆ ; it will then follow that p is a covariant equivalence.
Let f : X →` X denote the composition s ◦ p; we wish to show that the
map C[X / X S] induced by f is equivalent to the identity in hCat∆ .
We observe that f is homotopic to the identity idX via a homotopy
h : ∆1 × X → X. It will therefore suffice to show that h is a covariant
equivalence. But h admits a left inverse
X ' {0} × X ⊆ ∆1 × X,
which is left anodyne (Corollary 2.1.2.7) and therefore a covariant
equivalence by Lemma 2.1.4.6.

Proposition 2.1.4.8. The category (Set∆ )/S is a simplicial model category (with respect to the covariant model structure and the natural simplicial
structure).
Proof. We will deduce this from Proposition A.3.1.7. The only nontrivial
point is to verify that for any X ∈ (Set∆ )/S , the projection X × ∆n → X
is a covariant equivalence. But this map has a section X × {0} → X × ∆n ,which is left anodyne and therefore a covariant equivalence (Proposition
2.1.4.9).
We will refer to the model structure of Proposition 2.1.4.7 as the covariant model structure on (Set∆ )/S . We will prove later that the covariantly
fibrant objects of (Set∆ )/S are precisely the left fibrations X → S (Corollary 2.2.3.12). For the time being, we will be content to make a much weakerobservation:
Proposition 2.1.4.9. Let S be a simplicial set.
(1) Every left anodyne map in (Set∆ )/S is a trivial cofibration with respect
to the covariant model structure.
(2) Every covariant fibration in (Set∆ )/S is a left fibration of simplicial
sets.
(3) Every fibrant object of (Set∆ )/S determines a left fibration X → S.

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Proof. Assertion (1) follows from Lemma 2.1.4.6, and the implications (1) ⇒(2) ⇒ (3) are obvious.
Our next result expresses the idea that the covariant model structure on(Set∆ )/S depends functorially on S:
Proposition 2.1.4.10. Let j : S → S 0 be a map of simplicial sets. Let
j! : (Set∆ )/S → (Set∆ )/S 0 be the forgetful functor (given by composition withj) and let j ∗ : (Set∆ )/S 0 → (Set∆ )/S be its right adjoint, which is given by
the formulaj ∗ X 0 = X 0 ×S 0 S.
Then we have a Quillen adjunction(Set∆ )/S oj!
j∗/ (Set ) 0∆ /S(with the covariant model structures).
Proof. It is clear that j! preserves cofibrations. For X ∈ (Set∆ )/S , thepushout diagram of simplicial categories/ C[S 0 ]
C[S]

C[X /


`

X S]


/ C[X / ` S 0 ]X

is also a homotopy pushout diagram (the construction Y 7→ C[Y ] carries
monomorphisms of simplicial sets to cofibrations of simplicial categories; see
the proof of Theorem 2.2.5.1). Thus j! preserves covariant equivalences. It
follows that (j! , j ∗ ) is a Quillen adjunction.
Remark 2.1.4.11. Let j : S → S 0 be as in Proposition 2.1.4.10. If j isa categorical equivalence, then the Quillen adjunction (j! , j ∗ ) is a Quillen
equivalence. This follows from Theorem 2.2.1.2 and Proposition A.3.3.8.
Remark 2.1.4.12. Let S be a simplicial set. The covariant model structureon (Set∆ )/S is usually not self-dual. Consequently, we may define a newmodel structure on (Set∆ )/S as follows:(C) A map f in (Set∆ )/S is a contravariant cofibration if it is a monomorphism of simplicial sets.
(W ) A map f in (Set∆ )/S is a contravariant equivalence if f op is a covariant
equivalence in (Set∆ )/S op .
(F ) A map f in (Set∆ )/S is a contravariant fibration if f op is a covariant
fibration in (Set∆ )/S op .
We will refer to this model structure on (Set∆ )/S as the contravariant model
structure. Propositions 2.1.4.8, 2.1.4.9, and 2.1.4.10 have evident analogues
in the contravariant setting.

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2.2 SIMPLICIAL CATEGORIES AND ∞-CATEGORIESFor every topological category C and every pair of objects X, Y ∈ C, Theorem
1.1.5.13 asserts that the counit map
u : | MapC[N(C)] (X, Y )| → MapC (X, Y )is a weak homotopy equivalence of topological spaces. This result is the main
ingredient needed to establish the equivalence between the theory of topological categories and the theory of ∞-categories. The goal of this section is
to give a proof of Theorem 1.1.5.13 and to develop some of its consequences.
We first replace Theorem 1.1.5.13 by a statement about simplicial categories. Consider the compositionv

MapC[N(C)] (X, Y ) → Sing Map| C[N(C)]| (X, Y )Sing(u)→Sing MapC (X, Y ).

Classical homotopy theory ensures that v is a weak homotopy equivalence.
Moreover, u is a weak homotopy equivalence of topological spaces if and only
if Sing(u) is a weak homotopy equivalence of simplicial sets. Consequently, u
is a weak homotopy equivalence of topological spaces if and only if Sing(u)◦v
is a weak homotopy equivalence of simplicial sets. It will therefore suffice toprove the following simplicial analogue of Theorem 1.1.5.13:
Theorem 2.2.0.1. Let C be a fibrant simplicial category (that is, a simplicial
category in which each mapping space MapC (x, y) is a Kan complex) and let
x, y ∈ C be a pair of objects. The counit map
u : MapC[N(C)] (x, y) → MapC (x, y)
is a weak homotopy equivalence of simplicial sets.
The proof will be given in §2.2.4 (see Proposition 2.2.4.1). Our strategy is
as follows:(1) We will show that, for every simplicial set S, there is a close relationship between right fibrations S 0 → S and simplicial presheaves
F : C[S]op → Set∆ . This relationship is controlled by the straightening
and unstraightening functors which we introduce in §2.2.1.
(2) Suppose that S is an ∞-category. Then, for each object y ∈ S, theprojection S/y → S is a right fibration, which corresponds to a simplicial presheaf F : C[S]op → Set∆ . This simplicial presheaf F is related
to S/y in two different ways:(i) As a simplicial presheaf, F is weakly equivalent to the functor
x 7→ MapC[S] (x, y).
(ii) For each object x of S, there is a canonical homotopy equivalence F(x) → S/y ×S {x} ' HomR
S (x, y). Here the Kan complex
R
HomS (x, y) is defined as in §1.2.2.

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(3) Combining observations (i) and (ii), we will conclude that the mapping spaces HomR
S (x, y) are homotopy equivalent to the corresponding
mapping spaces HomC[S] (x, y).
(4) In the special case where S is the nerve of a fibrant simplicial category
C, there is a canonical map HomC (x, y) → HomR
S (x, y), which we willshow to be a homotopy equivalence in §2.2.2.
(5) Combining (3) and (4), we will obtain a canonical isomorphism
MapC (x, y) ' MapC[N(C)] (x, y)in the homotopy category of spaces. We will then show that this isomorphism is induced by the unit map appearing in the statement ofTheorem 2.2.0.1.
We will conclude this section with §2.2.5, where we apply Theorem 2.2.0.1
to construct the Joyal model structure on Set∆ and to establish a more refinedversion of the equivalence between ∞-categories and simplicial categories.
2.2.1 The Straightening and Unstraightening Constructions (Unmarked Case)In §2.1.1, we asserted that a left fibration X → S can be viewed as a functorfrom S into a suitable ∞-category of Kan complexes. Our goal in this section
is to make this idea precise. For technical reasons, it will be somewhat moreconvenient to phrase our results in terms of the dual theory of right fibrations
X → S. Given any functor φ : C[S]op → C between simplicial categories, wewill define an unstraightening functor Unφ : SetC
∆ → (Set∆ )/S . If F : C →Set∆ is a diagram taking values in Kan complexes, then the associated map
Unφ F → S is a right fibration whose fiber at a point s ∈ S is homotopy
equivalent to the Kan complex F(φ(s)).
Fix a simplicial set S, a simplicial category C, and a functor φ : C[S] → Cop .
Given an object X ∈ (Set∆ )/S , let v denote the cone point of X . . We canview the simplicial categorya op
M = C[X . ]C
C[X]

as a correspondence from C
functorop

to {v}, which we can identify with a simplicial

Stφ X : C → Set∆ .
This functor is described by the formula(Stφ X)(C) = MapM (C, v).
We may regard Stφ as a functor from (Set∆ )/S to (Set∆ )C . We refer to
Stφ as the straightening functor associated to φ. In the special case where
C = C[S]op and φ is the identity map, we will write StS instead of Stφ .

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By the adjoint functor theorem (or by direct construction), the straightening functor Stφ associated to φ : C[S] → Cop has a right adjoint, which
we will denote by Unφ and refer to as the unstraightening functor. We now
record the obvious functoriality properties of this construction.
Proposition 2.2.1.1. (1) Let p : S 0 → S be a map of simplicial sets,
C a simplicial category, and φ : C[S] → Cop a simplicial functor,
and let φ0 : C[S 0 ] → Cop denote the composition φ ◦ C[p]. Let p! :(Set∆ )/S 0 → (Set∆ )/S denote the forgetful functor given by composition with p. There is a natural isomorphism of functors
Stφ ◦ p! ' Stφ0
from (Set∆ )/S 0 to SetC∆.
(2) Let S be a simplicial set, π : C → C0 a simplicial functor between
simplicial categories, and φ : C[S] → Cop a simplicial functor. Then
there is a natural isomorphism of functors
Stπop ◦φ ' π! ◦ Stφ
0

0

C
C
from (Set∆ )/S to SetC
∆ . Here π! : Set∆ → Set∆ is the left adjoint to0CC
the functor π ∗ : Set∆ → Set∆ given by composition with π.

Our main result is the following:
Theorem 2.2.1.2. Let S be a simplicial set, C a simplicial category, andφ : C[S] → Cop a simplicial functor. The straightening and unstraightening
functors determine a Quillen adjunction(Set∆ )/S o

Stφ
Unφ

/ SetC ,∆where (Set∆ )/S is endowed with the contravariant model structure and SetC∆
with the projective model structure. If φ is an equivalence of simplicial categories, then (Stφ , Unφ ) is a Quillen equivalence.
Proof. It is easy to see that Stφ preserves cofibrations and weak equivalences, so that the pair (Stφ , Unφ ) is a Quillen adjunction. The real content
of Theorem 2.2.1.2 is the final assertion. Suppose that φ is an equivalenceof simplicial categories; then we wish to show that (Stφ , Unφ ) is a Quillen
equivalence. We will prove this result in §2.2.3 as a consequence of Proposition 2.2.3.11.
2.2.2 Straightening Over a Point
In this section, we will study the behavior of the straightening functor StX
in the case where the simplicial set X = {x} consists of a single vertex. In
this case, we can view StX as a colimit-preserving functor from the categoryof simplicial sets to itself. We begin with a few general remarks about suchfunctors.

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Let ∆ denote the category of combinatorial simplices and Set∆ the category of simplicial sets, so that Set∆ may be identified with the categoryof presheaves of sets on ∆. If C is any category which admits small colimits, then any functor f : ∆ → C extends to a colimit-preserving functor
F : Set∆ → C (which is unique up to unique isomorphism). We may regard
f as a cosimplicial object C • of C. In this case, we shall denote the functor
F by
S 7→ |S|C • .
Remark 2.2.2.1. Concretely, one constructs |S|C • by taking the disjoint
union of Sn × C n and making the appropriate identifications along the“boundaries.” In the language of category theory, the geometric realization
is given by the coendZ
Sn × C n .
[n]∈∆

The functor S 7→ |S|C • has a right adjoint which we shall denote by
SingC • . It may be described by the formula
SingC • (X)n = HomC (C n , X).
Example 2.2.2.2. Let C be the category CG of compactly generated Hausdorff spaces and let C • be the cosimplicial space defined by
C n = {(x0 , . . . , xn ) ∈ [0, 1]n+1 : x0 + · · · + xn = 1}.
Then |S|C • is the usual geometric realization |S| of the simplicial set S, andSingC • = Sing is the functor which assigns to each topological space X its
singular complex.
Example 2.2.2.3. Let C be the category Set∆ and let C • be the standard
simplex (the cosimplicial object of Set∆ given by the Yoneda embedding):
C n = ∆n .
Then ||C • and SingC • are both (isomorphic to) the identity functor on Set∆ .
Example 2.2.2.4. Let C = Cat and let f : ∆ → Cat be the functor whichassociates to each finite nonempty linearly ordered set J the corresponding
category. Then SingC • = N is the functor which associates to each category
its nerve, and ||C • associates to each simplicial set S the homotopy categoryhS as defined in §1.2.3.
Example 2.2.2.5. Let C = Cat∆ and let C • be the cosimplicial object of C
given in Definitions 1.1.5.1 and 1.1.5.3. Then SingC • is the simplicial nerve
functor, and ||C • is its left adjoint
S 7→ C[S].

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Let us now return to the case of the straightening functor StX , whereX = {x} consists of a single vertex. The above remarks show that we canidentify StX with the geometric realization functor ||Q• : Set∆ → Set∆ forsome cosimplicial object Q• in Set∆ . To describe Q• more explicitly, let us
first define a cosimplicial simplicial set J • by the formulaa
J n = (∆n ? {y}) {x}.
∆nThe cosimplicial simplicial set Q• can then be described by the formula
Qn = MapC[J n ] (x, y).
In order to proceed with our analysis, we need to understand better the
cosimplicial object Q• of Set∆ . It admits the following description:• For each n ≥ 0, let P[n] denote the partially ordered set of nonempty
subsets of [n], and K[n] the simplicial set N(P ) (which may be identified
with a simplicial subset of the (n + 1)-cube (∆1 )n+1 ). The simplicial
set Qn is obtained by collapsing, for each 0 ≤ i ≤ n, the subset(∆1 ){j:0≤j<i} × {1} × (∆1 ){j:i<j≤n} ⊆ K[n]to its quotient (∆1 ){j:i<j≤n} .
• A map f : [n] → [m] determines a map Pf : P[n] → P[m] by setting
Pf (I) = f (I). The map Pf in turn induces a map of simplicial sets
K[n] → K[m] , which determines a map of quotients Qn → Qm when fis order-preserving.
Remark 2.2.2.6. Let Q• = |Q• | denote the cosimplicial space obtainedby applying the (usual) geometric realization functor to Q• . The space Qn
may be described as a quotient of the cube of all functions p : [n] → [0, 1]
satisfying p(0) = 1. This cube is to be divided by the following equivalence relation: p ' p0 if there exists a nonnegative integer i ≤ n such thatp|{i, . . . n} = p0 |{i, . . . , n} and p(i) = p0 (i) = 1.
Each Qn is homeomorphic to an n-simplex, and these homeomorphisms
may be chosen to be compatible with the face maps of the cosimplicial spaceQ• . However, Q• is not isomorphic to the standard simplex because it hasvery different degeneracies. For example, the product of the degeneracy mappings Qn → (Q1 )n is not injective for n ≥ 2.
Our goal for the remainder of this section is to study the functors SingQ•
and ||Q• and to prove that they are “close” to the identity functor. Moreprecisely, there is a map π : Q• → ∆• of cosimplicial objects of Set∆ . It is
induced by a map K[n] → ∆n , which is the nerve of the map of partiallyordered sets P[n] → [n] that carries each nonempty subset of [n] to its largest
element.
Proposition 2.2.2.7. Let S be a simplicial set. Then the map pS : |S|Q• →S induced by π is a weak homotopy equivalence.

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Proof. Consider the collection A of simplicial sets S for which the assertionof Proposition 2.2.2.7 holds. Since A is stable under filtered colimits, it will
suffice to prove that every simplicial set S having only finitely many nondegenerate simplices belongs to A. We prove this by induction on the dimension
n of S and the number of nondegenerate simplices of S of dimension n. If
S = ∅, there is nothing to prove; otherwise we may writea∆n
S ' S0∂ ∆n|S|Q• ' |S 0 |Q•

a|∆n |Q• .

| ∂ ∆n |Q•

Since both of these pushouts are homotopy pushouts, it suffices to show thatpS 0 , p∂ ∆n , and p∆n are weak homotopy equivalences. For pS 0 and p∂ ∆n , thisfollows from the inductive hypothesis; for p∆n , we need only observe thatboth ∆n and |∆n |Q• = Qn are weakly contractible.
Remark 2.2.2.8. The strategy used to prove Proposition 2.2.2.7 will reappear frequently throughout this book: it allows us to prove theorems about
arbitrary simplicial sets by reducing to the case of simplices.
Proposition 2.2.2.9. The adjoint functors Set∆ o||Q•

SingQ•/ Set∆

determine a

Quillen equivalence from the category Set∆ (endowed with the Kan model
structure) to itself.
Proof. We first show that the functors (||Q• , SingQ• ) determine a Quillen
adjunction from Set∆ to itself. For this, it suffices to prove that the functor S 7→ |S|Q• preserves cofibrations and weak equivalences. The case of
cofibrations is easy, and the second case follows from Proposition 2.2.2.7.
To complete the proof, it will suffice to show that the left derived functor
L||Q• determines an equivalence from the homotopy category H to itself.
This follows immediately from Proposition 2.2.2.7, which implies that L||Q•
is equivalent to the identity functor.
Corollary 2.2.2.10. Let X be a Kan complex. Then the counit map
v : | SingQ• X|Q• → X
is a weak homotopy equivalence.
Remark 2.2.2.11. Let S be a simplicial set containing a vertex s. Let C bea simplicial category, φ : C[S]op → C a simplicial functor, and C = φ(s) ∈ C.
For every simplicial functor F : C → Set∆ , there is a canonical isomorphism(Unφ F) ×S {s} ' SingQ• F(C).
In particular, we have a canonical map from F(C) to the fiber (Unφ F)s ,which is a homotopy equivalence if F(C) is fibrant.

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Remark 2.2.2.12. Let C and C0 be simplicial categories. Given a pair ofsimplicial functors F : C → Set∆ , F0 : C0 → Set∆ , we let F  F0 : C × C0 →Set∆ denote the functor described by the formula(F  F0 )(C, C 0 ) = F(C) × F0 (C 0 ).
Given a pair of simplicial functors φ : C[S]op → C, φ0 : C[S 0 ]op → C0 , we let
φ  φ0 denote the induced map C[S × S 0 ]op → C × C0 . We observe that there
is a canonical isomorphism of functors
Unφφ0 (F  F0 ) ' Unφ (F) × Unφ0 (F0 ).
Restricting our attention to the case where S 0 = ∆0 and φ0 is an isomorphism, we obtain an isomorphism
Unφ (F K) ' Unφ (F) × SingQ• K
for every simplicial set K. In particular, for every pair of functors F, G ∈SetC∆ , we have a chain of maps
HomSet∆ (K, MapSetC∆ (F, G)) ' HomSetC∆ (F K, G)→ Hom(Set∆ )/S (Unφ (F K), Unφ G)' Hom(Set∆ )/S (Unφ (F) × SingQ• K, Unφ G)→ Hom(Set∆ )/S (Unφ (F) × K, Unφ G)' HomSet∆ (K, Map(Set∆ )/S (Unφ (F), Unφ (G)).
This construction is natural in K and therefore determines a map of simplicial sets
MapSetC∆ (F, G) → Map(Set∆ )/S (Unφ F, Unφ G).
Together, these maps endow the unstraightening functor Unφ with the structure of a simplicial functor from SetC
∆ to (Set∆ )/S .
The cosimplicial object Q• of Set∆ will play an important role in our proofof Theorem 1.1.5.13. To explain this, let us suppose that C is a simplicial
category and S = N(C) is its simplicial nerve. For every pair of vertices
x, y ∈ S, we can consider the right mapping space HomR
S (x, y). By definition,
giving an n-simplex of HomR
(x,
y)
is
equivalenttogivinga map of simplicial
S
sets J n → S, which carries x to x and y to y. Using the identification
S ' N(C), we see that this is equivalent to giving a map C[J n ] into C, which
again carries x to x and y to y. This is simply the data of a map of simplicial
sets Qn → MapC (x, y). Moreover, this identification is natural with respect
to [n]; we therefore have the following result:
Proposition 2.2.2.13. Let C be a simplicial category and let X, Y ∈ C be
two objects. There is a natural isomorphism of simplicial sets
HomR
N(C) (X, Y ) ' SingQ• MapC (X, Y ).

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2.2.3 Straightening of Right Fibrations
Our goal in this section is to prove Theorem 2.2.1.2, which asserts that the
Quillen adjunction(Set∆ )/S o

Stφ
Unφ

/ SetC∆is a Quillen equivalence when φ : C[S] → Cop is an equivalence of simplicial
categories. We first treat the case where S is a simplex.
Lemma 2.2.3.1. Let n be a nonnegative integer, let [n] denote the linearlyordered set {0, . . . , n}, regarded as a (discrete) simplicial category, and let
φ : C[∆n ] → [n] be the canonical functor. Then the Quillen adjunction(Set∆ )/∆n o

Stφ
Unφ

/ Set[n]∆is a Quillen equivalence.
Proof. It follows from the definition of the contravariant model structure[n]that the left derived functor LStφ : h(Set∆ )/∆n → hSet∆ is conservative. It
will therefore suffice to show that the counit map LStφ ◦ R Unφ → id is an[n]isomorphism of functors from hSet∆ to itself. For this, we must show that
if F : [n] → Set∆ is projectively fibrant, then the counit map
Stφ Unφ F → F[n]is an equivalence in Set∆ . In other words, we may assume that F(i) is a Kancomplex for i ∈ [n], and we wish to prove that each of the induced maps
vi : (Stφ Unφ F)(i) → F(i)is a weak homotopy equivalence of simplicial sets.
Let ψ : [n] → [1] be defined by the formula(
0 if 0 ≤ j ≤ i0ψ (j) =
1 otherwise.
Then, for every object X ∈ (Set∆ )/∆n , we have isomorphisms(Stφ X)(i) ' (Stψ◦φ X)(0) ' |X ×∆n ∆{n−i,...,n} |Q• ,where the twisted geometric realization functor ||Q• is as defined in §2.2.2.
Taking X = Unφ F, we see that vi fits into a commutative diagram|X ×∆n {n − i}|Q•
_

∼


vi
|X ×∆n ∆{n−i,...,n} |Q•/ | SingQ• F(i)|Q•/ F(i).

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Here the upper horizontal map is an isomorphism supplied by Corollary
2.2.2.11, and the right vertical map is a weak homotopy equivalence by Corollary 2.2.2.10. Consequently, to prove that the map vi is a weak homotopy
equivalence, it will suffice to show that the left vertical map is a weak homotopy equivalence. In view of Proposition 2.2.2.7, it will suffice to show thatthe inclusion
X ×∆n {n − i} ⊆ X ×∆n ∆{n−i,...,n}is a weak homotopy equivalence. In fact, X ×∆n {n − i} is a deformation
retract of X ×∆n ∆{n−i,...,n} : this follows from the observation that the
projection X → ∆n is a right fibration (Proposition 2.1.4.9).
It will be convenient to restate Lemma 2.2.3.1 in a slightly modified form.
First, we need to introduce a bit of terminology.
Definition 2.2.3.2. Suppose we are given a commutative diagram of simplicial sets
/Y

f

X
p



S,



q

where p and q are right fibrations. We will say that f is a pointwise equivalence if, for each vertex s ∈ S, the induced map Xs → Ys is a homotopy
equivalence of Kan complexes.
Remark 2.2.3.3. In the situation of Definition 2.2.3.2, the following conditions are equivalent:(a) The map f is a pointwise equivalence of right fibrations over S.
(b) The map f is a contravariant equivalence in (Set∆ )/S .
(c) The map f is a categorical equivalence of simplicial sets.
The equivalence (a) ⇔ (b) follows from Corollary 2.2.3.13 (see below), andthe equivalence (a) ⇔ (c) from Proposition 3.3.1.5.
Lemma 2.2.3.4. Let S 0 ⊆ S be simplicial sets. Let p : X → S be any map
and let q : Y → S be a right fibration. Let X 0 = X ×S S 0 and let Y 0 = Y ×S S 0 .
The restriction map
φ : Map(Set∆ )/S (X, Y ) → Map(Set∆ )/S0 (X 0 , Y 0 )is a Kan fibration.
Proof. We first show that φ is a right fibration. It will suffice to show that φ
has the right lifting property with respect to every right anodyne inclusion

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A ⊆ B. This follows from the fact that q has the right lifting property withrespect to the induced inclusiona(A × X) ⊆ B × X
i : (B × X 0 )A×X 0

since i is again right anodyne (Corollary 2.1.2.7).
Applying the preceding argument to the inclusion ∅ ⊆ S 0 , we deduce thatthe projection map
Map(Set∆ )/S0 (X 0 , Y 0 ) → ∆0
is a right fibration. Proposition 1.2.5.1 implies that Map(Set∆ )/S0 (X 0 , Y 0 ) isa Kan complex. Lemma 2.1.3.3 now implies that φ is a Kan fibration, as
desired.
Lemma 2.2.3.5. Let U be a collection of simplicial sets. Suppose that thefollowing conditions are satisfied:(i) The collection U is stable under isomorphism. That is, if S ∈ U andS 0 ' S, then S 0 ∈ U.
(ii) The collection U is stable under the formation of disjoint unions.
(iii) Every simplex ∆n belongs to U.
(iv) Suppose we are given a pushout diagram
X

Y

f

/ X0

/ Y0

in which X, X 0 , and Y belong to U. If the map f is a monomorphism,
then Y 0 belongs to U.
(v) Suppose we are given a sequence of monomorphisms of simplicial sets
X(0) → X(1) → · · ·If each X(i) belongs to U, then the colimit lim X(i) belongs to U.
−→
Then every simplicial set belongs to U.
Proof. Let S be a simplicial set; we wish to show that S ∈ U. In view of(v), it will suffice to show that each skeleton skn S belongs to U. We may
therefore assume that S is finite dimensional. We now proceed by induction
on the dimension n of S. Let A denote the set of nondegenerate n-simplexesof S, so that we have a pushout diagram`n/ skn−1 S
α∈A ∂ ∆

`



α∈A ∆

n


/ S.

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Invoking `assumption (iv),`
we are reduced to proving that the simplicial setsskn−1 S, α∈A ∂ ∆n , and α∈A ∆n belong to U. For the first two this follows
from the inductive hypothesis, and for the last it follows from assumptions(ii) and (iii).
Lemma 2.2.3.6. Suppose we are given a commutative diagram of simplicial
sets
/Y

f

X
p



S,



q

where p and q are right fibrations. The following conditions are equivalent:(a) The map f is a pointwise equivalence.
(b) The map f is an equivalence in the simplicial category (Set∆ )/S (thatis, f admits a homotopy inverse).
(c) For every object A ∈ (Set∆ )/S , composition with f induces a homotopy
equivalence of Kan complexes
Map(Set∆ )/S (A, X) → Map(Set∆ )/S (A, Y ).
Proof. The implication (b) ⇒ (a) is clear (any homotopy inverse to f determines homotopy inverses for the maps fs : Xs → Ys for each vertex s ∈ S),
and the implication (c) ⇒ (b) follows from Proposition 1.2.4.1. We will provethat (a) ⇒ (c). Let U denote the collection of all simplicial sets A such that,for every map A → S, composition with f induces a homotopy equivalenceof Kan complexes
Map(Set∆ )/S (A, X) → Map(Set∆ )/S (A, Y ).
We will show that U satisfies the hypotheses of Lemma 2.2.3.5 and therefore
contains all simplicial sets. Conditions (i) and (ii) are obvious, and conditions (iv) and (v) follow from Lemma 2.2.3.4. It will therefore suffice toshow that every simplex ∆n belongs to U. For every map ∆n → S, we havea commutative diagram
Map(Set∆ )/S (∆n , X)

/ Map(Set ) (∆n , Y )∆ /SMap(Set∆ )/S ({n}, X)/ Map(Set ) ({n}, Y ).
∆ /SSince the inclusion {n} ⊆ ∆n is right anodyne, the vertical maps are trivial
Kan fibrations. It will therefore suffice to show that the bottom horizontal
map is a homotopy equivalence, which follows immediately from (a).

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Lemma 2.2.3.7. Let φ : C[∆n ] → [n] be as in Lemma 2.2.3.1. Suppose we[n]are given a right fibration X → ∆n , a projectively fibrant diagram F ∈ Set∆ ,and a weak equivalence of diagrams α : Stφ X → F. Then the adjoint map
X → Unφ F is a pointwise equivalence of right fibrations over ∆n .
Proof. For 0 ≤ i ≤ n, let X(i) = X ×∆n ∆{n−i,...,n} ⊆ X. We observethat (Stφ X)(i) is canonically isomorphic to the twisted geometric realization|X(i)|Q• , where Q• is defined as in §2.2.2. Since X → ∆n is a right fibration,
the fiber X ×∆n {i} is a deformation retract of X(i). Using Proposition
2.2.2.7, we conclude that the induced inclusion |X ×∆n {n−i}|Q• → |X(i)|Q•
is a weak homotopy equivalence. Since α is a weak equivalence, we get weakequivalences |X ×∆n {n−i}|Q• → F(i) for each 0 ≤ i ≤ n. Using Proposition
2.2.2.9, we deduce that the adjoint maps X ×∆n {n − i} → SingQ• F(i) areagain weak homotopy equivalences. The desired result now follows from theobservation that SingQ• F(i) ' (Unφ F) ×∆n {n − i} (Remark 2.2.2.11).
Notation 2.2.3.8. For every simplicial set S, we let RFib(S) denote thefull subcategory of (Set∆ )/S spanned by those maps X → S which are rightfibrations.
Proposition 2.1.4.9 implies that if p : X → S exhibits X as a fibrant objectof the contravariant model category (Set∆ )/S , then p is a right fibration. We
will prove the converse below (Corollary 2.2.3.12). For the moment, we will
be content with the following weaker result:Lemma 2.2.3.9. For every integer n ≥ 0, the inclusion i : (Set∆ )◦/∆n ⊆RFib(∆n ) is an equivalence of simplicial categories.
Proof. It is clear that i is fully faithful. To prove that i is essentially surjective, consider any right fibration X → ∆n . Let φ : C[∆n ] → [n] be defined as in Lemma 2.2.3.1 and choose a weak equivalence Stφ X → F, where[n]
F ∈ Set∆ is a projectively fibrant diagram. Lemma 2.2.3.7 implies that theadjoint map X → Unφ F is a pointwise equivalence of right fibrations in ∆nand therefore a homotopy equivalence in RFib(∆n ) (Lemma 2.2.3.6). It now
suffices to observe that Unφ F ∈ (Set∆ )◦/∆n .
Lemma 2.2.3.10. For each integer n ≥ 0, the unstraightening functorC[∆n ]Un∆n : (Set∆ )◦ → RFib(∆n ) is an equivalence of simplicial categories.
Proof. In view of Lemma 2.2.3.9 and Proposition A.3.1.10, it will suffice
to show that the Quillen adjunction (St∆n , Un∆n ) is a Quillen equivalence.
This follows immediately from Lemma 2.2.3.1 and Proposition A.3.3.8.
Proposition 2.2.3.11. For every simplicial set S, the unstraightening functor UnS induces an equivalence of simplicial categories
C[S]op ◦(Set∆) → RFib(S).

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C[S]Proof. For each simplicial set S, let (Set∆)f denote the category of proC[S]opjectively fibrant objects of Set∆
and let WS be the class of weak equivaC[S]oplences in (Set∆)f . Let WS0 be the collection of pointwise equivalences inRFib(S). We have a commutative diagram of simplicial categories:
C[S]op ◦

(Set∆

C[S]

)

(Set∆

/ RFib(S)φS

/ RFib[W 0 −1
S ]

ψS



op

UnS

)f [WS−1 ](see Notation A.3.5.1). Lemma A.3.6.17 implies that the left vertical map isan equivalence. Using Lemma 2.2.3.6 and Remark A.3.2.14, we deduce thatthe right vertical map is also an equivalence. It will therefore suffice to showthat φS is an equivalence.
Let U denote the collection of simplicial sets S for which φS is an equivalence. We will show that U satisfies the hypotheses of Lemma 2.2.3.5 andtherefore contains every simplicial set S. Conditions (i) and (ii) are obviously
satisfied, and condition (iii) follows from Lemma 2.2.3.10 and Proposition
A.3.1.10. We will verify condition (iv); the proof of (v) is similar.
Applying Corollary A.3.6.18, we deduce:
C[S]op(∗) The functor S 7→ (Set∆)f [WS−1 ] carries homotopy colimit diagrams
indexed by a partially ordered set to homotopy limit diagrams in Cat∆ .
Suppose we are given a pushout diagram
/ X0

X

Y

f


/ Y0

in which X, X 0 , Y ∈ U, where f is a cofibration. We wish to prove thatY 0 ∈ U. We have a commutative diagram
C[Y 0 ]op

(Set∆

)f [WY−10 ]φY 0/ RFib(Y 0 )[W 0 −1
Y0]


u

/ RFib(Y )[W 0 −1
Y ]

w

v
−1

RFib(X 0 )[W 0 X 0 ](
/ RFib(X)[W 0 −1
X ].

Using (∗) and Corollary A.3.2.28, we deduce that φY 0 is an equivalence if−1
and only if, for every pair of objects x, y ∈ RFib(Y 0 )[W 0 Y 0 ], the diagram ofsimplicial sets
MapRFib(Y 0 )[W 0 −10 ] (x, y)

/ Map


MapRFib(X 0 )[W 0 −10 ] (v(x), v(y))/ Map(u(x), u(y))
RFib(Y )[W 0 −1
Y ]

Y

X



(w(x), w(y))
RFib(X)[W 0 −1
X ]

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FIBRATIONS OF SIMPLICIAL SETS

is homotopy Cartesian. Since ψY 0 is a weak equivalence of simplicial categories, we may assume without loss of generality that x = ψY 0 (x) and thaty = ψY 0 (y) for some x, y ∈ (Set∆ )◦/Y 0 . It will therefore suffice to prove thatthe equivalent diagram/ MapRFib(Y ) (u(x), u(y))

MapRFib(Y 0 ) (x, y)
MapRFib(X 0 ) (v(x), v(y))

g/ MapRFib(X) (w(x), w(y))

is homotopy Cartesian. But this diagram is a pullback square, and the map
g is a Kan fibration by Lemma 2.2.3.4.
We can now complete the proof of Theorem 2.2.1.2. Suppose that φ :C[S] → Cop is an equivalence of simplicial categories; we wish to show thatthe adjoint functors (Stφ , Unφ ) determine a Quillen equivalence between(Set∆ )/S and SetC
∆ . Using Proposition A.3.3.8, we can reduce to the casewhere φ is an isomorphism. In view of Proposition A.3.1.10, it will suffice to
C[S]op ◦show that Unφ induces an equivalence of simplicial categories (Set∆) →(Set∆ )◦/S , which follows immediately from Proposition 2.2.3.11.
Corollary 2.2.3.12. Let p : X → S be a map of simplicial sets. The following conditions are equivalent:(1) The map p is a right fibration.
(2) The map p exhibits X as a fibrant object of (Set∆ )/S (with respect tothe contravariant model structure).
Proof. The implication (2) ⇒ (1) follows from Proposition 2.1.4.9. For theconverse, let us suppose that p is a right fibration. Proposition 2.2.3.11 imC[S]op ◦plies that the unstraightening functor UnS : (Set∆) → FunR (S) is essentially surjective. Since UnS factors through the inclusion i : (Set∆ )◦/S ⊆FunR (S), we deduce that i is essentially surjective. Consequently, we canchoose a simplicial homotopy equivalence f : X → Y in (Set∆ )/S , where Y
is fibrant. Let g be a homotopy inverse to X so that there exists a homotopy
h : X × ∆1 → X from idX to g ◦ f .
To prove that X is fibrant, we must show that every lifting problem
A _
j


B

e0

/X
>

e


/S

p

has a solution, provided that j is a trivial cofibration in the contravariant
model category (Set∆ )/S . Since Y is fibrant, the map f ◦ e0 can be extended
to a map e : B → Y in (Set∆ )/S . Let e0 = g ◦e. The maps e and h◦(e0 ×id∆1 )highertopoi

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CHAPTER 2

determine another lifting problem`(A × ∆1 ) A×{1} (B × {1})
_
E

j0

/6 X
p


B × ∆1

/ S.

Proposition 2.1.2.6 implies that j 0 is right anodyne. Since p is a right fibration, there exists an extension E as indicated in the diagram. The restriction
e = E|B × {0} is then a solution the original problem.
Corollary 2.2.3.13. Suppose we are given a diagram of simplicial sets
/Y

f

X
p



q

S,



where p and q are right fibrations. Then f is a contravariant equivalence in(Set∆ )/S if and only if f is a pointwise equivalence.
Proof. Since (Set∆ )/S is a simplicial model category, this follows immediately from Corollary 2.2.3.12 and Lemma 2.2.3.6.
Corollary 2.2.3.12 admits the following generalization:
Corollary 2.2.3.14. Suppose we are given a diagram of simplicial sets
/Y

f

X
p



S,

q



where p and q are right fibrations. Then f is a contravariant fibration in(Set∆ )/S if and only if f is a right fibration.
Proof. The map f admits a factorization
f0

f 00

X → X 0 → Y,
where f 0 is a contravariant equivalence and f 00 is a contravariant fibration(in (Set∆ )/S ). Proposition 2.1.4.9 implies that f 00 is a right fibration, so thecomposition q ◦ f 00 is a right fibration. Invoking Corollary 2.2.3.13, we conclude that for every vertex s ∈ S, the map f 0 induces a homotopy equivalenceof fibers Xs → Xs0 . Consider the diagram
Xs

fs0

Ys .

~

/ Xs0

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FIBRATIONS OF SIMPLICIAL SETS

The vertical maps in this diagram are right fibrations between Kan complexes and therefore Kan fibrations (Lemma 2.1.3.3). Since fs is a homotopy
equivalence, we conclude that the induced map of fibers fy0 : Xy → Xy0 is a
homotopy equivalence for each vertex y ∈ Y . Invoking Lemma 2.2.3.6, wededuce that f 0 is an equivalence in the simplicial category (Set∆ )/Y .
We can now repeat the proof of Corollary 2.2.3.12. Let g be a homotopy
inverse to f 0 in the simplicial category (Set∆ )/Y and let h : X × ∆1 → X
be a homotopy from idX to g ◦ f 0 (which projects to the identity on Y ). To
prove that f is a covariant fibration, we must show that every lifting problem
A _
j

e0
e

/X
>
f



/Y
B
has a solution provided that j is a trivial cofibration in the contravariant
model category (Set∆ )/S . Since f 00 is a contravariant fibration, the map
f 0 ◦ e0 can be extended to a map e : B → X 0 in (Set∆ )/Y . Let e0 = g ◦ e.
The maps e and h ◦ (e0 × id∆1 ) determine another lifting problem`/6 X(A × ∆1 ) A×{1} (B × {1})_j0

B × ∆1

E

f


/ Y.

Proposition 2.1.2.6 implies that j 0 is right anodyne. Since f is a right fibration, there exists an extension E as indicated in the diagram. The restriction
e = E|B × {0} is then a solution to the original problem.
We conclude this section with one more result which will be useful instudying the Joyal model structure on Set∆ . Suppose that f : X → S is
any map of simplicial sets and that {s} is a vertex of S. Let Q• denote thecosimplicial object of Set∆ defined in §2.2.2. Then we have a canonical map|Xs |Q• ' (St{s} Xs )(s) → (StS X)(s).
Proposition 2.2.3.15. Suppose that f : X → S is a right fibration of
simplicial sets. Then for each vertex s of S, the canonical map φ : |Xs |Q• →(StS X)(s) is a weak homotopy equivalence of simplicial sets.
Proof. Choose a weak equivalence StS X → F, where F : C[S]op → Set∆ is a
projectively fibrant diagram. Theorem 2.2.1.2 implies that the adjoint map
X → UnS (F) is a contravariant equivalence in (Set∆ )/S . Applying the “onlyif” direction of Corollary 2.2.3.12, we conclude that each of the induced maps
Xs → (UnS F)s ' SingQ• F(s)is a homotopy equivalence of Kan complexes. Using Proposition 2.2.2.9, we
deduce that the adjoint map |Xs |Q• → F(s) is a weak homotopy equivalence.
It follows from the two-out-of-three property that φ is also a weak homotopy
equivalence.

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2.2.4 The Comparison Theorem
Let S be an ∞-category containing a pair of objects x and y and let Q• denote the cosimplicial object of Set∆ described in §2.2.2. We have a canonicalmap of simplicial sets
f : | HomR
S (x, y)|Q• → MapC[S] (x, y).
Moreover, in the special case where S is the nerve of a fibrant simplicial
category C, the compositionf| HomR
S (x, y)|Q• → MapC[S] (x, y) → MapC (x, y)
can be identified with the counit map| SingQ• MapC (x, y)|Q• → MapC (X, Y )and is therefore a weak equivalence (Corollary 2.2.2.10). Consequently, we
may reformulate Theorem 2.2.0.1 in the following way:
Proposition 2.2.4.1. Let S be an ∞-category containing a pair of objects
x and y. Then the natural map
f : | HomR
S (x, y)|Q• → MapC[S] (x, y)
is a weak homotopy equivalence of simplicial sets.
`.
Proof. Let C = S/yS/y S and let v denote the image in C of the cone point.
of S/y. There is a canonical projection π : C → S, which induces a map ofsimplicial sets
f 00 : (StS S/y )(x) → MapC[S] (x, y).
The map f can be identified with the composition f 00 ◦ f 0 , where f 0 is themap| HomR
S (x, y)|Q• ' (St{x} S/y ×S {x})(x) → (StS S/y )(x).
Since the projection S/y → S is a right fibration, the map f 0 is a weak homotopy equivalence (Proposition 2.2.3.15). It will therefore suffice to show thatf 00 is a weak homotopy equivalence. To see this, we consider the commutative
diagramg(StS S/y )(x)7h(StS {y})(x)f 00'/ MapC[S] (x, y).

The inclusion i : {y} ⊆ S/y is a retract of the inclusiona(S/y × {1})({idy } × ∆1 ) ⊆ S/y × ∆1 ,{idy }×{1}which is right anodyne by Corollary 2.1.2.7. It follows that i is a contravariant equivalence in (Set∆ )/S (Proposition 2.1.4.9), so the map g is a weakhomotopy equivalence of simplicial sets. Since the map h is an isomorphism,
the map f 00 is also a weak homotopy equivalence by virtue of the two-outof-three property.

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FIBRATIONS OF SIMPLICIAL SETS

89

2.2.5 The Joyal Model Structure
The category of simplicial sets can be endowed with a model structure forwhich the fibrant objects are precisely the ∞-categories. The original construction of this model structure is due to Joyal, who uses purely combinatorial arguments ([44]). In this section, we will exploit the relationship between
simplicial categories and ∞-categories to give an alternative description ofthis model structure. Our discussion will make use of a model structure on
the category Cat∆ of simplicial categories, which we review in §A.3.2.
Theorem 2.2.5.1. There exists a left proper combinatorial model structureon the category of simplicial sets with the following properties:(C) A map p : S → S 0 of simplicial sets is a cofibration if and only if it isa monomorphism.
(W ) A map p : S → S 0 is a categorical equivalence if and only if the induced simplicial functor C[S] → C[S 0 ] is an equivalence of simplicial
categories.
Moreover, the adjoint functors (C, N) determine a Quillen equivalence between Set∆ (with the model structure defined above) and Cat∆ .
Our proof will make use of the theory of inner anodyne maps of simplicial
sets, which we will study in detail in §2.3. We first establish a simple lemma.
Lemma 2.2.5.2. Every inner anodyne map f : A → B of simplicial sets isa categorical equivalence.
Proof. It will suffice to prove that if f is inner anodyne, then the associatedmap C[f ] is a trivial cofibration of simplicial categories. The collection of allmorphisms f for which this statement holds is weakly saturated (Definition
A.1.2.2). Consequently, we may assume that f is an inner horn inclusionΛni ⊆ ∆n , 0 < i < n. We now explicitly describe the map C[f ]:• The objects of C[Λni ] are the objects of C[∆n ]: namely, elements of thelinearly ordered set [n] = {0, . . . , n}.
• For 0 ≤ j ≤ k ≤ n, the simplicial set MapC[Λni ] (j, k) is equal to
MapC[∆n ] (j, k) unless j = 0 and k = n. In the latter case,
MapC[Λni ] (j, k) = K ⊆ (∆1 )n−1 ' MapC[∆n ] (j, k),where K is the simplicial subset of the cube (∆1 )n−1 obtained byremoving the interior and a single face.
We observe that C[f ] is a pushout of the inclusion [1]K ⊆ [1](∆1 )n−1 (see§A.3.2 for an explanation of this notation). It now suffices to observe thatthe inclusion K ⊆ (∆1 )n−1 is a trivial cofibration of simplicial sets (withrespect to the usual model structure on Set∆ ).

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Proof of Theorem 2.2.5.1. We first show that C carries cofibrations of simplicial sets to cofibrations of simplicial categories. Since the class of all cofibrations of simplicial sets is generated by the inclusions ∂ ∆n ⊆ ∆n , it suffices
to show that each map C[∂ ∆n ] → C[∆n ] is a cofibration of simplicial categories. If n = 0, then the inclusion C[∂ ∆n ] ⊆ C[∆n ] is isomorphic to theinclusion ∅ ⊆ ∗ of simplicial categories, which is a cofibration. In the casewhere n > 0, we make use of the following explicit description of C[∂ ∆n ] asa subcategory of C[∆n ]:• The objects of C[∂ ∆n ] are the objects of C[∆n ]: namely, elements of
the linearly ordered set [n] = {0, . . . , n}.
• For 0 ≤ j ≤ k ≤ n, the simplicial set HomC[∂ ∆n ] (j, k) is equal to
HomC[∆n ] (j, k) unless j = 0 and k = n. In the latter case, the simplicial
set HomC[∂ ∆n ] (j, k) consists of the boundary of the cube(∆1 )n−1 ' HomC[∆n ] (j, k).
In particular, the inclusion C[∂ ∆n ] ⊆ C[∆n ] is a pushout of the inclusion[1]∂(∆1 )n−1 ⊆ [1](∆1 )n−1 , which is a cofibration of simplicial categories (see§A.3.2 for an explanation of our notation).
We now declare that a map p : S → S 0 of simplicial sets is a categoricalfibration if it has the right lifting property with respect to all maps which arecofibrations and categorical equivalences. We now claim that the cofibrations,
categorical equivalences, and categorical fibrations determine a left proper
combinatorial model structure on Set∆ . To prove this, it will suffice to showthat the hypotheses of Proposition A.2.6.15 are satisfied:(1) The class of categorical equivalences in Set∆ is perfect. This follows
from Corollary A.2.6.14 since the functor C preserves filtered colimits
and the class of equivalences between simplicial categories is perfect.
(2) The class of categorical equivalences is stable under pushouts by cofibrations. Since C preserves cofibrations, this follows immediately fromthe left properness of Cat∆ .
(3) A map of simplicial sets which has the right lifting property with respect to all cofibrations is a categorical equivalence. In other words,
we must show that if p : S → S 0 is a trivial fibration of simplicial
sets, then the induced functor C[p] : C[S] → C[S 0 ] is an equivalence ofsimplicial categories.
Since p is a trivial fibration, it admits a section s : S 0 → S. It is clearthat C[p] ◦ C[s] is the identity; it therefore suffices to show that
C[s] ◦ C[p] : C[S] → C[S]is homotopic to the identity.
Let K denote the simplicial set MapS 0 (S, S). Then K is a contractible
Kan complex containing vertices x and y which classify s ◦ p and idS .

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FIBRATIONS OF SIMPLICIAL SETS

We note the existence of a natural “evaluation map” e : K × S → Ssuch that s ◦ p = e ◦ ({x} × idS ), idS = e ◦ ({y} × idS ). It therefore
suffices to show that the functor C carries {x} × idS and {y} × idS into
homotopic morphisms. Since both of these maps section the projection
K × S → S, it suffices to show that the projection C[K × S] → C[S] isan equivalence of simplicial categories. Replacing S by S × K and S 0
by S, we are reduced to the special case where S = S 0 × K and K isa contractible Kan complex.
By the small object argument, we can find an inner anodyne map S 0 →V , where V is an ∞-category. The corresponding map S 0 ×K → V ×K
is also inner anodyne (Proposition 2.3.2.1), so the maps C[S 0 ] → C[V ]and C[S 0 ×K] → C[V ×K] are both trivial cofibrations (Lemma 2.2.5.2).
It follows that we are free to replace S 0 by V and S by V × K. In otherwords, we may suppose that S 0 is an ∞-category (and now we will haveno further need of the assumption that S is isomorphic to the productS 0 × K).
Since p is surjective on vertices, it is clear that C[p] is essentiallysurjective. It therefore suffices to show that for every pair of vertices x, y ∈ S0 , the induced map of simplicial sets MapC[S] (x, y) →MapC[S 0 ] (p(x), p(y)) is a weak homotopy equivalence. Using Propositions 2.2.4.1 and 2.2.2.7, it suffices to show that the map HomR
S (x, y) →(p(x),
p(y))
is
a
weak
homotopy
equivalence.
Thismap
is obviHomR
0
S
ously a trivial fibration if p is a trivial fibration.
By construction, the functor C preserves weak equivalences. We verified
above that C preserves cofibrations as well. It follows that the adjoint functors (C, N) determine a Quillen adjunction
Set∆ oCN/ Cat .
∆

To complete the proof, we wish to show that this Quillen adjunction is a
Quillen equivalence. According to Proposition A.2.5.1, we must show thatfor every simplicial set S and every fibrant simplicial category C, a map
u : S → N(C)is a categorical equivalence if and only if the adjoint map
v : C[S] → C
is an equivalence of simplicial categories. We observe that v factors as acomposition
C[u]w

C[S] −→ C[N(C)] → C .
By definition, u is a categorical equivalence if and only if C[u] is an equivalence of simplicial categories. We now conclude by observing that the counit
map w is an equivalence of simplicial categories (Theorem 2.2.0.1).

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CHAPTER 2

We now establish a few pleasant properties enjoyed by the Joyal model
structure on Set∆ . We first note that every object of Set∆ is cofibrant; inparticular, the Joyal model structure is left proper (Proposition A.2.4.2).
Remark 2.2.5.3. The Joyal model structure is not right proper. To see
this, we note that the inclusion Λ21 ⊆ ∆2 is a categorical equivalence, but it
does not remain so after pulling back via the fibration ∆{0,2} ⊆ ∆2 .
Corollary 2.2.5.4. Let f : A → B be a categorical equivalence of simplicial
sets and K an arbitrary simplicial set. Then the induced map A×K → B×K
is a categorical equivalence.
Proof. Choose an inner anodyne map B → Q, where Q is an ∞-category.
Then B × K → Q × K is also inner anodyne, hence a categorical equivalence(Lemma 2.2.5.2). It therefore suffices to prove that A × K → Q × K is a
categorical equivalence. In other words, we may suppose to begin with that
B is an ∞-category.
f0

f 00

Now choose a factorization A → R → B, where f 0 is an inner anodyne map
and f 00 is an inner fibration. Since B is an ∞-category, R is an ∞-category.
The map A × K → R × K is inner anodyne (since f 0 is) and therefore acategorical equivalence; consequently, it suffices to show that R×K → B ×K
is a categorical equivalence. In other words, we may reduce to the case where
A is also an ∞-category.
Choose an inner anodyne map K → S, where S is an ∞-category. ThenA × K → A × S and B × K → B × S are both inner anodyne and therefore
categorical equivalences. Thus, to prove that A × K → B × K is a categorical equivalence, it suffices to show that A × S → B × S is a categoricalequivalence. In other words, we may suppose that K is an ∞-category.
Since A, B, and K are ∞-categories, we have canonical isomorphismsh(A × K ) ' hA × hKh(B × K ) ' hB × hK .

It follows that A × K → B × K is essentially surjective provided that f isessentially surjective. Furthermore, for any pair of vertices (a, k), (a0 , k 0 ) ∈(A × K)0 , we have
R
R
0
0
0 0
HomR
A×K ((a, k), (a , k )) ' HomA (a, a ) × HomK (k, k )
R
R
0
0
0
0
HomR
B×K ((f (a), k), (f (a ), k )) ' HomB (f (a), f (a )) × HomK (k, k ).

This shows that A × K → B × K is fully faithful provided that f is fully
faithful, which completes the proof.
Remark 2.2.5.5. Since every inner anodyne map is a categorical equivalence, it follows that every categorical fibration p : X → S is a inner fibration(see Definition 2.0.0.3). The converse is false in general; however, it is true
when S is a point. In other words, the fibrant objects for the Joyal model
structure on Set∆ are precisely the ∞-categories. The proof will be given in§2.4.6 as Theorem 2.4.6.1. We will assume this result for the remainder ofthe section. No circularity will result from this because the proof of Theorem
2.4.6.1 will not use any of the results proven below.

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The functor C[•] does not generally commute with products. However,Corollary 2.2.5.4 implies that C commutes with products in the following
weak sense:
Corollary 2.2.5.6. Let S and S 0 be simplicial sets. The natural map
C[S × S 0 ] → C[S] × C[S 0 ]is an equivalence of simplicial categories.
Proof. Suppose first that there are fibrant simplicial categories C, C0 with
S = N(C), S 0 = N(C0 ). In this case, we have a diagramfg

C[S × S 0 ] → C[S] × C[S 0 ] → C × C0 .
By the two-out-of-three property, it suffices to show that g and g ◦ f areequivalences. Both of these assertions follow immediately from the fact thatthe counit map C[N(D)] → D is an equivalence for any fibrant simplicial
category D (Theorem 2.2.5.1).
In the general case, we may choose categorical equivalences S → T , S 0 →0
T , where T and T 0 are nerves of fibrant simplicial categories. Since S ×S 0 →T × T 0 is a categorical equivalence, we reduce to the case treated above.
Let K be a fixed simplicial set and let C be a simplicial set which isfibrant with respect to the Joyal model structure. Then C has the extension
property with respect to all inner anodyne maps and is therefore an ∞category. It follows that Fun(K, C) is also an ∞-category. We might call
two morphisms f, g : K → C homotopic if they are equivalent when viewed
as objects of Fun(K, C). On the other hand, the general theory of model
categories furnishes another notion of homotopy: f and g are left homotopic
if the map
a
a
f
g:K
K→C
can be extended over a mapping cylinder I for K.
Proposition 2.2.5.7. Let C be a ∞-category and K an arbitrary simplicial
set. A pair of morphisms f, g : K → C are homotopic if and only if they are
left-homotopic.
Proof. Choose a contractible Kan complex S containing a pair of distinct
vertices, x and y. We note that the inclusionaK
K ' K × {x, y} ⊆ K × S
exhibits K × S as a mapping cylinder
It follows that f and g are left`for K. `homotopic if and only if the map f g : K K → C admits an extension
to K × S. In other words, f and g are left homotopic if and only if there
exists a map h : S → CK such that h(x) = f and h(y) = g. We note thatany such map factors through Z, where Z ⊆ Fun(K, C) is the largest Kancomplex contained in CK . Now, by classical homotopy theory, the map hexists if and only if f and g belong to the same path component of Z. It isclear that this holds if and only if f and g are equivalent when viewed as
objects of the ∞-category Fun(K, C).

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We are now in a position to prove Proposition 1.2.7.3, which was asserted
without proof in §1.2.7. We first recall the statement.
Proposition. Let K be an arbitrary simplicial set.
(1) For every ∞-category C, the simplicial set Fun(K, C) is an ∞-category.
(2) Let C → D be a categorical equivalence of ∞-categories. Then the
induced map Fun(K, C) → Fun(K, D) is a categorical equivalence.
(3) Let C be an ∞-category and K → K 0 a categorical equivalence of simplicial sets. Then the induced map Fun(K 0 , C) → Fun(K, C) is a categorical equivalence.
Proof. We first prove (1). To show that Fun(K, C) is an ∞-category, it suffices to show that it has the extension property with respect to every inner
anodyne inclusion A ⊆ B. This is equivalent to the assertion that C has theright lifting property with respect to the inclusion A × K ⊆ B × K. But C
is an ∞-category and A × K ⊆ B × K is inner anodyne (Corollary 2.3.2.4).
Let hSet∆ denote the homotopy category of Set∆ taken with respect to
the Joyal model structure. For each simplicial set X, we let [X] denotethe same simplicial set considered as an object of hSet∆ . For every pair ofobjects X, Y ∈ Set∆ , [X × Y ] is a product of [X] and [Y ] in hSet∆ . This is a
general fact when X and Y are fibrant; in the general case, we choose fibrantreplacements X → X 0 , Y → Y 0 and apply the fact that the canonical map
X × Y → X 0 × Y 0 is a categorical equivalence (Corollary 2.2.5.4).
If C is an ∞-category, then C is a fibrant object of Set∆ (Theorem 2.4.6.1).
Proposition 2.2.5.7 allows us to identify HomhSet∆ ([X], [C]) with the set of
equivalence classes of objects in the ∞-category Fun(X, C). In particular, wehave canonical bijections
HomhSet∆ ([X] × [K], [C]) ' HomhSet∆ ([X × K], [C])' HomhSet∆ ([X], [Fun(K, C)]).
It follows that [Fun(K, C)] is determined up to canonical isomorphism by [K]and [C] (more precisely, it is an exponential [C][K] in the homotopy category
hSet∆ ), which proves (2) and (3).
Our description of the Joyal model structure on Set∆ is different from thedefinition given in [44]. Namely, Joyal defines a map f : A → B to be a weak
categorical equivalence if, for every ∞-category C, the induced map
hFun(B , C) → hFun(A, C)
is an equivalence (of ordinary categories). To prove that our definition agrees
with his, it will suffice to prove the following.
Proposition 2.2.5.8. Let f : A → B be a map of simplicial sets. Then f isa categorical equivalence if and only if it is a weak categorical equivalence.

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95

Proof. Suppose first that f is a categorical equivalence. If C is an arbitrary∞-category, Proposition 1.2.7.3 implies that the induced map Fun(B, C) →Fun(A, C) is a categorical equivalence, so that hFun(B , C) → hFun(A, C)
is an equivalence of categories. This proves that f is a weak categoricalequivalence.
Conversely, suppose that f is a weak categorical equivalence. We wish toshow that f induces an isomorphism in the homotopy category of Set∆ withrespect to the Joyal model structure. It will suffice to show that for any
fibrant object C, f induces a bijection [B, C] → [A, C], where [X, C] denotes
the set of homotopy classes of maps from X to C. By Proposition 2.2.5.7,[X, C] may be identified with the set of isomorphism classes of objects in the
category hFun(X , C). By assumption, f induces an equivalence of categories
hFun(B , C) → hFun(A, C) and therefore a bijection on isomorphism classesof objects.
Remark 2.2.5.9. The proof of Proposition 1.2.7.3 makes use of Theorem 2.4.6.1, which asserts that the (categorically) fibrant objects of Set∆
are precisely the ∞-categories. Joyal proves the analogous assertion for his
model structure in [44]. We remark that one cannot formally deduce Theorem 2.4.6.1 from Joyal’s result since we need Theorem 2.4.6.1 to prove that
Joyal’s model structure coincides with the one we have defined above. On
the other hand, our approach does give a new proof of Joyal’s theorem.
Remark 2.2.5.10. Proposition 2.2.5.8 permits us to define the Joyal model
structure without reference to the theory of simplicial categories (this is
Joyal’s original point of view [44]). Our approach is less elegant but allows
us to easily compare the theory of ∞-categories with other models of higher
category theory, such as simplicial categories. There is another approach toobtaining comparison results, due to Toën. In [77], he shows that if C isa model category equipped with a cosimplicial object C • satisfying certain
conditions, then C is (canonically) Quillen equivalent to Rezk’s category ofcomplete Segal spaces. Toën’s theorem applies in particular when C is thecategory of simplicial sets and C • is the “standard simplex” C n = ∆n . In
fact, Set∆ is in some sense universal with respect to this property because itis generated by C • under colimits and the class of categorical equivalences
is dictated by Toën’s axioms. We refer the reader to [77] for details.

2.3 INNER FIBRATIONSIn this section, we will study the theory of inner fibrations between simplicial
sets. The meaning of this notion is somewhat difficult to explain because it
has no counterpart in classical category theory: Proposition 1.1.2.2 implies
that every functor between ordinary categories C → D induces an inner
fibration of nerves N(C) → N(D).
In the case where S is a point, a map p : X → S is an inner fibration
if and only if X is an ∞-category. Moreover, the class of inner fibrations ishighertopoi

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stable under base change: if
X0

/X

p0

p



/S
S0
is a pullback diagram of simplicial sets and p is an inner fibration, then so is
p0 . It follows that if p : X → S is an arbitrary inner fibration, then each fiber
Xs = X ×S {s} is an ∞-category. We may therefore think of p as encoding
a family of ∞-categories parametrized by S. However, the fibers Xs depend
functorially on s only in a very weak sense.
Example 2.3.0.1. Let F : C → C0 be a functor between ordinary categories.
Then the map N(C) → N(C0 ) is an inner fibration. Yet the fibers N(C)C =N(C ×C0 {C}) and N(C)D = N(C ×C0 {D}) over objects C, D ∈ C0 can havewildly different properties even if C and D are isomorphic objects of C0 .
In order to describe how the different fibers of an inner fibration are related
to one another, we will introduce the notion of a correspondence between ∞categories. We review the classical theory of correspondences in §2.3.1 andexplain how to generalize this theory to the ∞-categorical setting.
In §2.3.2, we will prove that the class of inner anodyne maps is stable
under smash products with arbitrary cofibrations between simplicial sets.
As a consequence, we will deduce that the class of inner fibrations (andhence the class of ∞-categories) is stable under the formation of mapping
spaces.
In §2.3.3, we will study the theory of minimal inner fibrations, a generalization of Quillen’s theory of minimal Kan fibrations. In particular, we willdefine a class of minimal ∞-categories and show that every ∞-category C
is (categorically) equivalent to a minimal ∞-category C0 , where C0 is welldefined up to (noncanonical) isomorphism. We will apply this theory in §2.3.4
to develop a theory of n-categories for n < ∞.
2.3.1 Correspondences
Let C and C0 be categories. A correspondence from C to C0 is a functor
M : Cop × C0 → Set .
If M is a correspondence from C to C0 , we can define a new category C ?M C0
as follows. An object of C ?M C0 is either an object of C or an object of C0 .
For morphisms, we take
HomC (X, Y ) if X, Y ∈ CHom 0 (X, Y ) if X, Y ∈ C0
C
HomC ?M C0 (X, Y ) =

M
(X,
Y)
if X ∈ C, Y ∈ C0
∅
if X ∈ C0 , Y ∈ C .
Composition of morphisms is defined in the obvious way, using the composition laws in C and C0 and the functoriality of M (X, Y ) in X and Y .

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Remark 2.3.1.1. In the special case where F : Cop × C0 → Set is the constant functor taking the value ∗, the category C ?F C0 coincides with theordinary join C ? C0 .
For any correspondence M : C → C0 , there is an obvious functor F :C ?M C0 → [1] (here [1] denotes the linearly ordered set {0, 1} regarded as acategory in the obvious way) which is uniquely determined by the condition
that F −1 {0} = C and F −1 {1} = C0 . Conversely, given any category M
equipped with a functor F : M → [1], we can define C = F −1 {0}, C0 =F −1 {1}, and a correspondence M : C → C0 by the formula M (X, Y ) =HomM (X, Y ). We may summarize the situation as follows:
Fact 2.3.1.2. Giving a pair of categories C, C0 and a correspondence between
them is equivalent to giving a category M equipped with a functor M → [1].
Given this reformulation, it is clear how to generalize the notion of acorrespondence to the ∞-categorical setting.
Definition 2.3.1.3. Let C and C0 be ∞-categories. A correspondence fromC to C0 is a ∞-category M equipped with a map F : M → ∆1 and identifications C ' F −1 {0}, C0 ' F −1 {1}.
Remark 2.3.1.4. Let C and C0 be ∞-categories. Fact 2.3.1.2 generalizes tothe ∞-categorical setting in the following way: there is a canonical bijection
between equivalence classes of correspondences from C to C0 and equivalenceclasses of functors Cop × C0 → S, where S denotes the ∞-category of spaces.
In fact, it is possible to prove a more precise result (a Quillen equivalence
between certain model categories), but we will not need this.
To understand the relevance of Definition 2.3.1.3, we note the following:
Proposition 2.3.1.5. Let C be an ordinary category and let p : X → N(C)be a map of simplicial sets. Then p is an inner fibration if and only if X is
an ∞-category.
Proof. This follows from the fact that any map Λni → N(C), 0 < i < n,admits a unique extension to ∆n .
It follows readily from the definition that an arbitrary map of simplicial
sets p : X → S is an inner fibration if and only if the fiber of p over any
simplex of S is an ∞-category. In particular, an inner fibration p associates
to each vertex s of S an ∞-category Xs and to each edge f : s → s0 in S acorrespondence between the ∞-categories Xs and Xs0 . Higher-dimensional
simplices give rise to what may be thought of as compatible “chains” ofcorrespondences.
Roughly speaking, we might think of an inner fibration p : X → S asa functor from S into some kind of ∞-category of ∞-categories where themorphisms are given by correspondences. However, this description is notquite accurate because the correspondences are required to “compose” only

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in a weak sense. To understand the issue, let us return to the setting of
ordinary categories. If C and C0 are two categories, then the correspondencesfrom C to C0 themselves constitute a category, which we may denote by
M (C, C0 ). There is a natural “composition” defined on correspondences. If weview an object F ∈ M (C, C0 ) as a functor Cop × C0 → Set and G ∈ M (C0 , C00 ),then we can define (G ◦ F )(C, C 00 ) to be the coendZ
F (C, C 0 ) × G(C 0 , C 00 ).
C 0 ∈C0

If we view F as determining a category C ?F C0 and G as determining acategory C0 ?G C00 , then C ?G◦F C00 is obtained by forming the pushouta(C ?F C0 ) (C0 ?G C00 )C0and then discarding the objects of C0 .
Now, giving a category equipped with a functor to [2] is equivalent togiving a triple of categories C, C0 , C00 together with correspondences F ∈M (C, C0 ), G ∈ M (C0 , C00 ), H ∈ M (C, C00 ), and a map α : G ◦ F → H. But themap α need not be an isomorphism. Consequently, the above data cannot
literally be interpreted as a functor from [2] into a category (or even a higher
category) in which the morphisms are given by correspondences.
If C and C0 are categories, then a correspondence from C to C0 can be
regarded as a kind of generalized functor from C to C0 . More specifically,
for any functor f : C → C0 , we can define a correspondence Mf by the formula Mf (X, Y ) = HomC0 (f (X), Y ). This construction gives a fully faithfulembedding MapCat (C, C0 ) → M (C, C0 ). Similarly, any functor g : C0 → C
determines a correspondence Mg given by the formula
Mg (X, Y ) = HomC (X, g(Y ));we observe that Mf ' Mg if and only if the functors f and g are adjoint to
one another.
If an inner fibration p : X → S corresponds to a “functor” from S to ahigher category of ∞-categories with morphisms given by correspondences,then some special class of inner fibrations should correspond to functors
from S into an ∞-category of ∞-categories with morphisms given by actual
functors. This is indeed the case, and the appropriate notion is that of a(co)Cartesian fibration which we will study in §2.4.
2.3.2 Stability Properties of Inner Fibrations
Let C be an ∞-category and K an arbitrary simplicial set. In §1.2.7, we
asserted that Fun(K, C) is an ∞-category (Proposition 1.2.7.3). In the courseof the proof, we invoked certain stability properties of the class of inner
anodyne maps. The goal of this section is to establish the required propertiesand deduce some of their consequences. Our main result is the following
analogue of Proposition 2.1.2.6:

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Proposition 2.3.2.1 (Joyal [44]). The following collections all generate thesame class of morphisms of Set∆ :(1) The collection A1 of all horn inclusions Λni ⊆ ∆n , 0 < i < n.
(2) The collection A2 of all inclusionsa(∆m × Λ21 )(∂ ∆m × ∆2 ) ⊆ ∆m × ∆2 .
∂ ∆m ×Λ21(3) The collection A3 of all inclusionsa(S 0 × Λ21 )(S × ∆2 ) ⊆ S 0 × ∆2 ,S×Λ21

where S ⊆ S 0 .
Proof. We will employ the strategy that we used to prove Proposition 2.1.2.6,
though the details are slightly more complicated. Working cell by cell, we
conclude that every morphism in A3 belongs to the weakly saturated classof morphisms generated by A2 . We next show that every morphism in A1 isa retract of a morphism belonging to A3 . More precisely, we will show thatfor 0 < i < n, the inclusion Λni ⊆ ∆n is a retract of the inclusiona(∆n × Λ21 )(Λni × ∆2 ) ⊆ ∆n × ∆2 .
2
Λn
i ×Λ1

To prove this, we embed ∆n into ∆n × ∆2 via the map of partially orderedsets s : [n] → [n] × [2] given by(j, 0) if j < is(j) = (j, 1) if j = i(j, 2) if j > iand consider the retraction ∆n × ∆2 → ∆n given by the mapr : [n] × [2] → [n]j if j < i, k = 0r(j, k) = j if j > i, k = 2i otherwise.
We now show that every morphism in A2 is inner anodyne (that is, it liesin the weakly saturated class of morphisms generated by A1 ). Choose m ≥ 0.
For each 0 ≤ i ≤ j < m, we let σij denote the (m + 1)-simplex of ∆m × ∆2
corresponding to the map
fij : [m + 1] → [m] × [2]
if 0 ≤ k ≤ i
(k, 0)fij (k) = (k − 1, 1) if i + 1 ≤ k ≤ j + 1(k − 1, 2) if j + 2 ≤ k ≤ m + 1.

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For each 0 ≤ i ≤ j ≤ m, we let τij denote the (m + 2)-simplex of ∆m × ∆2
corresponding to the map
gij : [m + 2] → [m] × [2]
if 0 ≤ k ≤ i
(k, 0)gij (k) = (k − 1, 1) if i + 1 ≤ k ≤ j + 1(k − 2, 2) if j + 2 ≤ k ≤ m + 2.
`m
2
Let X(0) = (∆ × Λ1 ) ∂ ∆m ×Λ2 (∂ ∆m × ∆2 ). For 0 ≤ j < m, we let
1

X(j + 1) = X(j) ∪ σ0j ∪ · · · ∪ σjj .
We have a chain of inclusions
X(j) ⊆ X(j) ∪ σ0j ⊆ · · · ⊂ X(j) ∪ σ0j ∪ · · · ∪ σjj = X(j + 1),
each of which is a pushout of a morphism in A1 and therefore inner anodyne.
It follows that each inclusion X(j) ⊆ X(j + 1) is inner anodyne. Set Y (0) =X(m) so that the inclusion X(0) ⊆ Y (0) is inner anodyne. We now set
Y (j + 1) = Y (j) ∪ τ0j ∪ · · · ∪ τjj for 0 ≤ j ≤ m. As before, we have a chainof inclusions
Y (j) ⊆ Y (j) ∪ τ0j ⊆ · · · ⊆ Yj ∪ τ0j ∪ · · · ∪ τjj = Y (j + 1),
each of which is a pushout of a morphism belonging to A1 . It follows thateach inclusion Y (j) ⊆ Y (j +1) is inner anodyne. By transitivity, we conclude
that the inclusion X(0) ⊆ Y (m + 2) is inner anodyne. We conclude the proof
by observing that Y (m + 2) = ∆m × ∆2 .
Corollary 2.3.2.2 (Joyal [44]). A simplicial set C is an ∞-category if andonly if the restriction map
Fun(∆2 , C) → Fun(Λ21 , C)
is a trivial fibration.
Proof. By Proposition 2.3.2.1, C → ∗ is an inner fibration if and only if C
has the extension property with respect to each of the inclusions in the class
A2 .
Remark 2.3.2.3. In §1.1.2, we asserted that the main function of the weak
Kan condition on a simplicial set C is that it allows us to compose the edgesof C. We can regard Corollary 2.3.2.2 as an affirmation of this philosophy:
the class of ∞-categories C is characterized by the requirement that one
can compose morphisms in C, and the composition is well-defined up to acontractible space of choices.
Corollary 2.3.2.4 (Joyal [44]). Let i : A → A0 be an inner anodyne map ofsimplicial sets and let j : B → B 0 be a cofibration. Then the induced mapa(A × B 0 )(A0 × B) → A0 × B 0
A×B

is inner anodyne.

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Proof. This follows immediately from Proposition 2.3.2.1, which characterizes the class of inner anodyne maps as the class generated by A3 (which isstable under smash products with any cofibration).
Corollary 2.3.2.5 (Joyal [44]). Let p : X → S be an inner fibration and
let i : A → B be any cofibration of simplicial sets. Then the induced map
q : X B → X A ×S A S B is an inner fibration. If i is inner anodyne, then q isa trivial fibration. In particular, if X is an ∞-category, then so is X B for
any simplicial set B.
2.3.3 Minimal Fibrations
One of the aims of homotopy theory is to understand the classification of
spaces up to homotopy equivalence. In the setting of simplicial sets, thisproblem admits an attractive formulation in terms of Quillen’s theory ofminimal Kan complexes. Every Kan complex X is homotopy equivalent toa minimal Kan complex, and a map X → Y of minimal Kan complexes is a
homotopy equivalence if and only if it is an isomorphism. Consequently, theclassification of Kan complexes up to homotopy equivalence is equivalent tothe classification of minimal Kan complexes up to isomorphism. Of course, inpractical terms, this is not of much use for solving the classification problem.
Nevertheless, the theory of minimal Kan complexes (and, more generally,
minimal Kan fibrations) is a useful tool in the homotopy theory of simplicial
sets. The purpose of this section is to describe a generalization of the theoryof minimal models in which Kan fibrations are replaced by inner fibrations.
An exposition of this theory can also be found in [44].
We begin by introducing a bit of terminology. Suppose we are given a
commutative diagram
A
i

u

/X
>
p



v
/S
B
of simplicial sets, where p is an inner fibration, and suppose also that wehave a pair f, f 0 : B → X of candidates for the dotted arrow which renderthe diagram commutative. We will say that f and f 0 are homotopic relative
to A over S if they are equivalent when viewed as objects in the ∞-category
given by the fiber of the map
X B → X A ×S A S B .
Equivalently, f and f 0 are homotopic relative to A over S if there exists a
map F : B×∆1 → X such that F |B×{0} = f , F |B×{1} = f 0 , p◦F = v◦πB ,F ◦ (i × id∆1 ) = u ◦ πA , and F |{b} × ∆1 is an equivalence in the ∞-category
Xv(b) for every vertex b of B.
Definition 2.3.3.1. Let p : X → S be an inner fibration of simplicial sets.
We will say that p is minimal if f = f 0 for every pair of maps f, f 0 : ∆n → X
which are homotopic relative to ∂ ∆n over S.

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We will say that an ∞-category C is minimal if the associated inner fibration C → ∗ is minimal.
Remark 2.3.3.2. In the case where p is a Kan fibration, Definition 2.3.3.1
recovers the usual notion of a minimal Kan fibration. We refer the reader to[32] for a discussion of minimal fibrations in this more classical setting.
Remark 2.3.3.3. Let p : X → ∆n be an inner fibration. Then X is an∞-category. Moreover, p is a minimal inner fibration if and only if X is aminimal ∞-category. This follows from the observation that for any pairof maps f, f 0 : ∆m → X, a homotopy between f and f 0 is automaticallycompatible with the projection to ∆n .
Remark 2.3.3.4. If p : X → S is a minimal inner fibration and T → S is anarbitrary map of simplicial sets, then the induced map XT = X ×S T → T
is a minimal inner fibration. Conversely, if p : X → S is an inner fibrationand if X ×S ∆n → ∆n is minimal for every map σ : ∆n → S, then p
is minimal. Consequently, for many purposes the study of minimal inner
fibrations reduces to the study of minimal ∞-categories.
Lemma 2.3.3.5. Let C be a minimal ∞-category and let f : C → C be afunctor which is homotopic to the identity. Then f is a monomorphism of
simplicial sets.
Proof. Choose a homotopy h : ∆1 × C → C from idC to f . We prove by
induction on n that the map f induces an injection from the set of n-simplices
of C to itself. Let σ, σ 0 : ∆n → C be such that f ◦ σ = f ◦ σ 0 . By the inductive
hypothesis, we deduce that σ| ∂ ∆n = σ 0 | ∂ ∆n = σ0 . Consider the diagram(∆2 × ∂ ∆n )

`

G0

2
n
Λ22 ×∂ ∆n (Λ2 × ∆ )

_

/3 C

G


∆2 × ∆n ,where G0 |Λ22 ×∆n is given by amalgamating h◦(id∆1 ×σ) with h◦(id∆1 ×σ 0 )and G0 |∆2 × ∂ ∆n is given by the compositionσh∆2 × ∂ ∆n → ∆1 × ∂ ∆n →0 ∆1 × C → C .
Since h|∆1 × {X} is an equivalence for every object X ∈ C, Proposition
2.4.1.8 implies the existence of the map G indicated in the diagram. The
restriction G|∆1 × ∆n is a homotopy between σ and σ 0 relative to ∂ ∆n .
Since C is minimal, we deduce that σ = σ 0 .
Lemma 2.3.3.6. Let C be a minimal ∞-category and let f : C → C be afunctor which is homotopic to the identity. Then f is an isomorphism of
simplicial sets.

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Proof. Choose a homotopy h : ∆1 × C → C from idC to f . We prove by
induction on n that the map f induces a bijection from the set of n-simplices
of C to itself. The injectivity follows from Lemma 2.3.3.5, so it will suffice toprove the surjectivity. Choose an n-simplex σ : ∆n → C. By the inductive
hypothesis, we may suppose that σ| ∂ ∆n = f ◦ σ00 for some map σ00 : ∂ ∆n →C. Consider the diagram(∆1 × ∂ ∆n )`({1} × ∆
{1}×∂
 ∆n
_

n

G0

)

3/ C

G


∆ 1 × ∆n ,where G0 |∆1 × ∂ ∆n = h ◦ (id∆1 ×σ00 ) and G0 |{1} × ∆n = σ. If n > 0,then the existence of the map G as indicated in the diagram follows from
Proposition 2.4.1.8; if n = 0, it is obvious. Now let σ 0 = G|{0} × ∆n . To
complete the proof, it will suffice to show that f ◦ σ 0 = σ.
Consider now the diagram(Λ20 × ∆n )`

Λ20 ×∂ ∆n (∆2

H0× ∂ ∆n )

3/ C

H


∆2 ,
where H0 |∆{0,1} × ∆n = h ◦ (id∆1 ×σ 0 ), H0 |∆{1,2} × ∆n = G, and H0 |(∆2 ×∂ ∆n ) is given by the composition
σ0h∆2 × ∂ ∆n → ∆1 × ∂ ∆n →0 ∆1 × C → C .
The existence of the dotted arrow H follows once again from Proposition
2.4.1.8. The restriction H|∆{1,2} ×∆n is a homotopy from f ◦σ 0 to σ relativeto ∂ ∆n . Since C is minimal, we conclude that f ◦ σ 0 = σ, as desired.
Proposition 2.3.3.7. Let f : C → D be an equivalence of minimal ∞categories. Then f is an isomorphism.
Proof. Since f is a categorical equivalence, it admits a homotopy inverse
g : D → C. Now apply Lemma 2.3.3.6 to the compositions f ◦ g and g ◦ f .
The following result guarantees a good supply of minimal ∞-categories:
Proposition 2.3.3.8. Let p : X → S be an inner fibration of simplicial sets.
Then there exists a retraction r : X → X onto a simplicial subset X 0 ⊆ X
with the following properties:(1) The restriction p|X 0 : X 0 → S is a minimal inner fibration.
(2) The retraction r is compatible with the projection p in the sense thatp ◦ r = p.

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CHAPTER 2(3) The map r is homotopic over S to idX relative to X 0 .
(4) For every map of simplicial sets T → S, the induced inclusion
X 0 ×S T ⊆ X ×S T
is a categorical equivalence.
Proof. For every n ≥ 0, we define a relation on the set of n-simplices of X:
given two simplices σ, σ 0 : ∆n → X, we will write σ ∼ σ 0 if σ is homotopic toσ 0 relative to ∂ ∆n . We note that σ ∼ σ 0 if and only if σ| ∂ ∆n = σ 0 | ∂ ∆n andσ is equivalent to σ 0 , where both are viewed as objects in the ∞-category
given by a fiber of the mapnnn

X ∆ → X ∂ ∆ ×S ∂ ∆n S ∆ .
Consequently, ∼ is an equivalence relation.
Suppose that σ and σ 0 are both degenerate and σ ∼ σ 0 . From the equalityσ| ∂ ∆n = σ 0 | ∂ ∆n , we deduce that σ = σ 0 . Consequently, there is at mostone degenerate n-simplex of X in each ∼-class. Let Y (n) ⊆ Xn denote a setof representatives for the ∼-classes of n-simplices in X, which contains alldegenerate simplices. We now define the simplicial subset X 0 ⊆ X recursively
as follows: an n-simplex σ : ∆n → X belongs to X 0 if σ ∈ Y (n) and σ| ∂ ∆nfactors through X 0 .
Let us now prove (1). To show that p|X 0 is an inner fibration, it suffices
to prove that every lifting problem of the form
s

Λni
_

/ X0
>

σ


∆n


/ S,

with 0 < i < n has a solution f in X 0 . Since p is an inner fibration, thislifting problem has a solution σ 0 : ∆n → X in the original simplicial set X.
Let σ00 = di σ : ∆n−1 → X be the induced map. Then σ00 | ∂ ∆n−1 factors
through X 0 . Consequently, σ00 is homotopic (over S and relative to ∂ ∆n−1 )to some map σ0 : ∆n−1 → X 0 . Let g0 : ∆1 × ∆n−1 → X be a homotopy
from σ00 to σ0 and let g1 : ∆1 × ∂ ∆n → X be the result of amalgamating g0
with the identity homotopy from s to itself. Let σ1 = g1 |{1} × ∂ ∆n . Using
Proposition 2.4.1.8, we deduce that g1 extends to a homotopy from σ 0 to
some other map σ 00 : ∆n → X with σ 00 | ∂ ∆n = σ1 . It follows that σ 00 is
homotopic over S relative to ∂ ∆n to a map σ : ∆n → X with the desired
properties. This proves that p|X 0 is an inner fibration. It is immediate fromthe construction that p|X 0 is minimal.
We now verify (2) and (3) by constructing a map h : X × ∆1 → X suchthat h|X ×{0} is the identity, h|X ×{1} is a retraction r : X → X with image
X 0 , and h is a homotopy over S and relative to X 0 . Choose an exhaustionof X by a transfinite sequence of simplicial subsets
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where each X α is obtained fromX <α =[Xββ<αby adjoining a single nondegenerate simplex, provided that such a simplex
exists. We construct hα = h|X α ×∆1 by induction on α. By the inductive hy1
pothesis, we may suppose that we have already defined h<α = h|X <α` ×∆ . If
X = X <α , then we are done. Otherwise, we can write X α = X <α ∂ ∆n ∆ncorresponding to some nondegenerate simplex τ : ∆n → X, and it suffices
to define hα |∆n × ∆1 . If τ factors through X 0 , we define hα |∆n × ∆1 to be
the compositionσ∆n × ∆1 → ∆n → X.
Otherwise, we use Proposition 2.4.1.8 to deduce the existence of the dottedarrow h0 in the diagram(∆n × {0})`(∂ ∆∂ ∆n ×{0}n_× ∆1 )(τ,h<α )

2/ X

h0


∆n × ∆1

p


/ S.

p◦σ

Let τ 0 = h0 |∆n × {1}. Then τ 0 | ∂ ∆n factors through X 0 . It follows that
there is a homotopy h00 : ∆n × ∆{1,2} → X from τ 0 to τ 00 , which is over S
and relative to ∂ ∆n and such that τ 00 factors through X 0 . Now consider the
diagram(∆n × Λ21 )`2 (∂ ∆∂ ∆n ×Λ 1_n

× ∆2 )

2/ X

H0
H

p


∆n × ∆2/ S,

where H0 |∆n × ∆{0,1} = h0 , H0 |∆n × ∆{1,2} = h00 , and H0 | ∂ ∆n × ∆2 isgiven by the compositionh<α∂ ∆n × ∆2 → ∂ ∆n × ∆1 → X.
Using the fact that p is an inner fibration, we deduce that there exists a
dotted arrow H rendering the diagram commutative. We may now definehα |∆n × ∆1 = H|∆n × ∆{0,2} ; it is easy to see that this extension has all
the desired properties.
We now prove (4). Using Proposition 3.2.2.8, we can reduce to the casewhere T = ∆n . Without loss of generality, we can replace S by T = ∆n ,so that X and X 0 are ∞-categories. The above constructions show thatr : X → X 0 is a homotopy inverse of the inclusion i : X 0 → X, so that i isan equivalence, as desired.

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2.3.4 n-Categories
The theory of ∞-categories can be regarded as a generalization of classical category theory: if C is an ordinary category, then its nerve N(C) is
an ∞-category which determines C up to canonical isomorphism. Moreover,
Proposition 1.1.2.2 provides a precise characterization of those ∞-categories
which can be obtained from ordinary categories. In this section, we will explain how to specialize the theory of ∞-categories to obtain a theory ofn-categories for every nonnegative integer n. (However, the ideas describedhere are appropriate for describing only those n-categories which have only
invertible k-morphisms for k ≥ 2.)
Before we can give the appropriate definition, we need to introduce a bitof terminology. Let f, f 0 : K → C be two diagrams in an ∞-category C andsuppose that K 0 ⊆ K is a simplicial subset such that f |K 0 = f 0 |K 0 = f0 . We
will say that f and f 0 are homotopic relative to K 0 if they are equivalent whenviewed as objects of the ∞-category Fun(K, C)×Fun(K 0 ,C) {f0 }. Equivalently,
f and f 0 are homotopic relative to K 0 if there exists a homotopy
h : K × ∆1 → C
with the following properties:(i) The restriction h|K 0 × ∆1 coincides with the compositionf0

K 0 × ∆1 → K 0 → C .
(ii) The restriction h|K × {0} coincides with f .
(iii) The restriction h|K × {1} coincides with f 0 .
(iv) For every vertex x of K, the restriction h|{x} × ∆1 is an equivalencein C.
We observe that if K 0 contains every vertex of K, then condition (iv)follows from condition (i).
Definition 2.3.4.1. Let C be a simplicial set and let n ≥ −1 be an integer.
We will say that C is an n-category if it is an ∞-category and the following
additional conditions are satisfied:(1) Given a pair of maps f, f 0 : ∆n → C, if f and f 0 are homotopic relativeto ∂ ∆n , then f = f 0 .
(2) Given m > n and a pair of maps f, f 0 : ∆m → C, if f | ∂ ∆m = f 0 | ∂ ∆m ,then f = f 0 .
It is sometimes convenient to extend Definition 2.3.4.1 to the case wheren = −2: we will say that a simplicial set C is a (−2)-category if it is a final
object of Set∆ : in other words, if it is isomorphic to ∆0 .

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107

Example 2.3.4.2. Let C be a (−1)-category. Using condition (2) of Definition 2.3.4.1, one shows by induction on m that C has at most one m-simplex.
Consequently, we see that up to isomorphism there are precisely two (−1)categories: ∆−1 ' ∅ and ∆0 .
Example 2.3.4.3. Let C be a 0-category and let X = C0 denote the set ofobjects of C. Let us write x ≤ y if there is a morphism φ from x to y in C.
Since C is an ∞-category, this relation is reflexive and transitive. Moreover,condition (2) of Definition 2.3.4.1 guarantees that the morphism φ is unique
if it exists. If x ≤ y and y ≤ x, it follows that the morphisms relating x andy are mutually inverse equivalences. Condition (1) then implies that x = y.
We deduce that (X, ≤) is a partially ordered set. It follows from Proposition
2.3.4.5 below that the map C → N(X) is an isomorphism.
Conversely, it is easy to see that the nerve of any partially ordered set(X, ≤) is a 0-category in the sense of Definition 2.3.4.1. Consequently, thefull subcategory of Set∆ spanned by the 0-categories is equivalent to thecategory of partially ordered sets.
Remark 2.3.4.4. Let C be an n-category and let m > n + 1. Then the
restriction map
θ : HomSet∆ (∆m , C) → HomSet∆ (∂ ∆m , C)is bijective. If n = −1, this is clear from Example 2.3.4.2; let us therefore
suppose that n ≥ 0, so that m ≥ 2. The injectivity of θ follows immediatelyfrom part (2) of Definition 2.3.4.1. To prove the surjectivity, we consider anarbitrary map f0 : ∂ ∆m → C. Let f : ∆m → C be an extension of f0 |Λm1(which exists since C is an ∞-category and 0 < 1 < m). Using condition (2)again, we deduce that θ(f ) = f0 .
The following result shows that, in the case where n = 1, Definition 2.3.4.1
recovers the usual definition of a category:
Proposition 2.3.4.5. Let S be a simplicial set. The following conditions
are equivalent:(1) The unit map u : S → N(hS ) is an isomorphism of simplicial sets.
(2) There exists a small category C and an isomorphism S ' N(C) of
simplicial sets.
(3) The simplicial set S is a 1-category.
Proof. The implications (1) ⇒ (2) ⇒ (3) are clear. Let us therefore assume
that (3) holds and show that f : S → N(hS ) is an isomorphism. We will
prove, by induction on n, that the map u is bijective on n-simplices.
For n = 0, this is clear. If n = 1, the surjectivity of u obvious. To prove
the injectivity, we note that if f (φ) = f (ψ), then the edges φ and ψ arehomotopic in S. A simple application of condition (2) of Definition 2.3.4.1
then shows that φ = ψ.

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Now suppose n > 1. The injectivity of u on n-simplices follows fromcondition (3) of Definition 2.3.4.1 and the injectivity of u on (n−1)-simplices.
To prove the surjectivity, let us suppose we are given a map s : ∆n →N(hS ). Choose 0 < i < n. Since u is bijective on lower-dimensional simplices,
the map s|Λni factors uniquely through S. Since S is an ∞-category, thisfactorization extends to a map se : ∆n → S. Since N(hS ) is the nerve of acategory, a pair of maps from ∆n into N(hS ) which agree on Λni must be
the same. We deduce that u ◦ se = s, and the proof is complete.
Remark 2.3.4.6. The condition that an ∞-category C be an n-category is
not invariant under categorical equivalence. For example, if D is a category
with several objects, all of which are uniquely isomorphic to one another,
then N(D) is categorically equivalent to ∆0 but is not a (−1)-category.
Consequently, there can be no intrinsic characterization of the class of ncategories itself. Nevertheless, there does exist a convenient description forthe class of ∞-categories which are equivalent to n-categories; see Proposition 2.3.4.18.
Our next goal is to establish that the class of n-categories is stable under
the formation of functor categories. In order to do so, we need to reformulate Definition 2.3.4.1 in a more invariant manner. Recall that for any
simplicial set X, the n-skeleton skn X is defined to be the simplicial subsetof X generated by all the simplices of X having dimension ≤ n.
Proposition 2.3.4.7. Let C be an ∞-category and let n ≥ −1. The following
are equivalent:(1) The ∞-category C is an n-category.
(2) For every simplicial set K and every pair of maps f, f 0 : K → C suchthat f | skn K and f 0 | skn K are homotopic relative to skn−1 K, we havef = f 0.
Proof. The implication (2) ⇒ (1) is obvious. Suppose that (1) is satisfied
and let f, f 0 : K → C be as in the statement of (2). To prove that f = f 0 ,it suffices to show that f and f 0 agree on every nondegenerate simplex of
K. We may therefore reduce to the case where K = ∆m . We now workby induction on m. If m < n, there is nothing to prove. In the case wherem = n, the assumption that C is an n-category immediately implies thatf = f 0 . If m > n, the inductive hypothesis implies that f | ∂ ∆m = f 0 | ∂ ∆m ,so that (1) implies that f = f 0 .
Corollary 2.3.4.8. Let C be an n-category and X a simplicial set. Then
Fun(X, C) is an n-category.
Proof. Proposition 1.2.7.3 asserts that Fun(X, C) is an ∞-category. We willshow that Fun(X, C) satisfies condition (2) of Proposition 2.3.4.7. Suppose
we are given a pair of maps f, f 0 : K → Fun(X, C) such that f | skn K and
f 0 | skn K are homotopic relative to f | skn−1 K. We wish to show that f = f 0 .

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We may identify f and f 0 with maps F, F 0 : K × X → C. Since C is an ncategory, to prove that F = F 0 it suffices to show that F | skn (K × X) andF 0 | skn (K ×X) are homotopic relative to skn−1 (K ×X). This follows at once
because skp (K × X) ⊆ (skp K) × X for every integer p.
When n = 1, Proposition 1.1.2.2 asserts that the class of n-categories canbe characterized by the uniqueness of certain horn fillers. We now prove a
generalization of this result.
Proposition 2.3.4.9. Let n ≥ 1 and let C be an ∞-category. Then C is an
n-category if and only if it satisfies the following condition:• For every m > n and every diagram
Λm
i _

f0

/C
>

f


∆m ,
where 0 < i < m, there exists a unique dotted arrow f as indicatedwhich renders the diagram commutative.
Proof. Suppose first that C is an n-category. Let f, f 0 : ∆m → C be two
0 mmaps with f |Λmi = f |Λi , where 0 < i < m and m > n. We wish to prove0mthat f = f . Since Λi contains the (n − 1)-skeleton of ∆m , it will suffice (byProposition 2.3.4.7) to show that f and f 0 are homotopic relative to Λmi .
mThis follows immediately from the fact that the inclusion Λm
⊆
∆
is
a
i
categorical equivalence.
Now suppose that every map f0 : Λmi → C, where 0 < i < m and n <m, extends uniquely to an m-simplex of C. We will show that C satisfies
conditions (1) and (2) of Definition 2.3.4.1. Condition (2) is obvious: if f, f 0 :∆m → C are two maps which coincide on ∂ ∆m , then they coincide on Λm1
and are therefore equal to one another (here we use the fact that m > 1
because of our assumption that n ≥ 1). Condition (1) is a bit more subtle.
Suppose that f, f 0 : ∆n → C are homotopic via a homotopy h : ∆n ×∆1 → C
which is constant on ∂ ∆n × ∆1 . For 0 ≤ i ≤ n, let σi denote the (n + 1)simplex of C obtained by composing h with the map[n + 1] → [n] × [1]((j, 0)
if j ≤ ij 7→(j − 1, 1) if j > i.
If i < n, then we observe that σi |Λn+1
i+1 is equivalent to the restriction(si di σi )|Λn+1.
Applying
ourhypothesis,
we conclude that σi = si di σi , so
i+1that di σi = di+1 σi . A dual argument establishes the same equality for 0 < i.
Since n > 0, we conclude that di σi = di+1 σi for all i. Consequently, we havea chain of equalities
f 0 = d0 σ0 = d1 σ0 = d1 σ1 = d2 σ1 = · · · = dn σn = dn+1 σn = f,so that f 0 = f , as desired.

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Corollary 2.3.4.10. Let C be an n-category and let p : K → C be a diagram.
Then C/p is an n-category.
Proof. If n ≤ 0, this follows easily from Examples 2.3.4.2 and 2.3.4.3. We
may therefore suppose that n ≥ 1. Proposition 1.2.9.3 implies that C/p is
an ∞-category. According to Proposition 2.3.4.9, it suffices to show that forevery m > n, 0 < i < m, and every map f0 : Λmi → C/p , there exists a
unique map f : ∆m → C/p extending f . Equivalently, we must show that
there is a unique map g rendering the diagramΛmi ? _ K

g0

;/ C

g


∆m ? K
commutative. The existence of g follows from the fact that C/p is an ∞category. Suppose that g 0 : ∆m ? K → C is another map which extends g0 .
Proposition 1.1.2.2 implies that g 0 |∆m = g|∆m . We conclude that g and g 0
coincide on the n-skeleton of ∆m ? K. Since C is an n-category, we deducethat g = g 0 , as desired.
We conclude this section by introducing a construction which allows us
to pass from an arbitrary ∞-category C to its “underlying” n-category by
discarding information about k-morphisms for k > n. In the case wheren = 1, we have already introduced the relevant construction: we simply
replace C by (the nerve of) its homotopy category.
Notation 2.3.4.11. Let C be an ∞-category and let n ≥ 1. For every
simplicial set K, let [K, C]n ⊆ HomSet∆ (skn K, C) be the subset consistingof those diagrams skn K → C which extend to the (n + 1)-skeleton of K
(in other words, the image of the restriction map HomSet∆ (skn+1 K, C) →HomSet∆ (skn K, C)). We define an equivalence relation ∼ on [K, C]n as follows: given two maps f, g : skn K → C, we write f ∼ g if f and g are
homotopic relative to skn−1 K.
Proposition 2.3.4.12. Let C be an ∞-category and n ≥ 1.
(1) There exists a simplicial set hn C with the following universal mapping
property: HomSet∆ (K, hn C) ' [K, C]n / ∼.
(2) The simplicial set hn C is an n-category.
(3) If C is an n-category, then the natural map θ : C → hn C is an isomorphism.
(4) For every n-category D, composition with θ induces an isomorphismof simplicial sets
ψ : Fun(hn C, D) → Fun(C, D).

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Proof. To prove (1), we begin by defining hn C([m]) = [∆m , C]n / ∼, so thatthe desired universal property holds by definition whenever K is a simplex.
Unwinding the definitions, to check the universal property for a general
simplicial set K we must verify the following fact:(∗) Given two maps f, g : ∂ ∆n+1 → C which are homotopic relative toskn−1 ∆n+1 , if f extends to an (n + 1)-simplex of C, then g extends toan (n + 1)-simplex of C.
This follows easily from Proposition A.2.3.1.
We next show that hn C is an ∞-category. Let η0 : Λmi → hn C be a morphism, where 0 < i < m. We wish to show that η0 extends to an m-simplex
n+1 mη : ∆m → C. If m ≤ n + 2, then Λm
Λi , so that η0 can be written
i = skas a composition
θΛmi → C → hn C .
The existence of η now follows from our assumption that C is an ∞-category.
m
If m > n+2, then HomSet∆ (Λmi , hn C) ' HomSet∆ (∆ , hn C) by construction,
so there is nothing to prove.
We next prove that hn C is an n-category. It is clear from the constructionthat for m > n, any two m-simplices of hn C with the same boundary mustcoincide. Suppose next that we are given two maps f, f 0 : ∆n → hn C which
are homotopic relative to ∂ ∆n . Let F : ∆n × ∆1 → hn C be a homotopy
from f to f 0 . Using (∗), we deduce that F is the image under θ of a map
Fe : ∆n × ∆1 → hn C, where Fe| ∂ ∆n × ∆1 factors through the projection∂ ∆n × ∆1 → ∂ ∆n . Since n > 0, we conclude that Fe is a homotopy fromFe|∆n × {0} to Fe|∆n × {1}, so that f = f 0 . This completes the proof of (2).
To prove (3), let us suppose that C is an n-category; we prove by induction
on m that the map C → hn C is bijective on m-simplices. For m < n, thisis clear. When m = n, it follows from part (1) of Definition 2.3.4.1. Whenm = n + 1, surjectivity follows from the construction of hn C and injectivity
from part (2) of Definition 2.3.4.1. For m > n + 1, we have a commutative
diagram
HomSet∆ (∆m , C)

/ HomSet∆ (∆m , hn C)
HomSet∆ (∂ ∆m , C)
/ HomSet∆ (∂ ∆m , hn C),

where the bottom horizontal map is an isomorphism by the inductive hypothesis, the left vertical map is an isomorphism by construction, and theright vertical map is an isomorphism by Remark 2.3.4.4; it follows that theupper horizontal map is an isomorphism as well.
To prove (4), we observe that if D is an n-category, then the composition
Fun(C, D) → Fun(hn C, hn D) ' Fun(hn C, D)
is an inverse to φ, where the second isomorphism is given by (3).

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Remark 2.3.4.13. The construction of Proposition 2.3.4.12 does not quite
work if n ≤ 0 because there may exist equivalences in hn C which do not arise
from equivalences in C. However, it is a simple matter to give an alternative
construction in these cases which satisfies conditions (2), (3), and (4); weleave the details to the reader.
Remark 2.3.4.14. In the case n = 1, the ∞-category h1 C constructed in
Proposition 2.3.4.12 is isomorphic to the nerve of the homotopy category hC.
We now apply the theory of minimal ∞-categories (§2.3.3) to obtaina characterization of the class of ∞-categories which are equivalent to ncategories. First, we need a definition from classical homotopy theory.
Definition 2.3.4.15. Let k ≥ −1 be an integer. A Kan complex X is ktruncated if, for every i > k and every point x ∈ X, we have
πi (X, x) ' ∗.
By convention, we will also say that X is (−2)-truncated if X is contractible.
Remark 2.3.4.16. If X and Y are homotopy equivalent Kan complexes,
then X is k-truncated if and only if Y is k-truncated. In other words, we mayview k-truncatedness as a condition on objects in the homotopy category Hof spaces.
Example 2.3.4.17. A Kan complex X is (−1)-truncated if it is either empty
or contractible. It is 0-truncated if the natural map X → π0 X is a homotopy
equivalence (equivalently, X is 0-truncated if it is homotopy equivalent to adiscrete space).
Proposition 2.3.4.18. Let C be an ∞-category and let n ≥ −1. The following conditions are equivalent:(1) There exists a minimal model C0 ⊆ C such that C0 is an n-category.
(2) There exists a categorical equivalence D ' C, where D is an n-category.
(3) For every pair of objects X, Y ∈ C, the mapping space MapC (X, Y ) ∈H is (n − 1)-truncated.
Proof. It is clear that (1) implies (2). Suppose next that (2) is satisfied;we will prove (3). Without loss of generality, we may replace C by D andthereby assume that C is an n-category. If n = −1, the desired result follows immediately from Example 2.3.4.2. Choose m ≥ n and an elementη ∈ πm (MapC (X, Y ), f ). We can represent η by a commutative diagram ofsimplicial sets/ {f }∂ ∆ m_∆ms


/ HomR(X,
Y ).
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We can identify s with a map ∆m+1 → C whose restriction to ∂ ∆m+1 is
specified. Since C is an n-category, the inequality m + 1 > n shows that s is
uniquely determined. This proves that πm (MapC (X, Y ), f ) ' ∗, so that (3)is satisfied.
To prove that (3) implies (1), it suffices to show that if C is a minimal∞-category which satisfies (3), then C is an n-category. We must show thatthe conditions of Definition 2.3.4.1 are satisfied. The first of these conditions
follows immediately from the assumption that C is minimal. For the second,
we must show that if m > n and f, f 0 : ∂ ∆m → C are such that f | ∂ ∆m =f 0 | ∂ ∆m , then f = f 0 . Since C is minimal, it suffices to show that f and f 0 are
homotopic relative to ∂ ∆m . We will prove that there is a map g : ∆m+1 → Csuch that dm+1 g = f , dm g = f 0 , and di g = di sm f = di sm f 0 for 0 ≤ i < m.
Then the sequence (s0 f, s1 f, . . . , sm−1 f, g) determines a map ∆m × ∆1 → C
which gives the desired homotopy between f and f 0 (relative to ∂ ∆m ).
To produce the map g, it suffices to solve the lifting problem depicted in
the diagramg∂ ∆m+1_;/ C∆m+1 .
Choose a fibrant simplicial category D and an equivalence of ∞-categories
C → N(D). According to Proposition A.2.3.1, it will suffice to prove that we
can solve the associated lifting problemC[∂ ∆m+1
_ ]

G0

/
:D

G


C[∆m+1 ].
Let X and Y denote the initial and final vertices of ∆m+1 , regarded as
objects of C[∂ ∆m+1 ]. Note that G0 determines a map
e0 : ∂(∆1 )m ' MapC[∂ ∆m+1 ] (X, Y ) → MapD (G0 (X), G0 (Y ))and that giving the desired extension G is equivalent to extending e0 to amape : (∆1 )m ' MapC[∆m+1 ] (X, Y ) → MapD (G0 (X), G0 (Y )).
The obstruction to constructing e lies in πm−1 (MapD (G0 (X), G0 (Y )), p) for
an appropriately chosen base point p. Since (m − 1) > (n − 1), condition (3)implies that this homotopy set is trivial, so that the desired extension canbe found.
Corollary 2.3.4.19. Let X be a Kan complex. Then X is (categorically)equivalent to an n-category if and only if it is n-truncated.

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Proof. For n = −2 this is obvious. If n ≥ −1, this follows from characterization (3) of Proposition 2.3.4.18 and the following observation: a Kan complex
X is n-truncated if and only if, for every pair of vertices x, y ∈ X0 , the Kancomplex1{x} ×X X ∆ ×X {y}of paths from x to y is (n − 1)-truncated.
Corollary 2.3.4.20. Let C be an ∞-category and K a simplicial set. Suppose that, for every pair of objects C, D ∈ C, the space MapC (C, D) is ntruncated. Then the ∞-category Fun(K, C) has the same property.
Proof. This follows immediately from Proposition 2.3.4.18 and Corollary
2.3.4.8 because the functor
C 7→ Fun(K, C)
preserves categorical equivalences between ∞-categories.

2.4 CARTESIAN FIBRATIONS
Let p : X → S be an inner fibration of simplicial sets. Each fiber of p is an ∞category, and each edge f : s → s0 of S determines a correspondence between
the fibers Xs and Xs0 . In this section, we would like to study the case inwhich each of these correspondences is associated to a functor f ∗ : Xs0 → Xs .
Roughly speaking, we can attempt to construct f ∗ as follows: for each vertexy ∈ Xs0 , we choose an edge fe : x → y lifting f , and set f ∗ y = x. However,this recipe does not uniquely determine x, even up to equivalence, since there
might be many different choices for fe. To get a good theory, we need to makea good choice of fe. More precisely, we should require that fe be a p-Cartesianedge of X. In §2.4.1, we will introduce the definition of p-Cartesian edges
and study their basic properties. In particular, we will see that a p-Cartesianedge fe is determined up to equivalence by its target y and its image in S.
Consequently, if there is a sufficient supply of p-Cartesian edges of X, thenwe can use the above prescription to define the functor f ∗ : Xs0 → Xs . This
leads us to the notion of a Cartesian fibration, which we will study in §2.4.2.
In §2.4.3, we will establish a few basic stability properties of the class ofCartesian fibrations (we will discuss other results of this type in Chapter 3
after we have developed the language of marked simplicial sets). In §2.4.4,
we will show that if p : C → D is a Cartesian fibration of ∞-categories, thenwe can reduce many questions about C to similar questions about the base
D and about the fibers of p. This technique has many applications, which wewill discuss in §2.4.5 and §2.4.6. Finally, in §2.4.7, we will study the theory ofbifibrations, which is useful for constructing examples of Cartesian fibrations.

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2.4.1 Cartesian Morphisms
Let C and C0 be ordinary categories and let M : Cop × C0 → Set be a correspondence between them. Suppose that we wish to know whether or not M
arises as the correspondence associated to some functor g : C0 → C. This is
the case if and only if, for each object C 0 ∈ C0 , we can find an object C ∈ C
and a point η ∈ M (C, C 0 ) having the property that the “composition with
η” mapψ : HomC (D, C) → M (D, C 0 )is bijective for all D ∈ C. Note that η may be regarded as a morphism in the
category C ?M C0 . We will say that η is a Cartesian morphism in C ?M C0 if ψ
is bijective for each D ∈ C. The purpose of this section is to generalize thisnotion to the ∞-categorical setting and to establish its basic properties.
Definition 2.4.1.1. Let p : X → S be an inner fibration of simplicial sets.
Let f : x → y be an edge in X. We shall say that f is p-Cartesian if theinduced map
X/f → X/y ×S/p(y) S/p(f )is a trivial Kan fibration.
Remark 2.4.1.2. Let M be an ordinary category, let p : N(M) → ∆1 be
a map (automatically an inner fibration), and let f : x → y be a morphism
in M which projects isomorphically onto ∆1 . Then f is p-Cartesian in thesense of Definition 2.4.1.1 if and only if it is Cartesian in the classical sense.
We now summarize a few of the formal properties of Definition 2.4.1.1:
Proposition 2.4.1.3. (1) Let p : X → S be an isomorphism of simplicial
sets. Then every edge of X is p-Cartesian.
(2) Suppose we are given a pullback diagram
X0

/X

q

p0

p



/S
S0
of simplicial sets, where p (and therefore also p0 ) is an inner fibration.
Let f be an edge of X 0 . If q(f ) is p-Cartesian, then f is p0 -Cartesian.
(3) Let p : X → Y and q : Y → Z be inner fibrations and let f : x0 → x
be an edge of X such that p(f ) is q-Cartesian. Then f is p-Cartesian
if and only if f is (q ◦ p)-Cartesian.
Proof. Assertions (1) and (2) follow immediately from the definition. To
prove (3), we consider the commutative diagram
ψ

X/f
ψ

0

'
X/x ×Y/p(x) Y/p(f ) .

ψ00/ X/x ×Z
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The map ψ 00 is a pullback of
Y/p(f ) → Y/p(x) ×Z/(q◦p)(x) Z/(q◦p)(f )and therefore a trivial fibration in view of our assumption that p(f ) is qCartesian. If ψ 0 is a trivial fibration, it follows that ψ is a trivial fibration as
well, which proves the “only if” direction of (3).
For the converse, suppose that ψ is a trivial fibration. Proposition 2.1.2.1
implies that ψ 0 is a right fibration. According to Lemma 2.1.3.4, it will
suffice to prove that the fibers of ψ 0 are contractible. Let t be a vertex of
X/x ×Y/p(x) Y/p(f ) and let K = (ψ 00 )−1 {ψ 00 (t)}. Since ψ 00 is a trivial fibration,
K is a contractible Kan complex. Since ψ is a trivial fibration, the simplicial
set (ψ 0 )−1 K = ψ −1 {ψ 00 (t)} is also a contractible Kan complex. It follows
that the fiber of ψ 0 over the point t is weakly contractible, as desired.
Remark 2.4.1.4. Let p : X → S be an inner fibration of simplicial sets.
Unwinding the definition, we see that an edge f : ∆1 → X is p-Cartesian if
and only if for every n ≥ 2 and every commutative diagram∆{n−1,n}
_
f


Λnn
_
#

/X
;

p


∆n


/ S,

there exists a dotted arrow as indicated, rendering the diagram commutative.
In particular, we note that Proposition 1.2.4.3 may be restated as follows:(∗) Let C be a ∞-category and let p : C → ∆0 denote the projection fromC to a point. A morphism φ of C is p-Cartesian if and only if φ is anequivalence.
In fact, it is possible to strengthen assertion (∗) as follows:
Proposition 2.4.1.5. Let p : C → D be an inner fibration between ∞categories and let f : C → C 0 be a morphism in C. The following conditions
are equivalent:(1) The morphism f is an equivalence in C.
(2) The morphism f is p-Cartesian, and p(f ) is an equivalence in D.
Proof. Let q denote the projection from D to a point. We note that both (1)and (2) imply that p(f ) is an equivalence in D and therefore q-Cartesian by(∗). The equivalence of (1) and (2) now follows from (∗) and the third partof Proposition 2.4.1.3.

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Corollary 2.4.1.6. Let p : C → D be an inner fibration between ∞categories. Every identity morphism of C (in other words, every degenerateedge of C) is p-Cartesian.
We now study the behavior of Cartesian edges under composition.
Proposition 2.4.1.7. Let p : C → D be an inner fibration between ∞categories and let σ : ∆2 → C be a 2-simplex of C, which we will depict as adiagram
f

C0

> C1
h

g

!
/ C2 .

Suppose that g is p-Cartesian. Then f is p-Cartesian if and only if h is
p-Cartesian.
Proof. We wish to show that the map
i0 : C/h → C/C2 ×D/p(C2 ) D/p(h)is a trivial fibration if and only if
i1 : C/f → C/C1 ×D/p(C1 ) D/p(f )is a trivial fibration. The dual of Proposition 2.1.2.1 implies that both maps
are right fibrations. Consequently, by (the dual of) Lemma 2.1.3.4, it suffices
to show that the fibers of i0 are contractible if and only if the fibers of i1 arecontractible.
For any simplicial subset B ⊆ ∆2 , let XB = C/σ|B ×D/σ|B D/σ . We notethat XB is functorial in B in the sense that an inclusion A ⊆ B induces a map
jA,B : XB → XA (which is a right fibration, again by Proposition 2.1.2.1).
Observe that j∆{2} ,∆{0,2} is the base change of i0 by the map D/p(σ) → D/p(h)and that j∆{1} ,∆{0,1} is the base change of i1 by the map D/σ → D/p(f ) . The
maps
D/p(f ) ← D/p(σ) → D/p(h)are both surjective on objects (in fact, both maps have sections). Consequently, it suffices to prove that j∆{1} ,∆{0,1} has contractible fibers if andonly if j∆{2} ,∆{0,2} has contractible fibers. Now we observe that the compositions
X∆2 → X∆{0,2} → X∆{2}X∆2 → XΛ21 → X∆{1,2} → X∆{2}
coincide. By Proposition 2.1.2.5, jA,B is a trivial fibration whenever the
inclusion A ⊆ B is left anodyne. We deduce that j∆{2} ,∆{0,2} is a trivial
fibration if and only if j∆{1,2} ,Λ21 is a trivial fibration. Consequently, it suffices

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to show that j∆{1,2} ,Λ21 is a trivial fibration if and only if j∆{1} ,∆{0,1} is atrivial fibration.
Since j∆{1,2} ,Λ21 is a pullback of j∆{1} ,∆{0,1} , the “if” direction is obvious.
For the converse, it suffices to show that the natural map
C/g ×D/p(g) D/p(σ) → C/C1 ×D/p(C1 ) D/p(σ)
is surjective on vertices. But this map is a trivial fibration because the inclusion {1} ⊆ ∆{1,2} is left anodyne.
Our next goal is to reformulate the notion of a Cartesian morphism in aform which will be useful later. For convenience of notation, we will prove
this result in a dual form. If p : X → S is an inner fibration and f is an edgeof X, we will say that f is p-coCartesian if it is Cartesian with respect tothe morphism pop : X op → S op .
Proposition 2.4.1.8. Let p : Y → S be an inner fibration of simplicial sets
and let e : ∆1 → Y be an edge. Then e is p-coCartesian if and only if foreach n ≥ 1 and each diagram{0} × ∆1_e(∆n × {0})

`


(∂ ∆
∂ ∆n ×{0}n_× ∆1 )
h

f

/*4 Y
p



g
/S
∆n × ∆ 1
there exists a map h as indicated, rendering the diagram commutative.
Proof. Let us first prove the “only if” direction. We recall a bit of the notation used in the proof of Proposition 2.1.2.6; in particular, the filtration
X(n + 1) ⊆ · · · ⊆ X(0) = ∆n × ∆1of ∆n × ∆1 . We construct h|X(m) by descending induction on m. To begin,
we set h|X(n + 1) = f . Now, for each m the space X(m) is obtained from⊆ ∆m+1 . If m > 0, theX(m + 1) by pushout along a horn inclusion Λn+1m
desired extension exists because p is an inner fibration. If m = 0, the desired
extension exists because of the hypothesis that e is a p-coCartesian edge.
We now prove the “if” direction. Suppose that e satisfies the condition in
the statement of Proposition 2.4.1.8. We wish to show that e is p-coCartesian.
In other words, we must show that for every n ≥ 2 and every diagram∆{0,1}
_
e


Λn0
_
"

/X
<
p


∆n


/ S,

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there exists a dotted arrow as indicated, rendering the diagram commutative.
Replacing S by ∆n and Y by Y ×S ∆n , we may reduce to the case where S is
an ∞-category. We again make use of the notation (and argument) employedin the proof of Proposition 2.1.2.6. Namely, the inclusion Λn0 ⊆ ∆n is a retractof the inclusiona(Λn0 × ∆1 )(∆n × {0}) ⊆ ∆n × ∆1 .
Λn0 ×{0}The retraction is implemented by mapsjr

∆ n → ∆n × ∆1 → ∆n ,which were defined in the proof of Proposition 2.1.2.6. We now set F = f ◦ r,
G = g ◦ r.
Let K = ∆{1,2,...,n} ⊆ ∆n . Thena
F |(∂ K × ∆1 )(K × ∆1 )∂ K×{0}1carries {1} × ∆ into e. By assumption, there exists an extension of F to
K ×∆1 which is compatible with G. In other words, there exists a compatible
extension F 0 of F toa∂ ∆n × ∆1∆n × {0}.
∂ ∆n ×{0}01Moreover, F carries {0} × ∆ to a degenerate edge; such an edge is automatically coCartesian (this follows from Corollary 2.4.1.6 because S is an∞-category), and therefore there exists an extension of F 0 to all of ∆n × ∆1
by the first part of the proof.
Remark 2.4.1.9. Let p : X → S be an inner fibration of simplicial sets, let
x be a vertex of X, and let f : x0 → p(x) be an edge of S ending at p(x).
There may exist many p-Cartesian edges f : x0 → x of X with p(f ) = f .
However, there is a sense in which any two such edges having the same target
x are equivalent to one another. Namely, any p-Cartesian edge f : x0 → x
lifting f can be regarded as a final object of the ∞-category X/x ×S/p(x) {f }and is therefore determined up to equivalence by f and x.
We now spell out the meaning of Definition 2.4.1.1 in the setting of simplicial categories.
Proposition 2.4.1.10. Let F : C → D be a functor between simplicial
categories. Suppose that C and D are fibrant and that for every pair of objects
C, C 0 ∈ C, the associated map
MapC (C, C 0 ) → MapD (F (C), F (C 0 ))is a Kan fibration. Then the following assertions hold:(1) The associated map q : N(C) → N(D) is an inner fibration between∞-categories.

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CHAPTER 2(2) A morphism f : C 0 → C 00 in C is q-Cartesian if and only if, for every
object C ∈ C, the diagram of simplicial sets
/ MapC (C, C 00 )MapC (C, C 0 )MapD (F (C), F (C 0 ))
/ MapD (F (C), F (C 00 ))is homotopy Cartesian.
Proof. Assertion (1) follows from Remark 1.1.5.11. Let f be a morphism in
C. By definition, f : C 0 → C 00 is q-Cartesian if and only if
θ : N(C)/f → N(C)/C 00 ×N(D)/F (C 00 ) N(D)/F (f )is a trivial fibration. Since θ is a right fibration between right fibrations overC, f is q-Cartesian if and only if for every object C ∈ C, the induced map
θC : {C} ×N(C) N(C)/f → {C} ×N(C) N(C)/C 00 ×N(D)/F (C 00 ) N(D)/F (f )is a homotopy equivalence of Kan complexes. This is equivalent to the assertion that the diagram/ N(C)/C 00 ×N(C) {C}
N(C)/f ×C {C}/ N(D)/F (C 00 ) ×N(D) {F (C)}
N(D)/F (f ) ×N(D) {F (C)}is homotopy Cartesian. In view of Theorem 1.1.5.13, this diagram is equivalent to the diagram of simplicial sets
/ MapC (C, C 00 )MapC (C, C 0 )/ MapD (F (C), F (C 00 )).

MapD (F (C), F (C 0 ))This proves (2).

In some contexts, it will be convenient to consider a slightly larger classof edges:
Definition 2.4.1.11. Let p : X → S be an inner fibration and let e : ∆1 →X be an edge. We will say that e is locally p-Cartesian if it is a p0 -Cartesianedge of the fiber product X ×S ∆1 , where p0 : X ×S ∆1 → ∆1 denotes the
projection.
Remark 2.4.1.12. Suppose we are given a pullback diagram
X0
p0

f

/X
p



/S
S0
of simplicial sets, where p (and therefore also p0 ) is an inner fibration. Anedge e of X 0 is locally p0 -Cartesian if and only if its image f (e) is locallyp-Cartesian.

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121

2.4.2 Cartesian Fibrations
In this section, we will introduce the study of Cartesian fibrations between
simplicial sets. The theory of Cartesian fibrations is a generalization of thetheory of right fibrations studied in §2.1. Recall that if f : X → S is a rightfibration of simplicial sets, then the fibers {Xs }s∈S are Kan complexes, which
depend in a (contravariantly) functorial fashion on the choice of vertex s ∈ S.
The condition that f be a Cartesian fibration has a similar flavor: we stillrequire that Xs depend functorially on s but weaken the requirement thatXs be a Kan complex; instead, we merely require that it be an ∞-category.
Definition 2.4.2.1. We will say that a map p : X → S of simplicial sets is
a Cartesian fibration if the following conditions are satisfied:(1) The map p is an inner fibration.
(2) For every edge f : x → y of S and every vertex ye of X with p(ey ) = y,eethere exists a p-Cartesian edge f : x
e → ye with p(f ) = f .
We say that p is a coCartesian fibration if the opposite map pop : X op →S is a Cartesian fibration.
op

If a general inner fibration p : X → S associates to each vertex s ∈ S an∞-category Xs and to each edge s → s0 a correspondence from Xs to Xs0 ,then p is Cartesian if each of these correspondences arises from a (canonicallydetermined) functor Xs0 → Xs . In other words, a Cartesian fibration withbase S ought to be roughly the same thing as a contravariant functor from
S into an ∞-category of ∞-categories, where the morphisms are given byfunctors. One of the main goals of Chapter 3 is to give a precise formulation(and proof) of this assertion.
Remark 2.4.2.2. Let F : C → C0 be a functor between (ordinary) categories. The induced map of simplicial sets N(F ) : N(C) → N(C0 ) is automatically an inner fibration; it is Cartesian if and only if F is a fibration of
categories in the sense of Grothendieck.
The following formal properties follow immediately from the definition:
Proposition 2.4.2.3.
sian fibration.

(1) Any isomorphism of simplicial sets is a Carte-(2) The class of Cartesian fibrations between simplicial sets is stable underbase change.
(3) A composition of Cartesian fibrations is a Cartesian fibration.
Recall that an ∞-category C is a Kan complex if and only if every morphism in C is an equivalence. We now establish a relative version of thisstatement:

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Proposition 2.4.2.4. Let p : X → S be an inner fibration of simplicial
sets. The following conditions are equivalent:(1) The map p is a Cartesian fibration, and every edge in X is p-Cartesian.
(2) The map p is a right fibration.
(3) The map p is a Cartesian fibration, and every fiber of p is a Kancomplex.
Proof. In view of Remark 2.4.1.4, the assertion that every edge of X is pCartesian is equivalent to the assertion that p has the right lifting property
with respect to Λnn ⊆ ∆n for all n ≥ 2. The requirement that p be a Cartesianfibration further imposes the right lifting property with respect to Λ11 ⊆ ∆1 .
This proves that (1) ⇔ (2).
Suppose that (2) holds. Since we have established that (2) implies (1), we
know that p is Cartesian. Furthermore, we have already seen that the fibersof a right fibration are Kan complexes. Thus (2) implies (3).
We complete the proof by showing that (3) implies that every edge f :x → y of X is p-Cartesian. Since p is a Cartesian fibration, there exists a
p-Cartesian edge f 0 : x0 → y with p(f 0 ) = p(f ). Since f 0 is p-Cartesian, there
exists a 2-simplex σ : ∆2 → X which we may depict as a diagram
0

?x
g

x

f

f0

/ y,

where p(σ) = s0 p(f ). Then g lies in the fiber Xp(x) and is therefore an
equivalence (since Xp(x) is a Kan complex). It follows that f is equivalent tof 0 as objects of X/y ×S/p(y) {p(f )}, so that f is p-Cartesian, as desired.
Corollary 2.4.2.5. Let p : X → S be a Cartesian fibration. Let X 0 ⊆ X
consist of all those simplices σ of X such that every edge of σ is p-Cartesian.
Then p|X 0 is a right fibration.
Proof. We first show that p|X 0 is an inner fibration. It suffices to show thatp|X 0 has the right lifting property with respect to every horn inclusion Λni ,0 < i < n. If n > 2, then this follows immediately from the fact that p
has the appropriate lifting property. If n = 2, then we must show that if
f : ∆2 → X is such that f |Λ21 factors through X 0 , then f factors throughX 0 . To prove this, we may replace S by ∆2 , in which case the result follows
immediately from Proposition 2.4.1.7.
We now wish to complete the proof by showing that p is a right fibration.
According to Proposition 2.4.2.4, it suffices to prove that every edge of X 0
is p|X 0 -Cartesian. This follows immediately from the characterization given
in Remark 2.4.1.4 because every edge of X 0 is p-Cartesian (when regarded
as an edge of X).

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In order to verify that certain maps are Cartesian fibrations, it is often
convenient to work in a slightly more general setting.
Definition 2.4.2.6. A map p : X → S of simplicial sets is a locally Cartesianfibration if it is an inner fibration and, for every edge ∆1 → S, the pullback
X ×S ∆1 → ∆1 is a Cartesian fibration.
In other words, an inner fibration p : X → S is a locally Cartesian fibration
if and only if, for every vertex x ∈ X and every edge e : s → p(x) in S, there
exists a locally p-Cartesian edge s → x which lifts e.
Let p : X → S be an inner fibration of ∞-categories. It is clear thatevery p-Cartesian morphism of X is locally p-Cartesian. Moreover, Proposition 2.4.1.7 implies that the class of p-Cartesian edges of X is stable undercomposition. The following result can be regarded as a sort of converse:
Lemma 2.4.2.7. Let p : X → S be a locally Cartesian fibration of simplicial
sets and let f : x0 → x be an edge of X. The following conditions areequivalent:(1) The edge f is p-Cartesian.
(2) For every 2-simplex σ
>x

0

g

f

/ x

h

x00

in X, the edge g is locally p-Cartesian if and only if the edge h is locallyp-Cartesian.
(3) For every 2-simplex σ
>x
g

x00

0
f

h

/ x

in X, if g is locally p-Cartesian, then h is locally p-Cartesian.
Proof. We first show that (1) ⇒ (2). Pulling back via the composition p ◦ σ :∆2 → S, we can reduce to the case where S = ∆2 . In this case, g is locally pCartesian if and only if it is p-Cartesian, and likewise for h. We now conclude
by applying Proposition 2.4.1.7.
The implication (2) ⇒ (3) is obvious. We conclude by showing that (3) ⇒(1). We must show that η : X/f → X/x ×S/p(x) S/p(f ) is a trivial Kan fibration.
Since η is a right fibration, it will suffice to show that the fiber of η over any
vertex is contractible. Any such vertex determines a map σ : ∆2 → S withσ|∆{1,2} = p(f ). Pulling back via σ, we may suppose that S = ∆2 .
It will be convenient to introduce a bit of notation: for every map q : K →X let Y/q ⊆ X/q denote the full simplicial subset spanned by those vertices

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of X/q which map to the initial vertex of S. We wish to show that the natural
map Y/f → Y/x is a trivial fibration. By assumption, there exists a locallyp-Cartesian morphism g : x00 → x0 in X covering the edge ∆{0,1} ⊆ S. Since
X is an ∞-category, there exists a 2-simplex τ : ∆2 → X with d2 (τ ) = g andd0 (τ ) = f . Then h = d1 (τ ) is a composite of f and g, and assumption (3)guarantees that h is locally p-Cartesian. We have a commutative diagram
5 Y/h
(

Y/τ
ζ

"
Y/τ |Λ21=

Y/x/ Y/f .

Moreover, all of the maps in this diagram are trivial fibrations except possibly
ζ, which is known to be a right fibration. It follows that ζ is a trivial fibration
as well, which completes the proof.
In fact, we have the following:
Proposition 2.4.2.8. Let p : X → S be a locally Cartesian fibration. The
following conditions are equivalent:(1) The map p is a Cartesian fibration.
(2) Given a 2-simplex
/ x0

f

x
h

 ~
z,

g

if f and g are locally p-Cartesian, then h is locally p-Cartesian.
(3) Every locally p-Cartesian edge of X is p-Cartesian.
Proof. The equivalence (2) ⇔ (3) follows from Lemma 2.4.2.7, and the implication (3) ⇒ (1) is obvious. To prove that (1) ⇒ (3), let us supposethat e : x → y is a locally p-Cartesian edge of X. Choose a p-Cartesianedge e0 : x0 → y lifting p(e). The edges e and e0 are both p0 -Cartesian in
X 0 = X ×S ∆1 , where p0 : X 0 → ∆1 denotes the projection. It follows thate and e0 are equivalent in X 0 and therefore also equivalent in X. Since e0 is
p-Cartesian, we deduce that e is p-Cartesian as well.
Remark 2.4.2.9. If p : X → S is a locally Cartesian fibration, then we canassociate to every edge s → s0 of S a functor Xs0 → Xs , which is well-definedhighertopoi

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FIBRATIONS OF SIMPLICIAL SETS

up to homotopy. A 2-simplex
/ s0

s


s00



determines a triangle of ∞-categories
Xs a o
F

< Xs

0

G
H

Xs00
which commutes up to a (generally noninvertible) natural transformation
α : F ◦ G → H. Proposition 2.4.2.8 implies that p is a Cartesian fibration if
and only if every such natural transformation is an equivalence of functors.
Corollary 2.4.2.10. Let p : X → S be an inner fibration of simplicial sets.
Then p is a Cartesian fibration if and only if every pullback X ×S ∆n → ∆n
is a Cartesian fibration for n ≤ 2.
One advantage the theory of locally Cartesian fibrations holds over thetheory of Cartesian fibrations is the following “fiberwise” existence criterion:
Proposition 2.4.2.11. Suppose we are given a commutative diagram ofsimplicial sets
/Y

r

X
p



S



q

satisfying the following conditions:(1) The maps p and q are locally Cartesian fibrations, and r is an inner
fibration.
(2) The map r carries locally p-Cartesian edges of X to locally q-Cartesianedges of Y .
(3) For every vertex s of S, the induced map rs : Xs → Ys is a locally
Cartesian fibration.
Then r is a locally Cartesian fibration. Moreover, an edge e of X is locallyr-Cartesian if and only if there exists a 2-simplex σ
?x

0

e0

x
with the following properties:

e

e00

/ x00

highertopoi

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CHAPTER 2(i) In the simplicial set S, we have p(σ) = s0 (p(e)).
(ii) The edge e00 is locally p-Cartesian.
(iii) The edge e0 is locally rp(x) -Cartesian.
Proof. Suppose we are given a vertex x00 ∈ X and an edge e0 : y → p(x00 )in Y . It is clear that we can construct a 2-simplex σ in X satisfying (i)through (iii), with p(e) = q(e0 ). Moreover, σ is uniquely determined up toequivalence. We will prove that e is locally r-Cartesian. This will prove thatr is a locally Cartesian fibration and the “if” direction of the final assertion.
The converse will then follow from the uniqueness (up to equivalence) of
locally r-Cartesian lifts of a given edge (with specified terminal vertex).
To prove that e is locally r-Cartesian, we are free to pull back by theedge p(e) : ∆1 → S and thereby reduce to the case S = ∆1 . Then p andq are Cartesian fibrations. Since e00 is p-Cartesian and r(e00 ) is q-Cartesian,
Proposition 2.4.1.3 implies that e00 is r-Cartesian. Remark 2.4.1.12 implies
that e0 is locally p-Cartesian. It follows from Lemma 2.4.2.7 that e is locallyp-Cartesian as well.
Remark 2.4.2.12. The analogue of Proposition 2.4.2.11 for Cartesian fibrations is false.
Remark 2.4.2.13. Let p : X → S be an inner fibration of simplicial set,
and let f : x → y be an edge of X. Suppose that f is locally p-Cartesian, andthat there exists a p-Cartesian edge fe : x
e → y of X such that p(f ) = p(fe).
Then f is p-Cartesian. To prove this, we can reduce to the case where S isa simplex. Since fe is p-Cartesian, the map f factors as a composition
fe

g

x→x
e → y.
The assumption that f is locally p-Cartesian implies that g is an equivalencein the ∞-category Xp(x) , and therefore p-Cartesian. Applying Proposition
2.4.2.8, we deduce that f is p-Cartesian.
Remark 2.4.2.14. Let p : X → S be an inner fibration of simplicial sets,
and suppose we are given a 2-simplex
f

x

?y
h

g

/ z

in X, where g is p-Cartesian. Then:(a) If h is locally p-Cartesian and there exists a p-Cartesian edge fe : xe→yewith p(f ) = p(f ), then f is p-Cartesian.
(b) If f is locally p-Cartesian and there exists a p-Cartesian edge e
h:xe→z
with p(eh) = p(h), then h is p-Cartesian.

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FIBRATIONS OF SIMPLICIAL SETS

In particular, if p is a Cartesian fibration, then f is p-Cartesian if and only
if h is p-Cartesian.
To prove (a), it will suffice to show that the hypothesis implies that f
is locally p-Cartesian (Remark 2.4.2.13). For this, we can replace p by the
projection map X ×S ∆2 → ∆2 , in which case the desired result follows from
Proposition 2.4.2.8. The proof of (b) is similar.
2.4.3 Stability Properties of Cartesian Fibrations
In this section, we will prove the class of Cartesian fibrations is stable under
the formation of overcategories and undercategories. Since the definition of
a Cartesian fibration is not self-dual, we must treat these results separately,using slightly different arguments (Propositions 2.4.3.1 and 2.4.3.2).
Proposition 2.4.3.1. Let p : C → D be a Cartesian fibration of simplicial
sets and let q : K → C be a diagram. Then
(1) The induced map p0 : C/q → D/pq is a Cartesian fibration.
(2) An edge f of C/q is p0 -Cartesian if and only if the image of f in C is
p-Cartesian.
Proof. Proposition 2.1.2.1 implies that p0 is an inner fibration. Let us call anedge f of C/q special if its image in C is p-Cartesian. To complete the proof,
we will verify the following assertions:(i) Given a vertex q ∈ C/q and an edge fe : r0 → p0 (q), there exists a special
edge f : r → q with p0 (f ) = fe.
(ii) Every special edge of C/q is p0 -Cartesian.
To prove (i), let fe0 denote the image of fe in D and c the image of q in
C. Using the assumption that p is a Cartesian fibration, we can choose ap-Cartesian edge f 0 : c → d lifting fe0 . To extend this data to the desirededge f of C/q , it suffices to solve the lifting problem depicted in the diagram`/8 C({1} ? K) {1} ∆1
_
p

i


∆1 ? K/ D.

This lifting problem has a solution because p is an inner fibration and i is
inner anodyne (Lemma 2.1.2.3).
To prove (ii), it will suffice to show that if n ≥ 2, then any lifting problemof the form
Λnn ? K_

∆n ? K

g
G

;/ C
p


/D

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CHAPTER 2

has a solution provided that e = g(∆{n−1,n} ) is a p-Cartesian edge of C.
Consider the set P of pairs (K 0 , GK 0 ), where K 0 ⊆ K and GK 0 fits in acommutative diagram(Λnn ? K)

`

(∆
Λn
n ?K
 0
_

n

GK 0

? K 0)

/C
p


/ D.


∆n ? K

Because e is p-Cartesian, there exists an element (∅, G∅ ) ∈ P . We regard P
as partially ordered, where (K 0 , GK 0 ) ≤ (K 00 , GK 00 ) if K 0 ⊆ K 00 and GK 0
is a restriction of GK 00 . Invoking Zorn’s lemma, we deduce the existence ofa maximal element (K 0 , GK 0 ) of P . If K 0 = K, then the proof is complete.
Otherwise, it is possible to enlarge K 0 by adjoining a single nondegenerate msimplex of K. Since (K 0 , GK 00 ) is maximal, we conclude that the associatedlifting problem`/5 C(Λnn ? ∆m ) Λnn ?∂ ∆m (∆n ? ∂ ∆m )
_
σ

p


/D


∆n ? ∆m

has no solution. The left vertical map is equivalent to the inclusion Λn+m+1⊆
n+1∆n+m+1 , which is inner anodyne. Since p is an inner fibration by assumption,
we obtain a contradiction.
Proposition 2.4.3.2. Let p : C → D be a coCartesian fibration of simplicial
sets and let q : K → C be a diagram. Then
(1) The induced map p0 : C/q → D/pq is a coCartesian fibration.
(2) An edge f of C/q is p0 -coCartesian if and only if the image of f in C
is p-coCartesian.
Proof. Proposition 2.1.2.1 implies that p0 is an inner fibration. Let us callan edge f of C/q special if its image in C is p-coCartesian. To complete theproof, it will suffice to verify the following assertions:(i) Given a vertex q ∈ C/q and an edge fe : p0 (q) → r0 , there exists a special
edge f : q → r with p0 (f ) = fe.
(ii) Every special edge of C/q is p0 -coCartesian.
To prove (i), we begin with a commutative diagram∆0 ? _ K
∆1 ? K

q

fe

/C

/ D.

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FIBRATIONS OF SIMPLICIAL SETS

Let C ∈ C denote the image under q of the cone point of ∆0 ? K and choosea p-coCartesian morphism u : C → C 0 lifting fe|∆1 . We now consider thecollection P of all pairs (L, fL ), where L is a simplicial subset of K and fL
is a map fitting into a commutative diagram(∆0 ? K)fL

1
∆ 0_ ?L (∆ ? L)`


∆1 ? K

/C

/ D,

fe

where fL |∆1 = u and fL |∆0 ? K = q. We partially order the set P as
follows: (L, fL ) ≤ (L0 , fL0 ) if L ⊆ L0 and fL is equal to the restriction of fL0 .
The partially ordered set P satisfies the hypotheses of Zorn’s lemma andtherefore contains a maximal element (L, fL ). If L 6= K, then we can choose
a simplex σ : ∆n → K of minimal dimension which does not belong to L.
By maximality, we obtain a diagram
Λn+2
0 _

/
=C


∆n+2


/D

in which the indicated dotted arrow cannot be supplied. This is a contradiction since the upper horizontal map carries the initial edge of Λn+2
to a0
p-coCartesian edge of C. It follows that L = K, and we may take f = fL .
This completes the proof of (i).
The proof of (ii) is similar. Suppose we are given n ≥ 2 and let
Λn0 ? K
_

f0

/; C

f


∆n ? Kg/D

be a commutative diagram, where f0 |K = q and f0 |∆{0,1} is a p-coCartesianedge of C. We wish to prove the existence of the dotted arrow f indicated in
the diagram. As above, we consider the collection P of all pairs (L, fL ), where
L is a simplicial subset of K and fL extends f0 and fits into a commutative
diagram(Λn0 ? K)

`

(∆
Λn
0 ?L
_


∆n ? K

n

fL

? L)

g

/C

/ D.

We partially order P as follows: (L, fL ) ≤ (L0 , fL0 ) if L ⊆ L0 and fL is
a restriction of fL0 . Using Zorn’s lemma, we conclude that P contains amaximal element (L, fL ). If L 6= K, then we can choose a simplex σ : ∆m →

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K which does not belong to L, where m is as small as possible. Invoking the
maximality of (L, fL ), we obtain a diagram
Λn+m+10 _
∆n+m+1

h

/; C

/ D,

where the indicated dotted arrow cannot be supplied. However, the map
h carries the initial edge of ∆n+m+1 to a p-coCartesian edge of C, so we
obtain a contradiction. It follows that L = K, so that we can take f = fL
to complete the proof.
2.4.4 Mapping Spaces and Cartesian Fibrations
Let p : C → D be a functor between ∞-categories and let X and Y beobjects of C. Then p induces a map
φ : MapC (X, Y ) → MapD (p(X), p(Y )).
Our goal in this section is to understand the relationship between the fibersof p and the homotopy fibers of φ.
Lemma 2.4.4.1. Let p : C → D be an inner fibration of ∞-categories andR
let X, Y ∈ C. The induced map φ : HomR
C (X, Y ) → HomD (p(X), p(Y )) is a
Kan fibration.
Proof. Since p is an inner fibration, the induced map φe : C/Y → D/p(Y ) ×D C
is a right fibration by Proposition 2.1.2.1. We note that φ is obtained from φeby restricting to the fiber over the vertex X of C. Thus φ is a right fibration;
since the target of φ is a Kan complex, φ is a Kan fibration by Lemma
2.1.3.3.
Suppose the conditions of Lemma 2.4.4.1 are satisfied. Let us consider
the problem of computing the fiber of φ over a vertex e : p(X) → p(Y ) of0
HomR
D (p(X), p(Y )). Suppose that there is a p-Cartesian edge e : X → Y
lifting e. By definition, we have a trivial fibration
ψ : C/e → C/Y ×D/p(Y ) D/e .
Consider the 2-simplex σ = s1 (e) regarded as a vertex of D/e . Passing to
the fiber, we obtain a trivial fibration
F → φ−1 (e),where F denotes the fiber of C/e → D/e ×D C over the point (σ, X). On
the other hand, we have a trivial fibration C/e → D/e ×D/p(X) C/X 0 by
Proposition 2.1.2.5. Passing to the fiber again, we obtain a trivial fibration0
F → HomR
Cp(X) (X, X ). We may summarize the situation as follows:highertopoi

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FIBRATIONS OF SIMPLICIAL SETS

Proposition 2.4.4.2. Let p : C → D be an inner fibration of ∞-categories.
Let X, Y ∈ C, let e : p(X) → p(Y ) be a morphism in D, and let e : X 0 → Y be
a locally p-Cartesian morphism of C lifting e. Then in the homotopy category
H of spaces, there is a fiber sequence
MapCp(X) (X, X 0 ) → MapC (X, Y ) → MapD (p(X), p(Y )).
Here the fiber is taken over the point classified by e : p(X) → p(Y ).
Proof. The edge e defines a map ∆1 → D. Note that the fiber of the Kan
Rfibration HomR
C (X, Y ) → HomD (pX, pY ) does not change if we replace p by
the induced projection C ×D ∆1 → ∆1 . We may therefore assume without
loss of generality that e is p-Cartesian, and the desired result follows fromthe above analysis.
A similar assertion can be taken as a characterization of Cartesian morphisms:
Proposition 2.4.4.3. Let p : C → D be an inner fibration of ∞-categories
and let f : Y → Z be a morphism in C. The following are equivalent:(1) The morphism f is p-Cartesian.
(2) For every object X of C, composition with f gives rise to a homotopy
Cartesian diagram/ MapC (X, Z)
MapC (X, Y )MapD (p(X), p(Y ))/ MapD (p(X), p(Z)).

Proof. Let φ : C/f → C/Z ×D/p(Z) D/p(f ) be the canonical map; then (1) isequivalent to the assertion that φ is a trivial fibration. According to Proposition 2.1.2.1, φ is a right fibration. Thus, φ is a trivial fibration if and only
if the fibers of φ are contractible Kan complexes. For each object X ∈ C, let
φX : C/f ×C {X} → C/Z ×D/p(Z) D/p(f ) ×C {X}
be the induced map. Then φX is a right fibration between Kan complexes
and therefore a Kan fibration; it has contractible fibers if and only if it is a
homotopy equivalence. Thus (1) is equivalent to the assertion that φX is a
homotopy equivalence for every object X of C.
We remark that (2) is somewhat imprecise: although all the maps in the
diagram are well-defined in the homotopy category H of spaces, we need torepresent this by a commutative diagram in the category of simplicial sets
before we can ask whether or not the diagram is homotopy Cartesian. We
therefore rephrase (2) more precisely: it asserts that the diagram of Kancomplexes/ C/Z ×C {X}C/f ×C {X}
D/p(f ) ×D {p(X)}/ D/p(Z) ×D {p(X)}highertopoi

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is homotopy Cartesian. Lemma 2.4.4.1 implies that the right vertical map
is a Kan fibration, so the homotopy limit in question is given by the fiber
product
C/Z ×D/p(Z) D/p(f ) ×C {X}.
Consequently, assertion (2) is also equivalent to the condition that φX be a
homotopy equivalence for every object X ∈ C.
pq

Corollary 2.4.4.4. Suppose we are given maps C → D → E of ∞-categories
such that both q and q ◦ p are locally Cartesian fibrations. Suppose that p
carries locally (q ◦ p)-Cartesian edges of C to locally q-Cartesian edges of D
and that for every object Z ∈ E, the induced map CZ → DZ is a categoricalequivalence. Then p is a categorical equivalence.
Proof. Proposition 2.4.4.2 implies that p is fully faithful. If Y is any object ofD, then Y is equivalent in the fiber Dq(Y ) to the image under p of some vertex
of Cq(Y ) . Thus p is essentially surjective, and the proof is complete.
Corollary 2.4.4.5. Let p : C → D be a Cartesian fibration of ∞-categories.
Let q : D0 → D be a categorical equivalence of ∞-categories. Then the induced
map q 0 : C0 = D0 ×D C → C is a categorical equivalence.
Proof. Proposition 2.4.4.2 immediately implies that q 0 is fully faithful. We
claim that q 0 is essentially surjective. Let X be any object of C. Since q is fully
faithful, there exists an object y of T 0 and an equivalence e : q(Y ) → p(X).
Since p is Cartesian, we can choose a p-Cartesian edge e : Y 0 → X liftinge. Since e is p-Cartesian and p(e) is an equivalence, e is an equivalence. By
construction, the object Y 0 of S lies in the image of q 0 .
Corollary 2.4.4.6. Let p : C → D be a Cartesian fibration of ∞-categories.
Then p is a categorical equivalence if and only if p is a trivial fibration.
Proof. The “if” direction is clear. Suppose then that p is a categorical equivalence. We first claim that p is surjective on objects. The essential surjectivityof p implies that for each Y ∈ D, there is an equivalence Y → p(X) for some
object X of C. Since p is Cartesian, this equivalence lifts to a p-Cartesianedge Ye → X of S, so that p(Ye ) = Y .
Since p is fully faithful, the map MapC (X, X 0 ) → MapD (p(X), p(X 0 ))is a homotopy equivalence for any pair of objects X, X 0 ∈ C. Supposethat p(X) = p(X 0 ). Then, applying Proposition 2.4.4.2, we deduce that
MapCp(X) (X, X 0 ) is contractible. It follows that the ∞-category Cp(X) is
nonempty with contractible morphism spaces; it is therefore a contractible
Kan complex. Proposition 2.4.2.4 now implies that p is a right fibration.
Since p has contractible fibers, it is a trivial fibration by Lemma 2.1.3.4.
We have already seen that if an ∞-category S has an initial object, thenthat initial object is essentially unique. We now establish a relative versionof this result.

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Lemma 2.4.4.7. Let p : C → D be a Cartesian fibration of ∞-categories
and let C be an object of C. Suppose that D = p(C) is an initial object of D
and that C is an initial object of the ∞-category CD = C ×D {D}. Then C
is an initial object of C.
Proof. Let C 0 be any object of C and set D0 = p(C 0 ). Since D is an initial
object of D, the space MapD (D, D0 ) is contractible. In particular, theree → C 0 be a p-Cartesian lift
exists a morphism f : D → D0 in D. Let fe : D
of f . According to Proposition 2.4.4.2, there exists a fiber sequence in thehomotopy category H:
e → MapC (C, C 0 ) → MapD (D, D0 ).
MapC (C, D)
D

Since the first and last spaces in this sequence are contractible, we deducethat MapC (C, C 0 ) is contractible as well, so that C is an initial object ofC.
Lemma 2.4.4.8. Suppose we are given a diagram of simplicial sets∂ ∆ _n

f0
f


∆n

/X
<
p

g


/ S,

where p is a Cartesian fibration and n > 0. Suppose that f0 (0) is an initial
object of the ∞-category Xg(0) = X ×S {g(0)}. Then there exists a map
f : ∆n → X as indicated by the dotted arrow in the diagram, which renders
the diagram commutative.
Proof. Pulling back via g, we may replace S by ∆n and thereby reduce to thecase where S is an ∞-category and g(0) is an initial object of S. It follows
from Lemma 2.4.4.7 that f0 (v) is an initial object of S, which implies theexistence of the desired extension f .
Proposition 2.4.4.9. Let p : X → S be a Cartesian fibration of simplicial
sets. Assume that for each vertex s of S, the ∞-category Xs = X ×S {s} hasan initial object.
(1) Let X 0 ⊆ X denote the full simplicial subset of X spanned by thosevertices x which are initial objects of Xp(x) . Then p|X 0 is a trivial
fibration of simplicial sets.
(2) Let C = MapS (S, X) be the ∞-category of sections of p. An arbitrary
section q : S → X is an initial object of C if and only if q factors
through X 0 .
Proof. Since every fiber Xs has an initial object, the map p|X 0 has theright lifting property with respect to the inclusion ∅ ⊆ ∆0 . If n > 0, thenLemma 2.4.4.8 shows that p|X 0 has the right lifting property with respectto ∂ ∆n ⊆ ∆n . This proves (1). In particular, we deduce that there exists

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a map q : S → X 0 which is a section of p. In view of the uniqueness of
initial objects, (2) will follow if we can show that q is an initial object of C.
Unwinding the definitions, we must show that for n > 0, any lifting problem
S × ∂ _ ∆nS × ∆n

f

/X
:

/S

q

can be solved provided that f |S × {0} = q. The desired extension can beconstructed simplex by simplex using Lemma 2.4.4.8.
2.4.5 Application: Invariance of Undercategories
Our goal in this section is to complete the proof of Proposition 1.2.9.3 byproving the following assertion:(∗) Let p : C → D be an equivalence of ∞-categories and let j : K → C bea diagram. Then the induced map
Cj/ → Dpj/
is a categorical equivalence.
We will need a lemma.
Lemma 2.4.5.1. Let p : C → D be a fully faithful map of ∞-categories andlet j : K → C be any diagram in C. Then, for any object x of C, the map of
Kan complexes
Cj/ ×C {x} → Dpj/ ×D {p(x)}is a homotopy equivalence.
Proof. For any map r : K 0 → K of simplicial sets, let Cr = Cjr/ ×C {x} andDr = Dpjr/ ×D {p(x)}.
Choose a transfinite sequence of simplicial subsets Kα of K such that
Kα+1 isS the result of adjoining a single nondegenerate simplex to Kα andKλ = α<λ Kα whenever λ is a limit ordinal (we include the case where
λ = 0, so that K0 = ∅). Let iα : Kα → K denote the inclusion. We claim
the following:(1) For every ordinal α, the map φα : Ciα → Diα is a homotopy equivalenceof simplicial sets.
(2) For every pair of ordinals β ≤ α, the maps Ciα → Ciβ and Diα → Diβ
are Kan fibrations of simplicial sets.
We prove both of these claims by induction on α. When α = 0, (2) isobvious and (1) follows since both sides are isomorphic to ∆0 . If α is alimit ordinal, (2) is again obvious, while (1) follows from the fact that bothhighertopoi

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Ciα and Diα are obtained as the inverse limits of transfinite sequences offibrations, and the map φα is an inverse limit of maps which are individuallyhomotopy equivalences.
`
Assume that α = β + 1 is a successor ordinal, so that Kα ' Kβ ∂ ∆n ∆n .
Let f : ∆n → Kα be the induced map, so thatCiα = Ciβ ×Cf | ∂ ∆n Cf
Diα = Diβ ×Df | ∂ ∆n Df .
We note that the projections Cf → Df | ∂ ∆n and Cf → Df | ∂ ∆n are left
fibrations by Proposition 2.1.2.1 and therefore Kan fibrations by Lemma
2.1.3.3. This proves (2) since the class of Kan fibrations is stable underpullback. We also note that the pullback diagrams defining Xiα and Yiα are
also homotopy pullback diagrams. Thus, to prove that φα is a homotopy
equivalence, it suffices to show that φβ and the maps
Cf | ∂ ∆n → Df | ∂ ∆n
Cf → Df
are homotopy equivalences. In other words, we may reduce to the case where
K is a finite complex.
We now work by induction on the dimension of K. Suppose that thedimension of K is n and that the result is known for all simplicial setshaving smaller dimensions. Running through the above argument again, we
can reduce to the case where K = ∆n . Let v denote the final vertex of ∆n .
By Proposition 2.1.2.5, the maps
Cj → Cj|{v}

Dj → Dj|{v}

are trivial fibrations. Thus, it suffices to consider the case where K is asingle point {v}. In this case, we have Cj = HomLC (j(v), x) and Yj =HomLD (p(j(v)), p(x)). It follows that the map φ is a homotopy equivalencesince p is assumed to be fully faithful.
Proof of (∗). Let p : C → D be a categorical equivalence of ∞-categories
and j : K → C any diagram. We have a factorizationfg

Cj/ → Dpj/ ×D C → Dpj/ .
Lemma 2.4.5.1 implies that Cj/ and Dpj/ ×D C are fiberwise equivalent left
fibrations over C, so that f is a categorical equivalence by Corollary 2.4.4.4(we note that the map f automatically carries coCartesian edges to coCartesian edges because all edges of the target Dpj/ ×D C are coCartesian). The
map g is a categorical equivalence by Corollary 2.4.4.5. It follows that g ◦ f
is a categorical equivalence, as desired.

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2.4.6 Application: Categorical Fibrations over a Point
Our main goal in this section is to prove the following result:
Theorem 2.4.6.1. Let C be a simplicial set. Then C is fibrant for the Joyalmodel structure if and only if C is an ∞-category.
The proof will require a few technical preliminaries.
Lemma 2.4.6.2. Let p : C → D be a categorical equivalence of ∞-categories
and let m ≥ 2 be an integer. Suppose we are given maps f0 : ∂ ∆{1,...,m} → C{1,...,m}and h0 : Λm
= p ◦ f0 . Suppose further that the
0 → D with h0 | ∂ ∆{0,1}
restriction of h to ∆is an equivalence in D. Then there exist maps
f : ∆{1,...,m} → C, h : ∆m → D such that h|∆{1,...,m} = p ◦ f , f0 =f | ∂ ∆{1,...,m} , and h0 = h|Λm
0 .
Proof. We may regard h0 as a point of the simplicial set D/p◦f0 . Since p is a
categorical equivalence, Proposition 1.2.9.3 implies that p0 : C/f0 → D/p◦f0
is a categorical equivalence. It follows that h0 lies in the essential imageof p0 . Consider the linearly ordered set {0 < 00 < 1 < · · · < m} and the0corresponding simplex ∆{0,0 ,...,m} . By hypothesis, we can extend f0 to a00{0 ,...,m}
map f00 : Λ00→ C and h0 to a map h00 : ∆{0,0 } ? ∂ ∆{1,...,m} → D such00{0 ,...,m}that h00 |∆{0,0 } is an equivalence and h00 |Λ0= p ◦ f00 .
0{0,00 }
0{0,1}
Since h0 |∆and h0 |∆are both equivalences in D, we deduce that0h00 |∆{0 ,1} is an equivalence in D. Since p is a categorical equivalence, it
0follows that f00 |∆{0 ,1} is an equivalence in C. Proposition 1.2.4.3 implies
00that f0 extends to a map f 0 : ∆{0 ,...,m} → C. The union of p ◦ f 0 and h00
0{0,0 ,...,m}→ D; since D is an ∞-category, this extends
determines a map Λ00
0{0,00 ,...,m}to a map h : ∆→ D. We may now take f = f 0 |∆{1,...,m} andh = h0 |∆m .
Lemma 2.4.6.3. Let p : C → D be a categorical equivalence of ∞-categories
and let A ⊆ B be an inclusion of simplicial sets. Let f0 : A → C, g : B → D
be any maps and let h0 : A × ∆1 → D be an equivalence from g|A to p ◦ f0 .
Then there exists a map f : B → C and an equivalence h : B × ∆1 → Dfrom g to p ◦ f such that f0 = f |A and h0 = h|A × ∆1 .
Proof. Working simplex by simplex with the inclusion A ⊆ B, we may reduce to the case where B = ∆n , A = ∂ ∆n . If n = 0, the existence of thedesired extensions is a reformulation of the assumption that p is essentiallysurjective. Let us assume therefore that n ≥ 1.
We consider the task of constructing h : ∆n × ∆1 → D. Consider thefiltration
X(n + 1) ⊆ · · · ⊆ X(0) = ∆n × ∆1
described in the proof of Proposition 2.1.2.6. We note that the value ofh on X(n + 1) is uniquely prescribed by h0 and g. We extend the definition of h to X(i) by descending induction on i. We note that X(i) 'highertopoi

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X(i + 1) Λn+1 ∆n+1 . For i > 0, the existence of the required extension iskguaranteed by the assumption that D is an ∞-category. Since n ≥ 1, Lemma
2.4.6.2 allows us to extend h over the simplex σ0 and to define f so that thedesired conditions are satisfied.
`Lemma 2.4.6.4. Let C ⊆ D be an inclusion of simplicial sets which is alsoa categorical equivalence. Suppose further that C is an ∞-category. Then C
is a retract of D.
Proof. Enlarging D by an inner anodyne extension if necessary, we may
suppose that D is an ∞-category. We now apply Lemma 2.4.6.3 in the casewhere A = C, B = D.
Proof of Theorem 2.4.6.1. The “only if” direction has already been established (Remark 2.2.5.5). For the converse, we must show that if C is an∞-category, then C has the extension property with respect to every inclusion of simplicial sets A ⊆ B which is a categorical equivalence. Fix any
map A`→ C. Since the Joyal model structure is left proper, the inclusion
C ⊆ C A B is a categorical equivalence.
We now apply Lemma 2.4.6.4 to`conclude that C is a retract of C A B.
We can state Theorem 2.4.6.1 as follows: if S is a point, then p : X → S
is a categorical fibration (in other words, a fibration with respect to the
Joyal model structure on S) if and only if it is an inner fibration. However,the class of inner fibrations does not coincide with the class of categoricalfibrations in general. The following result describes the situation when T is
an ∞-category:
Corollary 2.4.6.5 (Joyal). Let p : C → D be a map of simplicial sets,
where D is an ∞-category. Then p is a categorical fibration if and only if
the following conditions are satisfied:(1) The map p is an inner fibration.
(2) For every equivalence f : D → D0 in D and every object C ∈ C withp(C) = D, there exists an equivalence f : C → C 0 in C with p(f ) = f .
Proof. Suppose first that p is a categorical fibration. Then (1) follows immediately (since the inclusions Λni ⊆ ∆n are categorical equivalences for
0 < i < n). To prove (2), we let D0 denote the largest Kan complex contained in D, so that the edge f belongs to D. There exists a contractible
e →D
e 0 and a map q : K → D such
Kan complex K containing an edge fe : D
e ⊆ K is a categorical equivalence,that q(fe) = f . Since the inclusion {D}our assumption that p is a categorical fibration allows us to lift q to a map
e = C. We can now take f = qe(fe); since fe is anqe : K → C such that qe(D)equivalence in K, f is an equivalence in C.

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Now suppose that (1) and (2) are satisfied. We wish to show that p is a
categorical fibration. Consider a lifting problem
g0

A _

/C
>

g

p

i


B

h


/ D,

where i is a cofibration and a categorical equivalence; we wish to show that
there exists a morphism g as indicated which renders the diagram commutative. We first observe that condition (1), together with our assumption that
D is an ∞-category, guarantees that C is an ∞-category. Applying Theorem
2.4.6.1, we can extend g0 to a map g 0 : B → C (not necessarily satisfying
h = p ◦ g 0 ). The maps h and p ◦ g 0 have the same restriction to A. Leta
H0 : (B × ∂ ∆1 )(A × ∆1 ) → D
A×∂ ∆1

be given by (p ◦ g 0 , h) on B × ∂ ∆1 and by the compositionh

A × ∆1 → A ⊆ B → D
on A×∆1 . Applying Theorem 2.4.6.1 once more, we deduce that H0 extends
to a map H : B × ∆1 → D. The map H carries {a} × ∆1 to an equivalencein D for every vertex a of A. Since the inclusion A ⊆ B is a categoricalequivalence, we deduce that H carries {b} × ∆1 to an equivalence for everyb ∈ B.
Leta
G0 : (B × {0})(A × ∆1 ) → C
A×{0}be the composition of the projection to B with the map g 0 . We have acommutative diagram(B × {0})1
A×{0} (A × ∆ )`GB × ∆1

H

G0

/6 C
p


/ D.

To complete the proof, it will suffice to show that we can supply a map G
as indicated, rendering the diagram commutative; in this case, we can solve
the original lifting problem by defining g = G|B × {1}.
We construct the desired extension G working simplex by simplex on B.
We start by applying assumption (2) to construct the map G|{b} × ∆1 forevery vertex b of B (that does not already belong to A); moreover, we ensurethat G|{b} × ∆1 is an equivalence in C.

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To extend G0 to simplices of higher dimension, we encounter lifting problems of the type`e
5/ C(∆n × {0}) ∂ ∆n ×{0} (∂ ∆n × ∆1 )_p

∆n × ∆1/ D.

According to Proposition 2.4.1.8, these lifting problems can be solved provided that e carries {0} × ∆1 to a p-coCartesian edge of C. This follows
immediately from Proposition 2.4.1.5.
2.4.7 Bifibrations
As we explained in §2.1.2, left fibrations p : X → S can be thought of as
covariant functors from S into an ∞-category of spaces. Similarly, right fibrations q : Y → T can be thought of as contravariant functors from T into
an ∞-category of spaces. The purpose of this section is to introduce a convenient formalism which encodes covariant and contravariant functoriality
simultaneously.
Remark 2.4.7.1. The theory of bifibrations will not play an important rolein the remainder of the book. In fact, the only result from this section that
we will actually use is Corollary 2.4.7.12, whose statement makes no mentionof bifibrations. A reader who is willing to take Corollary 2.4.7.12 on faith,or supply an alternative proof, may safely omit the material covered in thissection.
Definition 2.4.7.2. Let S, T , and X be simplicial sets and let p : X → S×T
be a map. We shall say that p is a bifibration if it is an inner fibration having
the following properties:• For every n ≥ 1 and every diagram of solid arrows
/X
;

Λn0
_

∆n

f


/ S×T

such that πT ◦ f maps ∆{0,1} ⊆ ∆n to a degenerate edge of T , there
exists a dotted arrow as indicated, rendering the diagram commutative.
Here πT denotes the projection S × T → T .
• For every n ≥ 1 and every diagram of solid arrows
/X
;

Λnn
_

∆n

f


/ S×T

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such that πS ◦ f maps ∆{n−1,n} ⊆ ∆n to a degenerate edge of T , there
exists a dotted arrow as indicated, rendering the diagram commutative.
Here πS denotes the projection S × T → S.
Remark 2.4.7.3. The condition that p be a bifibration is not a condition on
p alone but also refers to a decomposition of the codomain of p as a productS × T . We note also that the definition is not symmetric in S and T : instead,p : X → S × T is a bifibration if and only if pop : X op → T op × S op is abifibration.
Remark 2.4.7.4. Let p : X → S × T be a map of simplicial sets. If T = ∗,then p is a bifibration if and only if it is a left fibration. If S = ∗, then p isa bifibration if and only if it is a right fibration.
Roughly speaking, we can think of a bifibration p : X → S × T as abifunctor from S×T to an ∞-category of spaces; the functoriality is covariant
in S and contravariant in T .
Lemma 2.4.7.5. Let p : X → S × T be a bifibration of simplicial sets.
Suppose that S is an ∞-category. Then the composition q = πT ◦ p is a
Cartesian fibration of simplicial sets. Furthermore, an edge e of X is qCartesian if and only if πS (p(e)) is an equivalence.
Proof. The map q is an inner fibration because it is a composition of inner
fibrations. Let us say that an edge e : x → y of X is quasi-Cartesian if
πS (p(e)) is degenerate in S. Let y ∈ X0 be any vertex of X and let e : x →q(y) be an edge of S. The pair (e, s0 q(y)) is an edge of S ×T whose projection
to T is degenerate; consequently, it lifts to a (quasi-Cartesian) edge e : x → y
in X. It is immediate from Definition 2.4.7.2 that any quasi-Cartesian edgeof X is q-Cartesian. Thus q is a Cartesian fibration.
Now suppose that e is a q-Cartesian edge of X. Then e is equivalent to aquasi-Cartesian edge of X; it follows easily that πS (p(e)) is an equivalence.
Conversely, suppose that e : x → y is an edge of X and that πS (p(e))
is an equivalence. We wish to show that e is q-Cartesian. Choose a quasiCartesian edge e0 : x0 → y with q(e0 ) = q(e). Since e0 is q-Cartesian, there
exists a simplex σ ∈ X2 with d0 σ = e0 , d1 σ = e, and q(σ) = s0 q(e). Let
f = d2 (σ), so that πS (p(e0 )) ◦ πS (p(f )) ' πS p(e) in the ∞-category S. We
note that f lies in the fiber Xq(x) , which is left fibered over S; since f maps
to an equivalence in S, it is an equivalence in Xq(x) . Consequently, f is
q-Cartesian, so that e = e0 ◦ f is q-Cartesian as well.
pq

Proposition 2.4.7.6. Let X → Y → S × T be a diagram of simplicial
sets. Suppose that q and q ◦ p are bifibrations and that p induces a homotopy
equivalence X(s,t) → Y(s,t) of fibers over each vertex (s, t) of S × T . Then p
is a categorical equivalence.
Proof. By means of a standard argument (see the proof of Proposition
2.2.2.7), we may reduce to the case where S and T are simplices; in particular, we may suppose that S and T are ∞-categories. Fix t ∈ T0 andhighertopoi

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consider the map of fibers pt : Xt → Yt . Both sides are left fibered overS × {t}, so that pt is a categorical equivalence by (the dual of) Corollary
2.4.4.4. We may then apply Corollary 2.4.4.4 again (along with the characterization of Cartesian edges given in Lemma 2.4.7.5) to deduce that p is a
categorical equivalence.
Proposition 2.4.7.7. Let p : X → S × T be a bifibration, let f : S 0 →S, g : T 0 → T be categorical equivalences between ∞-categories, and let
X 0 = X ×S×T (S 0 × T 0 ). Then the induced map X 0 → X is a categoricalequivalence.
Proof. We will prove the result assuming that f is an isomorphism. A dual
argument will establish the result when g is an isomorphism and applying
the result twice we will deduce the desired statement for arbitrary f and g.
Given a map i : A → S, let us say that i is good if the induced map
X ×S×T (A × T 0 ) → X ×S×T (A × T 0 ) is a categorical equivalence. We wish
to show that the identity map S → S is good; it will suffice to show that allmaps A → S are good. Using the argument of Proposition 2.2.2.7, we canreduce to showing that every map ∆n → S is good. In other words, we may
assume that S = ∆n , and in particular that S is an ∞-category. By Lemma
2.4.7.5, the projection X → T is a Cartesian fibration. The desired result
now follows from Corollary 2.4.4.5.
We next prove an analogue of Lemma 2.4.6.3.
pq

Lemma 2.4.7.8. Let X → Y → S × T satisfy the hypotheses of Proposition
2.4.7.6. Let A ⊆ B be a cofibration of simplicial sets over S × T . Let f0 :A → X, g : B → Y be morphisms in (Set∆ )/S×T and let h0 : A × ∆1 → Y
be a homotopy (again over S × T ) from g|A to p ◦ f0 .
Then there exists a map f : B → X (of simplicial sets over S × T ) and ahomotopy h : B × ∆1 → T (over S × T ) from g to p ◦ f such that f0 = f |Aand h0 = h|A × ∆1 .
Proof. Working simplex by simplex with the inclusion A ⊆ B, we may reduceto the case where B = ∆n , A = ∂ ∆n . If n = 0, we may invoke the factthat p induces a surjection π0 X(s,t) → π0 Y(s,t) on each fiber. Let us assume
therefore that n ≥ 1. Without loss of generality, we may pull back along themaps B → S, B → T and reduce to the case where S and T are simplices.
We consider the task of constructing h : ∆n × ∆1 → T . We now employ
the filtration
X(n + 1) ⊆ · · · ⊆ X(0)described in the proof of Proposition 2.1.2.6. We note that the value ofh on X(n + 1) is uniquely prescribed by h0 and g. We extend the definition of `h to X(i) by descending induction on i. We note that X(i) 'X(i + 1) Λn+1 ∆n+1 . For i > 0, the existence of the required extension iskguaranteed by the assumption that Y is inner-fibered over S × T .

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We note that, in view of the assumption that S and T are simplices, any
extension of h over the simplex σ0 is automatically a map over S × T . Since
S and T are ∞-categories, Proposition 2.4.7.6 implies that p is a categoricalequivalence of ∞-categories; the existence of the desired extension of h (andthe map f ) now follows from Lemma 2.4.6.2.
pqProposition 2.4.7.9. Let X → Y → S × T satisfy the hypotheses of Proposition 2.4.7.6. Suppose that p is a cofibration. Then there exists a retractionr : Y → X (as a map of simplicial sets over S × T ) such that r ◦ p = idX .
Proof. Apply Lemma 2.4.7.8 in the case where A = X and B = Y .
Let q : M → ∆1 be an inner fibration, which we view as a correspondence
from C = q −1 {0} to D = q −1 {1}. Evaluation at the endpoints of ∆1 induces
maps Map∆1 (∆1 , M) → C, Map∆1 (∆1 , M) → D.
Proposition 2.4.7.10. For every inner fibration q : M → ∆1 as above, themap p : Map∆1 (∆1 , M) → C × D is a bifibration.
Proof. We first show that p is an inner fibration. It suffices to prove that q
has the right lifting property with respect toa(Λni × ∆1 )(∆n × ∂ ∆1 ) ⊆ ∆n × ∆1
1
Λni ×∂ ∆

for any 0 < i < n. But this is a smash product of ∂ ∆1 ⊆ ∆1 with the inner
anodyne inclusion Λni ⊆ ∆n .
To complete the proof that p is a bifibration, we verify that for every n ≥ 1,
f0 : Λ0n → X, and g : ∆n → S × T with g|Λn0 = p ◦ f0 , if (πS ◦ g)|∆{0,1} isdegenerate, then there exists f : ∆n → X with g = p ◦ f and f0 = f |Λn0 .
(The dual assertion, regarding extensions of maps Λnn → X, is verified in thesame way.) The pair (f0 , g) may be regarded as a map
ah0 : (∆n × {0, 1})(Λn0 × ∆1 ) → M,Λn0 ×{0,1}and our goal is to prove that h0 extends to a map h : ∆n × ∆1 → M.
Let {σi }0≤i≤n be the maximal-dimensional simplices of ∆n × ∆1 , as in
the proof of Proposition 2.1.2.6. We setaK(0) = (∆n × {0, 1})(Λn0 × ∆1 )Λn0 ×{0,1}

and, for 0 ≤ i ≤ n, let K(i + 1) = K(i) ∪ σi . We construct maps hi :Ki → M, with hi =`hi+1 |Ki , by induction on i. We note that for i <n, K(i + 1) ' K(i) Λn+1 ∆n+1 , so that the desired extension exists by
i+1
virtue of the assumption that M is an ∞-category.`If i = n, we have instead
an isomorphism ∆n × ∆1 = K(n + 1) ' K(n) Λn+1 ∆n+1 . The desired
0

extension of hn can be found using Proposition 1.2.4.3 because h0 |∆{0,1} ×{0} is an equivalence in C ⊆ M by assumption.

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143

Corollary 2.4.7.11. Let C be an ∞-category. Evaluation at the endpoints
gives a bifibration Fun(∆1 , C) → C × C.
Proof. Apply Proposition 2.4.7.10 to the correspondence C ×∆1 .
Corollary 2.4.7.12. Let f : C → D be a functor between ∞-categories. The
projection
p : Fun(∆1 , D) ×Fun({1},D) C → Fun({0}, D)
is a Cartesian fibration. Moreover, a morphism of Fun(∆1 , D) ×Fun({1},D) C
is p-Cartesian if and only if its image in C is an equivalence.
Proof. Combine Corollary 2.4.7.11 with Lemma 2.4.7.5.

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Chapter Three
The ∞-Category of ∞-Categories
The power of category theory lies in its role as a unifying language for mathematics: nearly every class of mathematical structures (groups, manifolds,
algebraic varieties, and so on) can be organized into a category. This language is somewhat inadequate in situations where the structures need to beclassified up to some notion of equivalence less rigid than isomorphism. For
example, in algebraic topology one wishes to study topological spaces up
to homotopy equivalence; in homological algebra one wishes to study chain
complexes up to quasi-isomorphism. Both of these examples are most naturally described in terms of higher category theory (for example, the theoryof ∞-categories developed in this book).
Another source of examples arises in category theory itself. In classicalcategory theory, it is generally regarded as unnatural to ask whether two
categories are isomorphic; instead, one asks whether or not they are equivalent. The same phenomenon arises in higher category theory. Throughout
this book, we generally regard two ∞-categories C and D as the same ifthey are categorically equivalent, even if they are not isomorphic to one another as simplicial sets. In other words, we are not interested in the ordinary
category of ∞-categories (a full subcategory of Set∆ ) but in an underlying∞-category which we now define.
Definition 3.0.0.1. The simplicial category Cat∆∞ is defined as follows:(1) The objects of Cat∆∞ are (small) ∞-categories.
(2) Given ∞-categories C and D, we define MapCat∆(C, D) to be the largest∞
Kan complex contained in the ∞-category Fun(C, D).
We let Cat∞ denote the simplicial nerve N(Cat∆∞ ). We will refer to Cat∞as the ∞-category of (small) ∞-categories.
Remark 3.0.0.2. By construction, Cat∞ arises as the nerve of a simplicial category Cat∆∞ , where composition is strictly associative. This is one
advantage of working with ∞-categories: the correct notion of functor is encoded by simply considering maps of simplicial sets (rather than homotopy
coherent diagrams, say), so there is no difficulty in composing them.
Remark 3.0.0.3. The mapping spaces in Cat∆∞ are Kan complexes, so that
Cat∞ is an ∞-category (Proposition 1.1.5.10) as suggested by the terminology.

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Remark 3.0.0.4. By construction, the objects of Cat∞ are ∞-categories,
morphisms are given by functors, and 2-morphisms are given by homotopies
between functors. In other words, Cat∞ discards all information about noninvertible natural transformations between functors. If necessary, we could
retain this information by forming an ∞-bicategory of (small) ∞-categories.
We do not wish to become involved in any systematic discussion of ∞bicategories, so we will be content to consider only Cat∞ .

Our goal in this chapter is to study the ∞-category Cat∞ . For example, we would like to show that Cat∞ admits limits and colimits. There
are two approaches to proving this assertion. We can attack the problem
directly by giving an explicit construction of the limits and colimits in question: see §3.3.3 and §3.3.4. Alternatively, we can try to realize Cat∞ as the∞-category underlying a (simplicial) model category A and deduce the existence of limits and colimits in Cat∞ from the existence of homotopy limits
and homotopy colimits in A (Corollary 4.2.4.8). The objects of Cat∞ canbe identified with the fibrant-cofibrant objects of Set∆ with respect to the
Joyal model structure. However, we cannot apply Corollary 4.2.4.8 directly
because the Joyal model structure on Set∆ is not compatible with the (usual)
simplicial structure. We will remedy this difficulty by introducing the cat+egory Set+
∆ of marked simplicial sets. We will explain how to endow Set∆
with the structure of a simplicial model category in such a way that there+ ◦is an equivalence of simplicial categories Cat∆∞ ' (Set∆ ) . This will allow
us to identify Cat∞ with the ∞-category underlying Set+
∆ , so that Corollary
4.2.4.8 can be invoked.
We will introduce the formalism of marked simplicial sets in §3.1. In particular, we will explain the construction of a model structure not only on+
Set+∆ itself but also for the category (Set∆ )/S of marked simplicial sets overa given simplicial set S. The fibrant objects of (Set+∆ )/S can be identified
with Cartesian fibrations X → S, which we can think of as contravariant
functors from S into Cat∞ . In §3.2, we will justify this intuition by introducing the straightening and unstraightening functors which will allow us to passback and forth between Cartesian fibrations over S and functors from S op to
Cat∞ . This correspondence has applications both to the study of Cartesianfibrations and to the study of the ∞-category Cat∞ ; we will survey some ofthese applications in §3.3.

Remark 3.0.0.5. In the later chapters of this book, it will be necessary
to undertake a systematic study of ∞-categories which are not small. For
this purpose, we introduce the following notational conventions: Cat∞ willd∞denote the simplicial nerve of the category of small ∞-categories, while Cat
denotes the simplicial nerve of the category of ∞-categories which are notnecessarily small.

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3.1 MARKED SIMPLICIAL SETS
The Joyal model structure on Set∆ is a powerful tool in the study of ∞categories. Nevertheless, in relative situations it is somewhat inconvenient.
Roughly speaking, a categorical fibration p : X → S determines a familyof ∞-categories Xs parametrized by the vertices s of S. However, we are
generally more interested in those cases where Xs can be regarded as afunctor of s. As we explained in §2.4.2, this naturally translates into theassumption that p is a Cartesian (or coCartesian) fibration. According to
Proposition 3.3.1.7, every Cartesian fibration is a categorical fibration, but
the converse is false. Consequently, it is natural to try to endow (Set∆ )/S
with some other model structure in which the fibrant objects are precisely
the Cartesian fibrations over S.
Unfortunately, this turns out to be an unreasonable demand. In order tohave a model category, we need to be able to form fibrant replacements: inother words, we need the ability to enlarge an arbitrary map p : X → S into
a commutative diagram
/Y

φ

X
p



S,



q

where q is a Cartesian fibration generated by p. A question arises: for whichedges f of X should φ(f ) be a q-Cartesian edge of Y ? This sort of informationis needed for the construction of Y ; consequently, we need a formalism inwhich certain edges of X have been distinguished.
Definition 3.1.0.1. A marked simplicial set is a pair (X, E), where X is a
simplicial set and E is a set of edges of X which contains every degenerateedge. We will say that an edge of X is marked if it belongs to E.
A morphism f : (X, E) → (X 0 , E0 ) of marked simplicial sets is a map
f : X → X 0 having the property that f (E) ⊆ E0 . The category of marked
simplicial sets will be denoted by Set+∆.
Every simplicial set S may be regarded as a marked simplicial set, usually
in many different ways. The two extreme cases deserve special mention: if
S is a simplicial set, we let S ] = (S, S1 ) denote the marked simplicial set
in which every edge of S has been marked and let S [ = (S, s0 (S0 )) denotethe marked simplicial set in which only the degenerate edges of S have beenmarked.
Notation 3.1.0.2. Let S be a simplicial set. We let (Set+∆ )/S denote thecategory of marked simplicial sets equipped with a map to S (which mightotherwise be denoted as (Set+∆ )/S ] ).
Our goal in this section is to study the theory of marked simplicial sets
and, in particular, to endow each (Set+∆ )/S with the structure of a model category. We will begin in §3.1.1 by introducing the notion of a marked anodyne

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morphism in Set+
∆ . In §3.1.2, we will establish a basic stability property of
the class of marked anodyne maps, which implies the stability of Cartesianfibrations under exponentiation (Proposition 3.1.2.1). In §3.1.3, we will introduce the Cartesian model structure on (Set+∆ )/S for every simplicial set S.
In §3.1.4, we will study these model categories; in particular, we will see thateach (Set+∆ )/S is a simplicial model category whose fibrant objects are precisely the Cartesian fibrations X → S (with Cartesian edges of X marked).
Finally, we will conclude with §3.1.5, where we compare the Cartesian model
structure on (Set+∆ )/S with other model structures considered in this book(such as the Joyal and contravariant model structures).
3.1.1 Marked Anodyne Morphisms
In this section, we will introduce the class of marked anodyne morphisms in
Set+∆ . The definition is chosen so that the condition that a map X → S havethe right lifting property with respect to all marked anodyne morphisms is
closely related to the condition that the underlying map of simplicial sets
X → S be a Cartesian fibration (we refer the reader to Proposition 3.1.1.6
for a more precise statement). The theory of marked anodyne maps is atechnical device which will prove useful when we discuss the Cartesian model
structure in §3.1.3: every marked anodyne morphism is a trivial cofibration
with respect to the Cartesian model structure, but not conversely. In thisrespect, the class of marked anodyne morphisms of Set+
∆ is analogous to theclass of inner anodyne morphisms of Set∆ .
Definition 3.1.1.1. The class of marked anodyne morphisms in Set+
∆ is thesmallest weakly saturated (see §A.1.2) class of morphisms with the following
properties:(1) For each 0 < i < n, the inclusion (Λni )[ ⊆ (∆n )[ is marked anodyne.
(2) For every n > 0, the inclusion(Λnn , E ∩(Λnn )1 ) ⊆ (∆n , E)is marked anodyne, where E denotes the set of all degenerate edges of∆n together with the final edge ∆{n−1,n} .
(3) The inclusion(Λ21 )]a(∆2 )[ → (∆2 )](Λ21 )[

is marked anodyne.
(4) For every Kan complex K, the map K [ → K ] is marked anodyne.
Remark 3.1.1.2. The definition of a marked simplicial set is self-dual.
However, Definition 3.1.1.1 is not self-dual: if A → B is marked anodyne,then the opposite morphism Aop → B op need not be marked anodyne. This
reflects the fact that the theory of Cartesian fibrations is not self-dual.

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Remark 3.1.1.3. In part (4) of Definition 3.1.1.1, it suffices to allow K
to range over a set of representatives for all isomorphism classes of Kancomplexes with only countably many simplices. Consequently, we deducethat the class of marked anodyne morphisms in Set+
∆ is of small generation,
so that the small object argument applies (see §A.1.2).
Remark 3.1.1.4. In Definition 3.1.1.1, we are free to replace (1) by(10 ) For every inner anodyne map A → B of simplicial sets, the induced
map A[ → B [ is marked anodyne.
Proposition 3.1.1.5. Consider the following classes of morphisms in Set+∆:
(2) All inclusions(Λnn , E ∩(Λnn )1 ) ⊆ (∆n , E),
where n > 0 and E denotes the set of all degenerate edges of ∆n together
with the final edge ∆{n−1,n} .
(20 ) All inclusionsa((∂ ∆n )[ × (∆1 )] )((∆n )[ × {1}] ) ⊆ (∆n )[ × (∆1 )] .

(∂ ∆n )[ ×{1}](200 ) All inclusionsa(A[ × (∆1 )] )(B [ × {1}] ) ⊆ B [ × (∆1 )] ,A[ ×{1}]

where A ⊆ B is an inclusion of simplicial sets.
The classes (20 ) and (200 ) generate the same weakly saturated class of morphisms of Set+
∆ which contains the weakly saturated class generated by (2).
Conversely, the weakly saturated class of morphisms generated by (1) and(2) from Definition 3.1.1.1 contains (20 ) and (200 ).
Proof. To see that each of the morphisms specified in (200 ) is contained in
the weakly saturated class generated by (20 ), it suffices to work simplex bysimplex with the inclusion A ⊆ B. The converse is obvious since the classof morphisms of type (20 ) is contained in the class of morphisms of type(200 ). To see that the weakly saturated class generated by (200 ) contains (2),it suffices to show that every morphism in (2) is a retract of a morphism in(200 ). For this, we consider mapsjr∆n → ∆ n × ∆1 → ∆n .
Here j is the composition of the identification ∆n ' ∆n × {0} with theinclusion ∆n × {0} ⊆ ∆n × ∆1 , and r may be identified with the map ofpartially ordered sets(n if m = n − 1, i = 1
r(m, i) =m otherwise.

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Now we simply observe that j and r exhibit the inclusion(Λnn , E ∩(Λnn )1 ) ⊆ (∆n , E)
as a retract of((Λnn , E ∩(Λnn )1 )×(∆1 )] )a((∆n , E)×{1}] ) ⊆ (∆n , E)×(∆1 )] ,n](Λnn ,E ∩(Λn )1 )×{1}

which is turn a pushout of the inclusiona((Λnn )[ × (∆1 )] )((∆n )[ × {1}] ) ⊆ (∆n )[ × (∆1 )] ,[](Λnn ) ×{1}To complete the proof, we must show that each of the inclusionsa((∂ ∆n )[ × (∆1 )] )((∆n )[ × {1}] ) ⊆ (∆n )[ × (∆1 )](∂ ∆n )[ ×{1}]of type (20 ) belongs to the weakly saturated class generated by (1) and (2).
To see this, consider the filtration
Yn+1 ⊆ · · · ⊆ Y0 = ∆n × ∆1
which is the opposite of the filtration defined in the proof of Proposition
2.1.2.6. We let Ei denote the class of all edges of Yi which are marked in(∆n )[ × (∆1 )] . It will suffice to show that each inclusion fi : (Yi+1 , Ei+1 ) ⊆(Yi , Ei ) lies in the weakly saturated class generated by (1) and (2). For i 6= 0,[
n+1 [the map fi is a pushout of (Λn+1) . For i = 0, fi is a pushout
n+1−i ) ⊆ (∆
of
n+1
n+1
(Λn+1
, E),
n+1 , E ∩(Λn+1 )1 ) ⊆ (∆where E denotes the set of all degenerate edges of ∆n+1 , together with∆{n,n+1} .
We now characterize the class of marked anodyne maps:
Proposition 3.1.1.6. A map p : X → S in Set+
∆ has the right lifting
property with respect to all marked anodyne maps if and only if the following
conditions are satisfied:(A) The map p is an inner fibration of simplicial sets.
(B) An edge e of X is marked if and only if p(e) is marked and e is pCartesian.
(C) For every object y of X and every marked edge e : x → p(y) in S, there
exists a marked edge e : x → y of X with p(e) = e.
Proof. We first prove the “only if” direction. Suppose that p has the right
lifting property with respect to all marked anodyne maps. By consideringmaps of the form (1) from Definition 3.1.1.1, we deduce that (A) holds.
Considering (2) in the case n = 1, we deduce that (C) holds. Consideringhighertopoi

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CHAPTER 3(2) for n > 1, we deduce that every marked edge of X is p-Cartesian. For
the converse, let us suppose that e : x → y is a p-Cartesian edge of X
and that p(e) is marked in S. Invoking (C), we deduce that there exists a
marked edge e0 : x0 → y with p(e) = p(e0 ). Since e0 is Cartesian, we canfind a 2-simplex σ of X with d0 (σ) = e0 , d1 (σ) = e, and p(σ) = s0 p(e).
Then d2 (σ) is an equivalence between x and x0 in the ∞-category Xp(x) .
Let K denote the largest Kan complex contained in Xp(x) . Since p has theright lifting property with respect to K [ → K ] , we deduce that every edgeof K is marked; in particular, d2 (σ) is marked. Since p has the right lifting
property with respect to the morphism described in (3) of Definition 3.1.1.1,
we deduce that d1 (σ) = e is marked.
Now suppose that p satisfies the hypotheses of the proposition. We mustshow that p has the right lifting property with respect to the classes ofmorphisms (1), (2), (3), and (4) of Definition 3.1.1.1. For (1), this follows
from the assumption that p is an inner fibration. For (2), this follows from(C) and from the assumption that every marked edge is p-Cartesian. For(3), we are free to replace S by (∆2 )] ; then p is a Cartesian fibration overan ∞-category S, and we can apply Proposition 2.4.1.7 to deduce that theclass of p-Cartesian edges is stable under composition.
Finally, for (4), we may replace S by K ] ; then S is a Kan complex andp is a Cartesian fibration, so the p-Cartesian edges of X are precisely theequivalences in X. Since K is a Kan complex, any map K → X carries theedges of K to equivalences in X.
By Quillen’s small object argument, we deduce that a map j : A → B in
Set+
∆ is marked anodyne if and only if it has the left lifting property withrespect to all morphisms p : X → S satisfying the hypotheses of Proposition
3.1.1.6. From this, we deduce:
Corollary 3.1.1.7. The inclusionai : (Λ22 )](∆2 )[ ,→ (∆2 )](Λ22 )[

is marked anodyne.
Proof. It will suffice to show that i has the left lifting property with respect toany of the morphisms p : X → S described in Proposition 3.1.1.6. Without
loss of generality, we may replace S by (∆2 )] ; we now apply Proposition
2.4.1.7.
Definition 3.1.1.8. Let p : X → S be a Cartesian fibration of simplicial
sets. We let X \ denote the marked simplicial set (X, E), where E is the setof p-Cartesian edges of X.
Remark 3.1.1.9. Our notation is slightly abusive because X \ depends not
only on X but also on the map X → S.

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Remark 3.1.1.10. According to Proposition 3.1.1.6, a map (Y, E) → S ] has
the right lifting property with respect to all marked anodyne maps if andonly if the underlying map Y → S is a Cartesian fibration and (Y, E) = Y \ .
We conclude this section with the following easy result, which will beneeded later:
Proposition 3.1.1.11. Let p : X → S be an inner fibration of simplicial
sets, let f : A → B be a marked anodyne morphism in Set+
∆ , let q : B → X bea map of simplicial sets which carries each marked edge of B to a p-Cartesianedge of X, and set q0 = q ◦ f . Then the induced map
X/q → X/q0 ×S/pq0 S/pq
is a trivial fibration of simplicial sets.
Proof. It is easy to see that the class of all morphisms f of Set+
∆ which satisfy
the desired conclusion is weakly saturated. It therefore suffices to prove that
this class contains a collection of generators for the weakly saturated classof marked anodyne morphisms. If f induces a left anodyne map on theunderlying simplicial sets, then the desired result is automatic. It therefore
suffices to consider the case where f is the inclusion(Λnn , E ∩(Λnn )1 ) ⊆ (∆n , E)
as described in (2) of Definition 3.1.1.1. In this case, a lifting problem/
7 X/q∂ ∆ m_∆m/ X/q ×S
0



/pq0

S/pq

can be reformulated as an equivalent lifting problem
Λn+m+1n+m+1_∆n+m+1

σ0

/X
;
p


/ S.

This lifting problem admits a solution since the hypothesis on q guarantees
that σ0 carries ∆{n+m,n+m+1} to a p-Cartesian edge of X.
3.1.2 Stability Properties of Marked Anodyne Morphisms
Our main goal in this section is to prove the following stability result:
Proposition 3.1.2.1. Let p : X → S be a Cartesian fibration of simplicial
sets and let K be an arbitrary simplicial set. Then
(1) The induced map pK : X K → S K is a Cartesian fibration.

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CHAPTER 3(2) An edge ∆1 → X K is pK -Cartesian if and only if, for every vertex kof K, the induced edge ∆1 → X is p-Cartesian.
We could easily have given an ad hoc proof of this result in §2.4.3. However,
we have opted instead to give a proof using the language of marked simplicial
sets.
Definition 3.1.2.2. A morphism (X, E) → (X 0 , E0 ) in Set+
∆ is a cofibration
if the underlying map X → X 0 of simplicial sets is a cofibration.
The main ingredient we will need to prove Proposition 3.1.2.1 is the following:
Proposition 3.1.2.3. The class of marked anodyne maps in Set+
∆ is stable
under smash products with arbitrary cofibrations. In other words, if f : X →X 0 is marked anodyne and g : Y → Y 0 is a cofibration, then the induced mapa(X × Y 0 )(X 0 × Y ) → X 0 × Y 0
X×Y

is marked anodyne.
Proof. The argument is tedious but straightforward. Without loss of generality we may suppose that f belongs either to the class (20 ) of Proposition
3.1.1.5, or to one of the classes specified in (1), (3), or (4) of Definition 3.1.1.1.
The class of cofibrations is generated by the inclusions (∂ ∆n )[ ⊆ (∆n )[ and(∆1 )[ ⊆ (∆1 )] ; thus we may suppose that g : Y → Y 0 is one of these maps.
There are eight cases to consider:(A1) Let f be the inclusion (Λni )[ ⊆ (∆n )[ and let g be the inclusion(∂ ∆n )[ → (∆n )[ , where 0 < i < n. Since the class of inner anodyne
maps between simplicial sets is stable under smash products with inclusions, the smash product of f and g is marked anodyne (see Remark
3.1.1.4).
(A2) Let f be the inclusion (Λni )[ → (∆n )[ , and let g be the map (∆1 )[ →(∆1 )] , where 0 < i < n. Then the smash product of f and g is an
isomorphism (since Λni contains all vertices of ∆n ).
(B1) Let f be the inclusiona({1}] × (∆n )[ )((∆1 )] × (∂ ∆n )[ ) ⊆ (∆1 )] × (∆n )[{1}] ×(∂ ∆n )[and let g be the inclusion (∂ ∆n )[ → (∆n )[ . Then the smash productof f and g belongs to the class (200 ) of Proposition 3.1.1.5.
(B2) Let f be the inclusion({1}] × (∆n )[ )a{1}] ×(∂ ∆n )[((∆1 )] × (∂ ∆n )[ ) ⊆ (∆1 )] × (∆n )[highertopoi

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and let g denote the map (∆1 )[ → (∆1 )] . If n > 0, then the smashproduct of f and g is an isomorphism. If n = 0, then the smash productmay be identified with the map (∆1 × ∆1 , E) → (∆1 × ∆1 )] , where E
consists of all degenerate edges together with {0} × ∆1 , {1} × ∆1 , and∆1 × {1}. This map may be obtained as a composition of two marked
anodyne maps: the first is of type (3) in Definition 3.1.1.1 (adjoining
the “diagonal” edge to E), and the second is the map described in
Corollary 3.1.1.7 (adjoining the edge ∆1 × {0} to E).
(C1) Let f be the inclusion(Λ21 )]a(∆2 )[ → (∆2 )](Λ21 )[and let g be the inclusion (∂ ∆n )[ ⊆ (∆n )[ . Then the smash product off and g is an isomorphism for n > 0 and is isomorphic to f for n = 0.
(C2) Let f be the inclusion(Λ21 )]a(∆2 )[ → (∆2 )](Λ21 )[and let g be the canonical map (∆1 )[ → (∆1 )] . Then the smash product of f and g is a pushout of the map f .
(D1) Let f be the map K [ → K ] , where K is a Kan complex, and let g be
the inclusion (∂ ∆n )[ ⊆ (∆n )[ . Then the smash product of f and g is
an isomorphism for n > 0, and isomorphic to f for n = 0.
(D2) Let f be the map K [ → K ] , where K is a Kan complex, and let g
be the map (∆1 )[ → (∆1 )] . The smash product of f and g can be
identified with the inclusion(K × ∆1 , E) ⊆ (K × ∆1 )] ,where E denotes the class of all edges e = (e0 , e00 ) of K × ∆1 for which
either e0 : ∆1 → K or e00 : ∆1 → ∆1 is degenerate. This inclusion canbe obtained as a transfinite composition of pushouts of the mapa(Λ21 )](∆2 )[ → (∆2 )] .
(Λ21 )[

We now return to our main objective:
Proof of Proposition 3.1.2.1. Since p is a Cartesian fibration, it induces a
map X \ → S ] which has the right lifting property with respect to all marked
anodyne maps. By Proposition 3.1.2.3, the induced map[[(X \ )K → (S ] )K = (S K )]has the right lifting property with respect to all marked anodyne morphisms.
The desired result now follows from Remark 3.1.1.10.

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3.1.3 The Cartesian Model Structure
Let S be a simplicial set. Our goal in this section is to introduce the Cartesianmodel structure on the category (Set+∆ )/S of marked simplicial sets over S. We
will eventually show that the fibrant objects of (Set+∆ )/S correspond precisely
to Cartesian fibrations X → S and that they encode (contravariant) functors
from S into the ∞-category Cat∞ .
The category Set+
∆ is Cartesian-closed; that is, for any two objects X, Y ∈
Set+
,
there
existsaninternal mapping object Y X equipped with an “evalu∆
Xation map” Y × X → Y which induces bijections
HomSet+ (Z, Y X ) → HomSet+ (Z × X, Y )∆∆[for every Z ∈ Set+
∆ . We let Map (X, Y ) denote the underlying simplicial set][
Xof Y and Map (X, Y ) ⊆ Map (X, Y ) the simplicial subset consisting of
all simplices σ ⊆ Map[ (X, Y ) such that every edge of σ is a marked edgeof Y X . Equivalently, we may describe these simplicial sets by the mapping
properties
HomSet∆ (K, Map[ (X, Y )) ' HomSet+ (K [ × X, Y )∆HomSet∆ (K, Map] (X, Y )) ' HomSet+ (K ] × X, Y ).
∆]]If X and Y are objects of (Set+∆ )/S , then we let MapS (X, Y ) ⊆ Map (X, Y )and Map[S (X, Y ) ⊆ Map[ (X, Y ) denote the simplicial subsets classifying
those maps which are compatible with the projections to S.

Remark 3.1.3.1. If X ∈ (Set+∆ )/S and p : Y → S is a Cartesian fibration,then Map[S (X, Y \ ) is an ∞-category and Map]S (X, Y \ ) is the largest Kancomplex contained in Map[S (X, Y \ ).
Lemma 3.1.3.2. Let f : C → D be a functor between ∞-categories. The
following are equivalent:(1) The functor f is a categorical equivalence.
(2) For every simplicial set K, the induced map Fun(K, C) → Fun(K, D)
is a categorical equivalence.
(3) For every simplicial set K, the functor Fun(K, C) → Fun(K, D) induces a homotopy equivalence from the largest Kan complex containedin Fun(K, C) to the largest Kan complex contained in Fun(K, D).
Proof. The implications (1) ⇒ (2) ⇒ (3) are obvious. Suppose that (3) issatisfied. Let K = D. According to (3), there exists an object x of Fun(K, C)
whose image in Fun(K, D) is equivalent to the identity map K → D. We
may identify x with a functor g : D → C having the property that f ◦ g ishomotopic to the identity idD . It follows that g also has the property assertedhighertopoi

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155

by (3), so the same argument shows that there is a functor f 0 : C → D suchthat g ◦ f 0 is homotopic to idC . It follows that f ◦ g ◦ f 0 is homotopic to bothf and f 0 , so that f is homotopic to f 0 . Thus g is a homotopy inverse to f ,which proves that f is an equivalence.
Proposition 3.1.3.3. Let S be a simplicial set and let p : X → Y be amorphism in (Set+∆ )/S . The following are equivalent:(1) For every Cartesian fibration Z → S, the induced map
Map[S (Y, Z \ ) → Map[S (X, Z \ )is an equivalence of ∞-categories.
(2) For every Cartesian fibration Z → S, the induced map
Map]S (Y, Z \ ) → Map]S (X, Z \ )is a homotopy equivalence of Kan complexes.
Proof. Since Map]S (M, Z \ ) is the largest Kan complex contained in the ∞category Map[S (M, Z \ ), it is clear that (1) implies (2). Suppose that (2) issatisfied and let Z → S be a Cartesian fibration. We wish to show that
Map[S (Y, Z \ ) → Map[S (X, Z \ )is an equivalence of ∞-categories. According to Lemma 3.1.3.2, it suffices toshow that
Map[S (Y, Z \ )K → Map[S (X, Z \ )K
induces a homotopy equivalence on the maximal Kan complexes containedin each side, for every simplicial set K. Let Z(K) = Z K ×S K S. Proposition
3.1.2.1 implies that Z(K) → S is a Cartesian fibration and that there is anatural identification
Map[S (M, Z(K)\ ) ' Map[S (M, Z \ )K .
We observe that the largest Kan complex contained in the right hand sideis Map]S (M, Z(K)\ ). On the other hand, the natural map
Map]S (Y, Z(K)\ ) → Map]S (X, Z(K)\ )is a homotopy equivalence by assumption (2).
We will say that a map X → Y in (Set+∆ )/S is a Cartesian equivalence if
it satisfies the equivalent conditions of Proposition 3.1.3.3.
Remark 3.1.3.4. Let f : X → Y be a morphism in (Set+∆ )/S which ismarked anodyne when regarded as a map of marked simplicial sets. Since
the smash product of f with any inclusion A[ ⊆ B [ is also marked anodyne,
we deduce that the map
φ : Map[S (Y, Z \ ) → Map[S (X, Z \ )is a trivial fibration for every Cartesian fibration Z → S. Consequently, f is
a Cartesian equivalence.

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Let S be a simplicial set and let X, Y ∈ (Set+∆ )/S . We will say that a pair ofmorphisms f, g : X → Y are strongly homotopic if there exists a contractible
Kan complex K and a map K → Map[S (X, Y ) whose image contains bothof the vertices f and g. If Y = Z \ , where Z → S is a Cartesian fibration,then this simply means that f and g are equivalent when viewed as objectsof the ∞-category Map[S (X, Y ).
pq

Proposition 3.1.3.5. Let X → Y → S be a diagram of simplicial sets,
where both q and q ◦ p are Cartesian fibrations. The following assertions areequivalent:(1) The map p induces a Cartesian equivalence X \ → Y \ in (Set+∆ )/S .
(2) There exists a map r : Y → X which is a strong homotopy inverse top, in the sense that p ◦ r and r ◦ p are both strongly homotopic to theidentity.
(3) The map p induces a categorical equivalence Xs → Ys for each vertex
s of S.
Proof. The equivalence between (1) and (2) is easy, as is the assertion that(2) implies (3). It therefore suffices to show that (3) implies (2). We will
construct r and a homotopy from r ◦ p to the identity. It then follows thatthe map r satisfies (3), so the same argument will show that r has a righthomotopy inverse; by general nonsense this right homotopy inverse will automatically be homotopic to p, and the proof will be complete.
Choose a transfinite S
sequence of simplicial subsets S(α) ⊆ S, where eachS(α) is obtained from β<α S(β) by adjoining a single nondegenerate simplex (if such a simplex exists). We construct rα : Y ×S S(α) → X and an
equivalence hα : (X ×S S(α)) × ∆1 → X ×S S(α) from rα ◦ p to the identity
by induction on α. By this device we may reduce to the case where S = ∆n ,and the maps
r0 : Y 0 → X
h0 : X 0 × ∆1 → X
are already specified, where Y 0 = Y ×∆n ∂ ∆n ⊆ Y and X 0 = X ×∆n ∂ ∆n ⊆X. We may regard r0 and h0 together as defining a map ψ0 : Z 0 → X, whereaaZ0 = Y 0(X 0 × ∆1 )X.
X 0 ×{0}X 0 ×{1}Let Z = Y X×{0} X × ∆1 ; then our goal is to solve the lifting problem
depicted in the diagram`

Z 0 _

ψ0

/X
=

ψ


Z


/ ∆n

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157

in such a way that ψ carries {x}×∆1 to an equivalence in X for every object
x of X. We note that this last condition is vacuous for n > 0.
If n = 0, the problem amounts to constructing a map Y → X which ishomotopy inverse to p: this is possible in view of the assumption that p isa categorical equivalence. For n > 0, we note that any map φ : Z → X
extending φ0 is automatically compatible with the projection to S (since S
is a simplex and Z 0 contains all vertices of Z). Since the inclusion Z 0 ⊆ Z isa cofibration between cofibrant objects in the model category Set∆ (with the
Joyal model structure) and X is a ∞-category (since q is an inner fibrationand ∆n is an ∞-category), Proposition A.2.3.1 asserts that it is sufficient
to show that the extension φ exists up to homotopy. Since Corollary 2.4.4.4
implies that p is an equivalence, we are free to replace the inclusion Z 0 ⊆ Zwith the weakly equivalent inclusiona(X × {1})(X ×∆n ∂ ∆n × {1}) ⊆ X × ∆1 .
X×∆n ∂ ∆n ×∆1

Since φ0 carries {x} × ∆1 to a (q ◦ p)-Cartesian edge of X, for every vertex
x of X, the existence of φ follows from Proposition 3.1.1.5.
Lemma 3.1.3.6. Let S be a simplicial set, let i : X → Y be a cofibration in(Set+∆ )/S , and let Z → S be a Cartesian fibration. Then the associated map
p : Map]S (Y, Z \ ) → Map]S (X, Z \ ) is a Kan fibration.
Proof. We first show that p is a right fibration. Let A ⊆ B be a rightanodyne inclusion of simplicial sets. We must show that p has the right
lifting property with respect to to the inclusion A ⊆ B. Equivalently, wemust show that Z \ → S has the right lifting property with respect to theinclusiona(B ] × X)(A] × Y ) ⊆ B ] × Y.
A] ×X

This follows from Proposition 3.1.2.3 since the inclusion A] ⊆ B ] is marked
anodyne.
Applying the above result to inclusion ∅ ,→ X, we deduce that the map
Map]S (X, Z \ ) → ∆0 is a right fibration of simplicial sets. Using Lemma
2.1.3.3, we see that Map]S (X, Z \ ) is a Kan complex. Applying Lemma 2.1.3.3
again, we conclude that p is a Kan fibration.
Proposition 3.1.3.7. Let S be a simplicial set. There exists a left proper
combinatorial model structure on (Set+∆ )/S which may be described as follows:(C) The cofibrations in (Set+∆ )/S are those morphisms p : X → Y in
(Set+
)
which
are
cofibrations
when regarded as morphisms of sim∆ /Splicial sets.
(W ) The weak equivalences in (Set+∆ )/S are the Cartesian equivalences.

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CHAPTER 3(F ) The fibrations in (Set+∆ )/S are those maps which have the right lifting
property with respect to every map which is simultaneously a cofibration
and a Cartesian equivalence.
Proof. It suffices to show that the hypotheses of Proposition A.2.6.15 aresatisfied by the class (C) of cofibrations and the class (W ).
(1) We first show that class (W ) of Cartesian equivalences is perfect (in thesense of Definition A.2.6.12). Let A denote the quotient of ∆3 which
corepresents the functor
HomSet∆ (A, X) = {σ ∈ X3 , e ∈ X1 : d1 σ = s0 e, d2 σ = s1 e},and let f denote the image of ∆{0,1} in A. More informally, a map
from A to an ∞-category C is a commutative diagram
f

?y

7/ x

id

id

f

'/ 

f

x

y

in C. Let A denote the marked simplicial set (A, s0 A0 ∪ {f }). For eachvertex s ∈ S, let A[s and As denote the objects of (Set+∆ )/S given by
the constant maps A[ → S ] ← A taking the value s ∈ S. Note that
if Z → S is a Cartesian fibration, then for any vertex s ∈ S, a map
A → Zs automatically carries the edge f to an equivalence in Zs . It
follows that the restriction map Map]S (As , Z) → Map]S (A[s , Z) is an
isomorphism of simplicial sets.
Using Proposition A.1.2.5, we deduce that there exists a functor Tfrom (Set+∆ )/S to itself and a (functorial) factorization
i

jX

X
X→
T (X) → S ] ,where iX is a transfinite composition of pushouts of morphisms oftypes (1), (2), and (3) of Definition 3.1.1.1 together with inclusions of
the form A[s ,→ As for s ∈ S, and jX has the right lifting property
with respect to all such morphisms. Moreover, the functor T can bechosen to commute with filtered colimits. In what follows, we will abusenotation by identifying T (X) with its underlying simplicial set, and jX
with a map of simplicial sets. Arguing as in the proof of Proposition
3.1.1.6, we deduce that jX is an inner fibration, that every marked edgeof T (X) is jX -Cartesian, and that for every vertex x of T (X) and everyedge e : s → jX (x) in S, there exists a marked edge e : s → x in T (X)lifting e. It follows in particular that jX is a Cartesian fibration. We
claim that the marked edges of T (X) are precisely the jX -Cartesianedges. To prove this, suppose we are given an arbitrary p-Cartesianedge e : y → x in T (X), and choose a marked edge e0 : y 0 → x of T (X)highertopoi

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THE ∞-CATEGORY OF ∞-CATEGORIES

lifting jX (e). Since e0 is jX -Cartesian, we can find a 2-simplex
@y

f0

0
e0

/ x

e

y

where jX (f0 ) is the degenerate edge at some vertex s ∈ S. To provethat e is marked, it will suffice to show that f0 is marked. Since T (X)
has the extension property with respect to the inclusion A[s ,→ As , itwill suffice to prove that f0 : ∆{0,1} → T (X)s can be extended to amap of simplicial sets u : A → T (X)s . By construction the simplicial
set A is weakly contractible, so that the composite mapι : ∆{0,1} → ∆3 → A
is a weak homotopy equivalence. Since e0 and e are both jX -Cartesian,the map f0 is an equivalence in T (X)s ; the desired result now follows
from the fact that the largest Kan complex contained in T (X)s has
the extension property with respect to ι.
Note that for each X ∈ (Set+∆ )/S , the map iX is a Cartesian equivalence. It follows from Proposition 3.1.3.5 that a map X → Y in
(Set+∆ )/S is a Cartesian equivalence if and only if, for each vertexs ∈ S, the induced map T (X)s → T (Y )s is a categorical equivalence.
Applying Corollary A.2.6.14, we deduce that (W ) is a perfect class ofmorphisms.
(2) The class of weak equivalences is stable under pushouts by cofibrations.
Suppose we are given a pushout diagram
X

p

/Y

i



p0
/ Y0
X0
where i is a cofibration and p is a Cartesian equivalence. We wish toshow that p0 is also a Cartesian equivalence. In other words, we mustshow that for any Cartesian fibration Z → S, the associated map
Map]S (Y 0 , Z \ ) → Map]S (X 0 , Z \ ) is a homotopy equivalence. Consider
the pullback diagram
Map]S (Y 0 , Z \ )/ Map] (X 0 , Z \ )Map]S (Y, Z \ )/ Map] (X, Z \ ).

S

SSince p is a Cartesian equivalence, the bottom horizontal arrow is a
homotopy equivalence. According to Lemma 3.1.3.6, the right vertical arrow is a Kan fibration; it follows that the diagram is homotopy
Cartesian, so that the top horizontal arrow is an equivalence as well.

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CHAPTER 3(3) A map p : X → Y in (Set+∆ )/S which has the right lifting property withrespect to every map in (C) belongs to (W ). Unwinding the definition,
we see that p is a trivial fibration of simplicial sets and that an edge
e of X is marked if and only if p(e) is a marked edge of Y . It follows
that p has a section s with s ◦ p fiberwise homotopic to idX . From this,we deduce easily that p is a Cartesian equivalence.

Warning 3.1.3.8. Let S be a simplicial set. We must be careful to distinguish between Cartesian fibrations of simplicial sets (in the sense of Definition 2.4.2.1) and fibrations with respect to the Cartesian model structureon (Set+∆ )/S (in the sense of Proposition 3.1.3.7). Though distinct, thesenotions are closely related: for example, the fibrant objects of (Set+∆ )/S are\precisely those objects of the form X , where X → S is a Cartesian fibration(Proposition 3.1.4.1).
Remark 3.1.3.9. The definition of the Cartesian model structure on the
category (Set+∆ )/S is not self-opposite. Consequently, we can define anothermodel structure on (Set+∆ )/S as follows:(C) The cofibrations in (Set+∆ )/S are precisely the monomorphisms.
(W ) The weak equivalences in (Set+∆ )/S are precisely the coCartesian equivalences: that is, those morphisms f : X → Y such that the inducedop
op
map f op : X → Y is a Cartesian equivalence in (Set+∆ )/S op .
(F ) The fibrations in (Set+∆ )/S are those morphisms which have the right
lifting property with respect to every morphism satisfying both (C)and (W ).
We will refer to this model structure on (Set+∆ )/S as the coCartesian model
structure.
3.1.4 Properties of the Cartesian Model Structure
In this section, we will establish some of the basic properties of Cartesianmodel structures on (Set+∆ )/S which was introduced in §3.1.3. In particular,
we will show that each (Set+∆ )/S is a simplicial model category and characterize its fibrant objects.
Proposition 3.1.4.1. An object X ∈ (Set+∆ )/S is fibrant (with respect to
the Cartesian model structure) if and only if X ' Y \ , where Y → S is a
Cartesian fibration.
Proof. Suppose first that X is fibrant. The small object argument implies
that there exists a marked anodyne map j : X → Z \ for some Cartesianfibration Z → S. Since j is marked anodyne, it is a Cartesian equivalence.

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Since X is fibrant, it has the extension property with respect to the trivial
cofibration j; thus X is a retract of Z \ . It follows that X is isomorphic to
Y \ , where Y is a retract of Z.
Now suppose that Y → S is a Cartesian fibration; we claim that Y \ hasthe right lifting property with respect to any trivial cofibration j : A → B]\
in (Set+∆ )/S . Since j is a Cartesian equivalence, the map η : MapS (B, Y ) →]MapS (A, Y \ ) is a homotopy equivalence of Kan complexes. Hence, for any
map f : A → Z \ , there is a map g : B → Z \ such that g|A and f are joined`by an edge e of Map]S (A, Z \ ). Let M = (A × (∆1 )] ) A×{1}] (B × {1}] ) ⊆B × (∆1 )] . We observe that e and g together determine a map M → Z \ .
Consider the diagram
/ Z\
:

M
F


B × (∆1 )]


/ S].

The left vertical arrow is marked anodyne by Proposition 3.1.2.3. Consequently, there exists a dotted arrow F as indicated. We note that F |B × {0}is an extension of f to B, as desired.
We now study the behavior of the Cartesian model structures with respect
to products.
Proposition 3.1.4.2. Let S and T be simplicial sets and let Z be an objectof (Set+∆ )/T . Then the functor+(Set+∆ )/S → (Set∆ )/S×T

X 7→ X × Z
preserves Cartesian equivalences.
Proof. Let f : X → Y be a Cartesian equivalence in (Set+∆ )/S . We wish to0
show that f × idZ is a Cartesian equivalence in (Set+)∆ /S×T . Let X → X+
0be a marked anodyne map, where X ∈ (Set∆ )/S is fibrant. Now choose a`marked anodyne map X 0 X Y → Y 0 , where Y 0 ∈ (Set+∆ )/S is fibrant. Since
the product maps X × Z → X 0 × Z and Y × Z → Y 0 × Z are also marked
anodyne (by Proposition 3.1.2.3), it suffices to show that X 0 × Z → Y 0 × Z
is a Cartesian equivalence. In other words, we may reduce to the situation
where X and Y are fibrant. By Proposition 3.1.3.5, f has a homotopy inverseg; then g × idZ is a homotopy inverse to f × idZ .
0Corollary 3.1.4.3. Let f : A → B be a cofibration in (Set+∆ )/S and f :+00
A → B a cofibration in (Set∆ )/T . Then the smash product mapa(A × B 0 )(B × A0 ) → A0 × B 0
A×B 0

is a cofibration in (Set+∆ )/S×T , which is trivial if either f or g is trivial.

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Corollary 3.1.4.4. Let S be a simplicial set and regard (Set+∆ )/S as a sim]
plicial category with mapping objects given by MapS (X, Y ). Then (Set+∆ )/Sis a simplicial model category.
Proof. Unwinding the definitions, we are reduced to proving the following:0
given a cofibration i : X → X 0 in (Set+∆ )/S and a cofibration j : Y → Y in
Set∆ , the induced cofibrationa]](X 0 × Y ] )(X × Y 0 ) ⊆ X 0 × Y 0
X×Y ]in (Set+∆ )/S is trivial if either i is a Cartesian equivalence or j is a weakhomotopy equivalence. If i is trivial, this follows immediately from Corollary
3.1.4.3. If j is trivial, the same argument applies provided that we can verify]
that Y ] → Y 0 is a Cartesian equivalence in Set+
∆ . Unwinding the definitions,
we must show that for every ∞-category Z, the restriction map]

θ : Map] (Y 0 , Z \ ) → Map] (Y ] , Z \ )is a homotopy equivalence of Kan complexes. Let K be the largest Kancomplex contained in Z, so that θ can be identified with the restriction map
MapSet∆ (Y 0 , K) → MapSet∆ (Y, K).
Since j is a weak homotopy equivalence, this map is a trivial fibration.
Remark 3.1.4.5. There is a second simplicial structure on (Set+∆ )/S , where[
the simplicial mapping spaces are given by MapS (X, Y ). This simplicial
structure is not compatible with the Cartesian model structure: for fixedX ∈ (Set+∆ )/S the functor
A 7→ A[ × X
does not carry weak homotopy equivalences (in the A-variable) to Cartesianequivalences. It does, however, carry categorical equivalences (in A) to Cartesian equivalences, and consequently (Set+∆ )/S is endowed with the structureof a Set∆ -enriched model category, where we regard Set∆ as equipped with
the Joyal model structure. This second simplicial structure reflects the factthat (Set+∆ )/S is really a model for an ∞-bicategory.
Remark 3.1.4.6. Suppose S is a Kan complex. A map p : X → S is a
Cartesian fibration if and only if it is a coCartesian fibration (this follows
in general from Proposition 3.3.1.8; if S = ∆0 , the main case of interest forus, it is obvious). Moreover, the class of p-coCartesian edges of X coincides
with the class of p-Cartesian edges of X: both may be described as the classof equivalences in X. Consequently, if A ∈ (Set+∆ )/S , thenMap[S (A, X \ ) ' Map[S op (Aop , (X op )\ )op ,where Aop is regarded as a marked simplicial set in the obvious way. It
follows that a map A → B is a Cartesian equivalence in (Set+∆ )/S if and only

highertopoi

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if Aop → B op is a Cartesian equivalence in (Set+∆ )/S op . In other words, theCartesian model structure on (Set+)
isself-dual
when S is a Kan complex.
∆ /S
In particular, if S = ∆0 , we deduce that the functor
A 7→ Aop+determines an autoequivalence of the model category Set+
∆ ' (Set∆ )/∆0 .

3.1.5 Comparison of Model Categories
Let S be a simplicial set. We now have a plethora of model structures on
categories of simplicial sets over S:(0) Let C0 denote the category (Set∆ )/S of simplicial sets over S endowedwith the Joyal model structure defined in §2.2.5: the cofibrations aremonomorphisms of simplicial sets, and the weak equivalences are categorical equivalences.
(1) Let C1 denote the category (Set+∆ )/S of marked simplicial sets overS endowed with the marked model structure of Proposition 3.1.3.7:
the cofibrations are maps (X, EX ) → (Y, EY ) which induce monomorphisms X → Y , and the weak equivalences are the Cartesian equivalences.
(2) Let C2 denote the category (Set+∆ )/S of marked simplicial sets over S
endowed with the following localization of the Cartesian model structure: a map f : (X, EX ) → (Y, EY ) is a cofibration if the underlying map X → Y is a monomorphism, and a weak equivalence if
f : X ] → Y ] is a marked equivalence in (Set+∆ )/S .
(3) Let C3 denote the category (Set∆ )/S of simplicial sets over S, whichis endowed with the contravariant model structure described in §2.1.4:
the cofibrations are the monomorphisms, and the weak equivalences
are the contravariant equivalences.
(4) Let C4 denote the category (Set∆ )/S of simplicial sets over S endowedwith the usual homotopy-theoretic model structure: the cofibrations
are the monomorphisms of simplicial sets, and the weak equivalences
are the weak homotopy equivalences of simplicial sets.
The goal of this section is to study the relationship between these fivemodel categories. We may summarize the situation as follows:
Theorem 3.1.5.1. There exists a sequence of Quillen adjunctions
F

F

F

F

G

G

G

G

C0 →0 C1 →1 C2 →2 C3 →3 C4
C0 ←0 C1 ←1 C2 ←2 C3 ←3 C4 ,which may be described as follows:highertopoi

164

April 9, 2017

CHAPTER 3(A0) The functor G0 is the forgetful functor from (Set+∆ )/S to (Set∆ )/S ,which ignores the collection of marked edges. The functor F0 is theleft adjoint to G0 , which is given by X 7→ X [ . The Quillen adjunction(F0 , G0 ) is a Quillen equivalence if S is a Kan complex.
(A1) The functors F1 and G1 are the identity functors on (Set+∆ )/S .
(A2) The functor F2 is the forgetful functor from (Set+∆ )/S to (Set∆ )/S which
ignores the collection of marked edges. The functor G2 is the rightadjoint to F2 , which is given by X 7→ X ] . The Quillen adjunction(F2 , G2 ) is a Quillen equivalence for every simplicial set S.
(A3) The functors F3 and G3 are the identity functors on (Set∆ )/S . The
Quillen adjunction (F3 , G3 ) is a Quillen equivalence whenever S is a
Kan complex.
The rest of this section is devoted to giving a proof of Theorem 3.1.5.1. We
will organize our efforts as follows. First, we verify that the model category
C2 is well-defined (the analogous results for the other model structures havealready been established). We then consider each of the adjunctions (Fi , Gi )in turn and show that it has the desired properties.
Proposition 3.1.5.2. Let S be a simplicial set. There exists a left proper
combinatorial model structure on the category (Set+∆ )/S which may be described as follows:(C) A map f : (X, EX ) → (Y, EY ) is a cofibration if and only if the underlying map X → Y is a monomorphism of simplicial sets.
(W ) A map f : (X, EX ) → (Y, EY ) is a weak equivalence if and only if theinduced map X ] → Y ] is a Cartesian equivalence in (Set+∆ )/S .
(F ) A map f : (X, EX ) → (Y, EY ) is a fibration if and only if it has theright lifting property with respect to all trivial cofibrations.
Proof. It suffices to show that the conditions of Proposition A.2.6.15 aresatisfied. We check them in turn:(1) The class (W ) of Cartesian equivalences is perfect (in the sense of
Definition A.2.6.12). This follows from Corollary A.2.6.14, since theclass of Cartesian equivalences is perfect and the functor (X, EX ) → X ]commutes with filtered colimits.
(2) The class of weak equivalences is stable under pushouts by cofibrations.
This follows from the analogous property of the Cartesian model structure because the functor (X, EX ) 7→ X ] preserves pushouts.
(3) A map p : (X, EX ) → (Y, EY ) which has the right lifting property withrespect to every cofibration is a weak equivalence. In this case, theunderlying map of simplicial sets is a trivial fibration, so the induced

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map X ] → Y ] has the right lifting property with respect to all trivial
cofibrations and is a Cartesian equivalence (as observed in the proofof Proposition 3.1.3.7).

Proposition 3.1.5.3. Let S be a simplicial set. Consider the adjoint functors(Set∆ )/S o

F0/ (Set+ )

∆ /S

G0

described by the formulas
F0 (X) = X [G0 (X, E) = X.
The adjoint functors (F0 , G0 ) determine a Quillen adjunction between the
category (Set∆ )/S (with the Joyal model structure) and the category (Set+∆ )/S(with the Cartesian model structure). If S is a Kan complex, then (F0 , G0 )is a Quillen equivalence.
Proof. To prove that (F0 , G0 ) is a Quillen adjunction, it will suffice to showthat F0 preserves cofibrations and trivial cofibrations. The first claim is obvious. For the second, we must show that if X ⊆ Y is a categorical equivalence of simplicial sets over S, then the induced map X [ → Y [ is a Cartesianequivalence in (Set+∆ )/S . For this, it suffices to show that for any Cartesianfibration p : Z → S, the restriction map
Map[S (Y [ , Z \ ) → Map[S (X [ , Z \ )is a trivial fibration of simplicial sets. In other words, we must show thatfor every inclusion A ⊆ B of simplicial sets, it is possible to solve any lifting
problem of the form
/ Map[S (Y [ , Z \ )
9

A _


/ Map[S (X [ , Z \ ).
`Replacing Y by Y × B and X by (X × B) X×A (Y × A), we may supposethat A = ∅ and B = ∗. Moreover, we may rephrase the lifting problem as the
problem of constructing the dotted arrow indicated in the following diagram:
/Z
X _
>

B


Y


/S

p

By Proposition 3.3.1.7, p is a categorical fibration, and the lifting problem
has a solution by virtue of the assumption that X ⊆ Y is a categoricalequivalence.

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Now suppose that S is a Kan complex. We want to prove that (F0 , G0 )is a Quillen equivalence. In other words, we must show that for any fibrantobject of (Set+∆ )/S corresponding to a Cartesian fibration Z → S, a map
X → Z in (Set∆ )/S is a categorical equivalence if and only if the associatedmap X [ → Z \ is a Cartesian equivalence.
Suppose first that X → Z is a categorical equivalence. Then the induced
map X [ → Z [ is a Cartesian equivalence by the argument given above. It
therefore suffices to show that Z [ → Z \ is a Cartesian equivalence. Since S
is a Kan complex, Z is an ∞-category; let K denote the largest Kan complex
contained in Z. The marked edges of Z \ are precisely the edges which belong
to K, so we have a pushout diagram
K[

/ K]


Z[


/ Z \.

It follows that Z [ → Z \ is marked anodyne and therefore a Cartesian equivalence.
Now suppose that X [ → Z \ is a Cartesian equivalence. Choose a factorizaf
gtion X → Y → Z, where f is a categorical equivalence and g is a categoricalfibration. We wish to show that g is a categorical equivalence. Proposition
3.3.1.8 implies that Z → S is a categorical fibration, so that Y → S isa categorical fibration. Applying Proposition 3.3.1.8 again, we deduce that
Y → S is a Cartesian fibration. Thus we have a factorization
X [ → Y [ → Y \ → Z \,where the first two maps are Cartesian equivalences by the arguments given
above and the composite map is a Cartesian equivalence. Thus Y \ → Z \ isan equivalence between fibrant objects of (Set+∆ )/S and therefore admits ahomotopy inverse. The existence of this homotopy inverse proves that g is a
categorical equivalence, as desired.
Proposition 3.1.5.4. Let S be a simplicial set and let F1 = G1 be theidentity functor from (Set+∆ )/S to itself. Then (F1 , G1 ) determines a Quillen
adjunction between C1 and C2 .
Proof. We must show that F1 preserves cofibrations and trivial cofibrations.
+
The first claim is obvious. For the second, let B : (Set+∆ )/S → (Set∆ )/S be
the functor defined by
B(M, EM ) = M ] .
We wish to show that if X → Y is a Cartesian equivalence in (Set+∆ )/S , then
B(X) → B(Y ) is a Cartesian equivalence.
We first observe that if X → Y is marked anodyne, then the induced map
B(X) → B(Y ) is also marked anodyne: by general nonsense, it suffices tocheck this for the generators described in Definition 3.1.1.1, for which it ishighertopoi

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obvious. Now return to the case of a general Cartesian equivalence p : X → Y
and choose a diagram
/ X0

i

X

q

p


Y

/ X0

`

XY

j

$

/ Y0

in which X 0 and Y 0 are (marked) fibrant and i and j are marked anodyne.
It follows that B(i) and B(j) are marked anodyne and therefore Cartesianequivalences. Thus, to prove that B(p) is a Cartesian equivalence, it suffices
to show that B(q) is a Cartesian equivalence. But q is a Cartesian equivalence
between fibrant objects of (Set+∆ )/S and therefore has a homotopy inverse.
It follows that B(q) also has a homotopy inverse and is therefore a Cartesianequivalence, as desired.
Remark 3.1.5.5. In the language of model categories, we may summarize
Proposition 3.1.5.4 by saying that the model structure of Proposition 3.1.5.2
is a localization of the Cartesian model structure on (Set+∆ )/S .
Proposition 3.1.5.6. Let S be a simplicial set and consider the adjunction
o
(Set+
∆ )/S

F2
G2

/ (Set )∆ /Sdetermined by the formulas
F2 (X, E) = X
G2 (X) = X ] .
The adjoint functors (F2 , G2 ) determine a Quillen equivalence between C2and C3 .
Proof. We first claim that F2 is conservative: that is, a map f : (X, EX ) →(Y, EY ) is a weak equivalence in C2 if and only if the induced map X → Y isa weak equivalence in C3 . Unwinding the definition, f is a weak equivalence
if and only if X ] → Y ] is a Cartesian equivalence. This holds if and only if,for every Cartesian fibration Z → S, the induced map
φ : Map]S (Y ] , Z \ ) → Map]S (X ] , Z \ )is a homotopy equivalence. Let Z 0 → S be the right fibration associated to
Z → S (see Corollary 2.4.2.5). We have natural identifications
Map]S (Y ] , Z \ ) ' MapS (Y, Z 0 )Map]S (X ] , Z \ ) ' MapS (X, Z 0 ).

Consequently, f is a weak equivalence if and only if, for every right fibration
Z 0 → S, the associated map
MapS (Y, Z 0 ) → MapS (X, Z 0 )highertopoi

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CHAPTER 3

is a homotopy equivalence. Since C3 is a simplicial model category for which
the fibrant objects are precisely the right fibrations Z 0 → S (Corollary
2.2.3.12), this is equivalent to the assertion that X → Y is a weak equivalencein C3 .
To prove that (F2 , G2 ) is a Quillen adjunction, it suffices to show that F2
preserves cofibrations and trivial cofibrations. The first claim is obvious, andthe second follows because F2 preserves all weak equivalences (by the above
argument).
To show that (F2 , G2 ) is a Quillen equivalence, we must show that theunit and counit
LF2 ◦ RG2 → id
id → RG2 ◦ LF2
are weak equivalences. In view of the fact that F2 = LF2 is conservative, thesecond assertion follows from the first. To prove the first, it suffices to showthat if X is a fibrant object of C3 , then the counit map (F2 ◦ G2 )(X) → X
is a weak equivalence. But this map is an isomorphism.
Proposition 3.1.5.7. Let S be a simplicial set and let F3 = G3 be theidentity functor from (Set∆ )/S to itself. Then (F3 , G3 ) gives a Quillen adjunction between C3 and C4 . If S is a Kan complex, then (F3 , G3 ) is a Quillen
equivalence (in other words, the model structures on C3 and C4 coincide).
Proof. To prove that (F3 , G3 ) is a Quillen adjunction, it suffices to prove that
F3 preserves cofibrations and weak equivalences. The first claim is obvious(the cofibrations in C3 and C4 are the same). For the second, we note thatboth C3 and C4 are simplicial model categories in which every object iscofibrant. Consequently, a map f : X → Y is a weak equivalence if and only
if, for every fibrant object Z, the associated map Map(Y, Z) → Map(X, Z) isa homotopy equivalence of Kan complexes. Thus, to show that F3 preserves
weak equivalences, it suffices to show that G3 preserves fibrant objects. A
map p : Z → S is fibrant as an object of C4 if and only if p is a Kan
fibration, and fibrant as an object of C3 if and only if p is a right fibration(Corollary 2.2.3.12). Since every Kan fibration is a right fibration, it follows
that F3 preserves weak equivalences. If S is a Kan complex, then the converse
holds: according to Lemma 2.1.3.4, every right fibration p : Z → S is a Kan
fibration. It follows that G3 preserves weak equivalences as well, so that thetwo model structures under consideration coincide.

3.2 STRAIGHTENING AND UNSTRAIGHTENING
Let C be a category and let χ : Cop → Cat be a functor from C to thecategory Cat of small categories. To this data, we can associate (by means ofe whichthe Grothendieck construction discussed in §2.1.1) a new category C
may be described as follows:highertopoi

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e are pairs (C, η), where C ∈ C and η ∈ χ(C).
• The objects of Ce a morphism from (C, η) to• Given a pair of objects (C, η), (C 0 , η 0 ) ∈ C,
0 0e
(C , η ) in C is a pair (f, α), where f : C → C 0 is a morphism in the
category C and α : η → χ(f )(η 0 ) is a morphism in the category χ(C).
• Composition is defined in the obvious way.
This construction establishes an equivalence between Cat-valued functors on
Cop and categories which are fibered over C. (To formulate the equivalence
precisely, it is best to view Cat as a bicategory, but we will not dwell on thistechnical point here.)
The goal of this section is to establish an ∞-categorical version of theequivalence described above. We will replace the category C by a simplicial
set S, the category Cat by the ∞-category Cat∞ , and the notion of fiberedcategory with the notion of Cartesian fibration. In this setting, we will obtain
an equivalence of ∞-categories, which arises from a Quillen equivalence ofsimplicial model categories. On one side, we have the category (Set+∆ )/S ,
equipped with the Cartesian model structure (a simplicial model category
whose fibrant objects are precisely the Cartesian fibrations X → S; see§3.1.4). On the other, we have the category of simplicial functors
C[S]op → Set+∆
equipped with the projective model structure (see §A.3.3) whose underlying ∞-category is equivalent to Fun(S op , Cat∞ ) (Proposition 4.2.4.4). The
situation may be summarized as follows:
Theorem 3.2.0.1. Let S be a simplicial set, C a simplicial category, andφ : C[S] → Cop a functor between simplicial categories. Then there exists apair of adjoint functors
o
(Set+
∆ )/S

St+
φ
Un+
φ

/ (Set+ )C∆

with the following properties:+(1) The functors (St+φ , Unφ ) determine a Quillen adjunction between the+
category (Set∆ )/S (with the Cartesian model structure) and the cateCgory (Set+∆ ) (with the projective model structure).
+(2) If φ is an equivalence of simplicial categories, then (St+φ , Unφ ) is a
Quillen equivalence.
+
We will refer to St+
φ and Unφ as the straightening and unstraightening
functors, respectively. We will construct these functors in §3.2.1 and establish part (1) of Theorem 3.2.0.1. Part (2) is more difficult and requires somepreliminary work; we will begin in §3.2.2 by analyzing the structure of Cartesian fibrations X → ∆n . We will apply these analyses in §3.2.3 to completehighertopoi

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CHAPTER 3

the proof of Theorem 3.2.0.1 when S is a simplex. In §3.2.4, we will deduce the general result by using formal arguments to reduce to the case ofa simplex.
In the case where C is an ordinary category, the straightening and unstraightening procedures of §3.2.1 can be substantially simplified. We will
discuss the situation in §3.2.5, where we provide an analogue of Theorem
3.2.0.1 (see Propositions 3.2.5.18 and 3.2.5.21).
3.2.1 The Straightening Functor
Let S be a simplicial set and let φ : C[S] → Cop be a functor between
simplicial categories, which we regard as fixed throughout this section. Our++ C
objective is to define the straightening functor St+
φ : (Set∆ )/S → (Set∆ ) and++
its right adjoint Unφ . The intuition is that an object X of (Set∆ )/S associates∞-categories to vertices of S in a homotopy coherent fashion, and the functorSt+
φ “straightens” this diagram to obtain an ∞-category valued functor on C.
The right adjoint Un+
φ should be viewed as a forgetful functor which takes astrictly commutative diagram and retains the underlying homotopy coherent
diagram.
+
The functors St+
φ and Unφ are more elaborate versions of the straightening
and unstraightening functors introduced in §2.2.1. We begin by recalling theunmarked version of the construction. For each object X ∈ (Set∆ )/S , forma pushout diagram of simplicial categories/ C[X . ]
C[X]


φ


/ Cop ,

Cop

X

where the left vertical map is given by composing φ with the map C[X] →C[S]. The functor Stφ X : C → Set∆ is defined by the formula(Stφ X)(C) = MapCop(C, ∗),
X
where ∗ denotes the cone point of X . .
We will define St+
φ by designating certain marked edges on the simplicial
sets (Stφ X)(C) which depend in a natural way on the marked edges of X.
In order to describe this dependence, we need to introduce a bit of notation.
Notation 3.2.1.1. Let X be an object of (Set∆ )/S . Given an n-simplex σof the simplicial set MapCop (C, D), we let σ ∗ : (Stφ X)(D)n → (Stφ X)(C)ndenote the associated map on n-simplices.
Let c be a vertex of X and let C = φ(c) ∈ C. We may identify c witha map c : ∆0 → X. Then c ? id∆0 : ∆1 → X . is an edge of X . and so
determines a morphism C → ∗ in Cop
X , which we can identify with a vertexec ∈ (Stφ X)(C).
Similarly, suppose that f : c → d is an edge of X corresponding to amorphism
F

C→Dhighertopoi

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THE ∞-CATEGORY OF ∞-CATEGORIES

in the simplicial category Cop . We may identify f with a map f : ∆1 → X.
Then f ? id∆0 : ∆2 → X . determines a map C[∆2 ] → CX , which we may
identify with a diagram (not strictly commutative)
/D

F

C
e
c



∗



de

together with an edge
fe : ec → de ◦ F = F ∗ dein the simplicial set MapCop(C, ∗) = (Stφ X)(C).
X
Definition 3.2.1.2. Let S be a simplicial set, C a simplicial category, andφ : C[S] → Cop a simplicial functor. Let (X, E) be an object of (Set+∆ )/S .
Then+St+
φ (X, E) : C → Set∆
is defined by the formulaSt+
φ (X, E)(C) = ((Stφ X)(C), Eφ (C)),
where Eφ (C) is the set of all edges of (Stφ X)(C) having the formG∗ fe,
where f : d → e is a marked edge of X, giving rise to an edge fe : de → F ∗ eein (Stφ X)(D), and G belongs to MapCop (C, D)1 .
Remark 3.2.1.3. The construction(X, E) 7→ St+
φ (X, E) = (Stφ X, Eφ )is obviously functorial in X. Note that we may characterize the subsets{Eφ (C) ⊆ (Stφ X)(C)1 } as the smallest collection of sets which contain fe forevery f ∈ E and depend functorially on C.
The following formal properties of the straightening functor follow immediately from the definition:
Proposition 3.2.1.4. (1) Let S be a simplicial set, C a simplicial category, and φ : C[S] → Cop a simplicial functor; then the associatedstraightening functor
+
+ C
St+
φ : (Set∆ )/S → (Set∆ )preserves colimits.
(2) Let p : S 0 → S be a map of simplicial sets, C a simplicial category, andφ : C[S] → Cop a simplicial functor, and let φ0 : C[S 0 ] → Cop denote the+composition φ ◦ C[p]. Let p! : (Set+∆ )/S 0 → (Set∆ )/S denote the forgetful
functor given by composition with p. There is a natural isomorphismof functors+St+φ ◦ p! ' Stφ0
+ C
from (Set+∆ )/S 0 to (Set∆ ) .

highertopoi

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CHAPTER 3(3) Let S be a simplicial set, π : C → C0 a simplicial functor between
simplicial categories, and φ : C[S] → Cop a simplicial functor. Then
there is a natural isomorphism of functors+St+π◦φ ' π! ◦ Stφ
0

0

+ C
+ C
+ C
from (Set+is the left∆ )/S to (Set∆ ) . Here π! : (Set∆ ) → (Set∆ )
0
+ C
+ C
∗
adjoint to the functor π : (Set∆ ) → (Set∆ ) given by composition
with π; see §A.3.3.

Corollary 3.2.1.5. Let S be a simplicial set, C a simplicial category, andφ : C[S] → Cop any simplicial functor. The straightening functor St+
φ has aright adjoint
+ C
+
Un+
φ : (Set∆ ) → (Set∆ )/S .

Proof. This follows from part (1) of Proposition 3.2.1.4 and the adjoint functor theorem. (Alternatively, one can construct Un+
φ directly; we leave thedetails to the reader.)Notation 3.2.1.6. Let S be a simplicial set, let C = C[S]op , and let φ :+C[S] → Cop be the identity map. In this case, we will denote St+
φ by StS and+Un+
φ by UnS .
Our next goal is to show that the straightening and unstraightening func+
tors (St+φ , Unφ ) give a Quillen adjunction between the model categories
+ C
+
(Set+
∆ )/S and (Set∆ ) . The first step is to show that Stφ preserves cofibrations.
Proposition 3.2.1.7. Let S be a simplicial set, C a simplicial category, andφ : C[S] → Cop a simplicial functor. The functor St+
φ carries cofibrations(with respect to the Cartesian model structure on (Set+∆ )/S ) to cofibrations+ C(with respect to the projective model structure on (Set∆ ) )).
Proof. Let j : A → B be a cofibration in (Set+∆ )/S ; we wish to show that
St+
(j)
is
a
cofibration.
Bygeneralnonsense,
we
may suppose that j is aφgenerating cofibration having either the form (∂ ∆n )[ ⊆ (∆n )[ or the form(∆1 )[ → (∆1 )] . Using Proposition 3.2.1.4, we may reduce to the case where
S = B, C = C[S] and φ is the identity map. The result now follows from astraightforward computation.
+
To complete the proof that (St+φ , Unφ ) is a Quillen adjunction, it suffices to+show that St+
φ preserves trivial cofibrations. Since every object of (Set∆ )/S iscofibrant, this is equivalent to the apparently stronger claim that if f : X →+Y is a Cartesian equivalence in (Set+∆ )/S , then Stφ (f ) is a weak equivalence+ C
in (Set∆ ) . The main step is to establish this in the case where f is marked
anodyne. First, we need a few lemmas.

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173

Lemma 3.2.1.8. Let E be the set of all degenerate edges of ∆n ×∆1 together
with the edge {n} × ∆1 . Let B ⊆ ∆n × ∆1 be the coproducta(∆n × {1})(∂ ∆n × ∆1 ).
∂ ∆n ×{1}Then the map
i : (B, E ∩B1 ) ⊆ (∆n × ∆1 , E)is marked anodyne.
Proof. We must show that i has the left lifting property with respect toevery map p : X → S satisfying the hypotheses of Proposition 3.1.1.6. Thisis simply a reformulation of Proposition 2.4.1.8.
Lemma 3.2.1.9. Let K be a simplicial set, K 0 ⊆ K a simplicial subset,
and A a set of vertices of K. Let E denote the set of all degenerate edgesof K`× ∆1 together with the edges {a} × ∆1 , where a ∈ A. Let B = (K 0 ×∆1 ) K 0 ×{1} (K × {1}) ⊆ K × ∆1 . Suppose that, for every nondegenerate
simplex σ of K, either σ belongs to K 0 or the final vertex of σ belongs to A.
Then the inclusion(B, E ∩B1 ) ⊆ (K × ∆1 , E)is marked anodyne.
Proof. Working simplex by simplex, we reduce to Lemma 3.2.1.8.
Lemma 3.2.1.10. Let X be a simplicial set, and let E ⊆ E0 be sets of edgesof X containing all degenerate edges. The following conditions are equivalent:(1) The inclusion (X, E) → (X, E0 ) is a trivial cofibration in Set+
∆ (withrespect to the Cartesian model structure).
(2) For every ∞-category C and every map f : X → C which carries eachedge of E to an equivalence in C, f also carries each edge of E0 to an
equivalence in C.
Proof. By definition, (1) holds if and only if for every ∞-category C, theinclusionj : Map[ ((X, E0 ), C\ ) → Map[ ((X, E), C\ )is a categorical equivalence. Condition (2) is the assertion that j is an isomorphism. Thus (2) implies (1). Suppose that (1) is satisfied and let f : X → C
be a vertex of Map[ ((X, E), C\ ). By hypothesis, there exists an equivalencef ' f 0 , where f 0 belongs to the image of j. Let e ∈ E0 ; then f 0 (e) is an equivalence in C. Since f and f 0 are equivalent, f (e) is also an equivalence in C.
Consequently, f also belongs to the image of j, and the proof is complete.

highertopoi

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Proposition 3.2.1.11. Let S be a simplicial set, C a simplicial category,and φ : C[S] → Cop a simplicial functor. The functor St+
φ carries marked
anodyne maps in (Set+)(withrespect
totheCartesianmodel
structure) to∆ /S+ C
trivial cofibrations in (Set∆ ) (with respect to the projective model structure).
Proof. Let f : A → B be a marked anodyne map in (Set+∆ )/S . We wish toprove that St+
(f
)
is
a
trivialcofibration.
Itwillsufficetoprovethis under theφassumption that f is one of the generators for the class of marked anodyne
maps given in Definition 3.1.1.1. Using Proposition 3.2.1.4, we may reduceto the case where S is the underlying simplicial set of B, C = C[S]op , and φis the identity. There are four cases to consider:(1) Suppose first that f is among the morphisms listed in (1) of Definition
3.1.1.1; that is, f is an inclusion (Λni )[ ⊆ (∆n )[ , where 0 < i < n.
Let vk denote the kth vertex of ∆n , which we may also think of as
an object of the simplicial category C. We note that St+
φ (f ) is an
isomorphism when evaluated at vk for k 6= 0. Let K denote the cube(∆1 ){j:0<j≤n,j6=i} , let K 0 = ∂ K, let A denote the set of all vertices ofK corresponding to subsets of {j : 0 < j ≤ n, j 6= i} which contain anelement > i, and let E denote the set of all degenerate edges of K × ∆1
1
together with all edges
A. Finally,` of the 0 form1{a} × ∆ , where a ∈let B = (K × {1}) K 0 ×{1} (K × ∆ ). The morphism St+(f)(vn ) is aφ1
pushout of g : (B, E ∩B1 ) ⊆ (K × ∆ , E). Since i > 0, we may apply
Lemma 3.2.1.9 to deduce that g is marked anodyne and therefore atrivial cofibration in Set+∆.
(2) Suppose that f is among the morphisms of part (2) in Definition
3.1.1.1; that is, f is an inclusion(Λnn , E ∩(Λnn )1 ) ⊆ (∆n , F),
where F denotes the set of all degenerate edges of ∆n together withthe final edge ∆{n−1,n} . If n > 1, then one can repeat the argument
given above in case (1), except that the set of vertices A needs to be
replaced by the set of all vertices of K which correspond to subsets of{j : 0 < j < n} which contain n − 1. If n = 1, then we observe that
]
1 ]
St+
φ (f )(vn ) is isomorphic to the inclusion {1} ⊆ (∆ ) , which is again
a marked anodyne map and therefore a trivial cofibration in Set+∆.
(3) Suppose next that f is the morphisma(Λ21 )](∆2 )[ → (∆2 )](Λ21 )[

specified in (3) of Definition 3.1.1.1. A simple computation shows that+St+φ (f )(vn ) is an isomorphism for n 6= 0, and Stφ (f )(v0 ) may be identified with the inclusion(∆1 × ∆1 , E) ⊆ (∆1 × ∆1 )] ,highertopoi

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175

THE ∞-CATEGORY OF ∞-CATEGORIES

where E denotes the set of all degenerate edges of ∆1 × ∆1 together
with ∆1 ×{0}, ∆1 ×{1}, and {1}×∆1 . This inclusion may be obtained
as a pushout ofa(Λ21 )](∆2 )[ → (∆2 )](Λ21 )[

followed by a pushout ofa(Λ22 )](∆2 )[ → (∆2 )] .

(Λ22 )[

The first of these maps is marked anodyne by definition; the second ismarked anodyne by Corollary 3.1.1.7.
(4) Suppose that f is the morphism K [ → K ] , where K is a Kan complex,[
as in (4) of Definition 3.1.1.1. For each vertex v of K, let St+
φ (K )(v) =+0]](Xv , Ev ), so that Stφ (K ) = Xv . For each g ∈ MapC[K] (v, v )n , we letg ∗ : Xv × ∆n → Xv0 denote the induced map. We wish to show thatthe natural map (Xv , Ev ) → Xv] is an equivalence in Set+
∆ . By Lemma
3.2.1.10, it suffices to show that for every ∞-category Z, if h : Xv → Z
carries each edge belonging to Ev into an equivalence, then h carries
every edge of Xv to an equivalence.
We first show that h carries ee to an equivalence for every edge e : v → v 0
in K. Let me : ∆1 → MapCop (v, v 0 ) denote the degenerate edge atthe vertex corresponding to e. Since K is a Kan complex, the edgee : ∆1 → K extends to a 2-simplex σ : ∆2 → K depicted as follows:
?v

0

e

e0

/ v.

idv

v

Let me0 : ∆1 → MapC (v 0 , v) denote the degenerate edge corresponding
to e0 . The map σ gives rise to a diagram
ve
idve


ve

e
e

/ e∗ ve0
m∗
e0
ee


/ e∗ (e0 )∗ ve

in the simplicial set Xv . Since h carries the left vertical arrow andthe bottom horizontal arrow into equivalences, it follows that h carries
the composition (m∗e ee0 ) ◦ ee to an equivalence in Z; thus h(ee) has aleft homotopy inverse. A similar argument shows that h(ee) has a righthomotopy inverse, so that h(ee) is an equivalence.
We observe that every edge of Xv has the form g ∗ ee, where g is an edgeof MapCop (v, v 0 ) and e : v 0 → v 00 is an edge of K. We wish to showthat h(g ∗ ee) is an equivalence in Z. Above, we have shown that this ishighertopoi

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CHAPTER 3

true if v = v 0 and g is the identity. We now consider the more generalcase where g is not necessarily the identity but is a degenerate edge
corresponding to some map v 0 → v in C. Let h0 denote the compositionh

Xv0 → Xv → Z.
Then h(g ∗ ee) = h0 (ee) is an equivalence in Z by the argument given
above.
Now consider the case where g : ∆1 → MapCop (v, v 0 ) is nondegenerate.
In this case, there is a simplicial homotopy G : ∆1 × ∆1 → MapC (v, v 0 )with g = G|∆1 × {0} and g 0 = G|∆1 × {1} a degenerate edge of
MapCop (v, v 0 ) (for example, we can arrange that g 0 is the constant edge
at an endpoint of g). The map G induces a simplicial homotopy G(e)from g ∗ ee to (g 0 )∗ ee. Moreover, the edges G(e)|{0} × ∆1 and G(e)|{1} ×∆1 belong to Ev and are therefore carried by h into equivalences in Z.
Since h carries (g 0 )∗ ee into an equivalence of Z, it carries g ∗ ee into an
equivalence of Z, as desired.

We now study the behavior of straightening functors with respect to products.
00
Notation 3.2.1.12. Given two simplicial functors F : C → Set+∆, F : C →00++Set∆ , we let F  F : C × C → Set∆ denote the functor described by theformula(F  F0 )(C, C 0 ) = F(C) × F0 (C 0 ).
Proposition 3.2.1.13. Let S and S 0 be simplicial sets, C and C0 simplicial
categories, and φ : C[S] → Cop , φ0 : C[S 0 ] → (C0 )op simplicial functors;
let φ  φ0 denote the induced functor C[S × S 0 ] → (C × C0 )op . For every+0M ∈ (Set+∆ )/S , M ∈ (Set∆ )/S 0 , the natural map++
00sM,M 0 : St+
φφ0 (M × M ) → Stφ (M )  Stφ0 (M )is a weak equivalence of functors C × C0 → Set+∆.
Proof. Since both sides are compatible with the formations of filtered colimits in M , we may suppose that M has only finitely many nondegenerate
simplices. We work by induction on the dimension n of M and the number ofn-dimensional simplices of M . If M = ∅, there is nothing to prove. If n 6= 1,
we may choose a nondegenerate
simplex of M having maximal dimension`
and thereby write M = N (∂ ∆n )[ (∆n )[ . By the inductive hypothesis we
may suppose that the result is known for N and (∂ ∆n )[ . The map sM,M 0
is a pushout of the maps sN,M 0 and s(∆n )[ ,M 0 over s(∂ ∆n )[ ,M 0 . Since Set+∆
is left proper, this pushout is a homotopy pushout; it therefore suffices toprove the result after replacing M by N , (∂ ∆n )[ , or (∆n )[ . In the first twocases, the inductive hypothesis implies that sM,M 0 is an equivalence; we arehighertopoi

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THE ∞-CATEGORY OF ∞-CATEGORIES

therefore reduced to the case M = (∆n )[ . If n = 0, the result is obvious. Ifn > 2, we setaaa
K = ∆{0,1}∆{1,2}···∆{n−1,n} ⊆ ∆n .
{1}

{2}

{n−1}

The inclusion K ⊆ ∆n is inner anodyne so that K [ ⊆ M is marked anodyne.
By Proposition 3.2.1.11, we deduce that sM,M 0 is an equivalence if and only
if sK [ ,M 0 is an equivalence, which follows from the inductive hypothesis since
K is 1-dimensional.
We may therefore suppose that n = 1. Using the above argument, we
may reduce to the case where M consists of a single edge, either marked
or unmarked. Repeating the above argument with the roles of M and M 0interchanged, we may suppose that M 0 also consists of a single edge. Applying Proposition 3.2.1.4, we may reduce to the case where S = M , S 0 = M 0 ,C = C[S]op , and C0 = C[S 0 ]op .
Let us denote the vertices of M by x and y, and the unique edge joining
them by e : x → y. Similarly, we let x0 and y 0 denote the vertices of M 0 ,and e0 : x0 → y 0 the edge which joins them. We note that the map sM,M 0
induces an isomorphism when evaluated on any object of C × C0 except (x, x0 ).
Moreover, the map
+
+
0
0
0
0
sM,M 0 (x, x0 ) : St+
φφ0 (M × M )(x, x ) → Stφ (M )(x) × Stφ0 (M )(x )is obtained from s(∆1 )[ ,(∆1 )[ by successive pushouts along cofibrations of
the form (∆1 )[ ⊆ (∆1 )] . Since Set+
∆ is left proper, we may reduce to thecase where M = M 0 = (∆1 )[ . The result now follows from a simple explicit
computation.
We now study the situation in which S = ∆0 , C = C[S], and φ is the+identity map. In this case, St+
φ may be regarded as a functor T : Set∆ →+Set∆ . The underlying functor of simplicial sets is familiar: we have
T (X, E) = (|X|Q• , E0 ),where Q denotes the cosimplicial object of Set∆ considered in §2.2.2. In that
section, we exhibited a natural map |X|Q• → X which we proved to be aweak homotopy equivalence. We now prove a stronger version of that result:
Proposition 3.2.1.14. For any marked simplicial set M = (X, E), thenatural map |X|Q• → X induces a Cartesian equivalence
T (M ) → M.
Proof. As in the proof of Proposition 3.2.1.13, we may reduce to the casewhere M consists of a simplex of dimension at most 1 (either marked or
unmarked). In these cases, the map T (M ) → M is an isomorphism in Set+∆.

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Corollary 3.2.1.15. Let S be a simplicial set, C a simplicial category, φ :C[S] → Cop a simplicial functor, and X ∈ (Set+∆ )/S an object. For every
K ∈ Set+
,
there
is
a
naturalequivalence∆+St+φ (M × K) → Stφ (M )  K

of functors from C to Set+∆.
Proof. Combine the equivalences of Proposition 3.2.1.14 (in the case whereS 0 = ∆0 , C0 = C[S 0 ]op , and φ0 is the identity) and Proposition 3.2.1.15.
+
We can now complete the proof that (St+φ , Unφ ) is a Quillen adjunction:

Corollary 3.2.1.16. Let S be a simplicial set, C a simplicial category, andφ : C[S]op → C a simplicial functor. The straightening functor St+
φ carries+
Cartesian equivalences in (Set∆ )/S to (objectwise) Cartesian equivalences in
C
(Set+
∆) .
Proof. Let f : M → N be a Cartesian equivalence in (Set+∆ )/S . Choose a00marked anodyne map
M
→
M
,
where
M
is
fibrant;
thenchoosea marked`
anodyne map M 0 M N → N 0 , with N 0 fibrant. Since St+
carriesmarkedφanodyne maps to equivalences by Proposition 3.2.1.11, it suffices to prove+00that the induced map St+
φ (M ) → Stφ (N ) is an equivalence. In other words,
00
we may replace M by M and N by N , thereby reducing to the case where
M and N are fibrant.
Since f is an Cartesian equivalence of fibrant objects, it has a homotopy+
inverse g. We claim that St+
φ (g) is an inverse to Stφ (f ) in the homotopy++
C
category of (Set+∆ ) . We will show that Stφ (f ) ◦ Stφ (g) is homotopic to theidentity; applying the same argument with the roles of f and g reversed willthen establish the desired result.
Since f ◦ g is homotopic to the identity, there is a map h : N × K ] → N ,where K is a contractible Kan complex containing vertices x and y, suchthat f ◦ g = h|N × {x} and idN = h|N × {y}. The map St+
φ (h) factors as
+
+
]
]
St+
φ (N × K ) → Stφ (N )  K → Stφ (N ),where the left map is an equivalence by Corollary 3.2.1.15 and the right map+
because K is contractible. Since St+
φ (f ◦ g) and Stφ (idN ) are both sectionsof St+
φ (h), they represent the same morphism in the homotopy category of
C
(Set+
∆) .
3.2.2 Cartesian Fibrations over a Simplex
A map of simplicial sets p : X → S is a Cartesian fibration if and only if thepullback map X ×S ∆n → ∆n is a Cartesian fibration for each simplex of S.
Consequently, we might imagine that Cartesian fibrations X → ∆n are the“primitive building blocks” out of which other Cartesian fibrations are built.
The goal of this section is to prove a structure theorem for these building

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179

blocks. This result has a number of consequences and will play a vital rolein the proof of Theorem 3.2.0.1.
Note that ∆n is the nerve of the category associated to the linearly orderedset[n] = {0 < 1 < · · · < n}.
Since a Cartesian fibration p : X → S can be thought of as giving a (contravariant) functor from S to ∞-categories, it is natural to expect a close relationship between Cartesian fibrations X → ∆n and composable sequencesof maps between ∞-categories
A0 ← A1 ← · · · ← An .
In order to establish this relationship, we need to introduce a few definitions.
Suppose we are given a composable sequence of maps
φ : A0 ← A1 ← · · · ← Anof simplicial sets. The mapping simplex M (φ) of φ is defined as follows. If
J is a nonempty finite linearly ordered set with greatest element j, thento specify a map ∆J → M (φ), one must specify an order-preserving map
f : J → [n] together with a map σ : ∆J → Af (j) . Given an order-preserving
map p : J → J 0 of partially ordered sets containing largest elements j and0
j 0 , there is a natural map M (φ)(∆J ) → M (φ)(∆J ) which carries (f, σ) to0(f ◦ p, e ◦ σ), where e : Af (j ) → Af (p(j)) is obtained from φ in the obvious
way.
Remark 3.2.2.1. The mapping simplex M (φ) is equipped with a natural
map p : M (φ) → ∆n ; the fiber of p over the vertex j is isomorphic to thesimplicial set Aj .
Remark 3.2.2.2. More generally, let f : [m] → [n] be an order-preserving
map, inducing a map ∆m → ∆n . Then M (φ)×∆n ∆m is naturally isomorphic
to M (φ0 ), where the sequence φ0 is given by
Af (0) ← · · · ← Af (m) .
Notation 3.2.2.3. Let φ : A0 ← · · · ← An be a composable sequence ofmaps of simplicial sets. To give an edge e of M (φ), one must give a pair of
integers 0 ≤ i ≤ j ≤ n and an edge e ∈ Aj . We will say that e is marked if
e is degenerate; let E denote the set of all marked edges of M (φ). Then thepair (M (φ), E) is a marked simplicial set which we will denote by M \ (φ).
Remark 3.2.2.4. There is a potential ambiguity between the terminology
of Definition 3.1.1.8 and that of Notation 3.2.2.3. Suppose that φ : A0 ←· · · ← An is a composable sequence of maps and that p : M (φ) → ∆n is a
Cartesian fibration. Then M (φ)\ (Definition 3.1.1.8) and M \ (φ) (Notation
3.2.2.3) do not generally coincide as marked simplicial sets. We feel that
there is little danger of confusion since it is very rare that p is a Cartesianfibration.

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Remark 3.2.2.5. The construction of the mapping simplex is functorial in
the sense that a commutative ladderφ : A0 o


··· o

An


··· o


Bn

fn

f0

ψ : B0 o

induces a map M (f ) : M (φ) → M (ψ). Moreover, if each fi is a categoricalequivalence, then f is a categorical equivalence (this follows by induction onn using the fact that the Joyal model structure is left proper).
Definition 3.2.2.6. Let p : X → ∆n be a Cartesian fibration and let
φ : A0 ← · · · ← An
be a composable sequence of maps. A map q : M (φ) → X is a quasiequivalence if it has the following properties:(1) The diagram
/X

q

M (φ)
#

∆

n

~

p

is commutative.
(2) The map q carries marked edges of M (φ) to p-Cartesian edges of S; inother words, q induces a map M \ (φ) → X \ of marked simplicial sets.
(3) For 0 ≤ i ≤ n, the induced map Ai → p−1 {i} is a categorical equivalence.
The goal of this section is to prove the following:
Proposition 3.2.2.7. Let p : X → ∆n be a Cartesian fibration.
(1) There exists a composable sequence of maps
φ : A0 ← A1 ← · · · ← An
and a quasi-equivalence q : M (φ) → X.
(2) Letφ : A0 ← A1 ← · · · ← An
be a composable sequence of maps and let q : M (φ) → X be a quasiequivalence. For any map T → ∆n , the induced map
M (φ) ×∆n T → X ×∆n T
is a categorical equivalence.

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We first show that, to establish (2) of Proposition 3.2.2.7, it suffices toconsider the case where T is a simplex:
Proposition 3.2.2.8. Suppose we are given a diagram
X→Y →Zof simplicial sets. For any map T → Z, we let XT denote X ×Z T and YT
denote Y ×Z T . The following statements are equivalent:(1) For any map T → Z, the induced map XT → YT is a categoricalequivalence.
(2) For any n ≥ 0 and any map ∆n → Z, the induced map X∆n → Y∆n
is a categorical equivalence.
Proof. It is clear that (1) implies (2). Let us prove the converse. Since theclass of categorical equivalences is stable under filtered colimits, it suffices toconsider the case where T has only finitely many nondegenerate simplices.
We now work by induction on the dimension of T and the number of nondegenerate simplices contained in T`. If T is empty, there is nothing to prove.
Otherwise, we may write T = T 0 ∂ ∆n ∆n . By the inductive hypothesis, themaps
XT 0 → YT 0
X∂ ∆n → Y∂ ∆n
are categorical equivalences, and by assumption, X∆n → Y∆n is a categoricalequivalence as well. We note thata
XT = XT 0
X∆n
X∂ ∆n

YT = YT 0

a

Y∆n .

Y∂ ∆ nSince the Joyal model structure is left proper, these pushouts are homotopy
pushouts and therefore categorically equivalent to one another.
Suppose p : X → ∆n is a Cartesian fibration and q : M (φ) → X is a quasiequivalence. Let f : ∆m → ∆n be any map. We note (see Remark 3.2.2.5)that M (φ) ×∆n ∆m may be identified with a mapping simplex M (φ0 ) andthat the induced map
M (φ0 ) → X ×∆n ∆mis again a quasi-equivalence. Consequently, to establish (2) of Proposition
3.2.2.7, it suffices to prove that every quasi-equivalence is a categorical equivalence. First, we need the following lemma.

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Lemma 3.2.2.9. Let
φ : A0 ← · · · ← An
be a composable sequence of maps between simplicial sets, where n > 0. Let y
be a vertex of An and let the edge e : y 0 → y be the image of ∆{n−1,n} × {y}under the map ∆n × An → M (φ). Let x be any vertex of M (φ) which does
not belong to the fiber An . Then composition with e induces a weak homotopy
equivalence of simplicial sets
MapC[M (φ)] (x, y 0 ) → MapC[M (φ)] (x, y).
Proof. Replacing φ by an equivalent diagram if necessary (using Remark
3.2.2.5), we may suppose that the map An → An−1 is a cofibration. Let φ0denote the composable subsequence
A0 ← · · · ← An−1 .
Let C = C[M (φ)] and let C− = C[M (φ0 )] ⊆ C. There is a pushout diagramin Cat∆C[An × ∆n−1 ]/ C[An × ∆n ]


C−


/ C.

This diagram is actually a homotopy pushout since Cat∆ is a left proper
model category and the top horizontal map is a cofibration. Now form thepushout`{n−1,n}/ C[An × (∆n−1
C[An × ∆n−1 ])]{n−1} ∆C−


/ C0 .

This diagram is also a homotopy pushout. Since the diagram of simplicial
sets{n − 1}/ ∆{n−1,n}


∆n−1


/ ∆n

is homotopy coCartesian (with respect to the Joyal model structure), we
deduce that the natural map C0 → C is an equivalence of simplicial categories. It therefore suffices to prove that composition with e induces a weakhomotopy equivalence
MapC0 (x, y 0 ) → MapC (x, y).

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THE ∞-CATEGORY OF ∞-CATEGORIESForm a pushout square/ C[An ] × C[∆{n−1,n} ]C[An × {n − 1, n}]C0
/ C0 .

F

The left vertical map is a cofibration (since An → An−1 is a cofibration of
simplicial sets), and the upper horizontal map is an equivalence of simplicial
categories (Corollary 2.2.5.6). Invoking the left properness of Cat∆ , we conclude that F is an equivalence of simplicial categories. Consequently, it will
suffice to prove that MapC0 (F (x), F (y 0 )) → MapC0 (F (x), F (y)) is a weakhomotopy equivalence. We now observe that this map is an isomorphism of
simplicial sets.
Proposition 3.2.2.10. Let p : X → ∆n be a Cartesian fibration, let
φ : A0 ← · · · ← An
be a composable sequence of maps of simplicial sets and let q : M (φ) → X
be a quasi-equivalence. Then q is a categorical equivalence.
Proof. We proceed by induction on n. The result is obvious if n = 0, so let
us assume that n > 0. Let φ0 denote the composable sequence of maps
A0 ← A1 ← · · · ← An−1
which is obtained from φ by omitting An . Let v denote the final vertex of∆n and let T = ∆{0,...,n−1} denote the face of ∆n which is opposite v. Let
Xv = X ×∆n {v} and XT = X `×∆n T .
We note that M (φ) = M (φ0 ) An ×T (An × ∆n ). We wish to show that thesimplicial functora
F : C ' C[M (φ)] ' C[M (φ0 )]C[An × ∆n ] → C[X]
C[An ×T ]is an equivalence of simplicial categories. We note that C decomposes naturally into full subcategories C+ = C[An × {v}] and C− = C[M (φ0 )], having
the property that MapC (X, Y ) = ∅ if x ∈ C+ , y ∈ C− .
Similarly, D = C[X] decomposes into full subcategories D+ = C[Xv ] andD− = C[XT ], satisfying MapD (x, y) = ∅ if x ∈ D+ and y ∈ D− . We observethat F restricts to give an equivalence between C− and D− by assumption
and gives an equivalence between C+ and D+ by the inductive hypothesis.
To complete the proof, it will suffice to show that if x ∈ C− and y ∈ C+ ,then F induces a homotopy equivalence
MapC (x, y) → MapD (F (x), F (y)).
We may identify the object y ∈ C+ with a vertex of An . Let e denotethe edge of M (φ) which is the image of {y} × ∆{n−1,n} under the maphighertopoi

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An × ∆n → M (φ). We let [e] : y 0 → y denote the corresponding morphism
in C. We have a commutative diagram/ MapC (x, y)
MapC (x, y 0 )−MapD− (F (x), F (y 0 ))
/ MapD (F (x), F (y)).

Here the left vertical arrow is a weak homotopy equivalence by the inductive
hypothesis, and the bottom horizontal arrow (which is given by composition
with [e]) is a weak homotopy equivalence because q(e) is p-Cartesian. Consequently, to complete the proof, it suffices to show that the top horizontal
arrow (given by composition with e) is a weak homotopy equivalence. This
follows immediately from Lemma 3.2.2.9.
To complete the proof of Proposition 3.2.2.7, it now suffices to show thatfor any Cartesian fibration p : X → ∆n , there exists a quasi-equivalence
M (φ) → X. In fact, we will prove something slightly stronger (in order tomake our induction work):
Proposition 3.2.2.11. Let p : X → ∆n be a Cartesian fibration of simplicial sets and A another simplicial set. Suppose we are given a commutative
diagram of marked simplicial sets/ X\sA[ × (∆n )]%|(∆n )] .

Then there exists a sequence of composable morphisms
φ : A0 ← · · · ← An ,a map A → An , and an extension/ M \ (φ)

A[ × (∆n )]

%

f

/ X\

 |
(∆n )]of the previous diagram, such that f is a quasi-equivalence.
Proof. The proof goes by induction on n. We begin by considering the fibers over the final vertex v of ∆n . The map sv : A → Xv = X ×∆n {v} admitsa factorizationgh

A → An → Sv ,where g is a cofibration and h is a trivial Kan fibration. The smash productinclusiona({v}] × (An )[ )((∆n )] × A[ ) ⊆ (∆n )] × (An )[{v}] ×A[

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THE ∞-CATEGORY OF ∞-CATEGORIES

is marked anodyne (Proposition 3.1.2.3). Consequently, we deduce the existence of a dotted arrow f0 as indicated in the diagramn ]A[ × (∆
_ )

/ \
8X
f0


(An )[ × (∆n )]/ (∆n )]of marked simplicial sets, where f0 |(An × {n}) = h.
If n = 0, we are now done. If n > 0, then we apply the inductive hypothesis
to the diagram(An )[ × (∆n−1 )]f0 |An ×∆n−1'(∆n−1 )]/ (X ×∆n ∆n−1 )\w

to deduce the existence of a composable sequence of maps
φ0 : A0 ← · · · ← An−1 ,a map An → An−1 , and a commutative diagram(An )[ × (∆n−1 )]/ M \ (φ0 )f0/ (X ×∆n ∆n−1 )\'w(∆n−1 )] ,where f 0 is a quasi-equivalence. We now define φ to be the result of appending
the map An → An−1 to the beginning of φ0 and f : M (φ) → X be the map
obtained by amalgamating f0 and f 0 .
Corollary 3.2.2.12. Let p : X → S be a Cartesian fibration of simplicial
sets and let q : Y → Z be a coCartesian fibration. Define new simplicial sets
Y 0 and Z 0 equipped with maps Y 0 → S, Z 0 → S via the formulas
HomS (K, Y 0 ) ' Hom(X ×S K, Y )HomS (K, Z 0 ) ' Hom(X ×S K, Z).
Then(1) Composition with q determines a coCartesian fibration q 0 : Y 0 → Z 0 .
(2) An edge ∆1 → Y 0 is q 0 -coCartesian if and only if the induced map∆1 ×S X → Y carries p-Cartesian edges to q-coCartesian edges.
Proof. Let us say that an edge of Y 0 is special if it satisfies the hypothesis of(2). Our first goal is to show that there is a sufficient supply of special edges
in Y 0 . More precisely, we claim that given any edge e : z → z 0 in Z 0 and anyhighertopoi

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CHAPTER 3

vertex ze ∈ Y 0 covering z, there exists a special edge ee : ze → ze0 of Y 0 which
covers e.
Suppose that the edge e covers an edge e0 : s → s0 in S. We can identify ze
with a map from Xs to Y . Using Proposition 3.2.2.7, we can choose a morphism φ : Xs0 ← Xs0 0 and a quasi-equivalence M (φ) → X ×S ∆1 . Composingwith ze, we obtain a map Xs0 → Y . Using Propositions 3.3.1.7 and A.2.3.1,
we may reduce to the problem of providing a dotted arrow in the diagram
/Y
Xs0
=
_
q


M (φ)


/Z

which carries the marked edges of M \ (φ) to q-coCartesian edges of Y . This
follows from the fact that q Xs : Y Xs → Z Xs is a coCartesian fibration andthe description of the q Xs -coCartesian edges (Proposition 3.1.2.1).
To complete the proofs of (1) and (2), it will suffice to show that q 0 is an
inner fibration and that every special edge of Y 0 is q 0 -coCartesian. For this,
we must show that every lifting problem
Λni
_

/ Y0
>

σ0

q0


/ Z0


∆n

has a solution provided that either 0 < i < n or i = 0, n ≥ 2, and σ0 |∆{0,1}is special. We can reformulate this lifting problem using the diagram/: Y
X ×S Λni
_
q




/ Z.

X ×S ∆nUsing Proposition 3.2.2.7, we can choose a composable sequence of morphisms
ψ : X00 ← · · · ← Xn0
and a quasi-equivalence M (ψ) → X ×S ∆n . Invoking Propositions 3.3.1.7
and A.2.3.1, we may reduce to the associated mapping problem
M (ψ) ×∆n Λn
9/ Y
i

q


M (ψ)


/ Z.

Since i < n, this is equivalent to the mapping problem/Y
Xn0 × Λni_q
Xn0 × ∆n
/ Z,highertopoi

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THE ∞-CATEGORY OF ∞-CATEGORIES

which admits a solution by virtue of Proposition 3.1.2.1.
Corollary 3.2.2.13. Let p : X → S be a Cartesian fibration of simplicial
sets, and let q : Y → S be a coCartesian fibration. Define a new simplicial
set T equipped with a map T → S by the formula
HomS (K, T ) ' HomS (X ×S K, Y ).
Then:
(1) The projection r : T → S is a coCartesian fibration.
(2) An edge ∆1 → Z is r-coCartesian if and only if the induced map∆1 ×S X → ∆1 ×S Y carries p-Cartesian edges to q-coCartesian edges.
Proof. Apply Corollary 3.2.2.12 in the case where Z = S.
We conclude by noting the following property of quasi-equivalences (whichis phrased using the terminology of §3.1.3):
Proposition 3.2.2.14. Let S = ∆n , let p : X → S be a Cartesian fibration,
let
φ : A0 ← · · · ← An
be a composable sequence of maps, and let q : M (φ) → X be a quasiequivalence. The induced map M \ (φ) → X \ is a Cartesian equivalence in(Set+∆ )/S .
Proof. We must show that for any Cartesian fibration Y → S, the induced
map of ∞-categories
Map[S (X \ , Y \ ) → Map[S (M \ (φ), Y \ )is a categorical equivalence. Because S is a simplex, the left side may be
identified with a full subcategory of Y X and the right side with a full subcategory of Y M (φ) . Since q is a categorical equivalence, the natural map
Y X → Y M (φ) is a categorical equivalence; thus, to complete the proof, itsuffices to observe that a map of simplicial sets f : X → Y is compatible
with the projection to S and preserves marked edges if and only if q ◦ f has
the same properties.
3.2.3 Straightening over a Simplex
Let S be a simplicial set, C a simplicial category, and φ : C[S]op → C a simplicial functor. In §3.2.1, we introduced the straightening and unstraightening
functors
o
(Set+
∆ )/S

St+
φ
Un+
φ

/ (Set+ )C .
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In this section, we will prove that (St+φ , Unφ ) is a Quillen equivalence provided that φ is a categorical equivalence and S is a simplex (the case of ageneral simplicial set S will be treated in §3.2.4).
Our first step is to prove the result in the case where S is a point and φis an isomorphism of simplicial categories. We can identify the functor St+∆0+with the functor T : Set+→
Set
studiedin§3.2.1.
Consequently,
Theorem∆∆
3.2.0.1 is an immediate consequence of Proposition 3.2.1.14:+
Lemma 3.2.3.1. The functor T : Set+
∆ → Set∆ has a right adjoint U , andthe pair (T, U ) is a Quillen equivalence from Set+
∆ to itself.

Proof. We have already established the existence of the unstraightening functor U in §3.2.1 and proved that (T, U ) is a Quillen adjunction. To complete
the proof, it suffices to show that the left derived functor of T (which we may
identify with T because every object of Set+
∆ is cofibrant) is an equivalencefrom the homotopy category of Set+
∆ to itself. But Proposition 3.2.1.14 asserts that T is isomorphic to the identity functor on the homotopy categoryof Set+∆.
Let us now return to the case of a general equivalence φ : C[S] → Cop .
++Since we know that (St+φ , Unφ ) give a Quillen adjunction between (Set∆ )/S+ C
and (Set∆ ) , it will suffice to prove that the unit and counit+u : id → R Un+
φ ◦LStφ+v : LSt+φ ◦ R Unφ → id

are weak equivalences. Our first step is to show that R Un+
φ detects weakequivalences: this reduces the problem of proving that v is an equivalence tothe problem of proving that u is an equivalence.
Lemma 3.2.3.2. Let S be a simplicial set, C a simplicial category, and φ :C[S] → Cop an essentially surjective functor. Let p : F → G be a map between+++
C(weakly) fibrant objects of (Set+∆ ) . Suppose that Unφ (p) : Unφ F → Unφ G
is a Cartesian equivalence. Then p is an equivalence.
Proof. Since φ is essentially surjective, it suffices to prove that F(C) → F(D)is a Cartesian equivalence for every object C ∈ C which lies in the image ofφ. Let s be a vertex of S with ψ(s) = C. Let i : {s} → S denote the inclusion+
and let i∗ : (Set+∆ )/S → Set∆ denote the functor of passing to the fiber overs:i∗ X = Xs = X ×S ] {s}] .
Let i! denote the left adjoint to i∗ . Let {C} denote the trivial category
with one object (and only the identity morphism), and let j : {C} → C be
the simplicial functor corresponding to the inclusion of C as an object of C.
According to Proposition 3.2.1.4, we have a natural identification of functorsSt+φ ◦ i! ' j! ◦ T.

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Passing to adjoints, we get another identification∗i∗ ◦ Un+
φ 'U ◦j+Cfrom (Set+∆ ) to Set∆ . Here U denotes the right adjoint of T .
According to Lemma 3.2.3.1, the functor U detects equivalences between∗∗
fibrant objects of Set+
∆ . It therefore suffices to prove that U (j F) → U (j G)
is a Cartesian equivalence. Using the identification above, we are reduced toproving that+Un+
φ (F)s → Unφ (G)s+is a Cartesian equivalence. But Un+
φ (F) and Unφ (G) are fibrant objects of(Set+∆ )/S and therefore correspond to Cartesian fibrations over S: the desired
result now follows from Proposition 3.1.3.5.

We have now reduced the proof of Theorem 3.2.0.1 to the problem ofshowing that if φ : C[S] → Cop is an equivalence of simplicial categories,
then the unit transformation+u : id → R Un+
φ ◦Stφis an isomorphism of functors from the homotopy category h(Set+∆ )/S toitself.
Our first step is to analyze the effect of the straightening functor St+
φ on a+mapping simplex. We will need a bit of notation. For any X ∈ (Set∆ )/S andany vertex s of S, we let Xs denote the fiber X ×S ] {s}] and let is denotethe composite functorφ{s} ,→ C[S] → Copof simplicial categories. According to Proposition 3.2.1.4, there is a natural
identificationsSt+
φ (Xs ) ' i! T (Xs )which induces a map
ψsX : T (Xs ) → St+
φ (X)(s).
Lemma 3.2.3.3. Let
θ : A0 ← · · · ← An
be a composable sequence of maps of simplicial sets and let M \ (θ) ∈ (Set+∆ )∆nbe its mapping simplex. For each 0 ≤ i ≤ n, the map
M \ (θ)ψi\: T (Ai )[ → St+∆n (M (θ))(i)is a Cartesian equivalence in Set+∆.

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M \ (θ)Proof. The proof proceeds by induction on n. We first observe that ψn
is an isomorphism; we may therefore restrict our attention to i < n. Let θ0be the composable sequence
A0 ← · · · ← An−1
and M \ (θ0 ) its mapping simplex, which we may regard as an object of either+(Set+∆ )/∆n or (Set∆ )/∆n−1 .
For i < n, we have a commutative diagram
M \ (θ 0 )
ψi

\ 0
St+
∆n7 (M (θ ))(i)fi(/ St+ n (M \ (θ))(i).

T ((Ai )[ )

∆

+
\ 0
\ 0
n−1
By Proposition 3.2.1.4, St+]→∆n M (θ ) ' j! St∆n−1 M (θ ), where j : C[∆nC[∆ ] denotes the inclusion. Consequently, the inductive hypothesis implies
that the maps
\ 0
T (Ai )[ → St+∆n−1 (M (θ ))(i)are Cartesian equivalences for i < n. It now suffices to prove that fi is a
Cartesian equivalence for i < n.
We observe that there is a (homotopy) pushout diagram(An )[ × (∆n−1 )]/ (An )[ × (∆n )]
M \ (θ0 )
/ M \ (θ).

Since St+∆n is a left Quillen functor, it induces a homotopy pushout diagram
n [
n−1 ]
St+
))
∆n ((A ) × (∆g

/ St+ n ((An )[ × (∆n )] )∆/ St+ n M \ (θ)


\ 0
St+
∆n M (θ )∆Cin (Set+∆ ) . We are therefore reduced to proving that g induces a Cartesianequivalence after evaluation at any i < n.
According to Proposition 3.2.1.13, the vertical maps of the diagram
n [
n−1 ]
St+
))
∆n ((A ) × (∆/ St+ n ((An )[ × (∆n )] )n−1 ]T (An )[  St+
)
∆n (∆


/ T (An )[  St+ n (∆n )]∆∆are Cartesian equivalences. To complete the proof we must show that
n−1 ]
n ]
St+
) → St+∆n (∆∆n (∆ )

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THE ∞-CATEGORY OF ∞-CATEGORIES

induces a Cartesian equivalence when evaluated at any i < n. Consider the
diagram{n − 1}]/ (∆n−1 )](∆{n−1,n} )]/ (∆n )] .

The horizontal arrows are marked anodyne. It therefore suffices to show that+]{n−1,n} ]St+)∆n {n − 1} → St∆n (∆induces Cartesian equivalences when evaluated at any i < n. This follows
from an easy computation.
Proposition 3.2.3.4. Let n ≥ 0. Then the Quillen adjunctiono(Set+∆ )/∆n

St+
∆n
Un+
∆n

/ (Set+ )C[∆n ]∆is a Quillen equivalence.
Proof. As we have argued above, it suffices to show that the unit+
id → R Un+
φ ◦St∆nis an isomorphism of functors from h(Set+∆ )∆n to itself. In other words, wemust show that given an object X ∈ (Set+∆ )/∆n and a weak equivalenceSt+∆n X → F,nC[∆ ]where F ∈ (Set+is fibrant, the adjoint map∆)

j : X → Un+∆n F
is a Cartesian equivalence in (Set+∆ )/∆n .
Choose a fibrant replacement for X: that is, a Cartesian equivalence X →Y \ , where Y → ∆n is a Cartesian fibration. According to Proposition 3.2.2.7,
there exists a composable sequence of maps
θ : A0 ← · · · ← An
and a quasi-equivalence M \ (θ) → Y \ . Proposition 3.2.2.14 implies thatM \ (θ) → Y \ is a Cartesian equivalence. Thus, X is equivalent to M \ (θ)
in the homotopy category of (Set+∆ )/∆n and we are free to replace X by\
M (θ), thereby reducing to the case where X is a mapping simplex.
We wish to prove that j is a Cartesian equivalence. Since Un+∆n F is fibrant, Proposition 3.2.2.14 implies that it suffices to show that j is a quasiequivalence: in other words, we need to show that the induced map of fibersnjs : Xs → (Un+∆n F)s is a Cartesian equivalence for each vertex s of ∆ .
As in the proof of Lemma 3.2.3.2, we may identify (Un+
F)
with
U
(F(s)),
s∆n
where U is the right adjoint to T . By Lemma 3.2.3.1, Xs → U (F(s)) is ahighertopoi

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CHAPTER 3

Cartesian equivalence if and only if the adjoint map T (Xs ) → F(s) is a
Cartesian equivalence. This map factors as a composition
T (Xs ) → St+∆n (X)(s) → F(s).
The map on the left is a Cartesian equivalence by Lemma 3.2.3.3, and themap on the right also a Cartesian equivalence, by virtue of the assumptionthat St+∆n X → F is a weak equivalence.
3.2.4 Straightening in the General Case
Let S be a simplicial set and φ : C[S] → Cop an equivalence of simplicial
categories. Our goal in this section is to complete the proof of Theorem++
3.2.0.1 by showing that (St+φ , Unφ ) is a Quillen equivalence between (Set∆ )/SC
and (Set+∆ ) . In §3.2.3, we handled the case where S is a simplex (and φ+an isomorphism) by verifying that the unit map id → R Un+
φ ◦Stφ is an+
isomorphism of functors from h(Set∆ )/S to itself.
Here is the idea of the proof. Without loss of generality, we may supposethat φ is an isomorphism (since the pair (φ! , φ∗ ) is a Quillen equivalence
C[S]opC
between (Set+and (Set+∆)∆ ) by Proposition A.3.3.8). We wish to show+
Cthat Unφ induces an equivalence from the homotopy category of (Set+∆ ) tothe homotopy category of (Set+∆ )/S . According to Proposition 3.2.3.4, thisis true whenever S is a simplex. In the general case, we would like to regard
+
C
(Set+
∆ ) and (Set∆ )/S as somehow built out of pieces which are associated tosimplices and deduce that Un+
φ is an equivalence because it is an equivalenceon each piece. In order to make this argument work, it is necessary to work+C
not just with the homotopy categories of (Set+∆ ) and (Set∆ )/S but also withthe simplicial categories which give rise to them.
+C
We recall that both (Set+∆ ) and (Set∆ )/S are simplicial model categories
with respect to the simplicial mapping spaces defined by
HomSet∆ (K, Map(Set+ )C (F, G)) = Hom(Set+ )C (F K ] , G)∆∆

HomSet∆ (K, Map(Set+ )/S (X, Y )) = Hom(Set+ )/S (X × K ] , Y ).
∆∆The functor St+
φ is not a simplicial functor. However, it is weakly compatible
with the simplicial structure in the sense that there is a natural map
+
]
]
St+
φ (X  K ) → (Stφ X)  K

for any X ∈ (Set+∆ )/S , K ∈ Set∆ (according to Corollary 3.2.1.15, this map
Cis a weak equivalence in (Set+∆ ) ). Passing to adjoints, we get natural maps+
Map(Set+ )C (F, G) → Map]S (Un+φ F, Unφ G).
∆

In other words, Un+φ does have the structure of a simplicial functor. We now
invoke Proposition A.3.1.10 to deduce the following:
Lemma 3.2.4.1. Let S be a simplicial set, C a simplicial category, andφ : C[S] → Cop a simplicial functor. The following are equivalent:highertopoi

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THE ∞-CATEGORY OF ∞-CATEGORIES+(1) The Quillen adjunction (St+φ , Unφ ) is a Quillen equivalence.

(2) The functor Un+
φ induces an equivalence of simplicial categories
+ C ◦
+
◦
◦
(Un+
φ ) : ((Set∆ ) ) → ((Set∆ )/S ) ,+ C
C ◦where ((Set+∆ ) ) denotes the full (simplicial) subcategory of ((Set∆ ) )+◦consisting of fibrant-cofibrant objects and ((Set∆ )/S ) denotes the full
(simplicial) subcategory of (Set+∆ )/S consisting of fibrant-cofibrant objects.

Consequently, to complete the proof of Theorem 3.2.0.1, it will suffice to◦show that if φ is an equivalence of simplicial categories, then (Un+φ ) is an◦equivalence of simplicial categories. The first step is to prove that (Un+φ ) isfully faithful.
Lemma 3.2.4.2. Let S 0 ⊆ S be simplicial sets and let p : X → S, q : Y → S
be Cartesian fibrations. Let X 0 = X ×S S 0 and Y 0 = Y ×S S 0 . The restriction
map
\

\

Map]S (X \ , Y \ ) → Map]S 0 (X 0 , Y 0 )is a Kan fibration.
Proof. It suffices to show that the map Y \ → S has the right lifting property
with respect to the inclusiona\(X 0 × B ] )(X \ × A] ) ⊆ X \ × B ]X 0 \ ×A]

for any anodyne inclusion of simplicial sets A ⊆ B.
\
But this is a smash product of a marked cofibration X 0 → X \ (in++]](Set∆ )/S ) and a trivial marked cofibration A → B (in Set∆ ) and is therefore a trivial marked cofibration. We conclude by observing that Y \ is a
fibrant object of (Set+∆ )/S (Proposition 3.1.4.1).
C[S]op

Proof of Theorem 3.2.0.1. For each simplicial set S, let (Set+denote∆ )f+ C[S]op
the category of projectively fibrant objects of (Set∆ )and let WS be
C[S]op0
the class of weak equivalences in (Set+
)
.
Let
W
be
the collection
S
∆ f
+ ◦
of pointwise equivalences in (Set∆ )/S . We have a commutative diagram ofsimplicial categories
op

C[S]
)◦
((Set+∆)C[S]op(Set+∆ )f[WS−1 ]Un+S

/ (Set+ )◦

∆ /S

φS



ψS

/ (Set+ )◦ [W 0 −1
S ]
∆ /S

(see Notation A.3.5.1). In view of Lemma 3.2.4.1, it will suffice to show thatthe upper horizontal map is an equivalence of simplicial categories. Lemmahighertopoi

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CHAPTER 3

A.3.6.17 implies that the left vertical map is an equivalence. Using Lemma
2.2.3.6 and Remark A.3.2.14, we deduce that the right vertical map is also
an equivalence. It will therefore suffice to show that φS is an equivalence.
Let U denote the collection of simplicial sets S for which φS is an equivalence. We will show that U satisfies the hypotheses of Lemma 2.2.3.5 andtherefore contains every simplicial set S. Conditions (i) and (ii) are obviously
satisfied, and condition (iii) follows from Lemma 3.2.4.1 and Proposition
3.2.3.4. We will verify condition (iv); the proof of (v) is similar.
Applying Corollary A.3.6.18, we deduce:
C[S]op(∗) The functor S 7→ (Set+[WS−1 ] carries homotopy colimit diagrams∆ )findexed by a partially ordered set to homotopy limit diagrams in Cat∆ .
Suppose we are given a pushout diagram
/ X0

X

Y

f


/ Y0

in which X, X 0 , Y ∈ U, where f is a cofibration. We wish to prove thatY 0 ∈ U. We have a commutative diagram
C[Y 0 ]op

(Set+
∆ )f

[WY−1
0 ]
φY 0

(
◦
0 −1
(Set+
∆ )/Y 0 [W Y 0 ]u/ (Set+ )◦ [W 0 −1
Y ]
∆ /Y

w

v

◦
0 −1
(Set+
∆ )/X 0 [W X 0 ]
(/ (Set+ )◦ [W 0 −1
X ].
∆ /X

Using (∗) and Corollary A.3.2.28, we deduce that φY 0 is an equivalence if◦
0 −1
and only if, for every pair of objects x, y ∈ (Set+∆ )/Y 0 [W Y 0 ], the diagram ofsimplicial sets
Map(Set+ )◦ 0 [W 0 −10 ] (x, y)

/ Map


Map(Set+ )◦ 0 [W 0 −10 ] (v(x), v(y))

/ Map

∆ /Y

∆ /X

◦
0 −1 (u(x), u(y))(Set+∆ )/Y [W Y ]

Y

X



◦
0 −1 (w(x), w(y))(Set+∆ )/X [W X ]is homotopy Cartesian. Since ψY 0 is a weak equivalence of simplicial categories, we may assume without loss of generality that x = ψY 0 (x) and◦y = ψY 0 (y) for some x, y ∈ (Set+∆ )/Y 0 . It will therefore suffice to prove thathighertopoi

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THE ∞-CATEGORY OF ∞-CATEGORIES

the equivalent diagram/ Map] (u(x), u(y))

Map]Y 0 (x, y)Map]X 0 (v(x), v(y))

Yg/ Map] (w(x), w(y))
X

is homotopy Cartesian. But this diagram is a pullback square, and the map
g is a Kan fibration by Lemma 3.2.4.2.
3.2.5 The Relative Nerve
In §3.1.3, we defined the straightening and unstraightening functors, whichgive rise to a Quillen equivalence of model categories
o
(Set+
∆ )/S

St+
φ
Un+
φ

/ (Set+ )C∆

whenever φ : C[S] → Cop is a weak equivalence of simplicial categories. For
many purposes, these constructions are unnecessarily complicated. For exam+
ple, suppose that F : C → Set+
∆ is a (weakly) fibrant diagram, so that Unφ (F)is a fibrant object of (Set+∆ )/S corresponding to a Cartesian fibration of simplicial sets X → S. For every vertex s ∈ S, the fiber Xs is an ∞-category
which is equivalent to F(φ(s)) but usually not isomorphic to F(φ(s)). In thespecial case where C is an ordinary category and φ : C[N(C)op ] → Cop is thecounit map, there is another version of unstraightening construction Un+φwhich does not share this defect. Our goal in this section is to introduce thissimpler construction, which we call the marked relative nerve F 7→ N+
F (C),and to study its basic properties.
Remark 3.2.5.1. To simplify the exposition which follows, the relative
nerve functor introduced below will actually be an alternative to the oppositeof the unstraightening functor
op op
F 7→ (Un+φ F ) ,

which is a right Quillen functor from the projective model structure on
+
C
(Set+
∆ ) to the coCartesian model structure on (Set∆ )/ N(C) .
Definition 3.2.5.2. Let C be a small category and let f : C → Set∆ be afunctor. We define a simplicial set Nf (C), the nerve of C relative to f , asfollows. For every nonempty finite linearly ordered set J, a map ∆J → Nf (C)consists of the following data:(1) A functor σ from J to C.
(2) For every nonempty subset J 0 ⊆ J having a maximal element j 0 , a
0map τ (J 0 ) : ∆J → f (σ(j 0 )).

highertopoi

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CHAPTER 3(3) For nonempty subsets J 00 ⊆ J 0 ⊆ J, with maximal elements j 00 ∈ J 00 ,j 0 ∈ J 0 , the diagram00τ (J 00 )
0∆Jτ (J 0 )∆J _/ f (σ(j 00 ))/ f (σ(j 0 ))is required to commute.
Remark 3.2.5.3. Let I denote the linearly ordered set [n], regarded asa category, and let f : I → Set∆ correspond to a composable sequence ofmorphisms φ : X0 → · · · → Xn . Then Nf (I) is closely related to the mapping
simplex M op (φ) introduced in §3.2.2. More precisely, there is a canonical map
Nf (I) → M op (φ) compatible with the projection to ∆n , which induces anisomorphism on each fiber.
Remark 3.2.5.4. The simplicial set Nf (C) of Definition 3.2.5.2 depends
functorially on f . When f takes the constant value ∆0 , there is a canonicalisomorphism Nf (C) ' N(C). In particular, for any functor f , there is a
canonical map Nf (C) → N(C); the fiber of this map over an object C ∈ C
can be identified with the simplicial set f (C).
Remark 3.2.5.5. Let C be a small category. The construction f 7→ Nf (C)determines a functor from (Set∆ )C to (Set∆ )/ N(C) . This functor admits a left
adjoint, which we will denote by X 7→ FX (C) (the existence of this functor
follows from the adjoint functor theorem). If X → N(C) is a left fibration,then FX (C) is a functor C → Set∆ which assigns to each C ∈ C a simplicial
set which is weakly equivalent to the fiber XC = X ×N(C) {C}; this follows
from Proposition 3.2.5.18 below.
Example 3.2.5.6. Let C be a small category and regard N(C) as an object of(Set∆ )/ N(C) via the identity map. Then FN(C) (C) ∈ (Set∆ )C can be identified
with the functor C 7→ N(C/C ).
Remark 3.2.5.7. Let g : C → D be a functor between small categories
and let f : D → Set∆ be a diagram. There is a canonical isomorphism of
simplicial sets Nf ◦g (C) ' Nf (D) ×N(D) N(C). In other words, the diagram ofcategories(Set∆ )DN• (D)g∗/ (Set∆ )CN• (C)N(g)∗/ (Set∆ )/ N(C)
(Set∆ )/ N(D)commutes up to canonical isomorphism. Here g ∗ denotes the functor given
by composition with g, and N(g)∗ the functor given by pullback along themap of simplicial sets N(g) : N(C) → N(D).

highertopoi

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THE ∞-CATEGORY OF ∞-CATEGORIES

Remark 3.2.5.8. Combining Remarks 3.2.5.5 and 3.2.5.7, we deduce thatfor any functor g : C → D between small categories, the diagram of left
adjoints(Set∆ )D oOg!

(Set∆ )CO

F• (D)F• (C)(Set∆ )/ N(D) o(Set∆ )/ N(C)commutes up to canonical isomorphism; here g! denotes the functor of left
Kan extension along g, and the bottom arrow is the forgetful functor given
by composition with N(g) : N(C) → N(D).
Notation 3.2.5.9. Let C be a small category and let f : C → Set∆ be afunctor. We let f op denote the functor C → Set∆ described by the formulaf op (C) = f (C)op . We will use a similar notation in the case where f is afunctor from C to the category Set+
∆ of marked simplicial sets.
Remark 3.2.5.10. Let C be a small category, let S = N(C)op , and let
φ : C[S] → Cop be the counit map. For each X ∈ (Set∆ )/ N(C) , there is a
canonical map
αC (X) : Stφ X op → FX (C)op .
The collection of maps {αC (X)} is uniquely determined by the following
requirements:(1) The morphism αC (X) depends functorially on X. More precisely, suppose we are given a commutative diagram of simplicial sets
/Y

f

X

"
|
N(C).
Then the diagram
Stφ X opαC (X)/ FX (C)op

Stφ f opFf (C)op
αC (Y )/ FY (C)op
Stφ Y opcommutes.

(2) The transformation αC depends functorially on C in the following sense:
for every functor g : C → D, if φ0 : C[(N D)op ] → Dop denotes thecounit map and X ∈ (Set∆ )/ N(C) , then the diagram
Stφ X op
Stφ0 X op

g! αC/ g! FX (C)opαD/ FX (D)ophighertopoi

April 9, 2017198

CHAPTER 3

commutes, where the vertical arrows are the isomorphisms provided
by Remark 3.2.5.8 and Proposition 2.2.1.1.
(3) Let C be the category associated to a partially ordered set P and let
X = N(C), regarded as an object of (Set∆ )/ N(C) via the identity map.
Then (Stφ X op ) ∈ (Set∆ )C can be identified with the functor p 7→ N Xp ,where for each p ∈ P we let Xp denote the collection of nonempty finite
chains in P having largest element p. Similarly, Example 3.2.5.6 allows
us to identify FX (C) ∈ (Set∆ )C with the functor p 7→ N{q ∈ P : q ≤ p}.
The map αC (X) : (Stφ X op ) → FX (C)op is induced by the map ofpartially ordered sets Xp → {q ∈ P : q ≤ p} which carries every chain
to its smallest element.
To see that the collection of maps {αC (X)}X∈(Set∆ )/ N(C) is determinedby these properties, we first note that because the functors Stφ and F• (C)commute with colimits, any natural transformation βC : Stφ (•op ) → F• (C)opis determined by its values βC (X) : Stφ (X op ) → FX (C)op in the case whereX = ∆n is a simplex. In this case, any map X → N C factors through theisomorphism X ' N[n], so we can use property (2) to reduce to the casewhere the category C is a partially ordered set and the map X → N(C) is an
isomorphism. The behavior of the natural transformation αC is then dictatedby property (3). This proves the uniqueness of the natural transformations
αC ; the existence follows by a similar argument.
The following result summarizes some of the basic properties of the relative
nerve functor:
Lemma 3.2.5.11. Let I be a category and let α : f → f 0 be a naturaltransformation of functors f, f 0 : C → Set∆ .
(1) Suppose that, for each I ∈ C, the map α(I) : f (I) → f 0 (I) is an inner
fibration of simplicial sets. Then the induced map Nf (C) → Nf 0 (C) is
an inner fibration.
(2) Suppose that, for each I ∈ I, the simplicial set f (I) is an ∞-category.
Then Nf (C) is an ∞-category.
(3) Suppose that, for each I ∈ C, the map α(I) : f (I) → f 0 (I) is a categorical fibration of ∞-categories. Then the induced map Nf (C) → Nf 0 (C)is a categorical fibration of ∞-categories.
Proof. Consider a commutative diagramΛni_/ Nf (I)
<


∆n



p

/ Nf 0 (C)highertopoi

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THE ∞-CATEGORY OF ∞-CATEGORIES

and let I be the image of {n} ⊆ ∆n under the bottom map. If 0 ≤ i < n,then the lifting problem depicted in the diagram above is equivalent to theexistence of a dotted arrow in an associated diagram
Λni
_

g

/ f (I)
<
α(I)


∆n


/ f 0 (I).

If α(I) is an inner fibration and 0 < i < n, then we conclude that this lifting
problem admits a solution. This proves (1). To prove (2), we apply (1) in thespecial case where f 0 is the constant functor taking the value ∆0 . It follows
that Nf (C) → N(C) is an inner fibration, so that Nf (C) is an ∞-category.
We now prove (3). According to Corollary 2.4.6.5, an inner fibration
D → E of ∞-categories is a categorical fibration if and only if the following
condition is satisfied:(∗) For every equivalence e : E → E 0 in E and every object D ∈ D liftingE, there exists an equivalence e : D → D0 in D lifting e.
We can identify equivalences in Nf 0 (C) with triples (g : I → I 0 , X, e : X 0 →Y ), where g is an isomorphism in C, X is an object of f 0 (I), X 0 is the imageof X in f 0 (I 0 ), and e : X 0 → Y is an equivalence in f 0 (I 0 ). Given a lifting
X of X to f (I), we can apply the assumption that α(I 0 ) is a categorical0
fibration (and Corollary 2.4.6.5) to lift e to an equivalence e : X → Y in0f (I 0 ). This produces the desired equivalence (g : I → I 0 , X, e : X → Y ) in
Nf (C).
We now introduce a slightly more elaborate version of the relative nerve
construction.
Definition 3.2.5.12. Let C be a small category and F : C → Set+
∆ a functor.
We let N+(C)denotethemarkedsimplicial
set
(N
(C),
M
),
where
f denotesfFFthe composition C → Set+
∆ → Set∆ and M denotes the collection of all edges
e of Nf (C) with the following property: if e : C → C 0 is the image of e inN(C) and σ denotes the edge of f (C 0 ) determined by e, then σ is a marked
edge of F(C 0 ). We will refer to N+
F (C) as the marked relative nerve functor.
Remark 3.2.5.13. Let C be a small category. We will regard the construc++ Ction F 7→ N+
F (C) as determining a functor from (Set∆ ) to (Set∆ )/ N(C) (seeRemark 3.2.5.4). This functor admits a left adjoint, which we will denote by
X 7→ F+(C).
X
Remark 3.2.5.14. Remark 3.2.5.8 has an evident analogue for the functors
F+ : for any functor g : C → D between small categories, the diagram of left

highertopoi

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CHAPTER 3

adjoints
(Set+
)D o
∆
O

g!

(Set+
)C
O∆

F+
• (D)

F+
• (C)

o
(Set+
∆ )/ N(D)(Set+∆ )/ C

commutes up to canonical isomorphism.
Lemma 3.2.5.15. Let C be a small category. Then
(1) The functor X 7→ FX (C) carries cofibrations in (Set∆ )/ N(C) to cofibrations in (Set∆ )C (with respect to the projective model structure).
(2) The functor X 7→ F+(C) carries cofibrations (with respect to the coX+ C
Cartesian model structure on (Set+∆ )/ N(C) ) to cofibrations in (Set∆ )(with respect to the projective model structure).
Proof. We will give the proof of (2); the proof of (1) is similar. It will suffice to+ C+show that the right adjoint functor N+• (C) : (Set∆ ) → Set∆ N(C) preserves
0+ C
trivial fibrations. Let F → F be a trivial fibration in (Set∆ ) with respect
to the projective model structure, so that for each C ∈ C the induced map
F(C) → F0 (C) is a trivial fibration of marked simplicial sets. We wish to+prove that the induced map N+
F (C) → NF 0 (C) is also a trivial fibration ofFmarked simplicial sets. Let f denote the composition C → Set+
∆ → Set∆ andlet f 0 be defined likewise. We must verify two things:(1) Every lifting problem of the form
/ Nf (C)∂ ∆ _n∆nu
/ Nf 0 (C)admits a solution. Let C ∈ C denote the image of the final vertex of∆n under the map u. Then it suffices to solve a lifting problem of theform∂ ∆ _n/ f (C)∆n
/ f 0 (C),which is possible since the right vertical map is a trivial fibration of
simplicial sets.
+
0(2) If e is an edge of N+
F (C) whose image e in NF 0 (C) is marked, then e isitself marked. Let e : C → C 0 be the image of e in N(C) and let σ denotehighertopoi

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201

THE ∞-CATEGORY OF ∞-CATEGORIES

the edge of F(C 0 ) determined by e. Since e0 is a marked edge of N+
F 0 (C),the image of σ in F0 (C 0 ) is marked. Since the map F(C 0 ) → F0 (C 0 ) is atrivial fibration of marked simplicial sets, we deduce that σ is a marked
edge of F(C 0 ), so that e is a marked edge of N+
F (C) as desired.
Remark 3.2.5.16. Let C be a small category, let S = N(C)op , and let
φ : C[S] → Cop be the counit map. For every X = (X, M ) ∈ (Set+∆ )/ N(C) ,the morphism αC (X) : Stφ (X op ) → FX (C)op of Remark 3.2.5.10 induces a
op
+
→ F+
(C)op , which we will denote by αC
natural transformation St+
(X).
φX
X
+
We will regard the collection of morphisms {αC (X)}X∈(Set+ )/ N(C) as deter∆mining a natural transformation of functors
op+
op
αC : St+φ (• ) → F• (C) .
Lemma 3.2.5.17. Let C be a small category, let S = N(C)op , let φ : C[S] →Cop be the counit map, and let C ∈ C be an object. Then
(1) For every X ∈ (Set∆ )/ N(C) , the map αC (X) : Stφ (X op ) → FX (C)opof Remark 3.2.5.10 induces a weak homotopy equivalence of simplicial
sets Stφ (X op )(C) → FX (C)(C)op .
op

+
+
+
op
(2) For every X ∈ (Set+∆ )/ N(C) , the map αC (X) : Stφ (X ) → FX (C)opof Remark 3.2.5.16 induces a Cartesian equivalence St+
φ (X )(C) →+op
FX (C)(C) .

Proof. We will give the proof of (2); the proof of (1) is similar but easier. Let+
us say that an object X ∈ (Set+∆ )/ N(C) is good if the map αC (X) is a weakequivalence. We wish to prove that every object X = (X, M ) ∈ (Set+∆ )/ N(C)is good. The proof proceeds in several steps.
+(A) Since the functors St+
φ and F• (C) both commute with filtered colimits,
the collection of good objects of (Set+∆ )/ N(C) is stable under filteredcolimits. We may therefore reduce to the case where the simplicial set
X has only finitely many nondegenerate simplices.

(B) Suppose we are given a pushout diagram
X

Y

g

f

/ X0

/ Y0

in the category (Set+∆ )/ N(C) . Suppose that either f or g is a cofibra00tion and that the objects X, X , and Y are good. Then Y is good.
+
This follows from the fact that the functors Stφ and F+• (C) preserve
cofibrations (Proposition 3.2.1.7 and Lemma 3.2.5.15) together withC
the observation that the projective model structure on (Set+∆ ) is leftproper.

highertopoi

April 9, 2017202

CHAPTER 3+(C) Suppose that X ' ∆n for n ≤ 1. In this case, the map αC(X) is an
isomorphism (by direct calculation), so that X is good.

(D) We now work by induction on the number of nondegenerate marked
edges of X. If this number is nonzero, then there exists a pushout
diagram(∆1 )[/ (∆1 )]


Y


/ X,

where Y has fewer nondegenerate marked edges than X, so that Y isgood by the inductive hypothesis. The marked simplicial sets (∆1 )[ and(∆1 )] are good by virtue of (C), so that (B) implies that X is good. We
may therefore reduce to the case where X contains no nondegeneratemarked edges, so that X ' X [ .
(E) We now argue by induction on the dimension n of X and the numberof nondegenerate n-simplices of X. If X is empty, there is nothing toprove; otherwise, we have a pushout diagram
/ ∆n
∂ ∆n

Y


/ X.

The inductive hypothesis implies that (∂ ∆n )[ and Y [ are good. Invoking step (B), we can reduce to the case where X is an n-simplex. In
view of (C), we may assume that n ≥ 2.
```
Let Z = ∆{0,1} {1} ∆{1,2} {1} · · · {n−1} ∆{n−1,n} , so that Z ⊆ X
is an inner anodyne inclusion. We have a commutative diagramop [St+
φ (Z )

u

φ

v


op
F+
(C)
[
Z

/ St+ (X op )[w

/ F+ [ (C)op .
X

The inductive hypothesis implies that v is a weak equivalence, andProposition 3.2.1.11 implies that u is a weak equivalence. To complete
the proof, it will suffice to show that w is a weak equivalence.
(F ) The map X → N(C) factors as a compositiong∆n ' N([n]) → N(C).
Using Remark 3.2.5.14 (together with the fact that the left Kan extension functor g! preserves weak equivalences between projectively
cofibrant objects), we can reduce to the case where C = [n] and themap X → N(C) is an isomorphism.

highertopoi

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THE ∞-CATEGORY OF ∞-CATEGORIES(G) Fix an object i ∈ [n]. A direct computation shows that the map
F+(C)(i) → F+
(C)(i) can be identified with the inclusion
Z[
X[
a
a
a
(∆{0,1}∆{1,2}···∆{i−1,i} )op,[ ⊆ (∆i )op,[ .
{1}

{1}

{i−1}

This inclusion is marked anodyne and therefore an equivalence ofmarked simplicial sets, as desired.

Proposition 3.2.5.18. Let C be a small category. Then
(1) The functors F• (C) and N• (C) determine a Quillen equivalence between(Set∆ )/ N(C) (endowed with the covariant model structure) and (Set∆ )C(endowed with the projective model structure).
+(2) The functors F+• (C) and N• (C) determine a Quillen equivalence be+
tween (Set∆ )/ N(C) (endowed with the coCartesian model structure) andC(Set+∆ ) (endowed with the projective model structure).

Proof. We will give the proof of (2); the proof of (1) is similar but easier. We+
first show that the adjoint pair (F+• (C), N• (C)) is a Quillen adjunction. It+
will suffice to show that the functor F• (C) preserves cofibrations and weakequivalences. The case of cofibrations follows from Lemma 3.2.5.15, and thecase of weak equivalences from Lemma 3.2.5.17 and Corollary 3.2.1.16. To+
prove that (F+• (C), N• (C)) is a Quillen equivalence, it will suffice to showthat the left derived functor LF+• (C) induces an equivalence from the homo+ C
topy category h(Set+∆ )/ N(C) to the homotopy category h(Set∆ ) . In view ofLemma 3.2.5.17, it will suffice to prove an analogous result for the straightop
opening functor St+
φ , where φ denotes the counit map C[N(C) ] → C . We
now invoke Theorem 3.2.0.1.
Corollary 3.2.5.19. Let C be a small category and let α : f → f 0 be anatural transformation of functors f, f 0 : C → Set∆ . Suppose that, for eachC ∈ C, the induced map f (C) → f 0 (C) is a Kan fibration. Then the induced
map Nf (C) → Nf 0 (C) is a covariant fibration in (Set∆ )/ N(C) . In particular,
if each f (C) is Kan complex, then the map Nf (C) → N(C) is a left fibrationof simplicial sets.
Corollary 3.2.5.20. Let C be a small category and F : C → Set+
∆ a fibrantop
Cop
object of (Set+denote the counit∆ ) . Let S = N(C) and let φ : C[S ] → C+
map. Then the natural transformation αCof Remark 3.2.5.16 induces a+ op
op
weak equivalence NF (C) → (Unφ F ) (with respect to the Cartesian model
structure on (Set+∆ )/S op ).
+Proof. It suffices to show that αC
induces an isomorphism of right derived+
functors R N• (C)op → R(Unφ •op ), which follows immediately from Lemma
3.2.5.17.

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Proposition 3.2.5.21. Let C be a category and let f : C → Set∆ be afunctor such that f (C) is an ∞-category for each C ∈ C. Then
(1) The projection p : Nf (C) → N(C) is a coCartesian fibration of simplicial sets.
(2) Let e be an edge of Nf (C) covering a morphism C → C 0 in C. Thene is p-coCartesian if and only if the corresponding edge of f (C 0 ) is anequivalence.
(3) The coCartesian fibration p is associated to the functor N(f ) : N(C) →Cat∞ (see §3.3.2).
Proof. Let F : C → Set+
∆ be the functor described by the formula F(C) =C
f (C)\ . Then F is a projectively fibrant object of (Set+∆ ) . Invoking Propo+
sition 3.2.5.18, we deduce that NF (C) is a fibrant object of (Set+∆ )/ N(C) . Invoking Proposition 3.1.4.1, we deduce that the underlying map p : Nf (C) →N(C) is a coCartesian fibration of simplicial sets and that the p-coCartesianmorphisms of Nf (C) are precisely the marked wedges of N+
F (C). This proves(1) and (2). To prove (3), we let S = N(C) and φ : C[S]op → Cop be thecounit map. By definition, a coCartesian fibration X → N(C) is associatedto f if and only if it is equivalent to (Unφ f op )op ; the desired equivalence isfurnished by Corollary 3.2.5.20.

3.3 APPLICATIONS
The purpose of this section is to survey some applications of technology
developed in §3.1 and §3.2. In §3.3.1, we give some applications to the theoryof Cartesian fibrations. In §3.3.2, we will introduce the language of classifying
maps which will allow us to exploit the Quillen equivalence provided by
Theorem 3.2.0.1. Finally, in §3.3.3 and §3.3.4, we will use Theorem 3.2.0.1
to give explicit constructions of limits and colimits in the ∞-category Cat∞
(and also in the ∞-category S of spaces).
3.3.1 Structure Theory for Cartesian Fibrations
The purpose of this section is to prove that Cartesian fibrations between
simplicial sets enjoy several pleasant properties. For example, every Cartesian fibration is a categorical fibration (Proposition 3.3.1.7), and categoricalequivalences are stable under pullbacks by Cartesian fibrations (Proposition
3.3.1.3). These results are fairly easy to prove for Cartesian fibrations X → S
in the case where S is an ∞-category. Theorem 3.2.0.1 provides a method
for reducing to this special case:
Proposition 3.3.1.1. Let p : S → T be a categorical equivalence of simplicial sets. Then the forgetful functor+
p! : (Set+∆ )/S → (Set∆ )/Thighertopoi

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and its right adjoint p∗ induce a Quillen equivalence between (Set+∆ )/S and
(Set+
)
.
∆ /T
Proof. Let C = C[S]op and D = C[T ]op . Consider the following diagram ofmodel categories and left Quillen functors:
(Set+
∆ )/S

p!

∆

St+
S


C
(Set+
∆)

/ (Set+ )/T

C[p]!



St+
T

/ (Set+ )D .
∆According to Proposition 3.2.1.4, this diagram commutes (up to natural isomorphism). Theorem 3.2.0.1 implies that the vertical arrows are Quillen
equivalences. Since p is a categorical equivalence, C[p] is an equivalenceof simplicial categories, so that C[p]! is a Quillen equivalence (Proposition
A.3.3.8). It follows that (p! , p∗ ) is a Quillen equivalence as well.
Corollary 3.3.1.2. Let p : X → S be a Cartesian fibration of simplicial sets
and let S → T be a categorical equivalence. Then there exists a Cartesianfibration Y → T and an equivalence of X with S×T Y (of Cartesian fibrations
over S).
Proof. Proposition 3.3.1.1 implies that the right derived functor Rp∗ is essentially surjective.
As we explained in Remark 2.2.5.3, the Joyal model structure on Set∆ is
not right proper. In other words, it is possible to have a categorical fibration
X → S and a categorical equivalence T → S such that the induced map X×S
T → X is not a categorical equivalence. This poor behavior of categoricalfibrations is one of the reasons that they do not play a prominent role in thetheory of ∞-categories. Working with a stronger notion of fibration corrects
the problem:
Proposition 3.3.1.3. Let p : X → S be a Cartesian fibration and let T →S be a categorical equivalence. Then the induced map X ×S T → X is a
categorical equivalence.
Proof. We first suppose that the map T → S is inner anodyne. By meansof a simple argument, we may reduce to the case where T → S is a middle
horn inclusion Λni ⊆ ∆n , where 0 < i < n. According to Proposition 3.2.2.7,
there exists a sequence of maps
φ : A0 ← · · · ← An
and a map M (φ) → X which is a categorical equivalence, such that M (φ)×S
T → X ×S T is also a categorical equivalence. Consequently, it suffices toshow that the inclusion M (φ) ×S T ⊆ M (φ) is a categorical equivalence. But
this map is a pushout of the inclusion An × Λni ⊆ An × ∆n , which is inner
anodyne.

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We now treat the general case. Choose an inner anodyne map T` → T 0 ,where T 0 is an ∞-category. Then choose an inner anodyne map T 0 T S →S 0 , where S 0 is also an ∞-category. The map S → S 0 is inner anodyne; inparticular it is a categorical equivalence, so by Corollary 3.3.1.2 there is a
Cartesian fibration X 0 → S 0 and an equivalence X → X 0 ×S 0 S of Cartesianfibrations over S. We have a commutative diagram
X8 0 ×S 0 T

u0/ X 0 ×S 0 T 0
u00

u

$

X ×S T

:X

0

v 00

v

&

v0

X

/ X 0 ×S 0 S.

Consequently, to prove that v is a categorical equivalence, it suffices to showthat every other arrow in the diagram is a categorical equivalence. The mapsu and v 0 are equivalences of Cartesian fibrations and therefore categoricalequivalences. The other three maps correspond to special cases of the assertion we are trying to prove. For the map u00 , we have the special case of themap S 0 → T 0 , which is an equivalence of ∞-categories: in this case we simply
apply Corollary 2.4.4.5. For the maps u0 and v 00 , we need to know that theassertion of the proposition is valid in the special case of the maps S → S 0
and T → T 0 . Since these maps are inner anodyne, the proof is complete.
Corollary 3.3.1.4. Let
X

/ X0


S


/ S0

p0

be a pullback diagram of simplicial sets, where p0 is a Cartesian fibration.
Then the diagram is homotopy Cartesian (with respect to the Joyal model
structure).
Proof. Choose an inner-anodyne map S 0 → S 00 , where S 00 is an ∞-category.
Using Proposition 3.3.1.1, we may assume without loss of generality thatX 0 ' X 00 ×S 00 S 0 , where X 00 → S 00 is a Cartesian fibration. Now choose afactorization
θ0

θ 00

S → T → S 00 ,where θ0 is a categorical equivalence and θ00 is a categorical fibration. The
diagram
T → S 00 ← X 00
is fibrant. Consequently, the desired conclusion is equivalent to the assertion
that the map X → T ×S 00 X 00 is a categorical equivalence, which follows
immediately from Proposition 3.3.1.3.

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We now prove a stronger version of Corollary 2.4.4.4 which does not requirethat the base S is a ∞-category.
Proposition 3.3.1.5. Suppose we are given a diagram of simplicial sets
/Y

f

X
p



S,



q

where p and q are Cartesian fibrations and f carries p-Cartesian edges to
q-Cartesian edges. The following conditions are equivalent:(1) The map f is a categorical equivalence.
(2) For each vertex s of S, f induces a categorical equivalence Xs → Ys .
(3) The map X \ → Y \ is a Cartesian equivalence in (Set+∆ )/S .
Proof. The equivalence of (2) and (3) follows from Proposition 3.1.3.5. We
next show that (2) implies (1). By virtue of Proposition 3.2.2.8, we may
reduce to the case where S is a simplex. Then S is an ∞-category, and thedesired result follows from Corollary 2.4.4.4. (Alternatively, we could observethat (2) implies that f has a homotopy inverse.)
To prove that (1) implies (3), we choose an inner anodyne map j : S → S 0 ,where S 0 is an ∞-category. Let X \ denote the object of (Set+∆ )/S associated\to the Cartesian fibration p : X → S and let j! X denote the same marked
simplicial set, regarded as an object of (Set+∆ )/T . Choose a marked anodyne\
map j! X \ → X 0 , where X 0 → S 0 is a Cartesian fibration. By Proposition\
3.3.1.1, the map X \ → j ∗ X 0 is a Cartesian equivalence, so that X →0
X ×S 0 S is a categorical equivalence. According to Proposition 3.3.1.3, themap X 0 ×S 0 S → X 0 is a categorical equivalence; thus the composite map
X → X 0 is a categorical equivalence.
Similarly, we may choose a marked anodyne mapa\\X0
j! Y \ → Y 0
j! X \

for some Cartesian fibration Y 0 → S 0 . Since the Cartesian model structure is\left proper, the map j! Y \ → Y 0 is a Cartesian equivalence, so we may argue
as above to deduce that Y → Y 0 is a categorical equivalence. Now consider
the diagram
X

X0

f

f0

/Y

/ Y 0.

We have argued that the vertical maps are categorical equivalences. The
map f is a categorical equivalence by assumption. It follows that f 0 is ahighertopoi

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CHAPTER 3

categorical equivalence. Since S 0 is an ∞-category, we may apply Corollary
2.4.4.4 to deduce that Xs0 → Ys0 is a categorical equivalence for each object\\s of S 0 . It follows that X 0 → Y 0 is a Cartesian equivalence in (Set+∆ )/S , sothat we have a commutative diagram
X\

/ Y\


\
j∗X 0


/ j∗Y 0\where the vertical and bottom horizontal arrows are Cartesian equivalencesin (Set+∆ )/S . It follows that the top horizontal arrow is a Cartesian equivalence as well, so that (3) is satisfied.
Corollary 3.3.1.6. Let
W

/X


Y


/Z

/S

be a diagram of simplicial sets. Suppose that every morphism in this diagram
is a right fibration and that the square is a pullback. Then the diagram ishomotopy Cartesian with respect to the contravariant model structure on(Set∆ )/S .
Proof. Choose a fibrant replacement
X0 → Z0 ← Y 0
for the diagram
X→Z←Y
in (Set∆ )/S and let W 0 = X 0 ×Z 0 Y 0 . We wish to show that the induced map
i : W → W 0 is a covariant equivalence in (Set∆ )/S . According to Corollary
2.2.3.13, it will suffice to show that, for each vertex s of S, the map of fibers
Ws → Ws0 is a homotopy equivalence of Kan complexes. To prove this, weobserve that we have a natural transformation of diagrams from
Ws

/ Xs


Ys


/ Zs

Ws0

/ Xs0


Ys0


/ Zs0

to

highertopoi

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THE ∞-CATEGORY OF ∞-CATEGORIES

which induces homotopy equivalences
Xs → Xs0

Ys → Ys0

Zs → Zs0(Corollary 2.2.3.13), where both diagrams are homotopy Cartesian (Proposition 2.1.3.1).
Proposition 3.3.1.7. Let p : X → S be a Cartesian fibration of simplicial
sets. Then p is a categorical fibration.
Proof. Consider a diagram
/X
>

A _
f

p

i



g
/S
B
of simplicial sets, where i is an inclusion and a categorical equivalence. We
must demonstrate the existence of the indicated dotted arrow. Choose acategorical equivalence j : S → T , where T is an ∞-category. By Corollary
3.3.1.2, there exists a Cartesian fibration q : Y → T such that Y ×T S isequivalent to X. Thus there exist maps
u : X → Y ×T S
v : Y ×T S → X
such that u ◦ v and v ◦ u are homotopic to the identity (over S).
Consider the induced diagram
/Y
A _
>
i


B.

f0

Since Y is an ∞-category, there exists a dotted arrow f 0 making the diagram
commutative. Let g 0 = q ◦ f 0 : B → T . We note that g 0 |A = (j ◦ g)|A.
Since T is an ∞-category and i is a categorical equivalence, there exists a
homotopy B × ∆1 → T from g 0 to j ◦ g which is fixed on A. Since q is a
Cartesian fibration, this homotopy lifts to a homotopy from f 0 to some map
f 00 : B → Y , so that we have a commutative diagram
/Y
A _
>
i


B

q


/ T.

f 00

Consider the composite map(f 00 ,g)v

f 000 : B → Y ×T S → X.
Since f 0 is homotopic to f 00 and v◦u is homotopic to the identity, we conclude
that f 000 |A is homotopic to f0 (via a homotopy which is fixed over S). Since
p is a Cartesian fibration, we can extend h to a homotopy from f 000 to thedesired map f .

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In general, the converse to Proposition 3.3.1.7 fails: a categorical fibrationof simplicial sets X → S need not be a Cartesian fibration. This is clear since
the property of being a categorical fibration is self-dual, while the conditionof being a Cartesian fibration is not. However, in the case where S is a Kancomplex, the theory of Cartesian fibrations is self-dual, and we have thefollowing result:
Proposition 3.3.1.8. Let p : X → S be a map of simplicial sets, where S
is a Kan complex. The following assertions are equivalent:(1) The map p is a Cartesian fibration.
(2) The map p is a coCartesian fibration.
(3) The map p is a categorical fibration.
Proof. We will prove that (1) is equivalent to (3); the equivalence of (2) and(3) follows from a dual argument. Proposition 3.3.1.7 shows that (1) implies(3) (for this implication, the assumption that S is a Kan complex is notneeded).
Now suppose that (3) holds. Then X is an ∞-category. Since every edge of
S is an equivalence, the p-Cartesian edges of X are precisely the equivalencesin X. It therefore suffices to show that if y is a vertex of X and e : x → p(y)is an edge of S, then e lifts to an equivalence e : x → y in S. Since S is a Kancomplex, we can find a contractible Kan complex K and a map q : K → S
such that e is the image of an edge e0 : x0 → y 0 in K. The inclusion {y 0 } ⊆ K
is a categorical equivalence; since p is a categorical fibration, we can liftq to a map q : K → X with q(y 0 ) = y. Then e = q(e0 ) has the desired
properties.
3.3.2 Universal Fibrations
In this section, we will apply Theorem 3.2.0.1 to construct a universal Cartesian fibration. Recall that Cat∞ is defined to be the nerve of the simplicial+ ◦category Cat∆
∞ = (Set∆ ) of ∞-categories. In particular, we may regard the++ Cat∆∞.
inclusion Cat∆∞ ,→ Set∆ as a (projectively) fibrant object F ∈ (Set∆ )+Applying the unstraightening functor UnCatop, we obtain a fibrant object∞
op , which we may identify with a Cartesian fibration q : Z →of (Set+)∆ / Cat∞Catop∞ . We will refer to q as the universal Cartesian fibration. We observethat the objects of Cat∞ can be identified with ∞-categories and that thefiber of q over an ∞-category C can be identified with U (C), where U is
the functor described in Lemma 3.2.3.1. In particular, there is a canonicalequivalence of ∞-categories
C → U (C) = Z ×Catop{C}.
∞
Thus we may think of q as a Cartesian fibration which associates to eachobject of Cat∞ the associated ∞-category.

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April 9, 2017211

Remark 3.3.2.1. The ∞-categories Cat∞ and Z are large. However, theuniversal Cartesian fibration q is small in the sense that for any small simop
plicial set S and any map f : S → Catop∞ , the fiber product S ×Cat∞ Z issmall. This is because the fiber product can be identified with Un+(F|C[S]),
φ
where φ : C[S] → Set+isthecomposition
of
C[f
]
with
the
inclusion.
∆
Definition 3.3.2.2. Let p : X → S be a Cartesian fibration of simplicial
sets. We will say that a functor f : S → Catop∞ classifies p if there is an
S ' Un+equivalence of Cartesian fibrations X → Z ×CatopS f.
∞
Dually, if p : X → S is a coCartesian fibration, then we will say that afunctor f : S → Cat∞ classifies p if f op classifies the Cartesian fibration
pop : X op → S op .
Remark 3.3.2.3. Every Cartesian fibration X → S between small simplicial sets admits a classifying map φ : S → Catop∞ , which is uniquely determined up to equivalence. This is one expression of the idea that Z → Catop∞is a universal Cartesian fibration. However, it is not immediately obviousthat this property characterizes Cat∞ up to equivalence because Cat∞ is notitself small. To remedy the situation, let us consider an arbitrary uncountable regular cardinal κ, and let Cat∞ (κ) denote the full subcategory of Cat∞
spanned by the κ-small ∞-categories. We then deduce the following:(∗) Let p : X → S be a Cartesian fibration between small simplicial sets.
Then p is classified by a functor χ : S → Cat∞ (κ)op if and only if, forevery vertex s ∈ S, the fiber Xs is essentially κ-small. In this case, χis determined uniquely up to homotopy.
Enlarging the universe and applying (∗) in the case where κ is the supremum
of all small cardinals, we deduce the following property:(∗0 ) Let p : X → S be a Cartesian fibration between simplicial sets which
are not necessarily small. Then p is classified by a functor χ : S → Catop∞if and only if, for every vertex s ∈ S, the fiber Xs is essentially small.
In this case, χ is determined uniquely up to homotopy.
This property evidently determines the ∞-category Cat∞ (and the Cartesianfibration q : Z → Catop∞ ) up to equivalence.
Warning 3.3.2.4. The terminology of Definition 3.3.2.2 has the potential
to cause confusion in the case where p : X → S is both a Cartesian fibration
and a coCartesian fibration. In this case, p is classified both by a functor
S → Catop∞ (as a Cartesian fibration) and by a functor S → Cat∞ (as acoCartesian fibration).
The category Kan of Kan complexes can be identified with a full (simplicial) subcategory of Cat∆
∞ . Consequently we may identify the ∞-category
S of spaces with the full simplicial subset of Cat∞ , spanned by the vertices
which represent ∞-groupoids. We let Z0 = Z ×CatopSop be the restriction∞of the universal Cartesian fibration. The fibers of q 0 : Z0 → Sop are Kanhighertopoi

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CHAPTER 3

complexes (since they are equivalent to the ∞-categories represented by thevertices of S). It follows from Proposition 2.4.2.4 that q 0 is a right fibration.
We will refer to q 0 as the universal right fibration.
Proposition 2.4.2.4 translates immediately into the following characterization of right fibrations:
Proposition 3.3.2.5. Let p : X → S be a Cartesian fibration of simplicial
sets. The following conditions are equivalent:(1) The map p is a right fibration.
op(2) Every functor f : S → Catop⊆∞ which classifies p factors through S
op
Cat∞ .

(3) There exists a functor f : S → Sop which classifies p.
Consequently, we may speak of right fibrations X → S being classified byfunctors S → Sop and left fibrations being classified by functors S → S.
The ∞-category ∆0 corresponds to a vertex of Cat∞ which we will denoteby ∗. The fiber of q over this point may be identified with U ∆0 ' ∆0 ;consequently, there is a unique vertex ∗Z of Z lying over ∗. We note that∗ and ∗Z belong to the subcategories S and Z0 . Moreover, we have thefollowing:
Proposition 3.3.2.6. Let q 0 : Z0 → Sop be the universal right fibration.
The vertex ∗Z is a final object of the ∞-category Z0 .
Proof. Let n > 0 and let f0 : ∂ ∆n → Z0 have the property that f0 carries
the final vertex of ∆n to ∗Z . We wish to show that there exists an extension∂ ∆ _n

f0

/
<Z

f


∆n
(in which case the map f automatically factors through Z0 ).
Let D denote the simplicial category containing Sop∆ as a full subcategory
together with one additional object X, with the morphisms given by
MapD (K, X) = K
MapD (X, X) = ∗MapD (X, K) = ∅op
for all K ∈ S∆ . Let C = C[∆n ? ∆0 ] and let C0 denote the subcategory C0 =C[∂ ∆n ? ∆0 ] ⊆ C. We will denote the objects of C by {v0 , . . . , vn+1 }. Giving

the map f0 is tantamount to giving a simplicial functor F0 : C0 → D withF0 (vn+1 ) = X, and constructing f amounts to giving a simplicial functor
F : C → D which extends F0 .

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We note that the inclusion MapC0 (vi , vj ) → MapC (vi , vj ) is an isomorphism unless i = 0 and j ∈ {n, n + 1}. Consequently, to define F , it suffices
to find extensions
MapC0 (v 0 , vn )_j

/ MapD (F0 v0 , F0 vn )5MapC (v0 , vn )MapC0 (v0 , vn+1 )_j0/ MapD (F0 v0 , F0 vn+1 )5MapC (v0 , vn+1 )such that the following diagram commutes:
MapC (v0 , vn ) × MapC (vn , vn+1 )/ MapC (v0 , vn+1 )MapD (F0 v0 , F0 vn ) × MapD (F0 vn , F0 vn+1 )/ MapD (F0 v0 , F0 vn+1 ).

We note that MapC (vn , vn+1 ) is a point. In view of the assumption thatf0 carries the final vertex of ∆n to ∗Z , we see that MapD (F vn , F vn+1 ) isa point. It follows that, for any fixed choice of j 0 , there is a unique choiceof j for which the above diagram commutes. It therefore suffices to showthat j 0 exists. Since MapD (F0 v0 , X) is a Kan complex, it will suffice to showthat the inclusion MapC0 (v0 , vn+1 ) → MapC (v0 , vn+1 ) is an anodyne map ofsimplicial sets. In fact, it is isomorphic to the inclusiona({1} × (∆1 )n−1 )(∆1 × ∂(∆1 )n−1 ) ⊆ ∆1 × ∆n−1 ,{1}×∂(∆1 )n−1

which is the smash product of the cofibration ∂(∆1 )n−1 ⊆ (∆1 )n−1 and the
anodyne inclusion {1} ⊆ ∆1 .
Corollary 3.3.2.7. The universal right fibration q 0 : Z0 → Sop is represented by a final object of S.
Proof. Combine Propositions 3.3.2.6 and 4.4.4.5.
Corollary 3.3.2.8. Let p : X → S be a left fibration between small simplicial
sets. Then there exists a map S → S and an equivalence of left fibrations
X ' S ×S S∗/ .
Proof. Combine Corollary 3.3.2.7 with Remark 3.3.2.3.

highertopoi

April 9, 2017214

CHAPTER 3

3.3.3 Limits of ∞-Categories◦
The ∞-category Cat∞ can be identified with the simplicial nerve of (Set+∆) .
It follows from Corollary 4.2.4.8 that Cat∞ admits (small) limits and colimits, which can be computed in terms of homotopy (co)limits in the model
category Set+
∆ . For many applications, it is convenient to be able to constructlimits and colimits while working entirely in the setting of ∞-categories. We
will describe the construction of limits in this section; the case of colimits
will be discussed in §3.3.4.
Let p : S op → Cat∞ be a diagram in Cat∞ . Then p classifies a Cartesianfibration q : X → S. We will show (Corollary 3.3.3.2 below) that the limit
lim(p) ∈ Cat∞ can be identified with the ∞-category of Cartesian sections←−of q. We begin by proving a more precise assertion:

Proposition 3.3.3.1. Let K be a simplicial set, p : K . → Catop∞ a diagramin the ∞-category of spaces, X → K . a Cartesian fibration classified by p,
and X = X ×K . K. The following conditions are equivalent:(1) The diagram p is a limit of p = p|K.
(2) The restriction map\

θ : Map[K . ((K . )] , X ) → Map[K (K ] , X \ )is an equivalence of ∞-categories.
Proof. According to Proposition 4.2.3.14, there exists a small category C anda cofinal map f : N(C) → K; let C = C ?[0] be the category obtained from C
by adjoining a new final object and let f : N(C) → K . be the induced map(which is also cofinal). The maps f and f are contravariant equivalences in(Set∆ )/K . and therefore induce Cartesian equivalences
N(C)] → K ]N(C)] → (K . )] .

We have a commutative diagram\θ/ Map[K . (K ] , X \ )\
Map[K . (N(C)] , X )θ0/ Map[K . (N(C)] , X \ ).

Map[K . ((K . )] , X )The vertical arrows are categorical equivalences. Consequently, condition (2)holds for p : K . → Catop∞ if and only if condition (2) holds for the composition
N(C) → K . → Catop.
∞ We may therefore assume without loss of generalitythat K = N(C).
Using Corollary 4.2.4.7, we may further suppose that p is obtained as theop◦simplicial nerve of a functor F : C → (Set+∆ ) . Changing F if necessary, weop
may suppose that it is a strongly fibrant diagram in Set+
∆ . Let F = F| C . Letop
op
φ : C[K . ]op → C be the counit map and φ : C[K]op → C the restriction

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THE ∞-CATEGORY OF ∞-CATEGORIES

of φ. We may assume without loss of generality that X = St+
φ F. We have a(not strictly commutative) diagram of categories and functors
Set+
∆

×K ]

/ (Set+ )/K
∆

St+
φ

St+
∗


Set+
∆

δ /


Cop
(Set+
,
∆)

where δ denotes the diagonal functor. This diagram commutes up to a natural transformation
+
]
]
+
+
St+
φ (K × Z) → Stφ (K )  St∗ (Z) → δ(St∗ Z).

Here the first map is a weak equivalence by Proposition 3.2.1.13, and the second map is a weak equivalence because LSt+
φ is an equivalence of categories(Theorem 3.2.0.1) and therefore carries the final object K ] ∈ h(Set+∆ )/K toCopa final object of h(Set+).
We
therefore
obtainadiagramofrightderived∆functors
o
hSet+
O ∆

Γ

h(Set+
)
O∆ /K
R Un+
φ

R Un+
∗

o
hSet+
∆

C
h(Set+,∆)
op

C
which commutes up to natural isomorphism, where we regard (Set+as∆)
equipped with the injective model structure described in §A.3.3. Similarly,
we have a commutative diagram
op

o
hSet+
O ∆

Γ0

h(Set+∆ )/K .
O
R Un+

R Un+
∗

φ

o
hSet+
∆

op

C
h(Set+.
∆)

Condition (2) is equivalent to the assertion that the restriction map\

Γ0 (X ) → Γ(X \ )is an isomorphism in hSet+∆ . Since the vertical functors in both diagrams
are equivalences of categories (Theorem 3.2.0.1), this is equivalent to theassertion that the map
lim F → lim F←−←−
is a weak equivalence in Set+
∆ . Since C has an initial object v, (2) is equivalent◦
to the assertion that F exhibits F(v) as a homotopy limit of F in (Set+∆) .
Using Theorem 4.2.4.1, we conclude that (1) ⇔ (2), as desired.
It follows from Proposition 3.3.3.1 that limits in Cat∞ are computed byforming ∞-categories of Cartesian sections:highertopoi

April 9, 2017216

CHAPTER 3

Corollary 3.3.3.2. Let p : K → Catop∞ be a diagram in the ∞-category Cat∞of spaces and let X → K be a Cartesian fibration classified by p. There is anatural isomorphism
lim(p) ' Map[K (K ] , X \ )←−in the homotopy category hCat∞ .
0.
Proof. Let p : (K . )op → Catop∞ be a limit of p and let X → K be a
Cartesian fibration classified by p. Without loss of generality, we may suppose
X ' X 0 ×K . K. We have maps\\

Map[K (K ] , X \ ) ← Map[K . ((K . )] , X 0 ) → Map[K . ({v}] , X 0 ),where v denotes the cone point of K . . Proposition 3.3.3.1 implies that theleft map is an equivalence of ∞-categories. Since the inclusion {v}] ⊆ (K . )]is marked anodyne, the map on the right is a trivial fibration. We now\
conclude by observing that the space Map[K . ({v}] , X 0 ) ' X 0 ×K . {v} canbe identified with p(v) = lim(p).
←−Using Proposition 3.3.3.1, we can easily deduce an analogous characterization of limits in the ∞-category of spaces.
Corollary 3.3.3.3. Let K be a simplicial set, p : K / → S a diagram in
the ∞-category of spaces and X → K / a left fibration classified by p. The
following conditions are equivalent:(1) The diagram p is a limit of p = p|K.
(2) The restriction map
MapK / (K / , X) → MapK / (K, X)
is a homotopy equivalence of Kan complexes.
Proof. The usual model structure on Set∆ is a localization of the Joyal model
structure. It follows that the inclusion Kan ⊆ Cat∆∞ preserves homotopy limits (of diagrams indexed by categories). Using Theorem 4.2.4.1, Proposition
4.2.3.14, and Corollary 4.2.4.7, we conclude that the inclusion S ⊆ Cat∞
preserves (small) limits. The desired equivalence now follows immediatelyfrom Proposition 3.3.3.1.
Corollary 3.3.3.4. Let p : K → S be a diagram in the ∞-category S ofspaces, and let X → K be a left fibration classified by p. There is a natural
isomorphism
lim(p) ' MapK (K, X)←−
in the homotopy category H of spaces.
Proof. Apply Corollary 3.3.3.2.
Remark 3.3.3.5. It is also possible to adapt the proof of Proposition 3.3.3.1
to give a direct proof of Corollary 3.3.3.3. We leave the details to the reader.

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April 9, 2017217

THE ∞-CATEGORY OF ∞-CATEGORIES

3.3.4 Colimits of ∞-Categories
In this section, we will address the problem of constructing colimits in the∞-category Cat∞ . Let p : S op → Cat∞ be a diagram classifying a Cartesianfibration f : X → S. In §3.3.3, we saw that lim(p) can be identified with←−the ∞-category of Cartesian sections of f . To construct the colimit lim(p),−→we need to find an ∞-category which admits a map from each fiber Xs . The
natural candidate, of course, is X itself. However, because X is generally not
an ∞-category, we must take some care to formulate a correct statement.
Lemma 3.3.4.1. Let
X0

/X
p



q
/S
S0
be a pullback diagram of simplicial sets, where p is a Cartesian fibration and\q op is cofinal. The induced map X 0 → X \ is a Cartesian equivalence (in
Set+
∆ ).
Proof. Choose a cofibration S 0 → K, where K is a contractible Kan complex.
The map q factors as a composition
q0

q 00

S 0 → S × K → S.
It is obvious that the projection X \ × K ] → X \ is a Cartesian equivalence.
We may therefore replace S by S × K and q by q 0 , thereby reducing to thecase where q is a cofibration. Proposition 4.1.1.3 now implies that q is leftanodyne. It is easy to see that the collection of cofibrations q : S 0 → S for
which the desired conclusion holds is weakly saturated. We may therefore
reduce to the case where q is a horn inclusion Λni ⊆ ∆n , where 0 ≤ i < n.
We now apply Proposition 3.2.2.7 to choose a sequence of composablemaps
φ : A0 ← · · · ← An
and a quasi-equivalence M (φ) → X. We have a commutative diagram ofmarked simplicial sets:
M \ (φ) ×(∆ n )] (Λni )]_/ X 0 \_i

M \ (φ)/ X.

Using Proposition 3.2.2.14, we deduce that the horizontal maps are Cartesian equivalences. To complete the proof, it will suffice to show that i is a
Cartesian equivalence. We now observe that i is a pushout of the inclusion
i00 : (Λni )] × (An )[ ⊆ (∆n )] × (An )[ . It will therefore suffice to prove thati00 is a Cartesian equivalence. Using Proposition 3.1.4.2, we are reduced toproving that the inclusion (Λni )] ⊆ (∆n )] is a Cartesian equivalence. According to Proposition 3.1.5.7, this is equivalent to the assertion that the horn
inclusion Λni ⊆ ∆n is a weak homotopy equivalence, which is obvious.

highertopoi

April 9, 2017218

CHAPTER 3

Proposition 3.3.4.2. Let K be a simplicial set, p : K / → Catop∞ be adiagram in the ∞-category Cat∞ , X → K / a Cartesian fibration classified
by p, and X = X ×K / K. The following conditions are equivalent:(1) The diagram p is a limit of p = p|K.
\

(2) The inclusion X \ ⊆ X is a Cartesian equivalence in (Set+∆ )/K / .
\(3) The inclusion X \ ⊆ X is a Cartesian equivalence in Set+∆.
Proof. Using the small object argument, we can construct a factorizationji

X → Y → K /,where j is a Cartesian fibration, i induces a marked anodyne map X \ → Y \ ,and X ' Y ×K / K. Since i is marked anodyne, we can solve the lifting
problem
/ X\
<

X \_
q
i


Y\


/ (K / )] .

Since i is a Cartesian equivalence in (Set+∆ )/K / , condition (2) is equivalent tothe assertion that q is an equivalence of Cartesian fibrations over K / . Sinceq induces an isomorphism over each vertex of K, this is equivalent to thefollowing assertion:(20 ) The map qv : Yv → X v is an equivalence of ∞-categories, where v
denotes the cone point of K / .
We have a commutative diagram
Yv\
_

Y

\

qv

q

/ X \v_/ X \.

Lemma 3.3.4.1 implies that the vertical maps are Cartesian equivalences. It
follows that (20 ) ⇔ (3), so that (2) ⇔ (3).
To complete the proof, we will show that (1) ⇔ (2). According to Proposition 4.2.3.14, there exists a small category C and a map p : N(C) → K suchthat pop is cofinal. Let C = [0] ? C be the category obtained by adjoining an
initial object to C. Consider the diagram/ (X ×K / N(C))\(X ×K N(C))\ 
 

X\


/ X \.

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April 9, 2017219

THE ∞-CATEGORY OF ∞-CATEGORIES

Lemma 3.3.4.1 implies that the vertical maps are Cartesian equivalences (inSet+∆ ). It follows that the upper horizontal inclusion is a Cartesian equivalence if and only if the lower horizontal inclusion is a Cartesian equivalence.
Consequently, it will suffice to prove the equivalence (1) ⇔ (2) after replacing
K by N(C).
Using Corollary 4.2.4.7, we may further suppose that p is the nerve of a◦/functor F : C → (Set+∆ ) . Let φ : C[K ] → C be the counit map and let
φ : C[K] → C be the restriction of φ. Without loss of generality, we may
suppose that X = Unφ F. We have a commutative diagram of homotopy
categories and right derived functorsCh(Set+∆)G/ h(Set+ )C
∆

R Un+
φ

h(Set+∆ )/(K / )

G0



R Un+
φ

/ h(Set+ )/K ,∆0

where G and G are restriction functors. Let F and F 0 be the left adjoints
to G and G0 , respectively. According to Theorem 4.2.4.1, assumption (1) isequivalent to the assertion that F lies in the essential image of F . Since eachof the vertical functors is an equivalence of categories (Theorem 3.2.0.1), thisis equivalent to the assertion that X lies in the essential image of F 0 . Since
F 0 is fully faithful, this is equivalent to the assertion that the counit map
F 0 G0 X → X
is an isomorphism in h(Set+∆ )/K / ) , which is clearly a reformulation of (2).
Corollary 3.3.4.3. Let p : K op → Cat∞ be a diagram classifying a Cartesian fibration X → K. Then there is a natural isomorphism lim(p) ' X \ in−→the homotopy category in hCat∞ .
Proof. Let p : (K op ). → Cat∞ be a colimit of p, which classifies a Cartesianfibration X → K / . Let v denote the cone point of K / , so that lim(p) ' X v .
−→We now observe that the inclusions\\

X v ,→ X ←- X \are both Cartesian equivalences (Lemma 3.3.4.1 and Proposition 3.3.4.2).
Warning 3.3.4.4. In the situation of Corollary 3.3.4.3, the marked simplicial set X \ is usually not a fibrant object of Set+
∆ even when K is an∞-category.
Using exactly the same argument, we can establish a version of Proposition
3.3.4.2 which describes colimits in the ∞-category of spaces:
Proposition 3.3.4.5. Let K be a simplicial set, p : K . → S a diagramin the ∞-category of spaces, X → K . a left fibration classified by p, and
X = X ×K . K. The following conditions are equivalent:highertopoi

April 9, 2017220

CHAPTER 3(1) The diagram p is a colimit of p = p|K.
(2) The inclusion X ⊆ X is a covariant equivalence in (Set∆ )/K . .
(3) The inclusion X ⊆ X is a weak homotopy equivalence of simplicial
sets.
Proof. Using the small object argument, we can construct a factorizationij

X ,→ Y → K . ,where i is left anodyne, j is a left fibration, and the inclusion X ⊆ Y ×K . K
is an isomorphism. Choose a dotted arrow q as indicated in the diagram
/X
=

X _
q


Y

i


/ K ..

Since i is a covariant equivalence in (Set∆ )/K . , condition (2) is equivalent
to the assertion that q is an equivalence of left fibrations over K . . Sinceq induces an isomorphism over each vertex of K, this is equivalent to theassertion that qv : Yv → X v is an equivalence, where v denotes the conepoint of K . . We have a commutative diagram
Yv

Y

qv

q

/ Xv

/ X.

Proposition 4.1.2.15 implies that the vertical maps are right anodyne andtherefore weak homotopy equivalences. Consequently, qv is a weak homotopy equivalence if and only if q is a weak homotopy equivalence. Since theinclusion X ⊆ Y is a weak homotopy equivalence, this proves that (2) ⇔ (3).
To complete the proof, we will show that (1) ⇔ (2). According to Proposition 4.2.3.14, there exists a small category C and a cofinal map N(C) → K.
Let C = C ?[0] be the category obtained from C by adjoining a new final
object. Consider the diagram/ X × . N(C)X × N(C) 
K

K

 
X


/ X.

Proposition 4.1.2.15 implies that X → K . is smooth, so that the verticalarrows in the above diagram are cofinal. In particular, the vertical arrows
are weak homotopy equivalences, so that the upper horizontal inclusion isa weak homotopy equivalence if and only if the lower horizontal inclusion
is a weak homotopy equivalence. Consequently, it will suffice to prove theequivalence (1) ⇔ (2) after replacing K by N(C).

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221

THE ∞-CATEGORY OF ∞-CATEGORIES

Using Corollary 4.2.4.7, we may further suppose that p is obtained as thenerve of a functor F : C → Kan. Let φ : C[K . ] → C be the counit map andlet φ : C[K] → C be the restriction of φ. Without loss of generality, we may
opsuppose that X = Unφ F. We have a commutative diagram of homotopy
categories and right derived functorsGh(Set∆ )C/ h(Set∆ )C

R Unφ


h(Set∆ )/(K . )op

R Unφ
G0



/ h(Set∆ )/K ,where G and G0 are restriction functors. Let F and F 0 be the left adjoints
to G and G0 , respectively. According to Theorem 4.2.4.1, assumption (1) isequivalent to the assertion that F lies in the essential image of F . Since eachof the vertical functors is an equivalence of categories (Theorem 2.2.1.2), thisop
is equivalent to the assertion that X lies in the essential image of F 0 . Since0
F is fully faithful, this is equivalent to the assertion that the counit map
F 0 G0 X

op

→X

op

is an isomorphism in h(Set∆ )/(K . )op , which is clearly equivalent to (2). Thisshows that (1) ⇔ (2) and completes the proof.
Corollary 3.3.4.6. Let p : K → S be a diagram which classifies a left
e → K and let X ∈ S be a colimit of p. Then there is a natural
fibration K
isomorphisme 'X
Kin the homotopy category H.
e0 →Proof. Let p : K . → S be a colimit diagram which extends p and let K.
K be a left fibration classified by p. Without loss of generality, we may
e =K
e 0 ×K . K and X = K
e 0 ×K . {v}, where v denotes the
suppose that K.
cone point of K . Since the inclusion {v} ⊆ K . is right anodyne and the map
e 0 → K . is a left fibration, Proposition 4.1.2.15 implies that the inclusionK
e 0 is right anodyne and therefore a weak homotopy equivalence. On
X ⊆K
e ⊆K
e 0 is athe other hand, Proposition 3.3.4.5 implies that the inclusion K
weak homotopy equivalence. The composition
e0 ' Ke
X'K
is the desired isomorphism in H.

highertopoi

April 9, 2017

Chapter Four
Limits and Colimits
This chapter is devoted to the study of limits and colimits in the setting of ∞categories. Our goal is to provide tools for proving the existence of limits andcolimits, for analyzing them, and for comparing them to the (perhaps morefamiliar) notion of homotopy limits and colimits in simplicial categories. We
will generally confine our remarks to colimits; analogous results for limits
can be obtained by passing to the opposite ∞-categories.
We begin in §4.1 by introducing the notion of a cofinal map between
simplicial sets. If f : A → B is a cofinal map of simplicial sets, then we
can identify colimits of a diagram p : B → C with colimits of the induced
diagram p◦f : A → C. This is a basic maneuver which will appear repeatedly
in the later chapters of this book. Consequently, it is important to have agood supply of cofinal maps. This is guaranteed by Theorem 4.1.3.1, which
can be regarded as an ∞-categorical generalization of Quillen’s Theorem A.
In §4.2, we introduce a battery of additional techniques for analyzing colimits. We will explain how to analyze colimits of complicated diagrams interms of colimits of simpler diagrams. Using these ideas, we can often reduce
questions about the behavior of arbitrary colimits to questions about a few
basic constructions, which we will analyze explicitly in §4.4. We will also
explain the relationship between the ∞-categorical theory of colimits andthe more classical theory of homotopy colimits, which can be studied very
effectively using the language of model categories.
The other major topic of this chapter is the theory of Kan extensions,which can be viewed as relative versions of limits and colimits. We will
study the properties of Kan extensions in §4.3 and prove some fundamental
existence theorems which we will need throughout the later chapters of thisbook.

4.1 COFINALITYLet C be an ∞-category and let p : K → C be a diagram in C indexed
by a simplicial set K. In §1.2.13, we introduced the definition of a colimit
lim(p) for the diagram p. In practice, it is often possible to replace p by a−→simpler diagram without changing the colimit lim(p). In this section, we will−→
introduce a general formalism which will allow us to make replacements ofthis sort: the theory of cofinal maps between simplicial sets. We begin in§4.1.1 with a definition of the class of cofinal maps and show (Proposition

highertopoi

April 9, 2017223

LIMITS AND COLIMITS

4.1.1.8) that, if a map q : K 0 → K is cofinal, then there is an equivalencelim(p) ' lim(p ◦ q) (provided that either colimit exists). In §4.1.2, we will−→−→reformulate the definition of cofinality using the formalism of contravariant
model categories (§2.1.4). We conclude in §4.1.3 by establishing an important
recognition criterion for cofinal maps in the special case where K is an ∞category. This result can be regarded as a refinement of Quillen’s Theorem
A.
4.1.1 Cofinal Maps
The goal of this section is to introduce the definition of a cofinal map p :S → T of simplicial sets and study the basic properties of this notion. Our
main result is Proposition 4.1.1.8, which characterizes cofinality in terms of
the behavior of T -indexed colimits.
Definition 4.1.1.1 (Joyal [44]). Let p : S → T be a map of simplicial sets.
We shall say that p is cofinal if, for any right fibration X → T , the induced
map of simplicial sets
MapT (T, X) → MapT (S, X)
is a homotopy equivalence.
Remark 4.1.1.2. The simplicial set MapT (S, X) parametrizes sections of
the right fibration X → T . It may be described as the fiber of the induced
map X S → T S over the vertex of T S corresponding to the map p. Since
X S → T S is a right fibration, the fiber MapT (S, X) is a Kan complex.
Similarly, MapT (T, X) is a Kan complex.
We begin by recording a few simple observations about the class of cofinal
maps:
Proposition 4.1.1.3.

(1) Any isomorphism of simplicial sets is cofinal.

(2) Let f : K → K 0 and g : K 0 → K 00 be maps of simplicial sets. Supposethat f is cofinal. Then g is cofinal if and only if g ◦ f is cofinal.
(3) If f : K → K 0 is a cofinal map between simplicial sets, then f is aweak homotopy equivalence.
(4) An inclusion i : K ⊆ K 0 of simplicial sets is cofinal if and only if it isright anodyne.
Proof. Assertions (1) and (2) are obvious. We prove (3). Let S be a Kancomplex. Since f is cofinal, the composition
MapSet∆ (K 0 , S) = MapK 0 (K 0 , S × K 0 )→ MapK 0 (K, S × K 0 )= MapSet∆ (K, S)

highertopoi

April 9, 2017224

CHAPTER 4

is a homotopy equivalence. Passing to connected components, we deducethat K and K 0 corepresent the same functor in the homotopy category Hof spaces. It follows that f is a weak homotopy equivalence, as desired.
We now prove (4). Suppose first that i is right anodyne. Let X → K 0 be
a right fibration. Then the induced map MapK 0 (K 0 , X) → MapK 0 (K, X) isa trivial fibration and, in particular, a homotopy equivalence.
Conversely, suppose that i is a cofinal inclusion of simplicial sets. We wish
to show that i has the left lifting property with respect to any right fibration.
In other words, we must show that given any diagram of solid arrows
K _

s


K0

/X
=

K 0,

for which the right vertical map is a right fibration, there exists a dottedarrow as indicated, rendering the diagram commutative. Since i is cofinal,
the map s is homotopic to a map which extends over K 0 . In other words,
there exists a map
as0 : (K × ∆1 )(K 0 × {1}) → X
K×{1}compatible with the projection to K 0 , such that s0 |K × {0} coincides withs. Since the inclusiona(K × ∆1 )(K 0 × {1}) ⊆ K 0 × ∆1
K×{1}is right anodyne, there exists a map s00 : K 0 × ∆1 → X which extends s0
and is compatible with the projection to K 0 . The map s00 |K 0 × {0} has thedesired properties.
Warning 4.1.1.4. The class of cofinal maps does not satisfy the two-outof-three property. If f : K → K 0 and g : K 0 → K 00 are such that g ◦ f and g
are cofinal, then f need not be cofinal.
Our next goal is to establish a characterization of cofinality in terms of
the behavior of colimits (Proposition 4.1.1.8). First, we need a lemma.
Lemma 4.1.1.5. Let C be an ∞-category and let p : K → C and q : K 0 → C
be diagrams. Define simplicial sets M and N by the formulas
Hom(X, M ) = {f : (X × K) ? K 0 → C : f |(X × K) = p ◦ πK , f |K 0 = q}
Hom(X, N ) = {g : K ? (X × K 0 ) → C : f |K = p, f |(X × K 0 ) = q ◦ πK 0 }.
Here πK and πK 0 denote the projection from a product to the factor indicatedby the subscript.
Then M and N are Kan complexes, which are (naturally) homotopy equivalent to one another.

highertopoi

April 9, 2017225

LIMITS AND COLIMITSProof. We define a simplicial set D as follows. For every finite nonempty
linearly ordered set J, to give a map ∆J → D is to supply the following
data:J• A map ∆`→ ∆1 corresponding to a decomposition of J as a disjoint
union J− J+ , where J− ⊆ J is closed downward and J+ ⊆ J is closed
upward.

• A map e : (K × ∆J− ) ? (K 0 × ∆J+ ) → C such that e|K × ∆J− = p ◦ πK
and e|K 0 × ∆J+ = q ◦ πK 0 .
We first claim that D is an ∞-category. Fix a finite linearly ordered set
J as above and let j ∈ J be neither the largest nor the smallest element of
J. Let f0 : ΛJj → D be any map; we wish to show that there exists a map
f : ∆J → D which extends f0 . We first observe that the induced projectionJ1ΛJj → ∆1 extends uniquely` to ∆ (since ∆ is isomorphic to the nerve ofa category). Let J = J− J+ be the induced decomposition of J. Without
loss of generality, we may suppose that j ∈ J− . In this case, we may identify
f0 with a map
aJ((K ×Λj − )?(K 0 ×∆J+ ))((K ×∆J− )?(K 0 ×∂ ∆J+ )) → C,J

(K×Λj − )?(K 0 ×∂ ∆J+ )and our goal is to find an extensionf : (K × ∆J− ) ? (K 0 × ∆J+ ) → C .
Since C is an ∞-category, it will suffice to show that the inclusion`
J(K × Λj − ) ? (K 0 × ∆J+ ) (K×ΛJ− )?(K 0 ×∂ ∆J+ ) (K × ∆J− ) ? (K 0 × ∂ ∆J+ )j
(K × ∆J− ) ? (K 0 × ∆J+ )is inner anodyne. According to Lemma 2.1.2.3, it suffices to check that theJ
inclusion K × Λj − ⊆ K × ∆J− is right anodyne. This follows from CorollaryJ

2.1.2.7 since Λj − ⊆ ∆J− is right anodyne.
The ∞-category D has just two objects, which we will denote by x and y.
L
We observe that M = HomR
D (x, y) and N = HomD (x, y). Proposition 1.2.2.3
implies that M and N are Kan complexes. Propositions 2.2.2.7 and 2.2.4.1
imply that each of these Kan complexes is weakly homotopy equivalent to
MapC[D] (x, y), so that M and N are homotopy equivalent to one another,as desired.
Remark 4.1.1.6. In the situation of Lemma 4.1.1.5, the homotopy equivalence between M and N is furnished by the composition of a chain of weakhomotopy equivalences
M ← |M |Q• → HomC[D] (x, y) ← |N |Q• → N,
which is functorial in the triple (C, p : K → C, q : K 0 → C).

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Proposition 4.1.1.7. Let v : K 0 → K be a cofinal map and p : K → C adiagram in an ∞-category C. Then the map φ : Cp/ → Cpv/ is an equivalenceof left fibrations over C: in other words, it induces a homotopy equivalenceof Kan complexes after passing to the fiber over every object x of C.
Proof. We wish to prove that the map
Cp/ ×C {x} → Cpv/ ×C {x}is a homotopy equivalence of Kan complexes. Lemma 4.1.1.5 implies thatthe left hand side is homotopy equivalent to MapC (K, C/x ). Similarly, theright hand side can be identified with MapC (K 0 , C/x ). Using the functoriality
implicit in the proof of Lemma 4.1.1.5 (see Remark 4.1.1.6), it suffices toshow that the restriction map
MapC (K, C/x ) → MapC (K 0 , C/x )is a homotopy equivalence. Since v is cofinal, this follows immediately fromthe fact that the projection C/x → C is a right fibration.
Proposition 4.1.1.8. Let v : K 0 → K be a map of (small) simplicial sets.
The following conditions are equivalent:(1) The map v is cofinal.
(2) Given any ∞-category C and any diagram p : K → C, the induced map
Cp/ → Cp0 / is an equivalence of ∞-categories, where p0 = p ◦ v.
(3) For every ∞-category C and every diagram p : K . → C which is a.
colimit of p = p|K, the induced map p0 : K 0 → C is a colimit ofp0 = p0 |K 0 .
Proof. Suppose first that (1) is satisfied. Let p : K → C be as in (2). Proposition 4.1.1.7 implies that the induced map Cp/ → Cp0 / induces a homotopy
equivalence of Kan complexes after passing to the fiber over any object of
C. Since both Cp/ and Cp0 / are left-fibered over C, Corollary 2.4.4.4 implies
that Cp/ → Cp0 / is a categorical equivalence. This proves that (1) ⇒ (2).
Now suppose that (2) is satisfied and let p : K . → C be as in (3). Then
we may identify p with an initial object of the ∞-category Cp/ . The induced
map Cp/ → Cp0 / is an equivalence and therefore carries the initial object p
to an initial object p0 of Cp0 / ; thus p0 is a colimit of p0 . This proves that(2) ⇒ (3).
It remains to prove that (3) ⇒ (1). For this, we make use of the theory of classifying right fibrations (§3.3.2). Let X → K be a right fibration.
We wish to show that composition with v induces a homotopy equivalence
MapK (K, X) → MapK (K 0 , X). It will suffice to prove this result after replacing X by any equivalent right fibration. Let S denote the ∞-category of
spaces. According to Corollary 3.3.2.8, there is a classifying map p : K → Sop
and an equivalence of right fibrations between X and (S∗/ )op ×Sop K, where∗ denotes a final object of S.

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LIMITS AND COLIMITS

The ∞-category S admits small limits (Corollary 4.2.4.8). It follows that
there exists a map p : K . → Sop which is a colimit of p = p|K. Let y.
denote the image in S of the cone point of K . . Let p0 : K 0 → Sop be the000
induced map. Then, by hypothesis, p is a colimit of p = p |K 0 . According
to Lemma 4.1.1.5, there is a (natural) chain of weak homotopy equivalences
relating MapK (K, X) with (Sop )p/ ×Sop {y}. Similarly, there is a chain of
weak homotopy equivalences connecting MapK (K 0 , X) with (Sop )p0 / ×Sop{y}. Consequently, we are reduced to proving that the left vertical map in
the diagram(Sop )p/ ×Sop {y} o(Sop )p/ ×Sop {y}/ (Sop )x/ ×Sop {y}(Sop )p0 / ×Sop {y} o(Sop )p0 / ×Sop {y}/ (Sop )x/ ×Sop {y}is a homotopy equivalence. Since p and p0 are colimits of p and p0 , theleft horizontal maps are trivial fibrations. Since the inclusions of the cone.
points into K . and K 0 are right anodyne, the right horizontal maps are also
trivial fibrations. It therefore suffices to prove that the right vertical map is a
homotopy equivalence. But this map is an isomorphism of simplicial sets.
Corollary 4.1.1.9. Let p : K → K 0 be a map of simplicial sets and q :K 0 → K 00 a categorical equivalence. Then p is cofinal if and only if q ◦ p iscofinal. In particular (taking p = idK 0 ), q itself is cofinal.
Proof. Let C be an ∞-category, let r00 : K 00 → C be a diagram, and set
r0 = r00 ◦ q, r = r0 ◦ p. Since q is a categorical equivalence, Cr00 / → Cr0 / isa categorical equivalence. It follows that Cr00 / → Cr/ is a categorical equivalence if and only if Cr0 / → Cr/ is a categorical equivalence. We now apply
the characterization (2) of Proposition 4.1.1.8.
Corollary 4.1.1.10. The property of cofinality is homotopy invariant. In
other words, if two maps f, g : K → K 0 have the same image in the homotopy
category of Set∆ obtained by inverting all categorical equivalences, then f iscofinal if and only if g is cofinal.
Proof. Choose a categorical equivalence K 0 → C, where C is an ∞-category.
In view of Corollary 4.1.1.9, we may replace K 0 by C and thereby assume
that K 0 is itself an ∞-category. Since f and g are homotopic, there exists
a cylinder object S equipped with a trivial fibration p : S → K, a map
q : S → C, and two sections s, s0 : K → S of p, such that f = q ◦ s, g = q ◦ s0 .
Since p is a categorical equivalence, so is every section of p. Consequently,s and s0 are cofinal. We now apply Proposition 4.1.1.3 to deduce that f iscofinal if and only if q is cofinal. Similarly, g is cofinal if and only if q is
cofinal.
Corollary 4.1.1.11. Let p : X → S be a map of simplicial sets. The following are equivalent:highertopoi

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CHAPTER 4(1) The map p is a cofinal right fibration.
(2) The map p is a trivial fibration.
Proof. Clearly any trivial fibration is a right fibration. Furthermore, any trivial fibration is a categorical equivalence, hence cofinal by Corollary 4.1.1.9.
Thus (2) implies (1). Conversely, suppose that p is a cofinal right fibration. Since p is cofinal, the natural map MapS (S, X) → MapS (X, X) is a
homotopy equivalence of Kan complexes. In particular, there exists a section
f : S → X of p such that f ◦ p is (fiberwise) homotopic to the identity mapof X. Consequently, for each vertex s of S, the fiber Xs = X ×S {s} is acontractible Kan complex (since the identity map Xs → Xs is homotopic tothe constant map with value f (s)). The dual of Lemma 2.1.3.4 now showsthat p is a trivial fibration.
Corollary 4.1.1.12. A map X → Z of simplicial sets is cofinal if and only
if it admits a factorizationfg

X → Y → Z,
where X → Y is right anodyne and Y → Z is a trivial fibration.
Proof. The “if” direction is clear: if such a factorization exists, then f iscofinal (since it is right anodyne), g is cofinal (since it is a categorical equivalence), and consequently g ◦ f is cofinal (since it is a composition of cofinalmaps).
For the “only if” direction, let us suppose that X → Z is a cofinal map.
By the small object argument (Proposition A.1.2.5), there is a factorizationfg

X→Y →Z
where f is right anodyne and g is a right fibration. The map g is cofinal by
Proposition 4.1.1.3 and therefore a trivial fibration by Corollary 4.1.1.11.
Corollary 4.1.1.13. Let p : S → S 0 be a cofinal map and K any simplicial
set. Then the induced map K × S → K × S 0 is cofinal.
Proof. Using Corollary 4.1.1.12, we may suppose that p is either right anodyne or a trivial fibration. Then the induced map K × S → K × S 0 has thesame property.
4.1.2 Smoothness and Right Anodyne Maps
In this section, we explain how to characterize the classes of right anodyne
and cofinal morphisms in terms of the contravariant model structures studiedin §2.1.4. We also introduce a third class of maps between simplicial sets,
which we call smooth.
We begin with the following characterization of right anodyne maps:
Proposition 4.1.2.1. Let i : A → B be a map of simplicial sets. The
following conditions are equivalent:highertopoi

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LIMITS AND COLIMITS(1) The map i is right anodyne.
(2) For any map of simplicial sets j : B → C, the map i is a trivial cofibration with respect to the contravariant model structure on (Set∆ )/C .
(3) The map i is a trivial cofibration with respect to the contravariant model
structure on (Set∆ )/B .
Proof. The implication (1) ⇒ (2) follows immediately from Proposition
2.1.4.9, and the implication (2) ⇒ (3) is obvious. Suppose that (3) holds. To
prove (1), it suffices to show that given any diagram
/X
>

A _
f
i


/Y


B

p

such that p is a right fibration, one can supply the dotted arrow f as indicated. Replacing p : X → Y by the pullback X ×Y B → B, we may reduceto the case where Y = B. Corollary 2.2.3.12 implies that X is a fibrantobject of (Set∆ )/B (with respect to contravariant model structure) so thatthe desired map f can be found.
ijCorollary 4.1.2.2. Suppose we are given maps A → B → C of simplicial
sets. If i and j ◦ i are right anodyne and j is a cofibration, then j is rightanodyne.
Proof. By Proposition 4.1.2.1, i and j ◦ i are contravariant equivalences in
the category (Set∆ )/C . It follows that j is a trivial cofibration in (Set∆ )/C ,so that j is right anodyne (by Proposition 4.1.2.1 again).
Corollary 4.1.2.3. Let
A0 o

u

f0

A
f


B0 o

v


B

/ A00
f 00


/ B 00be a diagram of simplicial sets. Suppose that u and v are monomorphisms
and that f, f 0 , and f 00 are right anodyne. Then the induced mapaaA0
A00 → B 0
B 00
A

B

is right anodyne.
Proof. According to Proposition 4.1.2.1, each of the maps f , f 0 , and f 00 is acontravariant equivalence in the`category (Set∆ )/B 0 `B B 00 . The assumption
on u and v guarantees that `f 0 f f 00 is also a contravariant equivalence in0`(Set∆ )/B 0 B B 00 , so that f f f 00 is right anodyne by Proposition 4.1.2.1
again.

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Corollary 4.1.2.4. The collection of right anodyne maps of simplicial sets
is stable under filtered colimits.
Proof. Let f : A → B be a filtered colimit of right anodyne morphisms
fα : Aα → Bα . According to Proposition 4.1.2.1, each fα is a contravariant
equivalence in (Set∆ )/B . Since contravariant equivalences are stable under filtered colimits, we conclude that f is a contravariant equivalence in (Set∆ )/B ,so that f is right anodyne by Proposition 4.1.2.1.
Proposition 4.1.2.1 has an analogue for cofinal maps:
Proposition 4.1.2.5. Let i : A → B be a map of simplicial sets. The
following conditions are equivalent:(1) The map i is cofinal.
(2) For any map j : B → C, the inclusion i is a contravariant equivalencein (Set∆ )/C .
(3) The map i is a contravariant equivalence in (Set∆ )/B .
Proof. Suppose (1) is satisfied. By Corollary 4.1.1.12, i admits a factorization
as a right anodyne map followed by a trivial fibration. Invoking Proposition
4.1.2.1, we conclude that (2) holds. The implication (2) ⇒ (3) is obvious. If(3) holds, then we can choose a factorization
i0

i00

A → A0 → B
of i, where i0 is right anodyne and i00 is a right fibration. Then i00 is acontravariant fibration (in Set∆/B ) and a contravariant weak equivalence
and is therefore a trivial fibration of simplicial sets. We now apply Corollary
4.1.1.12 to conclude that i is cofinal.
Corollary 4.1.2.6. Let p : X → S be a map of simplicial sets, where S
is a Kan complex. Then p is cofinal if and only if it is a weak homotopy
equivalence.
Proof. By Proposition 4.1.2.5, p is cofinal if and only if it is a contravariant
equivalence in (Set∆ )/S . If S is a Kan complex, then Proposition 3.1.5.7
asserts that the contravariant equivalences are precisely the weak homotopy
equivalences.
Corollary 4.1.2.7. Suppose we are given a pushout diagram
A
f


B

g

/ A0
f0


/ B0

of simplicial sets. If f is cofinal and either f or g is a cofibration, then f 0 is
cofinal.

highertopoi

April 9, 2017231

LIMITS AND COLIMITSProof. Combine Proposition 4.1.2.5 with the left-properness of the contrvariant model structure.
Let p : X → Y be an arbitrary map of simplicial sets. In §2.1.4, we showedthat p induces a Quillen adjunction (p! , p∗ ) between the contravariant model
categories (Set∆ )/X and (Set∆ )/Y . The functor p∗ itself has a right adjoint,
which we will denote by p∗ ; it is given byp∗ (M ) = MapY (X, M ).
∗The adjoint functors p and p∗ are not Quillen adjoints in general. Instead
we have the following result:
Proposition 4.1.2.8. Let p : X → Y be a map of simplicial sets. The
following conditions are equivalent:(1) For any right anodyne map i : A → B in (Set∆ )/Y , the induced map
A ×Y X → B ×Y X is right anodyne.
(2) For every Cartesian diagram
/X

X0
p0

p


Y0


/ Y,

∗

the functor p0 : (Set∆ )/Y 0 → (Set∆ )/X 0 preserves contravariant equivalences.
(3) For every Cartesian diagram
/X

X0
p0

p


Y0


/ Y,

∗

the adjoint functors (p0 , p0∗ ) give rise to a Quillen adjunction between
the contravariant model categories (Set∆ )/Y 0 and (Set∆ )/X 0 .
Proof. Suppose that (1) is satisfied; let us prove (2). Since property (1) isclearly stable under base change, we may suppose that p0 = p. Let u : M →N be a contravariant equivalence in (Set∆ )/Y . If M and N are fibrant, thenu is a homotopy equivalence, so that p∗ (u) : p∗ M → p∗ N is also a homotopy
equivalence. In the general case, we may select a diagram
i

M

/ M0
v

u


N

i0

/N`



M M

0

j

%
/ N 0,

highertopoi

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CHAPTER 4

where M 0 and N 0 are fibrant and the maps i and j are right anodyne (andtherefore i0 is also right anodyne). Then p∗ (v) is a contravariant equivalence,
while the maps p∗ (i), p∗ (j), and p∗ (i0 ) are all right anodyne; by Proposition
4.1.2.1 they are contravariant equivalences as well. It follows that p∗ (u) is acontravariant equivalence.
∗
To prove (3), it suffices to show that p0 preserves cofibrations and trivial
cofibrations. The first statement is obvious, and the second follows imme∗
diately from (2). Conversely, the existence of a Quillen adjunction (p0 , p∗ )0∗
implies that p preserves contravariant equivalences between cofibrant objects. Since every object of (Set∆ )/Y 0 is cofibrant, we deduce that (3) implies(2).
Now suppose that (2) is satisfied and let i : A → B be a right anodyne
map in (Set∆ )/Y as in (1). Then i is a contravariant equivalence in (Set∆ )/B .
Let p0 : X ×Y B → B be base change of p; then (2) implies that the induced∗∗
map i0 : p0 A → p0 B is a contravariant equivalence in (Set∆ )/B×Y X . By
Proposition 4.1.2.1, the map i0 is right anodyne. Now we simply note thati0 may be identified with the map A ×Y X → B ×Y X in the statement of(1).
Definition 4.1.2.9. We will say that a map p : X → Y of simplicial sets issmooth if it satisfies the (equivalent) conditions of Proposition 4.1.2.8.
Remark 4.1.2.10. Let
X0

S0

f0

f

/X

/S

p

be a pullback diagram of simplicial sets. Suppose that p is smooth and that f
is cofinal. Then f 0 is cofinal: this follows immediately from characterization(2) of Proposition 4.1.2.8 and characterization (3) of Proposition 4.1.2.5.
We next give an alternative characterization of smoothness. Let
X0

q0

p0


Y0

q

/X

/Y

p

be a Cartesian diagram of simplicial sets. Then we obtain an isomorphism∗∗
Rp0 Rq ∗ ' Rq 0 Rp∗ of right derived functors, which induces a natural transformation∗

ψp,q : Lq!0 Rp0 → Rp∗ Lq! .
Proposition 4.1.2.11. Let p : X → Y be a map of simplicial sets. The
following conditions are equivalent:(1) The map p is smooth.

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LIMITS AND COLIMITS(2) For every Cartesian rectangle
X 00


q0

/ X0
p0

p00

Y 00

/X
p


/ Y0

q


/ Y,

the natural transformation ψp0 ,q is an isomorphism of functors from thehomotopy category of (Set∆ )/Y 00 to the homotopy category of (Set∆ )/X 0(here we regard all categories as endowed with the contravariant model
structure).
Proof. Suppose that (1) is satisfied and consider any Cartesian rectangle as∗in (2). Since p is smooth, p0 and p00 are also smooth. It follows that p0 and
00 ∗p preserve weak equivalences, so they may be identified with their rightderived functors. Similarly, q! and q!0 preserve weak equivalences, so they
may be identified with their left derived functors. Consequently, the naturaltransformation ψp0 ,q is simply obtained by passage to the homotopy category
from the natural transformation∗∗

q!0 p00 → p0 q! .
But this is an isomorphism of functors before passage to the homotopy categories.
Now suppose that (2) is satisfied. Let q : Y 00 → Y 0 be a right anodyne mapin (Set∆ )/Y and form the Cartesian square as in (2). Let us compute the∗∗value of the functors Lq!0 Rp00 and Rp0 Lq! on the object Y 00 of (Set∆ )/Y 00 .
∗0
00 ∗The composite Lq! Rp is easy: because Y 00 is fibrant and X 00 = p00 Y 00 is
00
cofibrant, the result is X , regarded as an object of (Set∆ )/X 0 . The othercomposition is slightly trickier: Y 00 is cofibrant, but q! Y 00 is not fibrant whenviewed as an object of (Set∆ )/Y 0 . However, in view of the assumption that q isright anodyne, Proposition 4.1.2.1 ensures that Y 0 is a fibrant replacement for∗∗
q! Y 0 ; thus we may identify Rp0 Lq! with the object p0 Y 0 = X 0 of (Set∆ )/X 0 .
00
Condition (2) now implies that the natural map X → X 0 is a contravariant
equivalence in (Set∆ )/X 0 . Invoking Proposition 4.1.2.1, we deduce that q 0 isright anodyne, as desired.
Remark 4.1.2.12. The terminology “smooth” is suggested by the analogyof Proposition 4.1.2.11 with the smooth base change theorem in the theoryof étale cohomology (see, for example, [28]).
Proposition 4.1.2.13. Suppose we are given a commutative diagram
X

/ X0

i
p


X 00

p

00

! 
/S

p0

of simplicial sets. Assume that i is`a cofibration and that p, p0 , and p00 aresmooth. Then the induced map X 0 X X 00 → S is smooth.

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Proof. This follows immediately from Corollary 4.1.2.3 and characterization(1) of Proposition 4.1.2.8.
Proposition 4.1.2.14. The collection of smooth maps p : X → S is stable
under filtered colimits in (Set∆ )/S .
Proof. Combine Corollary 4.1.2.4 with characterization (1) of Proposition
4.1.2.8.
Proposition 4.1.2.15. Let p : X → S be a coCartesian fibration of simplicial sets. Then p is smooth.
Proof. Let i : B 0 → B be a right anodyne map in (Set∆ )/S ; we wish to showthat the induced map B 0 ×S X → B ×S X is right anodyne. By general
nonsense, we may reduce ourselves to the case where i is an inclusion Λni ⊆∆n , where 0 < i ≤ n. Making a base change, we may suppose that S = B.
By Proposition 3.2.2.7, there exists a composable sequence of maps
φ : A0 → · · · → An
and a quasi-equivalence M op (φ) → X. Consider the diagramn
M op (φ) × _ ∆n Λ iM op (φ)

f/ X ×∆n Λ n
_ i
h

g


/( X.

The left vertical map is right anodyne since it is a pushout of the inclusion
A0 × Λni ⊆ A0 × ∆n . It follows that f is cofinal, being a composition ofa right anodyne map and a categorical equivalence. Since g is cofinal (being a categorical equivalence), we deduce from Proposition 4.1.1.3 that h iscofinal. Since h is a monomorphism of simplicial sets, it is right anodyne by
Proposition 4.1.1.3.
Proposition 4.1.2.16. Let p : X → S × T be a bifibration. Then the composite map πS ◦ p : X → S is smooth.
Proof. For every map T 0 → T , let XT 0 = X ×T T 0 . We note that X isa filtered colimit of XT 0 as T 0 ranges over the finite simplicial subsets of
T . Using Proposition 4.1.2.14, we can reduce to the case where T is finite.
Working by induction on the dimension and `the number of nondegenerate
simplices of T , we may suppose that T = T 0 ∂ ∆n ∆n , where the result isknown for T 0 and for ∂ ∆n . Applying Proposition 4.1.2.13, we can reduceto the case T = ∆n . We now apply Lemma 2.4.7.5 to deduce that p is a
coCartesian fibration and therefore smooth (Proposition 4.1.2.15).
Lemma 4.1.2.17. Let C be an ∞-category containing an object C and let
f : X → Y be a covariant equivalence in (Set∆ )/ C . The induced map
X ×C C/C → Y ×C C/C
is also a covariant equivalence in C/C .

highertopoi

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235

LIMITS AND COLIMITSProof. It will suffice to prove that for every object Z → C of (Set∆ )/ C , thefiber product Z ×C C/C is a homotopy product of Z with C/C in (Set∆ )/ C(with respect to the covariant model structure). Choose a factorizationji

Z → Z 0 → C,
where i is left anodyne and j is a left fibration. According to Corollary
2.2.3.12, we may regard Z 0 as a fibrant replacement for Z in (Set∆ )/ C . It
therefore suffices to prove that the map i0 : Z ×C C/C → Z 0 ×C C/C isa covariant equivalence. According to Proposition 4.1.2.5, it will suffice toprove that i0 is left anodyne. The map i0 is a base change of i by the projection
p : C/C → C; it therefore suffices to prove that pop is smooth. This follows
from Proposition 4.1.2.15 since p is a right fibration of simplicial sets.
Proposition 4.1.2.18. Let C be an ∞-category and
/Y

f

X
p

q


C
a commutative diagram of simplicial sets. Suppose that p and q are smooth.
The following conditions are equivalent:(1) The map f is a covariant equivalence in (Set∆ )/ C .
(2) For each object C ∈ C, the induced map of fibers XC → YC is a weakhomotopy equivalence.
Proof. Suppose that (1) is satisfied and let C be an object of C. We have acommutative diagram of simplicial sets
/ YC
XC


/ Y ×C C/C .
X ×C C/C
Lemma 4.1.2.17 implies that the bottom horizontal map is a covariant equivalence. The vertical maps are both pullbacks of the right anodyne inclusion{C} ⊆ C/C along smooth maps and are therefore right anodyne. In particular, the vertical arrows and the bottom horizontal arrow are all weakhomotopy equivalences; it follows that the map XC → YC is a weak homotopy equivalence as well.
Now suppose that (2) is satisfied. Choose a commutative diagram
/Y

f

X

X0


/ Y0

f0
p0

C



q0

highertopoi

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CHAPTER 4

in (Set∆ )/ C , where the vertical arrows are left anodyne and the maps p0
and q 0 are left fibrations. Using Proposition 4.1.2.15, we conclude that p0
and q 0 are smooth. Applying (1), we deduce that for each object C ∈ C,
the maps XC → XC0 and YC → YC0 are weak homotopy equivalences. It
follows that each fiber fC0 : XC0 → YC0 is a homotopy equivalence of Kancomplexes, so that f 0 is an equivalence of left fibrations and therefore a
covariant equivalence. Inspecting the above diagram, we deduce that f isalso a covariant equivalence, as desired.
4.1.3 Quillen’s Theorem A for ∞-Categories
Suppose that f : C → D is a functor between ∞-categories and that wewish to determine whether or not f is cofinal. According to Proposition
4.1.1.8, the cofinality of f is equivalent to the assertion that for any diagram
p : D → E, f induces an equivalencelim(p) ' lim(p ◦ f ).
−→−→One can always define a morphism
φ : lim(p ◦ f ) → lim(p)−→−→(provided that both sides are defined); the question is whether or not we
can define an inverse ψ = φ−1 . Roughly speaking, this involves defining acompatible family of maps ψD : p(D) → lim(p ◦ f ) indexed by D ∈ D. The−→
only reasonable candidate for ψD is a compositionp(D) → (p ◦ f )(C) → lim(p ◦ f ),−→where the first map arises from a morphism D → f (C) in C. Of course, theexistence of C is not automatic. Moreover, even if C exists, it is usually notunique. The collection of candidates for C is parametrized by the ∞-category
CD/ = C ×D DD/ . In order to make the above construction work, we need
the ∞-category CD/ to be weakly contractible. More precisely, we will prove
the following result:
Theorem 4.1.3.1 (Joyal [44]). Let f : C → D be a map of simplicial sets,
where D is an ∞-category. The following conditions are equivalent:(1) The functor f is cofinal.
(2) For every object D ∈ D, the simplicial set C ×D DD/ is weakly contractible.
We first need to establish the following lemma:
Lemma 4.1.3.2. Let p : U → S be a Cartesian fibration of simplicial
sets. Suppose that for every vertex s of S, the fiber Us = p−1 {s} is weakly
contractible. Then p is cofinal.

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237

LIMITS AND COLIMITSProof. Let q : N → S be a right fibration. For every map of simplicial
sets T → S, let XT = MapS (T, N ) and YT = MapS (T ×S U, N ). Our
goal is to prove that the natural map XS → YS is a homotopy equivalenceof Kan complexes. We will prove, more generally, that for any map T →S, the map φT : XT → YT is a homotopy equivalence. The proof uses
induction on the (possibly infinite) dimension of T . Choose a transfinite
sequence ofSsimplicial subsets T (α) ⊆ T , where each T (α) is obtained fromT (< α) = β<α T (β) by adjoining a single nondegenerate simplex of T (ifsuch a simplex exists). We prove that φT (α) is a homotopy equivalence by
induction on α. Assuming that φT (β) is a homotopy equivalence for every
β < α, we deduce that φT (<α) is the homotopy inverse limit of a tower of
equivalences and therefore a homotopy equivalence. `If T (α) = T (< α), weare done. Otherwise, we may write T (α) = T (< α) ∂ ∆n ∆n . Then φT (α)
can be written as a homotopy pullback of φT (<α) with φ∆n over φ∂ ∆n . The
third map is a homotopy equivalence by the inductive hypothesis. It therefore
suffices to prove that φ∆n is an equivalence. In other words, we may reduceto the case T = ∆n .
By Proposition 3.2.2.7, there exists a composable sequence of maps
θ : A0 ← · · · ← An
and a quasi-equivalence f : M (θ) → N ×S T , where M (θ) denotes the
mapping simplex of the sequence θ. Given a map T 0 → T , we let ZT 0 =MapS (M (θ) ×T T 0 , N ). Proposition 3.3.1.7 implies that q is a categoricalfibration. It follows that for any map T 0 → T , the categorical equivalence
M (θ)×T T 0 → U ×S T 0 induces another categorical equivalence ψT 0 = YT 0 →ZT 0 . Since YT 0 and ZT 0 are Kan complexes, the map ψT 0 is a homotopy
equivalence. Consequently, to prove that φT is an equivalence, it suffices toshow that the composite map
XT → YT → ZT
is an equivalence.
Consider the composition
u0

u00

u000

u : X∆n−1 → Z∆n−1 → MapS (∆n−1 × An , N ) → MapS ({n − 1} × An , N ).
Using the fact that q is a right fibration and that An is weakly contractible,
we deduce that u and u000 are homotopy equivalences. The inductive hypothesis implies that u0 is a homotopy equivalence. Consequently, u00 is also ahomotopy equivalence. The space ZT fits into a homotopy Cartesian diagram/ Z∆n−1
ZT
v 00

MapS (∆n × An , N )

u00/ MapS (∆n−1 × An , N ).

It follows that v 00 is a homotopy equivalence. Now consider the composition
v0

v 00

v 000

v : X∆n → Z∆n → MapS (∆n × An , N ) → MapS ({n} × An , N ).

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Again, because q is a right fibration and An is weakly contractible, the maps
v and v 000 are homotopy equivalences. Since v 00 is a homotopy equivalence,
we deduce that v 0 is a homotopy equivalence, as desired.
Proof of Theorem 4.1.3.1. Using the small object argument, we can factor
f as a composition
f0

f 00

C → C0 → D,
where f 0 is a categorical equivalence and f 00 is an inner fibration. Then f 00
is cofinal if and only if f is cofinal (Corollary 4.1.1.10). For every D ∈ D,
the map DD/ → D is a left fibration, so the induced map CD/ → C0 D/ is a
categorical equivalence (Proposition 3.3.1.3). Consequently, it will suffice toprove that (1) ⇔ (2) for the morphism f 00 : C0 → D. In other words, we may
assume that the simplicial set C is an ∞-category.
Suppose first that (1) is satisfied and choose D ∈ D. The projection
DD/ → D is a left fibration and therefore smooth (Proposition 4.1.2.15).
Applying Remark 4.1.2.10, we deduce that the projection C ×D DD/ →DD/ is cofinal and therefore a weak homotopy equivalence (Proposition
4.1.1.3). Since DD/ has an initial object, it is weakly contractible. Therefore C ×D DD/ is weakly contractible, as desired.
We now prove that (2) ⇒ (1). Let M = Fun(∆1 , D) ×Fun({1},D) C. Thenthe map f factors as a composition
f0

f 00

C → M → D,
where f 0 is the obvious map and f 00 is given by evaluation at the vertex{0} ⊆ ∆1 . Note that there is a natural projection map π : M → C, that f 0
is a section of π, and that there is a simplicial homotopy h : ∆1 × M → M
from idM to f 0 ◦ π which is compatible with the projection to C. It follows
from Proposition 2.1.2.11 that f 0 is right anodyne.
Corollary 2.4.7.12 implies that f 00 is a Cartesian fibration. The fiber off 00 over an object D ∈ D is isomorphic to C ×D DD/ , which is equivalent
to C ×D DD/ and therefore weakly contractible (Proposition 4.2.1.5). By
assumption, the fibers of f 00 are weakly contractible. Lemma 4.1.3.2 assertsthat f 00 is cofinal. It follows that f , as a composition of cofinal maps, is also
cofinal.
Using Theorem 4.1.3.1, we can easily deduce the following classical resultof Quillen:
Corollary 4.1.3.3 (Quillen’s Theorem A). Let f : C → D be a functor
between ordinary categories. Suppose that for every object D ∈ D, the fiber
product category C ×D DD/ has a weakly contractible nerve. Then f induces
a weak homotopy equivalence of simplicial sets N(C) → N(D).
Proof. The assumption implies that N(f ) : N(C) → N(D) satisfies the hypotheses of Theorem 4.1.3.1. It follows that N(f ) is a cofinal map of simplicial
sets and therefore a weak homotopy equivalence (Proposition 4.1.1.3).

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4.2 TECHNIQUES FOR COMPUTING COLIMITSIn this section, we will introduce various techniques for computing, analyzing, and manipulating colimits. We begin in §4.2.1 by introducing a variant
on the join construction of §1.2. The new join construction is (categorically) equivalent to the version we are already familiar with but has better
formal behavior in some contexts. For example, it permits us to define aparametrized version of overcategories and undercategories, which we will
analyze in §4.2.2.
In §4.2.3, we address the following question: given a diagram p : K → C
and a decomposition of K into “pieces,” how is the colimit lim(p) related−→
to the colimits of those pieces? For example, if K = A ∪ B, then it seems
reasonable to expect an equation of the forma(lim p|B).
lim(p) = (lim p|A)−→−→−→lim(p|A∩B)−→
Of course there are many variations on this theme; we will lay out a generalframework in §4.2.3 and apply it to specific situations in §4.4.
Although the ∞-categorical theory of colimits is elegant and powerful, it
can be be difficult to work with because the colimit lim(p) of a diagram−→
p is well-defined only up to equivalence. This problem can sometimes be
remedied by working in the more rigid theory of model categories, where
the notion of an ∞-categorical colimit should be replaced by the notion of ahomotopy colimit (see §A.3.3). In order to pass smoothly between these twosettings, we need to know that the ∞-categorical theory of colimits agrees
with the more classical theory of homotopy colimits. A precise statement of
this result (Theorem 4.2.4.1) will be formulated and proved in §4.2.4.
4.2.1 Alternative Join and Slice Constructions
In §1.2.8, we introduced the join functor ? on simplicial sets. In [44], Joyal
introduces a closely related operation  on simplicial sets. This operation isequivalent to ? (Proposition 4.2.1.2) but is technically more convenient incertain contexts. In this section we will review the definition of the operation
 and establish some of its basic properties (see also [44] for a discussion).
Definition 4.2.1.1 ([44]). Let X and Y be simplicial sets. The simplicial
set X  Y is defined to be pushoutaaX(X × Y × ∆1 )Y.
X×Y ×{0}X×Y ×{1}We note that since X × Y × (∂ ∆1 ) → X × Y × ∆1 is a monomorphism, thepushout diagram defining X Y is a homotopy pushout in Set∆ (with respect
to the Joyal model structure). Consequently, we deduce that categoricalequivalences X → X 0 , Y → Y 0 induce a categorical equivalence X  Y →X 0  Y 0.

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The simplicial set X  Y admits a map p : X  Y → ∆1 with X ' p−1 {0}and Y ' p−1 {1}. Consequently, there is a unique map X  Y → X ? Y whichis compatible with the projection to ∆1 and induces the identity maps on
X and Y .
Proposition 4.2.1.2. For any simplicial sets X and Y , the natural map
φ : X  Y → X ? Y is a categorical equivalence.
Proof. Since both sides are compatible with the formation of filtered colimits
in X, we may suppose that X contains only finitely many nondegenerate simplices. If X is empty, then φ is an isomorphism and the result is obvious.
Working by induction on the dimension of X and the number of nondegenerate simplices in X, we may writea
X = X0∆n ,∂ ∆nand we may assume that the statement is known for the pairs (X 0 , Y ) and(∂ ∆n , Y ). Since the Joyal model structure on Set∆ is left proper, we have a
map of homotopy pushoutsaa(X 0  Y )(∆n  Y ) → (X 0 ? Y )(∆n ? Y ),∂ ∆n Y∂ ∆n ?Yand we are therefore reduced to proving the assertion in the case whereX = ∆n . The inclusionaa∆{0,1}···∆{n−1,n} ⊆ ∆n
{1}

{n−1}

is inner anodyne. Thus if n > 1, we can conclude by induction. Thus we may
suppose that X = ∆0 or X = ∆1 . By a similar argument, we may reduce tothe case where Y = ∆0 or Y = ∆1 . The desired result now follows from an
explicit calculation.
Corollary 4.2.1.3. Let S → T and S 0 → T 0 be categorical equivalences ofsimplicial sets. Then the induced map
S ? S0 → T ? T 0
is a categorical equivalence.
Proof. This follows immediately from Proposition 4.2.1.2 since the operation
 has the desired property.
Corollary 4.2.1.4. Let X and Y be simplicial sets. Then the natural map
C[X ? Y ] → C[X] ? C[Y ]is an equivalence of simplicial categories.
Proof. Using Corollary 4.2.1.3, we may reduce to the case where X and Y
are ∞-categories. We note that C[X ? Y ] is a correspondence from C[X]
to C[Y ]. To complete the proof, it suffices to show that MapC[X?Y ] (x, y) ishighertopoi

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weakly contractible for any pair of objects x ∈ X, y ∈ Y . Since X ? Y
is an ∞-category, we can apply Theorem 1.1.5.13 to deduce that the mapping space MapC[X?Y ] (x, y) is weakly homotopy equivalent to HomR
X?Y (x, y),which consists of a single point.
For fixed X, the functor
Y 7→ X  Y
Set∆ → (Set∆ )X/
preserves all colimits. By the adjoint functor theorem (or by direct construction), this functor has a right adjoint. In other words, for every map ofsimplicial sets p : X → C, there exists a simplicial set Cp/ with the following
universal property: for every simplicial set Y , there is a canonical bijection
HomSet∆ (Y, Cp/ ) ' Hom(Set∆ )X/ (X  Y, C).
Since the functor Y 7→ X  Y preserves cofibrations and categorical equivalences, we deduce that the passage from C to Cp/ preserves categorical fibrations and categorical equivalences between ∞-categories. Moreover, Proposition 4.2.1.2 has the following consequence:
Proposition 4.2.1.5. Let C be an ∞-category and let p : X → C be adiagram. Then the natural map
Cp/ → Cp/
is an equivalence of ∞-categories.
According to Definition 1.2.13.4, a colimit for a diagram p : X → C is an
initial object of the ∞-category Cp/ . In view of the above remarks, an objectof Cp/ is a colimit for p if and only if its image in Cp/ is an initial object; inother words, we can replace Cp/ by Cp/ (and ? by ) in Definition 1.2.13.4.
By Proposition 2.1.2.1, for any ∞-category C and any map p : X → C,
the induced map Cp/ → C is a left fibration. We now show that Cp/ has thesame property:
Proposition 4.2.1.6. Suppose we are given a diagram of simplicial setspq

K0 ⊆ K → X → S,
where q is a categorical fibration. Let r = q ◦ p : K → S, p0 = p|K0 , andr0 = r|K0 . Then the induced map
φ : X p/ → X p0 / ×S r0 / S r/
is a left fibration.
Proof. We must show that φ has the right lifting property with respect toevery left anodyne inclusion A0 ⊆ A. Unwinding the definition, this amounts
to proving that q has the right lifting property with respect to the inclusionai : (A0  K)
(A  K0 ) ⊆ A  K.
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Since q is a categorical fibration, it suffices to show that i is a categoricalequivalence. The above pushout is a homotopy pushout, so it will suffice toprove the analogous statement for the weakly equivalent inclusiona(A ? K0 ) ⊆ A ? K.
(A0 ? K)
A0 ?K0

But this map is inner anodyne (Lemma 2.1.2.3).
Corollary 4.2.1.7. Let C be an ∞-category and let p : K → C be any
diagram. For every vertex v of C, the map Cp/ ×C {v} → Cp/ ×C {v} is a
homotopy equivalence of Kan complexes.
Proof. The map Cp/ → Cp/ is a categorical equivalence of left fibrations over
C; now apply Proposition 3.3.1.5.
Corollary 4.2.1.8. Let C be an ∞-category containing vertices x and y.
The mapsL
HomR
C (x, y) → HomC (x, y) ← HomC (x, y)

are homotopy equivalences of Kan complexes (see §1.2.2 for an explanationof this notation).
Proof. Apply Corollary 4.2.1.7 (the dual of Corollary 4.2.1.7) to the casewhere p is the inclusion {x} ⊆ C (the inclusion {y} ⊆ C).
Remark 4.2.1.9. The above ideas dualize in an evident way; given a mapof simplicial sets p : K → X, we can define a simplicial set X /p with theuniversal mapping property
HomSet∆ (K 0 , X /p ) = Hom(Set∆ )K/ (K 0  K, X).

4.2.2 Parametrized Colimits
Let p : K → C be a diagram in an ∞-category C. The goal of this section
is to make precise the idea that the colimit lim(p) depends functorially on−→p (provided that lim(p) exists). We will prove this in a very general context−→
where not only the diagram p but also the simplicial set K is allowed to vary.
We begin by introducing a relative version of the -operation.
Definition 4.2.2.1. Let S be a simplicial set and let X, Y ∈ (Set∆ )/S . We
defineaa
X S Y = X(X ×S Y × ∆1 )Y ∈ (Set∆ )/S .
X×S Y ×{0}X×S Y ×{1}

highertopoi

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LIMITS AND COLIMITSWe observe that the operation S is compatible with base change in thefollowing sense: for any map T → S of simplicial sets and any objects X, Y ∈(Set∆ )/S , there is a natural isomorphism(XT T YT ) ' (X S Y )T ,where we let ZT denote the fiber product Z ×S T . We also note that in thecase where S is a point, the operation S coincides with the operation 
introduced in §4.2.1.
Fix K ∈ (Set∆ )/S . We note that functor (Set∆ )/S → ((Set∆ )/S )K/ definedby
X 7→ X S S
has a right adjoint; this right adjoint associates to a diagram
/Y

pS

K


S



the simplicial set Y pS / , defined by the property that HomS (X, Y pS / ) classifies commutative diagramsK _K S X

pS

/Y
;

/ S.

The base change properties of the operation S imply similar base change
properties for the relative slice construction: given a map pS : K → Y in
(Set∆ )/S and any map T → S, we have a natural isomorphism
Y pS / ×S T ' (Y ×S T )pT / ,where pT denotes the induced map KT → YT . In particular, the fiber of Y pS /over a vertex s of S can be identified with the absolute slice constructionp /Ys s studied in §4.2.1.
Remark 4.2.2.2. Our notation is somewhat abusive: the simplicial set Y pS /depends not only on the map pS : K → Y but also on the simplicial set S.
We will attempt to avoid confusion by always indicating the simplicial set
S by a subscript in the notation for the map in question; we will omit thissubscript only in the case S = ∆0 , in which case the functor described above
coincides with the functor defined in §4.2.1.
Lemma 4.2.2.3. Let n > 0 and leta
B = (Λnn × ∆1 )(∆n × ∂ ∆1 ) ⊆ ∆n × ∆1 .
1
Λnn ×∂ ∆

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Suppose we are given a diagram of simplicial sets
/9 Y

f0

A × _ B

f

q




/S

A × ∆n × ∆ 1

in which q is a Cartesian fibration, and that f0 carries {a}×∆{n−1,n} ×{1} toa q-Cartesian edge of Y for each vertex a of A. Then there exists a morphism
f rendering the diagram commutative.
Proof. Invoking Proposition 3.1.2.1, we may replace q : Y → S by the induced map Y A → S A and thereby reduce to the case where A = ∆0 . We
now recall the notation introduced in the proof of Proposition 2.1.2.6: more
specifically, the family {σi }0≤i≤n of nondegenerate simplices of ∆n ×∆1 . Let
B(0) = B and more generally set B(n) = B ∪ σn ∪ · · · ∪ σn+1−i so that wehave a filtration
B(0) ⊆ · · · ⊆ B(n + 1) = ∆n × ∆1 .
A map f0 : B(0) → Y has been prescribed for us already; we constructextensions fi : B(i) → Y by induction on i. For i < n, there is a pushout
diagram
Λn+1
n−i
_

/ B(i)
_


∆n+1


/ B(i + 1).

Thus the extension fi+1 can be found by virtue of the assumption that q is
an inner fibration. For i = n, we obtain instead a pushout diagram
Λn+1
n+1
_

/ B(n)
_


∆n+1


/ B(n + 1),

and the desired extension can be found by virtue of the assumption that f0
carries the edge ∆{n−1,n} × {1} to a q-Cartesian edge of Y .
Proposition 4.2.2.4. Suppose we are given a diagram of simplicial sets
K

pS

/X
t

q

/Y
' 
S.

Let p0S = q ◦ pS , and assume that the following conditions are satisfied:(1) The map q is a Cartesian fibration.

highertopoi

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245

LIMITS AND COLIMITS(2) The map t is a coCartesian fibration.
0

Then the induced map r : X pS / → Y pS / is a Cartesian fibration; moreover,an edge of X pS / is r-Cartesian if and only if its image in X is q-Cartesian.
Proof. We first show that r is an inner fibration. Suppose we are given
0 < i < n and a diagramΛni
_/ X pS /<∆n/ Y p0S / ,we must show that it is possible to provide the dotted arrow. Unwinding
the definitions, we see that it suffices to produce the indicated arrow in the
diagram
K S Λni
_

/
;X




/ Y.

q

K S ∆nSince q is a Cartesian fibration, it is a categorical fibration by Proposition
3.3.1.7. Consequently, it suffices to show that the inclusion
K S Λni ⊆ K S ∆n
is a categorical equivalence. In view of the definition of K S M as a pushoutaaK(K ×S M × ∆1 )M,
K×S M ×{0}K×S M ×{1}

it suffices to verify that the inclusions
Λni ⊆ ∆n
K ×S Λni ⊆ K ×S ∆n
are categorical equivalences. The first statement is obvious; the second follows from (the dual of) Proposition 3.3.1.3.
Let us say that an edge of X pS / is special if its image in X is q-Cartesian.
To complete the proof, it will suffice to show that every special edge of X pS /is r-Cartesian and that there are sufficiently many special edges of X pS / .
More precisely, consider any n ≥ 1 and any diagram
Λnn
_

∆n

h

/ X pS /</ Y p0S / .

We must verify the following assertions:highertopoi

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CHAPTER 4• If n = 1, then there exists a dotted arrow rendering the diagram commutative, classifying a special edge of X pS / .
• If n > 1 and h|∆{n−1,n} classifies a special edge of X pS / , then there
exists a dotted arrow rendering the diagram commutative.
Unwinding the definitions, we have a diagram
K S Λnn
_

/X

f0

;

f

q




/ Y,

K  S ∆n

and we wish to prove the existence of the indicated arrow f . As a first step,
we consider the restricted diagramΛnn
_

f0 |Λn
n

/X
>

f1


∆n

q


/ Y.

By assumption, f0 |Λnn carries ∆{n−1,n} to a q-Cartesian edge of X (if n > 1),
so there exists a map f1 rendering the diagram commutative (and classifying
a q-Cartesian edge of X if n = 1). It now suffices to produce the dotted arrowin the diagram`/8 X(K S Λnn ) Λn ∆n
n
_
f



q

i


/ Y,

K  S ∆nwhere the top horizontal arrow is the result of amalgamating f0 and f1 .
Without loss of generality, we may replace S by ∆n . By (the dual of)
Proposition 3.2.2.7, there exists a composable sequence of maps
φ : A0 → · · · → An
and a quasi-equivalence M op (φ) → K. We have a commutative diagram`/ (K S Λnn ) ` n ∆n
(M op (φ) S Λnn ) Λnn ∆n
Λn
_
i0

i


M op (φ) S ∆n/ K S ∆n .

Since q is a categorical fibration, Proposition A.2.3.1 shows that it suffices
to produce a dotted arrow f 0 in the induced diagram`/7 X(M op (φ) S Λnn ) Λnn ∆n
_
0
f

i


M op (φ) S ∆n

q


/ Y.

highertopoi

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247

LIMITS AND COLIMITSLet B be as in the statement of Lemma 4.2.2.3; then we have a pushout
diagram/ (M op (φ) S Λnn ) ` n ∆nA0 × _ BΛni00

A0 × ∆ n × ∆ 1

/ M op (φ) S ∆n .

Consequently, it suffices to prove the existence of the map f 00 in the diagram
8/ X

g

A0 × _ B
f

i00

00

q


A0 × ∆ n × ∆ 1/ Y.

Here the map g carries {a} × ∆{n−1,n} × {1} to a q-Cartesian edge of Y foreach vertex a of A0 . The existence of f 00 now follows from Lemma 4.2.2.3.
Remark 4.2.2.5. In most applications of Proposition 4.2.2.4, we will have0
Y = S. In that case, Y pS / can be identified with S, and the conclusion isthat the projection X pS / → S is a Cartesian fibration.
Remark 4.2.2.6. The hypothesis on s in Proposition 4.2.2.4 can be weakened: all we need in the proof is the existence of maps M op (φ) → K ×S
∆n which are universal categorical equivalences (that is, induce categoricalequivalences M op (φ) ×∆n T → K ×S T for any T → ∆n ). Consequently,
Proposition 4.2.2.4 remains valid when K ' S × K 0 for any simplicial set
K 0 (not necessarily an ∞-category). It seems likely that Proposition 4.2.2.4
remains valid whenever s is a smooth map of simplicial sets, but we havenot been able to prove this.
We can now express the idea that the colimit of a diagram should depend
functorially on the diagram (at least for “smoothly parametrized” familiesof diagrams):
Proposition 4.2.2.7. Let q : Y → S be a Cartesian fibration and let pS :K → Y be a diagram. Suppose that the following conditions are satisfied:(1) For each vertex s of S, the restricted diagram ps : Ks → Ys has acolimit in the ∞-category Ys .
(2) The composition q ◦ pS is a coCartesian fibration.
There exists a map p0S rendering the diagram
K _

pS
p0S


K S S

/Y
;

/S

q

highertopoi

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CHAPTER 4

commutative and having the property that for each vertex s of S, the restriction p0s : Ks  {s} → Ys is a colimit of ps . Moreover, the collection of all suchmaps is parametrized by a contractible Kan complex.
Proof. Apply Proposition 2.4.4.9 to the Cartesian fibration Y pS / and observethat the collection of sections of a trivial fibration constitutes a contractible
Kan complex.
4.2.3 Decomposition of Diagrams
Let C be an ∞-category and p : K → C a diagram indexed by a simplicial
set K. In this section, we will try to analyze the colimit lim(p) (if it exists)−→
in terms of the colimits {lim(p|KI )}, where {KI } is some family of simplicial−→
subsets of K. In fact, it will be useful to work in slightly more generality:
we will allow each KI to be an arbitrary simplicial set mapping to K (notnecessarily via a monomorphism).
Throughout this section, we will fix a simplicial set K, an ordinary category I, and a functor F : I → (Set∆ )/K . It may be helpful to imagine that
I is a partially ordered set and that F is an order-preserving map from I to
the collection of simplicial subsets of K; this will suffice for many but not
all of our applications. We will denote F (I) by KI and the tautological map
KI → K by πI .
Our goal is to show that, under appropriate hypotheses, we can recover the
colimit of a diagram p : K → C in terms of the colimits of diagrams p ◦ πI :
KI → C. Our first goal is to show that the construction of these colimits issuitably functorial in I. For this, we need an auxiliary construction.
Notation 4.2.3.1. We define a simplicial set KF as follows. A map ∆n →KF is determined by the following data:(i) A map ∆n → ∆1 corresponding to a decomposition [n] = {0, . . . , i} ∪{i + 1, . . . , n}.
(ii) A map e− : ∆{0,...,i} → K.
(iii) A map e+ : ∆{i+1,...,n} → N(I) which we may view as a chain ofcomposable morphisms
I(i + 1) → · · · → I(n)in the category I.
(iv) For each j ∈ {i + 1, . . . , n}, a map ej which fits into a commutative
diagramej∆{0,...,i}

e−

:

KI(j)
πI(j)


/ K.

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Moreover, for j ≤ k, we require that ek is given by the compositionej∆{0,...,i} → KI(j) → KI(k) .
Remark 4.2.3.2. In the case where i < n, the maps e− and {ej }j>i arecompletely determined by ei+1 , which can be arbitrary.
The simplicial set KF is equipped with a map KF → ∆1 . Under this map,the preimage of the vertex {0} is K ⊆ KF and the preimage of the vertex{1} is N(I) ⊆ KF . For I ∈ I, we will denote the corresponding vertex ofN(I) ⊆ KF by XI . We note that, for each I ∈ I, there is a commutative
diagram
KI

KI.

πI

/K

πI0


/ KF ,

where πI0 carries the cone point of KI. to the vertex XI of KF .
Let us now suppose that p : K → C is a diagram in an ∞-category C.
Our next goal is to prove Proposition 4.2.3.4, which will allow us to extend
p to a larger diagram KF → C which carries each vertex XI to a colimit ofp ◦ πI : KI → C. First, we need a lemma.
Lemma 4.2.3.3. Let C be an ∞-category and let σ : ∆n → C be a simplex
having the property that σ(0) is an initial object of C. Let ∂ σ = σ| ∂ ∆n . The
natural map Cσ/ → C∂ σ/ is a trivial fibration.
Proof. Unwinding the definition, we are reduced to solving the extension
problem depicted in the diagram
`
f0
n
m
4/ C
(∂ ∆n ? ∆m ) ∂ ∆n ?∂ _ ∆m (∆ ? ∂ ∆ )f∆n ? ∆m .
We can identify the domain of f0 with ∂ ∆n+m+1 . Our hypothesis guarantees
that f0 (0) is an initial object of C, which in turn guarantees the existence off.
Proposition 4.2.3.4. Let p : K → C be a diagram in an ∞-category C, let
I be an ordinary category, and let F : I → (Set∆ )/K be a functor. Suppose
that, for each I ∈ I, the induced diagram pI = p ◦ πI : KI → C has a colimit
qI : KI. → C.
There exists a map q : KF → C such that q ◦ πI0 = qI and q|K = p.
Furthermore, for any such q, the induced map Cq/ → Cp/ is a trivial fibration.
Proof. For each X ⊆ N(I), we let KX denote the simplicial subset of KF
consisting of all simplices σ ∈ KF such that σ ∩ N(I) ⊆ X. We note thatK∅ = K and that KN(I) = KF .

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Define a transfiniteSsequence Yα of simplicial subsets of N(I) as follows. Let
Y0 = ∅ and let Yλ = γ<λ Yγ when λ is a limit ordinal. Finally, let Yα+1 beobtained from Yα by adjoining a single nondegenerate simplex provided thatsuch a simplex exists. We note that for α sufficiently large, such a simplex
will not exist, and we set Yβ = Yα for all β > α.
We define a sequence of maps qβ : KYβ → C so that the following conditions are satisfied:(1) We have q0 = p : K∅ = K → C.
(2) If α < β, then qα = qβ |KYα .
(3) If {XI } ⊆ Yα , then qα ◦ πI0 = qI : KI. → C.
Provided that such a sequence can be constructed, we may conclude theproof by setting q = qα for α sufficiently large.
The construction of qα goes by induction on α. If α = 0, then qα isdetermined by condition (1); if α is a (nonzero) limit ordinal, then qα isdetermined by condition (2). Suppose that qα has been constructed; we givea construction of qα+1 .
There are two cases to consider. Suppose first that Yα+1 is obtained fromYα by adjoining a vertex XI . In this case, qα+1 is uniquely determined byconditions (2) and (3).
Now suppose that Yα+1 is obtained from Yα by adjoining a nondegenerate
simplex σ of positive dimension corresponding to a sequence of composable
maps
I0 → · · · → In
in the category I. We note that the inclusion KYα ⊆ KYα+1 is a pushout of
the inclusion
KI0 ? ∂ σ ⊆ KI0 ? σ.
Consequently, constructing the map qα+1 is tantamount to finding an extension of a certain map s0 : ∂ σ → CpI / to the whole of the simplex σ. By
assumption, s0 carries the initial vertex of σ to an initial object of CpI / , sothat the desired extension s can be found. For use below, we record a further
property of our construction: the projection Cqα+1 / → Cqα / is a pullback of
the map (CpI / )s/ → (CpI / )s0 / , which is a trivial fibration.
We now wish to prove that, for any extension q with the above properties,
the induced map Cq/ → Cp/ is a trivial fibration. We first observe that themap q can be obtained by the inductive construction given above: namely, wetake qα to be the restriction of q to KYα . It will therefore suffice to show that,for every pair of ordinals α ≤ β, the induced map Cqβ / → Cqα / is a trivial
fibration. The proof proceeds by induction on β: the case β = 0 is clear, andif β is a limit ordinal, we observe that the inverse limit of a transfinite towerof trivial fibrations is itself a trivial fibration. We may therefore suppose thatβ = γ + 1 is a successor ordinal. Using the factorization
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and the inductive hypothesis, we are reduced to proving this in the casewhere β is the successor of α, which was treated above.
Let us now suppose that we are given diagrams p : K → C, F : I →(Set∆ )/K as in the statement of Proposition 4.2.3.4 and let q : KF → C bea map which satisfies its conclusions. Since Cq/ → Cp/ is a trivial fibration,
we may identify colimits of the diagram q with colimits of the diagram p(up to equivalence). Of course, this is not useful in itself since the diagram
q is more complicated than p. Our objective now is to show that, under theappropriate hypotheses, we may identify the colimits of q with the colimitsof q| N(I). First, we need a few lemmas.
Lemma 4.2.3.5 (Joyal [44]). Let f : A0 ⊆ A and g : B0 ⊆ B be inclusionsof simplicial sets and suppose that g is a weak homotopy equivalence. Thenthe induced mapah : (A0 ? B)
(A ? B0 ) ⊆ A ? B
A0 ?B0is right anodyne.
Proof. Our proof follows the pattern of Lemma 2.1.2.3. The collection of allmaps f which satisfy the conclusion (for any choice of g) forms a weakly
saturated class of morphisms. It will therefore suffice to prove that the h isright anodyne when f is the inclusion ∂ ∆n ⊆ ∆n . Similarly, the collection of
all maps g which satisfy the conclusion (for fixed f ) forms a weakly saturated
class. We may therefore reduce to the case where g is a horn inclusion Λmi ⊆∆m . In this case, we may identify h with the horn inclusion Λm+n+1
i+n+1 ⊆
∆m+n+1 , which is clearly right anodyne.
Lemma 4.2.3.6. Let A0 ⊆ A be an inclusion of simplicial sets and let B
be weakly contractible. Then the inclusion A0 ? B ⊆ A ? B is right anodyne.
Proof. As above, we may suppose that the inclusion A0 ⊆ A is identified
with ∂ ∆n ⊆ ∆n . If B is a point, then the inclusion A0 ? B ⊆ A ? B isn+1
isomorphic to Λn+1
, which is clearly right anodyne.
n+1 ⊆ ∆In the general case, B is nonempty, so we may choose a vertex b of B.
Since B is weakly contractible, the inclusion {b} ⊆ B is a weak homotopy
equivalence. We have already shown that A0 ?{b} ⊆ A?{b} is right anodyne.
It follows that the pushout inclusiona
A0 ? B ⊆ (A ? {b})(A0 ? B)
A0 ?{b}is right anodyne. To complete the proof, we apply Lemma 4.2.3.5 to deducethat the inclusiona(A ? {b})(A0 ? B) ⊆ A ? B
A0 ?{b}is right anodyne.

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Notation 4.2.3.7. Let σ ∈ Kn be a simplex of K. We define a categoryIσ as follows. The objects of Iσ are pairs (I, σ 0 ), where I ∈ I, σ 0 ∈ (KI )n ,and πI (σ 0 ) = σ. A morphism from (I 0 , σ 0 ) to (I 00 , σ 00 ) in Iσ consists of amorphism α : I 0 → I 00 in I with the property that F (α)(σ 0 ) = σ 00 . We let
I0σ ⊆ Iσ denote the full subcategory consisting of pairs (I, σ 0 ), where σ 0 is adegenerate simplex in KI . Note that if σ is nondegenerate, I0σ is empty.
Proposition 4.2.3.8. Let K be a simplicial set, I an ordinary category,and F : I → (Set∆ )/K a functor. Suppose that the following conditions are
satisfied:(1) For each nondegenerate simplex σ of K, the category Iσ is acyclic (thatis, the simplicial set N(Iσ ) is weakly contractible).
(2) For each degenerate simplex σ of K, the inclusion N(I0σ ) ⊆ N(Iσ ) is aweak homotopy equivalence.
Then the inclusion N(I) ⊆ KF is right anodyne.
Proof. Consider any family of subsets {Ln ⊆ Kn } which is stable underthe “face maps” di on K (but not necessarily the degeneracy maps si , sothat the family {Ln } does not necessarily have the structure of a simplicial
set). We define a simplicial subset LF ⊆ KF as follows: a nondegenerate
simplex ∆n → KF belongs to LF if and only if the corresponding (possibly
degenerate) simplex ∆{0,...,i} → K belongs to Li ⊆ Ki (see Notation 4.2.3.1).
We note that if L = ∅, then LF = N(I). If L = K, then LF = KF (so that
our notation is unambiguous). Consequently, it will suffice to prove that, for
any L ⊆ L0 , the inclusion LF ⊆ L0F is right anodyne. By general nonsense,
we may reduce to the case where L0 is obtained from L by adding a single
simplex σ ∈ Kn .
We now have two cases to consider. Suppose first that the simplex σ
is nondegenerate. In this case, it is not difficult to see that the inclusion
LF ⊆ L0F is a pushout of ∂ σ ? N(Iσ ) ⊆ σ ? N(Iσ ). By hypothesis, N(Iσ )is weakly contractible, so that the inclusion LF ⊆ L0F is right anodyne byLemma 4.2.3.6.
In the case where σ is degenerate, we observe that LF ⊆ L0F is a pushoutof the inclusiona(∂ σ ? N(Iσ ))(σ ? N(I0σ )) ⊆ σ ? N(Iσ ),∂ σ?N(I0σ )which is right anodyne by Lemma 4.2.3.5.
Remark 4.2.3.9. Suppose that I is a partially ordered set and that F is anorder-preserving map from I to the collection of simplicial subsets of K. In
this case, we observe that I0σ = Iσ whenever σ is a degenerate simplex of K
and that Iσ = {I ∈ I : σ ∈ KI } for any σ. Consequently, the conditions of
Proposition 4.2.3.8 hold if and only if each of the partially ordered subsets
Iσ ⊆ S
I has a contractible nerve. This holds automatically if I is directed and
K = I∈I KI .

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LIMITS AND COLIMITSCorollary 4.2.3.10. Let K be a simplicial set, I a category, and F : I →(Set∆ )/K a functor which satisfies the hypotheses of Proposition 4.2.3.8. Let
C be an ∞-category, let p : K → C be any diagram, and let q : KF → C be
an extension of p which satisfies the conclusions of Proposition 4.2.3.4. The
natural maps
Cp/ ← Cq/ → Cq| N(I)/are trivial fibrations. In particular, we may identify colimits of p with colimitsof q| N(I).
Proof. This follows immediately from Proposition 4.2.3.8, since the right anodyne inclusion N(I) ⊆ KF is cofinal and therefore induces a trivial fibration
Cq/ → Cq| N(I)/ by Proposition 4.1.1.8.
We now illustrate the usefulness of Corollary 4.2.3.10 by giving a sample
application. First, a bit of terminology. If κ and τ are regular cardinals, wewill write τ  κ if, for any cardinals τ0 < τ , κ0 < κ, we have κτ00 < κ(we refer the reader to Definition A.2.6.3 and the surrounding discussion for
more details concerning this condition).
Corollary 4.2.3.11. Let C be an ∞-category and τ  κ regular cardinals.
Then C admits κ-small colimits if and only if C admits τ -small colimits andcolimits indexed by (the nerves of ) κ-small τ -filtered partially ordered sets.
Proof. The “only if” direction is obvious. Conversely, let p : K → C be any
κ-small diagram. Let I denote the partiallyordered set of τ -small simplicial
S
subsets of K. Then I is directed and I∈I KI = K, so that the hypothesesof Proposition 4.2.3.8 are satisfied. Since each pI = p ◦ πI has a colimit inC, there exists a map q : KF → C satisfying the conclusions of Proposition
4.2.3.4. Because Cq/ → Cp/ is an equivalence of ∞-categories, p has a colimit
if and only if q has a colimit. By Corollary 4.2.3.10, q has a colimit if andonly if q| N(I) has a colimit. It is clear that I is a τ -filtered partially orderedset. Furthermore, it is κ-small provided that τ  κ.
The following result can be proven by the same argument:
Corollary 4.2.3.12. Let f : C → C0 be a functor between ∞-categories andlet τ  κ be regular cardinals. Suppose that C admits κ-small colimits. Thenf preserves κ-small colimits if and only if it preserves τ -small colimits andall colimits indexed by (the nerves of) κ-small τ -filtered partially ordered sets.
We will conclude this section with another application of Proposition
4.2.3.8 in which I is not a partially ordered set and the maps πI : KI → K
are not (necessarily) injective. Instead, we take I to be the category of simplices of K. In other words, an object I ∈ I consists of a map σI : ∆n → K,
and a morphism from I to I 0 is given by a commutative diagram/ ∆ n0
∆n
σI 0

σI

!

K.

}

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For each I ∈ I, we let KI denote the domain ∆n of σI , and we let πI = σI :KI → K.
Lemma 4.2.3.13. Let K be a simplicial set and let I denote the categoryof simplices of K (as defined above). Then there is a retraction r : KF → K
which fixes K ⊆ KF .
Proof. Given a map e : ∆n → KF , we will describe the composite mapr ◦ e : ∆n → K. The map e classifies the following data:(i) A decomposition [n] = {0, . . . , i} ∪ {i + 1, . . . , n}.
(ii) A map e− : ∆i → K.
(iii) A string of morphisms∆mi+1 → · · · → ∆mn → K.
(iv) A compatible family of maps {ej : ∆i → ∆mj }j>i having the property
ej
that each composition ∆i → ∆mj → K coincides with e− .
If i = n, we set r ◦ e = e− . Otherwise, we let r ◦ e denote the compositionf∆n → ∆mn → K
where f : ∆n → ∆mn is defined as follows:• The restriction f |∆i coincides with en .
• For i < j ≤ n, we let f (j) denote the image in ∆mn of the final vertexof ∆mj .

Proposition 4.2.3.14. For every simplicial set K, there exists a category
I and a cofinal map f : N(I) → K.
Proof. We take I to be the category of simplices of K, as defined above, andf to be the composition of the inclusion N(I) ⊆ KF with the retraction rof Lemma 4.2.3.13. To prove that f is cofinal, it suffices to show that theinclusion N(I) ⊆ KF is right anodyne and that the retraction r is cofinal.
To show that N(I) ⊆ KF is right anodyne, it suffices to show that the
hypotheses of Proposition 4.2.3.8 are satisfied. Let σ : ∆J → K be a simplexof K. We observe that the category Iσ may be described as follows: its objects
consist of pairs of maps (s : ∆J → ∆M , t : ∆M → K) with t ◦ s = σ. A
morphism from (s, t) to (s0 , t0 ) consists of a map
α : ∆M → ∆M0with s0 = α ◦ s and t = t0 ◦ α. In particular, we note that Iσ has an initial
object (id∆J , σ). It follows that N(Iσ ) is weakly contractible for any simplex
σ of K. It will therefore suffice to show that N(I0σ ) is weakly contractible

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whenever σ is degenerate. Let I00σ denote the full subcategory of I0σ spannedby those objects for which the map s is surjective. The inclusion I00σ ⊆ I0σ
has a right adjoint, so that N(I00σ ) is a deformation retract of N(I0σ ). It will
therefore suffice to prove that N(I00σ ) is weakly contractible. For this, west
simply observe that N(I00σ ) has a final object ∆J → ∆M → K, characterized
by the property that t is a nondegenerate simplex of K.
We now show that r is cofinal. According to Proposition 4.1.1.8, it suffices
to show that for any ∞-category C and any map p : K → C, the induced
map Cq/ → Cp/ is a categorical equivalence, where q = p ◦ r. This follows
from Proposition 4.2.3.4.
Variant 4.2.3.15. Proposition 4.2.3.14 can be strengthened as follows:(∗) For every simplicial set K, there exists a cofinal map φ : K 0 → K,
where K 0 is the nerve of a partially ordered set.
Moreover, the map φ can be chosen to depend functorially on K. We will
construct φ as a composition of five cofinal maps
φ5

φ4

φ3

φ2

φ1

K 0 = K (5) → K (4) → K (3) → K (2) → K (1) → K,
which are defined as follows:(1) The simplicial set K (1) is the nerve N(I1 ), where I1 denotes the category of simplices of K, and the morphism φ1 is the cofinal map described in Proposition 4.2.3.14.
(2) The simplicial set K (2) is the nerve N(I2 )op , where I2 denotes thecategory of simplices of K (1) . The map φ2 is induced by the functor
I2 → I1 which carries a chain of morphisms C0 → C1 → · · · → Cn in I1
to the object C0 . We claim that φ2 is cofinal. To prove this, it will suffice(by virtue of Theorem 4.1.3.1) to prove that for every object C ∈ I1 ,the category Iop
2 ×I1 (I1 )C/ has weakly contractible nerve. Indeed, thisis the opposite of the category J of simplices of N(I1 )C/ . Proposition
4.2.3.14 supplies a cofinal map N(J) → N(I1 )C/ , so that N(J) is weakly
homotopy equivalent to N(I1 )C/ , which is weakly contractible (since it
has an initial object).
(3) Let I3 denote the subcategory of I2 consisting of injective maps between simplices of K (1) , and let K (3) = N(I3 )op ⊆ N(I2 )op . We have apullback diagram
N(I3 )op/ N(I2 )opN(∆s )op/ N(∆)op

where lower horizontal map is the cofinal inclusion of Lemma 6.5.3.7.
The vertical maps are left fibrations and therefore smooth (Proposition
4.1.2.15), so that the inclusion φ3 : K (3) ⊆ K (2) is cofinal.

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CHAPTER 4(4) Let K (4) denote the nerve N(I4 ), where I4 is the category of simplices
of K (3) , and let φ4 denote the cofinal map described in Proposition
4.2.3.14.
(5) Let K (5) denote the nerve N(I5 ) ⊆ N(I4 ), where I5 is the full subcategory spanned by the nondegenerate simplices of K (3) . We observethat the category I3 has the following property: the collection of nonidentity morphisms in I3 is stable under composition. It follows thatevery face of a nondegenerate simplex of K (3) is again nondegenerate.
Consequently, the inclusion I5 ⊆ I4 admits a left adjoint, so that the inclusion φ5 : N(I5 ) ⊆ N(I4 ) is cofinal (this follows easily from Theorem
4.1.3.1). We conclude by observing that the category I5 is equivalent
to a partially ordered set, because if σ is a simplex of K (3) , then any
face of σ is uniquely determined by the vertices that it contains.
Variant 4.2.3.16. If K is a finite simplicial set, then the we can arrangethat the simplicial set K 0 appearing in Variant 4.2.3.15 is again finite (though
our construction is not functorial in K). First suppose that K satisfies thefollowing condition:(∗) Every nondegenerate simplex σ : ∆n → K is a monomorphism of
simplicial sets.
Let I denote the category of simplices of K, and let I0 denote the full subcategory spanned by the nondegenerate simplices. Condition (∗) guarantees thatthe inclusion I0 ⊆ I admits a left adjoint, so that Theorem 4.1.3.1 implies
that the inclusion N(I0 ) ⊆ N(I) is cofinal. Combining this with Proposition
4.2.3.14, we deduce that the map N(I0 ) → K is cofinal. Moreover, N(I0 ) canbe identified with the nerve of the partially ordered set of simplicial subsets
K0 ⊆ K such that K0 is isomorphic to a simplex. In particular, this partiallyordered set is finite.
To handle the general case, it will suffice to establish the following claim:e → K,(∗0 ) For every finite simplicial set K, there exists a cofinal map K
ewhere K is a finite simplicial set satisfying (∗).
The proof proceeds by induction on the number of nondegenerate simplices
of K. If K is empty, the result is obvious; otherwise, we have a pushout
diagram
/ ∆n
∂ ∆n

K0


/ K.

e 0 → K0 satThe inductive hypothesis guarantees theexistence of a map K
e = (Ke 0 × ∆n ) ` n ∆n . It follows from Corollary
isfying (∗). Now define K∂∆
e → K is cofinal.
4.1.2.7 (and the weak contractibility of ∆n ) that the map KnneMoreover, since the map ∂ ∆ → K0 × ∆ is a monomorphism, we conclude
e satisfies condition (∗), as desired.
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4.2.4 Homotopy Colimits
Our goal in this section is to compare the ∞-categorical theory of colimits
with the more classical theory of homotopy colimits in simplicial categories(see Remark A.3.3.13). Our main result is the following:
Theorem 4.2.4.1. Let C and I be fibrant simplicial categories and F : I → C
a simplicial functor. Suppose we are given an object C ∈ C and a compatible
family of maps {ηI : F (I) → C}I∈I . The following conditions are equivalent:(1) The maps ηI exhibit C as a homotopy colimit of the diagram F .
(2) Let f : N(I) → N(C) be the simplicial nerve of F and f : N(I). → N(C)the extension of f determined by the maps {ηI }. Then f is a colimit
diagram in N(C).
Remark 4.2.4.2. For an analogous result (in a slightly different setting),
we refer the reader to [39].
The proof of Theorem 4.2.4.1 will occupy the remainder of this section.
We begin with a convenient criterion for detecting colimits in ∞-categories:
Lemma 4.2.4.3. Let C be an ∞-category, K a simplicial set, and p : K . →C a diagram. The following conditions are equivalent:(i) The diagram p is a colimit of p = p|K.
(ii) Let X ∈ C denote the image under p of the cone point of K . , letδ : C → Fun(K, C) denote the diagonal embedding, and let α : p → δ(X)denote the natural transformation determined by p. Then, for every
object Y ∈ C, composition with α induces a homotopy equivalenceφY : MapC (X, Y ) → MapFun(K,C) (p, δ(Y )).
Proof. Using Corollary 4.2.1.8, we can identify MapFun(K,C) (p, δ(Y )) withthe fiber Cp/ ×C {Y } for each object Y ∈ C. Under this identification, themap φY can be identified with the fiber over Y of the composition
φ0

φ00

CX/ → Cp/ → Cp/ ,where φ0 is a section to the trivial fibration Cp/ → CX/ . The map φ00 is a left
fibration (Proposition 4.2.1.6). Condition (i) is equivalent to the requirementthat φ00 be a trivial Kan fibration, and condition (ii) is equivalent to therequirement that each of the maps
φ00Y : Cp/ ×C {Y } → Cp/ ×C {Y }is a homotopy equivalence of Kan complexes (which, in view of Lemma
2.1.3.3, is equivalent to the requirement that φ00Y be a trivial Kan fibration).
The equivalence of these two conditions now follows from Lemma 2.1.3.4.

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The key to Theorem 4.2.4.1 is the following result, which compares theconstruction of diagram categories in the ∞-categorical and simplicial settings:
Proposition 4.2.4.4. Let S be a small simplicial set, C a small simplicial
category, and u : C[S] → C an equivalence. Suppose that A is a combinatorial simplicial model category and let U be a C-chunk of A (see Definition
A.3.4.9). Then the induced map
N((UC )◦ ) → Fun(S, N(U◦ ))is a categorical equivalence of simplicial sets.
Remark 4.2.4.5. In the statement of Proposition 4.2.4.4, it makes no difference whether we regard AC as endowed with the projective or the injectivemodel structure.
Remark 4.2.4.6. An analogous result was proved by Hirschowitz and Simpson; see [39].
Proof. Choose a regular cardinal κ such that S and C are κ-small. Using
Lemma A.3.4.15, we can write U as a κ-filtered colimit of small C-chunks
U0 contained in U. Since the collection of categorical equivalences is stable
under filtered colimits, it will suffice to prove the result after replacing U byeach U0 ; in other words, we may suppose that U is small.
According to Theorem 2.2.5.1, we may identify the homotopy categoryof Set∆ (with respect to the Joyal model structure) with the homotopy
category of Cat∆ . We now observe that because N(U◦ ) is an ∞-category, thesimplicial set Fun(S, N(U◦ )) can be identified with an exponential [N(U◦ )][S]in the homotopy category hSet∆ . We now conclude by applying Corollary
A.3.4.14.
One consequence of Proposition 4.2.4.4 is that every homotopy coherent
diagram in a suitable model category A can be “straightened,” as we indicated in Remark 1.2.6.2.
Corollary 4.2.4.7. Let I be a fibrant simplicial category, S a simplicial set,
and p : N(I) → S a map. Then it is possible to find the following:(1) A fibrant simplicial category C.
(2) A simplicial functor P : I → C.
(3) A categorical equivalence of simplicial sets j : S → N(C).
(4) An equivalence between j ◦ p and N(P ) as objects of the ∞-category
Fun(N(I), N(C)).
Proof. Choose an equivalence i : C[S] → C0 , where C0 is fibrant; let A denotethe model category of simplicial presheaves on C0 (endowed with the injectivemodel structure). Composing i with the Yoneda embedding of C0 , we obtainhighertopoi

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a fully faithful simplicial functor C[S] → A◦ , which we may alternatively
view as a morphism j0 : S → N(A◦ ).
We now apply Proposition 4.2.4.4 to the case where u is the counit map
C[N(I)] → I. We deduce that the natural map
N((AI )◦ ) → Fun(N(I), N(A◦ ))is an equivalence. From the essential surjectivity, we deduce that j0 ◦ p is
equivalent to N(P0 ), where P0 : I → A◦ is a simplicial functor.
We now take C to be the essential image of C[S] in A◦ and note that j0
and P0 factor uniquely through maps j : S → N(C), P : I → C which possessthe desired properties.
We now return to our main result.
Proof of Theorem 4.2.4.1: Let A denote the category SetC
∆ endowed withthe projective model structure. Let j : Cop → A denote the Yoneda embedding and let U denote the full subcategory of A spanned by those objects
which are weakly equivalent to j(C) for some C ∈ C, so that j induces anequivalence of simplicial categories Cop → U◦ . Choose a trivial injective cofiop
bration j ◦ F → F 0 , where F 0 is a injectively fibrant object of AI . Let◦
f 0 : N(I)op → N(U ) be the nerve of F 0 and let C 0 = j(C), so that the maps
{ηI : F (I) → C}I∈I induce a natural transformation α : δ(C 0 ) → f 0 , whereδ : N(U◦ ) → Fun(N(I)op , N(U◦ )) denotes the diagonal embedding. In viewof Lemma 4.2.4.3, condition (1) admits the following reformulation:(10 ) For every object A ∈ U◦ , composition with α induces a homotopy
equivalence
MapN(U◦ ) (A, C 0 ) → MapFun(N(I)op ,N(U◦ )) (δ(A), f 0 ).
Using Proposition 4.2.4.4, we can reformulate this condition again:(100 ) For every object A ∈ U◦ , the canonical map
MapA (A, C 0 ) → MapAIop (δ 0 (A), F 0 )is a homotopy equivalence, where δ 0 : A → AI
embedding.

opdenotes the diagonal

Let B ∈ A be a limit of the diagram F 0 , so we have a canonical map
β : C 0 → B between fibrant objects of A. Condition (2) is equivalent to theassertion that β is a weak equivalence in A, while condition (100 ) is equivalent
to the assertion that composition with β induces a homotopy equivalence
MapA (A, C 0 ) → MapA (A, B)
for each A ∈ U◦ . The implication (2) ⇒ (100 ) is clear. Conversely, supposethat (100 ) is satisfied. For each X ∈ C, the object j(X) belongs to U◦ , so thatβ induces a homotopy equivalence
C 0 (X) ' MapA (j(X), C 0 ) → MapA (j(X), B) ' B(X).
It follows that β is a weak equivalence in A, as desired.

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Corollary 4.2.4.8. Let A be a combinatorial simplicial model category. The
associated ∞-category S = N(A◦ ) admits (small) limits and colimits.
Proof. We give the argument for colimits; the case of limits follows by adual argument. Let p : K → S be a (small) diagram in S. By Proposition
4.2.3.14, there exists a (small) category I and a cofinal map q : N(I) → K.
Since q is cofinal, p has a colimit in S if and only if p ◦ q has a colimit in S;
thus we may reduce to the case where K = N(I).
Using Proposition 4.2.4.4, we may suppose that p is the nerve of a injectively fibrant diagram p0 : I → A◦ . Let p0 : I ?{x} → AI be a colimit of p0 , sothat p0 is a homotopy colimit diagram in A. Now choose a trivial fibration
p00 → p0 in AI , where p00 is cofibrant. The simplicial nerve of p00 determines
a colimit diagram f : N(I). → S by Theorem 4.2.4.1. We now observe thatf = f | N(I) is equivalent to p, so that p also admits a colimit in S.

4.3 KAN EXTENSIONSLet C and I be ordinary categories. There is an obvious “diagonal” functor
δ : C → CI , which carries an object C ∈ C to the constant diagram I → C
taking the value C. If C admits small colimits, then the functor δ has a left
adjoint CI → C. This left adjoint admits an explicit description: it carries
an arbitrary diagram f : I → C to the colimit lim(f ). Consequently, we−→
can think of the theory of colimits as the study of left adjoints to diagonal
functors.
More generally, if one is given a functor i : I → I0 between diagram0
categories, then composition with i induces a functor i∗ : CI → CI . Assumingthat C has a sufficient supply of colimits, one can construct a left adjoint toi∗ . We then refer to this left adjoint as the left Kan extension along i.
In this section, we will study the ∞-categorical analogue of the theory ofleft Kan extensions. In the extreme case where I0 is the one-object category∗, this theory simply reduces to the theory of colimits introduced in §1.2.13.
Our primary interest will be at the opposite extreme, when i is a fully faithfulembedding; this is the subject of §4.3.2. We will treat the general case in§4.3.3.
With a view toward later applications, we will treat not only the theoryof absolute left Kan extensions but also a relative notion which works over abase simplicial set S. The most basic example is the case of a relative colimit
which we study in §4.3.1.
4.3.1 Relative Colimits
In §1.2.13, we introduced the notions of limit and colimit for a diagram
p : K → C in an ∞-category C. For many applications, it is convenient tohave a relative version of these notions, which makes reference not to an∞-category C but to an arbitrary inner fibration of simplicial sets.

highertopoi

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LIMITS AND COLIMITSDefinition 4.3.1.1. Let f : C → D be an inner fibration of simplicial sets,
let p : K . → C be a diagram, and let p = p|K. We will say that p is an
f -colimit of p if the map
Cp/ → Cp/ ×Df p/ Df p/
is a trivial fibration of simplicial sets. In this case, we will also say that p isan f -colimit diagram.
Remark 4.3.1.2. Let f : C → D and p : K . → C be as in Definition 4.3.1.1.
Then p is an f -colimit of p = p|K if and only if the map
φ : Cp/ → Cp/ ×Df p/ Df p/
is a categorical equivalence. The “only if” direction is clear. The converse
follows from Proposition 2.1.2.1 (which implies that φ is a left fibration),
Proposition 3.3.1.7 (which implies that φ is a categorical fibration), and thefact that a categorical fibration which is a categorical equivalence is a trivial
Kan fibration.
Observe that Proposition 2.1.2.1 also implies that the map
Df p/ → Df p/
is a left fibration. Using Propositions 3.3.1.3 and 3.3.1.7, we conclude thatthe fiber product Cp/ ×Df p/ Df p/ is also a homotopy fiber product of Cp/
with Df p/ over Df p/ (with respect to the Joyal model structure on Set∆ ).
Consequently, we deduce that p is an f -colimit diagram if and only if the
diagram of simplicial sets
/ Df p/
Cp/

Cp/


/ Df p/is homotopy Cartesian.
Example 4.3.1.3. Let C be an ∞-category and f : C → ∗ the projection of
C to a point. Then a diagram p : K . → C is an f -colimit if and only if it isa colimit in the sense of Definition 1.2.13.4.
Example 4.3.1.4. Let f : C → D be an inner fibration of simplicial sets
and let e : ∆1 = (∆0 ). → C be an edge of C. Then e is an f -colimit if andonly if it is f -coCartesian.
The following basic stability properties follow immediately from the definition:
Proposition 4.3.1.5. (1) Let f : C → D be a trivial fibration of simplicial
sets. Then every diagram p : K . → C is an f -colimit.
(2) Let f : C → D and g : D → E be inner fibrations of simplicial sets andlet p : K . → C be a diagram. Suppose that f ◦ p is a g-colimit. Then p
is an f -colimit if and only if p is a (g ◦ f )-colimit.

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CHAPTER 4(3) Let f : C → D be an inner fibration of ∞-categories and let p, q :K . → C be diagrams which are equivalent when viewed as objects of
the ∞-category Fun(K . , C). Then p is an f -colimit if and only if q is
an f -colimit.
(4) Suppose we are given a Cartesian diagram
/C

g

C0
f0

f



/D
D0
of simplicial sets, where f (and therefore also f 0 ) is an inner fibration.
Let p : K . → C0 be a diagram. If g ◦ p is an f -colimit, then p is an
f 0 -colimit.
Proposition 4.3.1.6. Suppose we are given a commutative diagram of ∞categories
C
p


D

f

/ C0
p0


/ D0 ,

where the horizontal arrows are categorical equivalences and the vertical arrows are inner fibrations. Let q : K . → C be a diagram and let q = q|K.
Then q is a p-colimit of q if and only if f ◦ q is a p0 -colimit of f ◦ q.
Proof. Consider the diagramCq/Cq/ ×Dpq/ Dpq/

/ C0f q/

/ C0f q/ ×D0p0 f q/.

D0p0 f q/

According to Remark 4.3.1.2, it will suffice to show that the left vertical map
is a categorical equivalence if and only if the right vertical map is a categorical equivalence. For this, it suffices to show that both of the horizontal
maps are categorical equivalences. Proposition 1.2.9.3 implies that the maps
Cq/ → C0f q/ , Cq/ → C0f q/ , Dpq/ → D0p0 f q/ , and Dpq/ → D0p0 f q/ are categoricalequivalences. It will therefore suffice to show that the diagrams/ Cq// C0f q/Cq/ ×D Dpq/
C0f q/ ×D0
D0p0 f q/
pq/


Dpq/

p0 f q/ψ/ Dpq/

D0p0 f q/

ψ0
/ D0p0 f q/

are homotopy Cartesian (with respect to the Joyal model structure). This
follows from Proposition 3.3.1.3 because ψ and ψ 0 are coCartesian fibrations.

highertopoi

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LIMITS AND COLIMITSThe next pair of results can be regarded as a generalization of Proposition
4.1.1.8. They assert that, when computing relative colimits, we are free toreplace any diagram by a cofinal subdiagram.
Proposition 4.3.1.7. Let p : C → D be an inner fibration of ∞-categories,
let i : A → B be a cofinal map, and let q : B . → C be a diagram. Then q isa p-colimit if and only if q ◦ i. is a p-colimit.
Proof. Recall (Remark 4.3.1.2) that q is a relative colimit diagram if andonly if the diagram
Cq/

/ Cq/


Dq 0 /

/ Dq /
0is homotopy Cartesian with respect to the Joyal model structure. Since i
and i. are both cofinal, this is equivalent to the assertion that the diagram
Cqi. // Cqi/


Dq 0 i . // Dq i/0

is homotopy Cartesian, which (by Remark 4.3.1.2) is equivalent to the assertion that q ◦ i. is a relative colimit diagram.
Proposition 4.3.1.8. Let p : C → D be a coCartesian fibration of ∞categories, let i : A → B be a cofinal map, and let
q

B

/C
p


B.

q0


/D

be a diagram. Suppose that q ◦ i has a relative colimit lifting q 0 ◦ i. . Then q
has a relative colimit lifting q 0 .
Proof. Let q0 = q 0 |B. We have a commutative diagram
Cq/
Dq 0 /f/ Cqi/ ×Dpqi// Dq /0

Dpq// Cqi/
/ Dq i/ ,0where the horizontal maps are categorical equivalences (this follows fromthe fact that i is cofinal and Proposition 3.3.1.3). Proposition 2.4.3.1 implies
that the vertical maps are coCartesian fibrations and that f preserves coCartesian edges. Applying Proposition 3.3.1.5 to f , we deduce that the maphighertopoi

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φ : Cq/ ×Dq0 / {q 0 } → Cqi/ ×Dq0 i/ {q 0 i. } is a categorical equivalence. Since φ isessentially surjective, we conclude that there exists an extension q : B . → C
of q which covers q 0 , such that q ◦ i. is a p-colimit diagram. We now apply
Proposition 4.3.1.7 to conclude that q is itself a p-colimit diagram.
Let p : X → S be a coCartesian fibration. The following results will allow
us to reduce the theory of p-colimits to the theory of ordinary colimits inthe fibers of p.
Proposition 4.3.1.9. Let p : X → S be an inner fibration of ∞-categories,
K a simplicial set, and h : ∆1 × K . → X a natural transformation fromh0 = h|{0} × K . to h1 = h|{1} × K . . Suppose that
(1) For every vertex x of K . , the restriction h|∆1 × {x} is a p-coCartesianedge of X.
(2) The compositionhp∆1 × {∞} ⊆ ∆1 × K . → X → S
is a degenerate edge of S, where ∞ denotes the cone point of K . .
Then h0 is a p-colimit diagram if and only if h1 is a p-colimit diagram.
Proof. Let h = h|∆1 × K, h0 = h|{0} × K, and h1 = h|{1} × K. Consider
the diagram
Xh0 / oφψ
Xh0 / ×Sph0 / Sph0 / o

/X

Xh/

h1 /


Xh/ ×Sph/ Sph/

/ Xh / ×S
1



ph1 /

Sph1 / .

According to Remark 4.3.1.2, it will suffice to show that the left vertical map
is a categorical equivalence if and only if the right vertical map is a categoricalequivalence. For this, it will suffice to show that each of the horizontal arrows
is a categorical equivalence. Because the inclusions {1} × K ⊆ ∆1 × K and{1} × K . ⊆ ∆1 × K . are right anodyne, the horizontal maps on the right
are trivial fibrations. We are therefore reduced to proving that φ and ψ arecategorical equivalences.
Let f : x → y denote the edge of X obtained by restricting h to the conepoint of K . . The map φ fits into a commutative diagram
Xh/

Xf /

φ

/ Xh /
0/ Xx/ .

Since the inclusion of the cone point into K . is right anodyne, the verticalarrows are trivial fibrations. Moreover, hypotheses (1) and (2) guarantee thathighertopoi

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f is an equivalence in X, so that the map Xf / → Xx/ is a trivial fibration.
This proves that φ is a categorical equivalence.
The map ψ admits a factorization
ψ0

ψ 00

Xh/ ×Sph/ Sph/ → Xh0 / ×Sph0 / Sph/ → Xh0 ×Sph0 / Sph0 / .
To complete the proof, it will suffice to show that ψ 0 and ψ 00 are trivial
fibrations of simplicial sets. We first observe that ψ 0 is a pullback of the map
Xh/ → Xh0 / ×Sph0 / Sph/ ,which is a trivial fibration (Proposition 3.1.1.11). The map ψ 00 is a pullbackof the left fibration ψ000 : Sph/ → Sph0 / . It therefore suffices to show that ψ000is a categorical equivalence. To prove this, we consider the diagramSph/
Sp(f )/

ψ000

/S

ph0 /

ψ100


/ Sp(x)/ .

As above, we observe that the vertical arrows are trivial fibrations and that
ψ100 is a trivial fibration (because the morphism p(f ) is an equivalence in S).
It follows that ψ000 is a categorical equivalence, as desired.
Proposition 4.3.1.10. Let q : X → S be a locally coCartesian fibration of∞-categories, let s be an object of S, and let p : K . → Xs be a diagram.
The following conditions are equivalent:(1) The map p is a q-colimit diagram.
(2) For every morphism e : s → s0 in S, the associated functor e! : Xs →Xs0 has the property that e! ◦ p is a colimit diagram in the ∞-category
Xs0 .
Proof. Assertion (1) is equivalent to the statement that the map
θ : Xp/ → Xp/ ×Sqp/ Sqp/
is a trivial fibration of simplicial sets. Since θ is a left fibration, it will suffice
to show that the fibers of θ are contractible. Consider an arbitrary vertex
11of Sqp/ corresponding to a morphism` t : K1 ? ∆ → S. Since K ? ∆ iscategorically equivalent to (K ? {0}) {0} ∆ and t|K ? {0} is constant, we
may assume without loss of generality that t factors as a compositione

K ? ∆1 → ∆1 → S.
Here e : s → s0 is an edge of S. Pulling back by the map e, we can reduceto the problem of proving the following analogue of (1) in the case whereS = ∆1 :(10 ) The projection h0 : Xp/ ×S {s0 } → Xp/ ×S {s0 } is a trivial fibration of
simplicial sets.

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Choose a coCartesian transformation α : K . × ∆1 → X from p to p0 ,which covers the projection
K . × ∆1 → ∆1 ' S.
Consider the diagram
Xp/ ×S {s0 } oh0Xp/ ×S {s0 } o/ Xp0 / ×S {s0 }Xα/ ×S {s0 }

h1

h


/ Xp0 / ×S {s0 }.


Xα/ ×S {s0 }Note that the vertical maps are left fibrations (Proposition 2.1.2.1). Since theinclusion K . × {1} ⊆ K . × ∆1 is right anodyne, the upper right horizontal
map is a trivial fibration. Similarly, the lower right horizontal map is a trivial
fibration. Since α is a coCartesian transformation, we deduce that the left
horizontal maps are also trivial fibrations (Proposition 3.1.1.11). Condition(2) is equivalent to the assertion that h1 is a trivial fibration (for each edgee : s → s0 of the original simplicial set S). Since h1 is a left fibration andtherefore a categorical fibration (Proposition 3.3.1.7), this is equivalent to theassertion that h1 is a categorical equivalence. Chasing through the diagram,we deduce that (2) is equivalent to the assertion that h0 is a categoricalequivalence, which (by the same argument) is equivalent to the assertion
that h0 is a trivial fibration.
Corollary 4.3.1.11. Let p : X → S be a coCartesian fibration of ∞categories and let K be a simplicial set. Suppose that
(1) For each vertex s of S, the fiber Xs = X ×S {s} admits colimits for all
diagrams indexed by K.
(2) For each edge f : s → s0 , the associated functor Xs → Xs0 preserves
colimits of K-indexed diagrams.
Then for every diagram
q

K _

/X
=

q


K.

f


/S

p

there exists a map q as indicated, which is a p-colimit.
Proof. Consider the map K × ∆1 → K . which is the identity on K × {0}and carries K × {1} to the cone point of K . . Let F denote the compositionf

K × ∆1 → K . → S
and let Q : K × ∆1 → X be a coCartesian lifting of F to X, so that Q is anatural transformation from q to a map q 0 : K → Xs , where s is the image
under f of the cone point of K . . In view of assumption (1), there exists ahighertopoi

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map q 0 : K . → Xs which is a colimit of q 0 . Assumption (2) and Proposition
4.3.1.10 guarantee that q 0 is also a p-colimit diagram when regarded as amap from K . to X.
We have a commutative diagram(K × ∆1 )(Q,q 0 )(K . × {1})
K×{1}
_

`

r

/4 X
p




/ S.

(K × ∆1 ).

The left vertical map is an inner fibration, so there exists a morphism r as
indicated, rendering the diagram commutative. We now consider the map
K . × ∆1 → (K × ∆1 ). which is the identity on K × ∆1 and carries theother vertices of K . × ∆1 to the cone point of (K × ∆1 ). . Let Q denote thecompositionr

K . × ∆1 → (K × ∆1 ). → X
and let q = Q|K . ×{0}. Then Q can be regarded as a natural transformation
q → q 0 of diagrams K . → X. Since q 0 is a p-colimit diagram, Proposition
4.3.1.9 implies that q is a p-colimit diagram as well.
Proposition 4.3.1.12. Let p : X → S be a coCartesian fibration of ∞categories and let q : K . → X be a diagram. Assume that the following
conditions are satisfied:(1) The map q carries each edge of K to a p-coCartesian edge of K.
(2) The simplicial set K is weakly contractible.
Then q is a p-colimit diagram if and only if it carries every edge of K . toa p-coCartesian edge of X.
Proof. Let s denote the image under p ◦ q of the cone point of K . . Consider
the map K . × ∆1 → K . which is the identity on K . × {0} and collapses
K . × {1} to the cone point of K . . Let h denote the compositionqp

K . × ∆1 → K . → X → S,
which we regard as a natural transformation from p ◦ q to the constant map
with value s. Let H : q → q 0 be a coCartesian transformation from q toa diagram q 0 : K . → Xs . Using Proposition 2.4.1.7, we conclude that q 0
carries each edge of K to a p-coCartesian edge of X, which is therefore an
equivalence in Xs .
Let us now suppose that q carries every edge of K . to a p-coCartesianedge of X. Arguing as above, we conclude that q 0 carries each edge of K . toan equivalence in Xs . Let e : s → s0 be an edge of S and e! : Xs → Xs0 an
associated functor. The compositionq0e

K . → Xs →! Xs0

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carries each edge of K . to an equivalence in Xs , and is therefore a colimit
diagram in Xs0 (Corollary 4.4.4.10). Proposition 4.3.1.10 implies that q 0 isa p-colimit diagram, so that Proposition 4.3.1.9 implies that q is a p-colimit
diagram as well.
For the converse, let us suppose that q is a p-colimit diagram. Applying
Proposition 4.3.1.9, we conclude that q 0 is a p-colimit diagram. In particular,
q 0 is a colimit diagram in the ∞-category Xs . Applying Corollary 4.4.4.10,
we conclude that q 0 carries each edge of K . to an equivalence in Xs . Now
consider an arbitrary edge f : x → y of K . . If f belongs to K, then q(f ) is
p-coCartesian by assumption. Otherwise, we may suppose that y is the conepoint of K. The map H gives rise to a diagram
q(f )

q(x)

/ q(y)φ0φ
q 0 (f )/ q 0 (y)q 0 (x)in the ∞-category X ×S ∆1 . Here q 0 (f ) and φ0 are equivalences in Xs , sothat q(f ) and φ are equivalent as morphisms ∆1 → X ×S ∆1 . Since φ is
p-coCartesian, we conclude that q(f ) is p-coCartesian, as desired.
Lemma 4.3.1.13. Let p : C → D be an inner fibration of ∞-categories, let
C ∈ C be an object, and let D = p(C). Then C is a p-initial object of C if
and only if (C, idD ) is an initial object of C ×D DD/ .
Proof. We have a commutative diagram
/ CC/

ψ

CC/ ×DD/ DidD /φC ×D DD/

φ0

C ×D DD/ ,where the vertical arrows are left fibrations and therefore categorical fibrations (Proposition 3.3.1.7). We wish to show that φ is a trivial fibration if
and only if φ0 is a trivial fibration. This is equivalent to proving that φ isa categorical equivalence if and only if φ0 is a categorical equivalence. For
this, it will suffice to show that ψ is a categorical equivalence. But ψ is apullback of the trivial fibration DidD / → DD/ and therefore itself a trivial
fibration.
Proposition 4.3.1.14. Suppose we are given a diagram of ∞-categories
/D

p

C
q



r

E,



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LIMITS AND COLIMITS

where p and r are inner fibrations, q is a Cartesian fibration, and p carries
q-Cartesian morphisms to r-Cartesian morphisms.
Let C ∈ C be an object, D = p(C), and E = q(C). Let CE = C ×E {E},DE = D ×E {E}, and pE : CE → DE be the induced map. Suppose that C is
a pE -initial object of CE . Then C is a p-initial object of C.
Proof. Our hypothesis, together with Lemma 4.3.1.13, implies that (C, idD )is an initial object of
CE ×DE (DE )D/ ' (C ×D DD/ ) ×EE/ {idE }.
We will prove that the map φ : C ×D DD/ → EE/ is a Cartesian fibration.
Since idE is an initial object of EE/ , Lemma 2.4.4.7 will allow us to conclude
that (C, idD ) is an initial object of C ×D DD/ . We can then conclude theproof by applying Lemma 4.3.1.13 once more.
It remains to prove that φ is a Cartesian fibration. Let us say that amorphism of C ×D DD/ is special if its image in C is q-Cartesian. Since φ is
obviously an inner fibration, it will suffice to prove the following assertions:(1) Given an object X of C ×D DD/ and a morphism f : Y → φ(X) in EE/ ,we can write f = φ(f ), where f is a special morphism of C ×D DD/ .
(2) Every special morphism in C ×D DD/ is φ-Cartesian.
To prove (1), we first identify X with a pair consisting of an object C 00 ∈ C
and a morphism D → p(C 00 ) in D, and f with a 2-simplex σ : ∆2 → E which
we depict as a diagram:
?E

0
g

"
/ q(C 00 ).

E

Since q is a Cartesian fibration, the morphism g can be written as q(g) forsome morphism g : C 0 → C 00 in C. We now have a diagramp(C 0 )p(g)#/ p(C 00 )D

in D. Since p carries q-Cartesian morphisms to r-Cartesian morphisms, weconclude that p(g) is r-Cartesian, so that the above diagram can be completed to a 2-simplex σ : ∆2 → D such that r(σ) = σ.
We now prove (2). Suppose that n ≥ 2 and that we have a commutative
diagram
Λnn
_

σ0
σ


∆n

/ C ×D DD/:
/ EE/ ,

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CHAPTER 4

where σ0 carries the final edge of Λnn to a special morphism of C ×D DD/ .
We wish to prove the existence of the morphism σ indicated in the diagram.
We first let τ0 denote the composite mapσΛnn →0 C ×D DD/ → C .
Consider the diagram
/
>C

τ0

Λnn
_

τ

q


/ E.


∆n

Since τ0 (∆{n−1,n} ) is q-Cartesian, there exists an extension τ as indicated in
the diagram. The morphisms τ and σ0 together determine a map θ0 whichfits into a diagram
/D
=

θ0

Λn+1
n+1
_

θ

r




/ E.

∆n+1

To complete the proof, it suffices to prove the existence of the indicatedarrow θ. This follows from the fact that θ0 (∆{n,n+1} ) = (p ◦ τ0 )(∆{n−1,n} )is an r-Cartesian morphism of D.
Proposition 4.3.1.14 immediately implies the following slightly stronger
statement:
Corollary 4.3.1.15. Suppose we are given a diagram of ∞-categories
/D

p

C
q



r

E,



where q and r are Cartesian fibrations, p is an inner fibration, and p carries
q-Cartesian morphisms to r-Cartesian morphisms.
Suppose we are given another ∞-category E0 equipped with a functor s :E0 → E. Set C0 = C ×E E0 , set D0 = D ×E E0 , and let p0 : C0 → D0 be thefunctor induced by p. Let f 0 : K . → C0 be a diagram, and let f denote thefcomposition K . →0 C0 → C. Then f 0 is a p0 -colimit diagram if and only if
f is a p-colimit diagram.
Proof. Let f0 = f 0 |K and f = f |K. Replacing our diagram by/ Dpf /Cf /"|Eqf / ,

highertopoi

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LIMITS AND COLIMITS

we can reduce to the case where K = ∅. Then f 0 determines an object
C ∈ C0 . Let E denote the image of C in E0 . We have a commutative diagram
s0

{E}
s

/ E0

00

!

E.

s

~

Consequently, to prove Corollary 4.3.1.15 for the map s, it will suffice toprove the analogous assertions for s0 and s00 ; these follow from Proposition
4.3.1.14.
Corollary 4.3.1.16. Let p : C → E be a Cartesian fibration of ∞-categories,
E ∈ E an object, and f : K . → CE a diagram. Then f is a colimit diagramin CE if and only if it is a p-colimit diagram in C.
Proof. Apply Corollary 4.3.1.15 in the case where D = E.
4.3.2 Kan Extensions along Inclusions
In this section, we introduce the theory of left Kan extensions. Let F : C → D
be a functor between ∞-categories and let C0 be a full subcategory of C.
Roughly speaking, the functor F is a left Kan extension of its restriction
F0 = F | C0 if the values of F are as “small” as possible given the values ofF0 . In order to make this precise, we need to introduce a bit of terminology.
Notation 4.3.2.1. Let C be an ∞-category and let C0 be a full subcategory.
If p : K → C is a diagram, we let C0/p denote the fiber product C/p ×C C0 . In
particular, if C is an object of C, then C0/C denotes the full subcategory of
C/C spanned by the morphisms C 0 → C where C 0 ∈ C0 .
Definition 4.3.2.2. Suppose we are given a commutative diagram of ∞categories
F0

C0 _

F

/D
>
p


/ D0 ,

C

where p is an inner fibration and the left vertical map is the inclusion of afull subcategory C0 ⊆ C.
We will say that F is a p-left Kan extension of F0 at C ∈ C if the induced
diagram(C0/C )
_

FC

/D
=
p



(C0/C ).

/ D0

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CHAPTER 4

exhibits F (C) as a p-colimit of FC .
We will say that F is a p-left Kan extension of F0 if it is a p-left Kan
extension of F0 at C for every object C ∈ C.
In the case where D0 = ∆0 , we will omit mention of p and simply say that
F is a left Kan extension of F0 if the above condition is satisfied.
Remark 4.3.2.3. Consider a diagram
C0 _

/D
>

F0
F

p


/ D0


C

as in Definition 4.3.2.2. If C is an object of C0 , then the functor FC :(C0/C ). → D is automatically a p-colimit. To see this, we observe that idC :C → C is a final object of C0/C . Consequently, the inclusion {idC } → (C0/C )is cofinal, and we are reduced to proving that F (idC ) : ∆1 → D is a colimitof its restriction to {0}, which is obvious.
Example 4.3.2.4. Consider a diagram
q

C _

/D
=

q

p



/ D0 .
C.
The map q is a p-left Kan extension of q if and only if it is a p-colimit of q.
The “only if” direction is clear from the definition, and the converse follows
immediately from Remark 4.3.2.3.
We first note a few basic stability properties for the class of left Kan
extensions.
Lemma 4.3.2.5. Consider a commutative diagram of ∞-categories
C0 _

F0

/D
>

F


C

p


/ D0

as in Definition 4.3.2.2. Let C and C 0 be equivalent objects of C. Then F isa p-left Kan extension of F0 at C if and only if F is a p-left Kan extensionof F0 at C 0 .
Proof. Let f : C → C 0 be an equivalence, so that the restriction maps
C/C ← C/f → C/C 0
are trivial fibrations of simplicial sets. Let C0/f = C0 ×C C/f , so that we havetrivial fibrations
g

g0

C0/C ← C0/f → C0/C 0 .

highertopoi

April 9, 2017273

LIMITS AND COLIMITSConsider the associated diagram(C0/C ).
:

FC

G

"

(C0/f ).
G

0

<D

FC 0

$
(C0/C 0 ). .
This diagram does not commute, but the functors FC ◦ G and FC 0 ◦ G0 are
0.
equivalent in the ∞-category D(C/f ) . Consequently, FC ◦ G is a p-colimit
0
diagram if and only if FC 0 ◦ G is a p-colimit diagram (Proposition 4.3.1.5).
Since g and g 0 are cofinal, we conclude that FC is a p-colimit diagram if andonly if FC 0 is a p-colimit diagram (Proposition 4.3.1.7).
Lemma 4.3.2.6. (1) Let C be an ∞-category, let p : D → D0 be an inner
fibration of ∞-categories, and let F, F 0 : C → D be two functors which
are equivalent in DC . Let C0 be a full subcategory of C. Then F is ap-left Kan extension of F | C0 if and only if F 0 is a p-left Kan extensionof F 0 | C0 .
(2) Suppose we are given a commutative diagram of ∞-categories
C0


G0

00

C

/C

F

/D

F0


/ D0

G


/ C0

p

p0

/E

/ E0 ,

where the left horizontal maps are inclusions of full subcategories, theright horizontal maps are inner fibrations, and the vertical maps are
categorical equivalences. Then F is a p-left Kan extension of F | C0 if
0and only if F 0 is a p0 -left Kan extension of F 0 |C0 .
Proof. Assertion (1) follows immediately from Proposition 4.3.1.5. Let us
prove (2). Choose an object C ∈ C and consider the diagram(C0/C ).

/D

0
(C0 /G(C) ).


/ D0

p

p0

/E

/ E0 .

We claim that the upper left horizontal map is a p-colimit diagram if andonly if the bottom left horizontal map is a p0 -colimit diagram. In view of
Proposition 4.3.1.6, it will suffice to show that each of the vertical maps isan equivalence of ∞-categories. For the middle and right vertical maps, thishighertopoi

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CHAPTER 4

holds by assumption. To prove that the left vertical map is a categoricalequivalence, we consider the diagram
C0/C/ C0 0/G(C)


C/C


/ C0/G(C) .

The bottom horizontal map is a categorical equivalence (Proposition 1.2.9.3),
and the vertical maps are inclusions of full subcategories. It follows that the
top horizontal map is fully faithful, and its essential image consists of thosemorphisms C 0 → G(C) where C 0 is equivalent (in C0 ) to the image of an0
object of C0 . Since G0 is essentially surjective, this is the whole of C0 /G(C) .
0

It follows that if F 0 is a p0 -left Kan extension of F 0 |C0 , then F is a p-left
Kan extension of F | C0 . Conversely, if F is a p-left Kan extension of F | C0 ,0then F 0 is a p0 -left Kan extension of F 0 |C0 at G(C) for every object C ∈ C.
Since G is essentially surjective, Lemma 4.3.2.5 implies that F 0 is a p0 -left0
Kan extension of F 0 |C0 at every object of C0 . This completes the proof of(2).
Lemma 4.3.2.7. Suppose we are given a diagram of ∞-categories
C0 _

F0
F


C

/D
>
p


/ D0

as in Definition 4.3.2.2, where p is a categorical fibration and F is a p-left
Kan extension of F0 . Then the induced map
DF/ → D0pF/ ×D0pF / DF0 /0is a trivial fibration of simplicial sets. In particular, we may identify pcolimits of F with p-colimits of F0 .
Proof. Let A be the collection of simplices σ : ∆n → C for which σ(0) ∈/ C0 .
For each σ ∈ A, let d(σ) denote the dimension of the simplex σ. Choose awell-ordering of A such that if d(σ) < d(τ ), then σ < τ . For every simplex
σ : ∆n → C which does not belong to C0 , there exists a least integer i ≤ nsuch that σ(i) ∈/ C0 . We let r(σ) ∈ A denote the simplex σ|∆{i,i+1,...,n} .
Let α be the order type of A, so that we have an order-preserving bijection
β 7→ σβ where β ranges over the set of ordinals {β : β < α}. For each β ≤ α,
we let Cβ denote the simplicial subset of C whose nondegenerate simplices σ
are those which either belong to C0 or satisfy r(σ) = σγ for some γ < β. Let
Fβ = F | Cβ . We will prove that for every β ≤ β 0 the projection
φβ 0 ,β : DFβ0 / → D0pFβ0 / ×D0pF / DFβ /βhighertopoi

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LIMITS AND COLIMITS

is a trivial fibration of simplicial sets. Taking β = 0 and β 0 = α, we willobtain the desired result.
The proof proceeds by induction on β 0 . If β 0 = β, then φβ 0 ,β is an isomorphism and there is nothing to prove. If β 0 > β is a limit ordinal, then the
inductive hypothesis implies that φβ 0 ,β is the inverse limit of a transfinite
tower of trivial fibrations and therefore a trivial fibration. It therefore suffices
to prove that if φβ 0 ,β is a trivial fibration, then φβ 0 +1,β is a trivial fibration.
We observe that φβ 0 +1,β = φ0β 0 ,β ◦ φβ 0 +1,β 0 , where φ0β 0 ,β is a pullback of φβ 0 ,βand therefore a trivial fibration by the inductive hypothesis. Consequently,
it will suffice to prove that φβ 0 +1,β 0 is a trivial fibration for β 0 < α.
There are two cases to consider. Suppose first that σ = σβ 0 : ∆n → C is a
nondegenerate simplex of C. Unwinding the definitions, we have a pushout
diagram
C0/σ ? ∂ ∆n

/ Cβ 0C0/σ ?∆n/ Cβ 0 +1 .

Let g denote the composition
F

C0/σ ?∆n → C → D
and g∂ = g| C0/σ ? ∂ ∆n . It follows that φα+1,α is a pullback of the inducedmapψ : Dg/ → D0pg/ ×D0pg / Dg∂ / .
∂To complete the proof, it will suffice to show that ψ is a trivial fibration of
simplicial sets. To prove this, let g0 = g| C0/σ , let E = Dg0 / , let E0 = D0p◦g0 / ,and let q : E → E0 be the induced map. We can identify g with a map
σ 0 : ∆n → E. Let σ00 = σ 0 | ∂ ∆n . Then we wish to prove that the map
ψ 0 : Eσ0 / → E0qσ0 / ×E0 0 Eqσ00 /
qσ0 /

0

is a trivial fibration. We show that ψ has the right lifting property withrespect to the inclusion ∂ ∆m ⊆ ∆m for each m ≥ 0. Unwinding the definitions, this amounts to the existence of a dotted arrow as indicated in the
diagram∂ ∆n+m+1
_

∆n+m+1

s

9/ E


q

/ E0 .

Our assumption that F is a p-left Kan extension of F0 guarantees that s
carries the initial vertex of ∆n+m+1 to a q-initial vertex of E, so that thedesired extension can be found.

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We now treat the case where σ = σβ 0 is degenerate. The map φβ 0 +1,β 0
is obviously a categorical fibration (even a left fibration). It will therefore
suffice to show that it is a categorical equivalence: that is, that the diagram/ DF 0 /DF 0 /
β +1

β
D0Fβ0 +1 /


/ D0F

β0 /

is homotopy Cartesian. We will prove that both of the horizontal maps are
categorical equivalences. For this, it suffices to show that the inclusion Cβ 0 ,→Cβ 0 +1 is a categorical equivalence. We define a simplicial subset K ⊆ C0/σ asfollows: an m-simplex τ of C0/σ belongs to K if and only if the induced map
00∆m ? ∆n → C0 factors through ∆m ? ∆n for surjective map ∆n → ∆n ,0n < n. We have a pushout diagram of simplicial sets`
/ Cβ 0(C0/σ ? ∂ ∆n ) K?∂ ∆n (K ? ∆n )i

C0/σ ?∆n/ Cβ 0 +1 .

It will therefore suffice to show that i is a categorical equivalence: that is,that the diagram/ K ? ∆nK ? ∂ ∆nC0/σ ? ∂ ∆n
/ C0/σ ?∆n

is a homotopy pushout square. In fact, the vertical maps in this diagram arecategorical equivalences. To prove this, it suffices to show that the inclusion
K ,→ C0/σ is a categorical equivalence. Let J be the category whose objects
are commutative diagrams/ ∆ n0



∆n
σ

C

~

τ

where  is surjective and n < n; we will identify the objects of J with0
simplices τ : ∆n → C. Unwinding the definitions, we see that K can be
identified with the colimit of the diagram θ : Jop → Set∆ given by τ 7→
C0/τ . Since the diagram is cofibrant, we are reduced to proving that themap K → C0/σ exhibits C0/σ as a homotopy colimit of the diagram θ. Our
assumption that σ is degenerate implies that J has a final object (given
0 τ

by the factorization of σ as ∆n → ∆n → C where τ is nondegenerate), it
will suffice to show that the diagram θ is weakly equivalent to the constant0highertopoi

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277

LIMITS AND COLIMITS

diagram with value C0/σ . In other words, we must show that for each τ ∈ J,
the canonical map C0/τ → C0/σ is a categorical equivalence. This follows fromthe two-out-of-three property, since both of the vertical maps in the diagram/ C0/σ

C0/τ

C0/C

~

are trivial Kan fibrations, where C = τ (0) = σ(0) ∈ C.
Proposition 4.3.2.8. Let F : C → D be a functor between ∞-categories,
p : D → D0 a categorical fibration of ∞-categories, and C0 ⊆ C1 ⊆ C fullsubcategories. Suppose that F | C1 is a p-left Kan extension of F | C0 . Then F
is a p-left Kan extension of F | C1 if and only if F is a p-left Kan extensionof F | C0 .
Proof. Let C be an object of C; we will show that F is a p-left Kan extensionof F | C0 at C if and only if F is a p-left Kan extension of F | C1 at C. Consider
the composition
F1C
D.
FC0 : (C0/C ). ⊆ (C1/C ). →We wish to show that FC0 is a p-colimit diagram if and only if FC1 is a pcolimit diagram. According to Lemma 4.3.2.7, it will suffice to show that
FC1 | C1/C is a left Kan extension of FC0 . Let f : C 0 → C be an object of C1/C .
We wish to show that the composite map
F00

C(C0/f ). → (C0/C 0 ). →D

is a p-colimit diagram. Since the projection C0/f → C0/C 0 is cofinal (in fact,
a trivial fibration), it will suffice to show that FC0 0 is a p-colimit diagram(Proposition 4.3.1.7). This follows from our hypothesis that F | C1 is a p-left
Kan extension of F | C0 .
Proposition 4.3.2.9. Let F : C × C0 → D denote a functor between ∞categories, p : D → D0 a categorical fibration of ∞-categories, and C0 ⊆ C afull subcategory. The following conditions are equivalent:(1) The functor F is a p-left Kan extension of F | C0 × C0 .
(2) For each object C 0 ∈ C0 , the induced functor FC 0 : C ×{C 0 } → D is ap-left Kan extension of FC 0 | C0 ×{C 0 }.
Proof. It suffices to show that F is a p-left Kan extension of F | C0 × C0
at an object (C, C 0 ) ∈ C × C0 if and only if FC 0 is a p-left Kan extensionof FD | C0 ×{D} at C. This follows from the observation that the inclusion
C0/C ×{idC 0 } ⊆ C0/C × C0/C 0 is cofinal (because idC 0 is a final object of C0/C 0 ).

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Lemma 4.3.2.10. Let m ≥ 0, n ≥ 1 be integers and let`
f0(∂ ∆m × ∆n ) ∂ ∆m ×∂ ∆n (∆m × ∂ ∆n )
_

3/ X


∆m × ∆n


/S

f

p

be a diagram of simplicial sets, where p is an inner fibration and f0 (0, 0) isa p-initial vertex of X. Then there exists a morphism f : ∆m × ∆n → X
rendering the diagram commutative.
Proof. Choose a sequence of simplicial setsa(∆m × ∂ ∆n ) = Y (0) ⊆ · · · ⊆ Y (k) = ∆m × ∆n ,(∂ ∆m × ∆n )∂ ∆m ×∂ ∆nwhere each Y (i+1) is obtained from Y (i) by adjoining a single nondegenerate
simplex whose boundary already lies in Y (i). We prove by induction on ithat f0 can be extended to a map fi such that the diagram
fi

Y (i)
_

/X
p


∆m × ∆n/S

is commutative. Having done so, we can then complete the proof by choosing
i = k.
If i = 0, there is nothing to prove. Let us therefore suppose that fi has been
constructed and consider the problem of constructing fi+1 which extends fi .
This is equivalent to the lifting problem
/X
<

σ0

∂ ∆ _ r

σ

p



/ S.
∆r
It now suffices to observe that where r > 0 and σ0 (0) = f0 (0, 0) is a p-initialvertex of X (since every simplex of ∆m × ∆n which violates one of these
conditions already belongs to Y (0)).
Lemma 4.3.2.11. Suppose we are given a diagram of simplicial sets
/Y

p

X


S,



where p is an inner fibration. Let K be a simplicial set, let qS ∈ MapS (K ×
S, X), and let qS0 = p ◦ qS . Then the induced map
0p0 : X qS / → Y qS /is an inner fibration (where the above simplicial sets are defined as in §4.2.2).

highertopoi

April 9, 2017279

LIMITS AND COLIMITSProof. Unwinding the definitions, we see that every lifting problem/ X qS /
=

A _
i


/ Y qS /B

is equivalent to a lifting problem`(A × (K  ∆0 ))  A×K (B × K)
_

/5 X
p

i0


B × (K  ∆0 )

/ Y.

We wish to show that this lifting problem has a solution provided that i is
inner anodyne. Since p is an inner fibration, it will suffice to prove that i0 is
inner anodyne, which follows from Corollary 2.3.2.4.
Lemma 4.3.2.12. Consider a diagram of ∞-categoriesp

C → D0 ← D,
where p is an categorical fibration. Let C0 ⊆ C be a full subcategory. Suppose
we are given n > 0 and a commutative diagram∂ ∆ _nf0/ FunD0 (C, D)
8

f


∆n

g


/ FunD0 (C0 , D)with the property that the functor F : C → D, determined by evaluating f0 at
the vertex {0} ⊆ ∂ ∆n , is a p-left Kan extension of F | C0 . Then there exists
a dotted arrow f rendering the diagram commutative.
Proof. Let A be as in the proof of Lemma 4.3.2.7, choose a well-ordering of A
such that σ < τ whenever the dimension of σ is smaller than the dimensionof τ , let α be the order type of A, and define Cβ for β ≤ α as in the proof
of Lemma 4.3.2.7. We will construct a compatible family of solutions to themapping problems/ FunD0 (Cβ , D)8∂ ∆ _n
fβ


∆n

g


/ FunD0 (C0 , D)using induction on β; when β = α, this will yield the desired result. If
β = 0, then Cβ = C0 and the result is obvious. If β is a limit ordinal,
then we invoke the observation that FunD0 (Cβ , D) ' limγ<β FunD0 (Cγ , D).
←−highertopoi

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CHAPTER 4

It therefore suffices to treat the case where β = γ + 1 is a successor ordinal:that is, we must show that the lifting problem∂ ∆ _n
fβ


∆n

fγ

/ FunD0 (Cβ , D)8
/ FunD0 (Cγ , D)admits a solution.
There are two cases to consider. Suppose first that the element σ = σγ ∈ A
is a nondegenerate simplex ∆m → C. We then have a pushout diagram
C0/σ ? ∂ ∆m

/ Cγ

C0/σ ?∆m/ Cβ .

It therefore suffices to solve the lifting problem depicted in the diagram
C0/σ ? ∂ ∆m_/ Fun(∆n , D)4

C0/σ ?∆m/ Fun(∆n , D0 ) ×Fun(∂ ∆n ,D0 ) Fun(∂ ∆n , D).

Let q : C0/σ → Fun(∆n , D) be the restriction of the upper horizontal map
and let q 0 : C0/σ → Fun(∆n , D0 ), q∂ : C0/σ → Fun(∂ ∆n , D), and q∂0 : C0/σ →Fun(∂ ∆n , D0 ) be defined by composition with q. We are therefore reduced
to proving that the lifting problem∂ ∆ m_/ Fun(∆n , D)q/
4


∆m


/ Fun(∆n , D0 )q0 / ×Fun(∂ ∆n ,D0 ) 0 Fun(∂ ∆n , D)q /∂q /∂has a solution. In view of Proposition A.2.3.1, it will suffice to solve theequivalent lifting problem
∂ ∆m

∆m

/ Fun(∆n , D)q/
4
/ Fun(∆n , D0 )q0 / ×
0Fun(∂ ∆n ,D0 )q∂ /Fun(∂ ∆n , D)q∂ / .

Let q∆n denote the map C0/σ ×∆n → D ×∆n determined by q and let0
X = (D ×∆n )q∆n / be the simplicial set constructed in §4.2.2. Let q∆n :highertopoi

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LIMITS AND COLIMITS0q∆n/

C0/σ ×∆n → D0 ×∆n and X0 = (D0 ×∆n )natural isomorphisms

be defined similarly. We have

Fun(∆n , D)q/ ' Map∆n (∆n , X)
Fun(∂ ∆n , D)q∂ / ' Map∆n (∂ ∆n , X)
0Fun(∆n , D0 )q / ' Map∆n (∆n , X0 )0Fun(∂ ∆n , D0 )q∂ / ' Map∆n (∂ ∆n , X0 ).
These identifications allow us to reformulate our lifting problem once more:(∂ ∆m × ∆n )`∂ ∆m ×∂ ∆n (∆

_

m

h0

× ∂ ∆n )

3/ X

h




∆m × ∆ nψ/ X0 .

We have a commutative diagram
/ X0

ψ

X

r0

r

!
}
∆n .
Proposition 4.2.2.4 implies that r and r0 are Cartesian fibrations and that ψ
carries r-Cartesian edges to r0 -Cartesian edges. Lemma 4.3.2.11 implies that
ψ is an inner fibration. Let ψ0 : X{0} → X0{0} be the diagram induced bytaking the fibers over the vertex {0} ⊆ ∆n . We have a commutative diagram
DC0/σ(0) / o

D0C0



θ

/σ(0)

/

o

DC0/σ

/ X{0}



D0C0 //σψ0/ X0{0}in which the horizontal arrows are categorical equivalences. We can lift
g0 (0, 0) ∈ X0{0} to a vertex of DC0/σ / whose image in DC0/σ(0) / is θ-initial(by virtue of our assumption that F is a p-left Kan extension of F | C0 ). It
follows that g0 (0, 0) is ψ0 -initial when regarded as a vertex of X{0} . Applying
Proposition 4.3.1.14, we deduce that g0 (0, 0) is ψ-initial when regarded asa vertex of X. Lemma 4.3.2.10 now guarantees the existence of the dottedarrow h.
Suppose now that the simplex σ = σγ ∈ A is degenerate. To prove theexistence of fβ in this case, it will suffice to show that the restriction map
FunD0 (Cβ , D) → FunD0 (Cγ , D) is a trivial Kan fibration. Since p is a categorical fibration, it will suffice to show that the inclusion Cγ ,→ Cβ is a categoricalequivalence, which follows exactly as in the proof of Lemma 4.3.2.7.

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The following result addresses the existence problem for left Kan extensions:Lemma 4.3.2.13. Suppose we are given a diagram of ∞-categories
F0

C0 _

/D
>

F

p



G /
C
D0 ,
where p is a categorical fibration and the left vertical arrow is the inclusionof a full subcategory. The following conditions are equivalent:(1) There exists a functor F : C → D rendering the diagram commutative,
such that F is a p-left Kan extension of F0 .
(2) For every object C ∈ C, the diagram given by the compositionF

C0/C → C0 →0 D
Gadmits a p-colimit (compatible with the evident map (C0/C ). → C →D0 ).
Proof. It is clear that (1) implies (2). Let us therefore suppose that (2) issatisfied; we wish to prove that F0 admits a p-left Kan extension. We will
follow the basic strategy used in the proofs of Lemmas 4.3.2.7 and 4.3.2.12.
Let A be as in the proof of Lemma 4.3.2.7, choose a well-ordering of A suchthat σ < τ whenever the dimension of σ is smaller than the dimension of τ ,let α be the order type of A, and define Cβ for β ≤ α as in the proof of Lemma
4.3.2.7. We will construct a compatible collection of maps Fβ : Cβ → D withp ◦ Fβ = G| Cβ using induction
S on β. If β = 0, set Fβ = F0 , and if β ≤ α isa limit ordinal we set Fβ = γ<β Fγ .
To carry out the inductive step, let us suppose that β < α and that Fβ has
already been constructed; we wish to extend Fβ to a map Fβ+1 : Cβ+1 → D.
There are three cases to consider:(a) Suppose first that σβ ∈ A consists of a single object C ∈ C. In thiscase, we have a pushout diagram
/ Cβ
C0/C
_

C0/σ ?∆0/ Cβ+1 .

To construct Fβ , it suffices to choose a solution to the associated lifting
problem
q
/D
C0/C
<
q



(C0/C ).

p
/ D0 .

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We will choose such a solution where q is a p-colimit diagram (this ispossible by virtue of assumption (2)).
(b) Suppose that σ = σβ ∈ A is a nondegenerate n-simplex of C for somen > 0. We have a pushout diagram
C0/σ ? ∂ ∆n
_

/ Cβ


C0/σ ?∆n/ Cβ+1 .

Consequently, to define Fβ+1 it will suffice to solve the lifting problem
depicted in the diagram
C0/σ ? ∂ ∆n
_

/
:D

f0
f

p


C0/σ ?∆n/ D0 .

Such a solution exists by virtue of the fact that f0 | C0/σ ?{0} is a pcolimit diagram (which is guaranteed by our treatment of case (a)).
(c) Suppose that σ = σβ ∈ A is a degenerate n-simplex of C. We wish toshow that there is a solution to the lifting problem
Fβ

Cβ

/D
<
p


/ D0 .

Cβ+1

Since p is a categorical fibration, it suffices to show that the inclusion
Cβ ,→ Cβ+1 is a categorical equivalence, which follows as in the proof
of Lemma 4.3.2.7.
Let F = Fα , so that we have a commutative diagram
C0 _

F0
F


C

/D
=
p

G


/ D0 .

It follows from our treatment of case (a) that F is a p-left Kan extension of
F0 , which completes the proof of (1).
Corollary 4.3.2.14. Let p : D → E be a coCartesian fibration of ∞categories. Suppose that each fiber of p admits small colimits and that forevery morphism E → E 0 in E the associated functor DE → DE 0 preserves
small colimits. Let C be a small ∞-category and C0 ⊆ C a full subcategory.
Then every functor F0 : C0 → D admits a left Kan extension relative to p.

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Proof. This follows easily from Lemma 4.3.2.13 and Corollary 4.3.1.11.
Combining Lemmas 4.3.2.12 and 4.3.2.13, we deduce the following result:
Proposition 4.3.2.15. Suppose we are given a diagram of ∞-categoriesp

C → D0 ← D,
where p is a categorical fibration. Let C0 be a full subcategory of C. Let K ⊆MapD0 (C, D) be the full subcategory spanned by those functors F : C → D
which are p-left Kan extensions of F | C0 . Let K0 ⊆ MapD0 (C0 , D) be the fullsubcategory spanned by those functors F0 : C0 → D with the property that,for each object C ∈ C, the induced diagram C0/C → D has a p-colimit. Thenthe restriction functor K → K0 is a trivial fibration of simplicial sets.
Corollary 4.3.2.16. Suppose we are given a diagram of ∞-categoriesp

C → D0 ← D,
where p is a categorical fibration. Let C0 be a full subcategory of C. Suppose
further that, for every functor F0 ∈ MapD0 (C0 , D), there exists a functor
F ∈ MapD0 (C, D) which is a p-left Kan extension of F0 . Then the restriction
map i∗ : MapD0 (C, D) → MapD0 (C0 , D) admits a section i! whose essential
image consists of precisely those functors F which are p-left Kan extensionsof F | C0 .
In the situation of Corollary 4.3.2.16, we will refer to i! as a left Kan
extension functor. We note that Proposition 4.3.2.15 proves not only theexistence of i! but also its uniqueness up to homotopy (the collection of all
such functors is parametrized by a contractible Kan complex). The following
characterization of i! gives a second explanation for its uniqueness:
Proposition 4.3.2.17. Suppose we are given a diagram of ∞-categoriesp

C → D0 ← D,
where p is a categorical fibration. Let i : C0 ⊆ C be the inclusion of a fullsubcategory and suppose that every functor F0 ∈ MapD0 (C0 , D) admits a pleft Kan extension. Then the left Kan extension functor i! : MapD0 (C0 , D) →MapD0 (C, D) is a left adjoint to the restriction functor i∗ : MapD0 (C, D) →MapD0 (C0 , D).
Proof. Since i∗ ◦ i! is the identity functor on MapD0 (C0 , D), there is an obvious candidate for the unitu : id → i∗ ◦ i!
of the adjunction: namely, the identity. According to Proposition 5.2.2.8, it
will suffice to prove that for every F ∈ MapD0 (C0 , D) and G ∈ MapD0 (C, D),
the composite map
MapMapD0 (C,D) (i! F, G) → MapMapD0 (C0 ,D) (i∗ i! F, i∗ G)u→ MapMapD0 (C0 ,D) (F, i∗ G)highertopoi

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285

is an isomorphism in the homotopy category H. This morphism in H isrepresented by the restriction map
R∗
HomR
MapD0 (C,D) (i! F, G) → HomMapD0 (C0 ,D) (F, i G),

which is a trivial fibration by Lemma 4.3.2.12.

Remark 4.3.2.18. Throughout this section we have focused our attention
on the theory of (relative) left Kan extensions. There is an entirely dual
theory of right Kan extensions in the ∞-categorical setting, which can beobtained from the theory of left Kan extensions by passing to opposite ∞categories.

4.3.3 Kan Extensions along General Functors
Our goal in this section is to generalize the theory of Kan extensions to thecase where the change of diagram category is not necessarily given by a fully
faithful inclusion C0 ⊆ C. As in §4.3.2, we will discuss only the theory of left
Kan extensions; a dual theory of right Kan extensions can be obtained bypassing to opposite ∞-categories.
The ideas introduced in this section are relatively elementary extensionsof the ideas of §4.3.2. However, we will encounter a new complication. Let
δ : C → C0 be a map of diagram ∞-categories, f : C → D a functor,and δ! (f ) : C0 → D its left Kan extension along δ (to be defined below).
Then one does not generally expect δ ∗ δ! (f ) to be equivalent to the originalfunctor f . Instead, one has only a unit transformation f → δ ∗ δ! (f ). To
set up the theory, this unit transformation must be taken as part of thedata. Consequently, the theory of Kan extensions in general requires moreelaborate notation and terminology than the special case treated in §4.3.2.
We will compensate for this by considering only the case of absolute left Kan
extensions. It is straightforward to set up a relative theory as in §4.3.2, but
we will not need such a theory in this book.
Definition 4.3.3.1. Let δ : K → K 0 be a map of simplicial sets, let D be an∞-category and let f : K → D be a diagram. A left extension of f along δ
consists of a map f 0 : K 0 → D and a morphism f → f 0 ◦ δ in the ∞-category
Fun(K, D).
Equivalently, we may view a left extension of f : K → D along δ : K → K 0
as a map F`: M op (δ) → D such that F |K = f , where M op (δ) = M (δ op )op =(K × ∆1 ) K×{1} K 0 denotes the mapping cylinder of δ.
Definition 4.3.3.2. Let δ : K → K 0 be a map of simplicial sets, and let
F : M op (δ) → D be a diagram in an ∞-category D (which we view as a left
extension of f = F |K along δ). We will say that F is a left Kan extension

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of f along δ if there exists a commutative diagram
M op (δ)

F 00

F0

/K

/D

p

# 
∆1

where F 00 is a categorical equivalence, K is an ∞-category, F = F 0 ◦ F 00 , andF 0 is a left Kan extension of F 0 | K ×∆1 {0}.
Remark 4.3.3.3. In the situation of Definition 4.3.3.2, the map p : K → ∆1
is automatically a coCartesian fibration. To prove this, choose a factorizationi

M (δ op )\ → (K0 )] → (∆1 )] ,where i is marked anodyne and K0 → ∆1 is a Cartesian fibration. Then i isa quasi-equivalence, so that Proposition 3.2.2.7 implies that M (δ op ) → K0
is a categorical equivalence. It follows that K is equivalent to (K0 )op (via anequivalence which respects the projection to ∆1 ), so that the projection p is
a coCartesian fibration.
The following result asserts that the condition of Definition 4.3.3.2 is essentially independent of the choice of K.
Proposition 4.3.3.4. Let δ : K → K 0 be a map of simplicial sets and let
F : M op (δ) → D be a diagram in an ∞-category D which is a left Kan
extension along δ. Let
M op

F 00

/K
p

" 
∆1

be a diagram where F 00 is both a cofibration and a categorical equivalence ofsimplicial sets. Then F = F 0 ◦ F 00 for some map F 0 : K → D which is a left
Kan extension of F 0 | K ×∆1 {0}.
Proof. By hypothesis, there exists a commutative diagram
M op (δ)
F 00


K

G00
r
p

/ K0
;

G0

/D

q


/ ∆1 ,

where K0 is an ∞-category, F = G0 ◦ G00 , G00 is a categorical equivalence,and G0 is a left Kan extension of G0 | K0 ×∆1 {0}. Since K0 is an ∞-category,there exists a map r as indicated in the diagram such that G00 = r ◦ F 00 . Wenote that r is a categorical equivalence, so that the commutativity of thelower triangle p = q ◦ r follows automatically. We now define F 0 = G0 ◦ r andnote that part (2) of Lemma 4.3.2.6 implies that F 0 is a left Kan extensionof F 0 | K ×∆1 {0}.

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We have now introduced two different definitions of left Kan extensions:Definition 4.3.2.2 which applies in the situation of an inclusion C0 ⊆ C of afull subcategory into an ∞-category C, and Definition 4.3.3.2 which applies
in the case of a general map δ : K → K 0 of simplicial sets. These two
definitions are essentially the same. More precisely, we have the following
assertion:
Proposition 4.3.3.5. Let C and D be ∞-categories and let δ : C0 → C
denote the inclusion of a full subcategory.
(1) Let f : C → D be a functor and f0 its restriction to C0 , so that (f, idf0 )can be viewed as a left extension of f0 along δ. Then (f, idf0 ) is a left
Kan extension of f0 along δ if and only if f is a left Kan extension of
f0 .
(2) A functor f0 : C0 → D has a left Kan extension if and only if it has aleft Kan extension along δ.
Proof. Let K denote the full subcategory of C ×∆1 spanned by the objects(C, {i}), where either C ∈ C0 or i = 1, so that we have inclusions
M op (δ) ⊆ K ⊆ C ×∆1 .
To prove (1), suppose that f : C → D is a left Kan extension of f0 = f | C0
and let F denote the composite mapf

K ⊆ C ×∆1 → C → D .
It follows immediately that F is a left Kan extension of F | C0 ×{0}, so that
F |M op (δ) is a left Kan extension of f0 along δ.
To prove (2), we observe that the “only if” direction follows from (1); theconverse follows from the existence criterion of Lemma 4.3.2.13.
Suppose that δ : K 0 → K 1 is a map of simplicial sets, that D is an ∞category, and that every diagram K 0 → D admits a left Kan extension along
δ. Choose a diagram/Kj

M op (δ)
#

∆1 ,

~

where j is inner anodyne and K is an ∞-category, which we regard as acorrespondence from K0 = K ×∆1 {0} to K1 = K ×∆1 {1}. Let C denotethe full subcategory of Fun(K, D) spanned by those functors F : K → D
such that F is a left Kan extension of F0 = F | K0 . The restriction map
0
p : C → Fun(K 0 , D) can be written as a composition of C → DK (a trivial
0
fibration by Proposition 4.3.2.15) and Fun(K , D) → Fun(K 0 , D) (a trivial
fibration since K 0 → K0 is inner anodyne) and is therefore a trivial fibration.

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Let δ ! be the composition of a section of p with the restriction map C ⊆Fun(K, D) → Fun(M op (δ), D) and let δ! denote the composition of δ ! withthe restriction map Fun(M op (δ), D) → Fun(K 1 , D). Then δ ! and δ! are welldefined up to equivalence, at least once K has been fixed (independence ofthe choice of K will follow from the characterization given in Proposition
4.3.3.7). We will abuse terminology by referring to both δ ! and δ! as left
Kan extensions along δ (it should be clear from the context which of thesefunctors is meant in a given situation). We observe that δ ! assigns to eachobject f0 : K 0 → D a left Kan extension of f0 along δ.
Example 4.3.3.6. Let C and D be ∞-categories and let i : C0 → C be theinclusion of a full subcategory. Suppose that i! : Fun(C0 , D) → Fun(C, D) isa section of i∗ , which satisfies the conclusion of Corollary 4.3.2.16. Then i!
is a left Kan extension along i in the sense defined above; this follows easily
from Proposition 4.3.3.5.
Left Kan extension functors admit the following characterization:
Proposition 4.3.3.7. Let δ : K 0 → K 1 be a map of simplicial sets, let D be
an ∞-category, let δ ∗ : Fun(K 1 , D) → Fun(K 0 , D) be the restriction functor,
and let δ! : Fun(K 0 , D) → Fun(K 1 , D) be a functor of left Kan extension
along δ. Then δ! is a left adjoint of δ ∗ .
Proof. The map δ can be factored as a compositionir

K 0 → M op (δ) → K 1
where r denotes the natural retraction of M op (δ) onto K 1 . Consequently,δ ∗ = i∗ ◦r∗ . Proposition 4.3.2.17 implies that the left Kan extension functor δ !
is a left adjoint to i∗ . By Proposition 5.2.2.6, it will suffice to prove that r∗ isa right adjoint to the restriction functor j ∗ : Fun(M op (δ), D) → Fun(K 1 , D).
Using Corollary 2.4.7.12, we deduce that j ∗ is a coCartesian fibration. Moreover, there is a simplicial homotopy Fun(M op (δ), D)×∆1 → Fun(M op (δ), D)
from the identity to r∗ ◦ j ∗ , which is a fiberwise homotopy over Fun(K 1 , D).
It follows that for every object F of Fun(K 1 , D), r∗ F is a final object of the∞-category Fun(M op (δ), D) ×Fun(K 1 ,D) {F }. Applying Proposition 5.2.4.3,
we deduce that r∗ is right adjoint to j ∗ , as desired.
Let δ : K 0 → K 1 be a map of simplicial sets and D an ∞-category forwhich the left Kan extension δ! : Fun(K 0 , D) → Fun(K 1 , D) is defined. In
general, the terminology “Kan extension” is perhaps somewhat unfortunate:
if F : K 0 → D is a diagram, then δ ∗ δ! F need not be equivalent to F .
If δ is fully faithful, then the unit map F → δ ∗ δ! F is an equivalence: thisfollows from Proposition 4.3.3.5. We will later need the following more precise
assertion:
Proposition 4.3.3.8. Let δ : C0 → C1 and f0 : C0 → D be functors between∞-categories and let f1 : C1 → D, α : f0 → δ ∗ f1 = f1 ◦ δ be a left Kan

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extension of f0 along δ. Let C be an object of C0 such that, for every C 0 ∈ C0 ,the functor δ induces an isomorphism
MapC0 (C 0 , C) → MapC1 (δC 0 , δC)
in the homotopy category H. Then the morphism α(C) : f0 (C) → f1 (δC) is
an equivalence in D.
Proof. Choose a diagram
M op (δ)

/M

G

#

F

/D


∆1 ,

where M is a correspondence from C0 to C1 associated to δ, F is a left Kan
extension of f0 = F | C0 , and F ◦ G is the map M op (δ) → D determined by
f0 , f1 , and α. Let u : C → δC be the morphism in M given by the image of{C} × ∆1 ⊆ M op (δ) under G. Then α(C) = F (u), so it will suffice to provethat F (u) is an equivalence. Since F is a left Kan extension of f0 at δC, thecompositionF(C0/δC ). → M → D
is a colimit diagram. Consequently, it will suffice to prove that u : C → δC
is a final object of C0/δC . Consider the diagramq

C0/C ← C0/u → C0/δC .
The ∞-category on the left has a final object idC , and the map on the left isa trivial fibration of simplicial sets. We deduce that s0 u is a final object of
C0/u . Since q(s0 u) = u ∈ C0/δC , it will suffice to show that q is an equivalenceof ∞-categories. We observe that q is a map of right fibrations over C0 .
According to Proposition 3.3.1.5, it will suffice to show that, for each object
C 0 in C0 , the map q induces a homotopy equivalence of Kan complexes
C0/u ×C0 {C 0 } → C0/δC ×C0 {C 0 }.
This map can be identified with the map
MapC0 (C 0 , C) → MapM (C 0 , δC) ' MapC1 (δC 0 , δC)
in the homotopy category H and is therefore a homotopy equivalence byassumption.
We conclude this section by proving that the construction of left Kan
extensions behaves well in families.
Lemma 4.3.3.9. Suppose we are given a commutative diagram
C0

/C

i
q

p

 
E

F

/D

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of ∞-categories, where p and q are coCartesian fibrations, i is the inclusionof a full subcategory, and i carries q-coCartesian morphisms of C0 to pcoCartesian morphisms of C. The following conditions are equivalent:(1) The functor F is a left Kan extension of F | C0 .
(2) For each object E ∈ E, the induced functor FE : CE → D is a left Kan
extension of FE | C0E .
Proof. Let C be an object of C and let E = p(C). Consider the composition
G.

F

C(C0E )./C → (C0/C ). →D.

We will show that FC is a colimit diagram if and only if FC ◦ G. is a colimit
diagram. For this, it suffices to show that the inclusion G : (C0E )/C ⊆ C0/C iscofinal. According to Proposition 2.4.3.2, the projection p0 : C/C → E/E is a
coCartesian fibration, and a morphism/ C 00

f

C0

C

~

in C/C is p0 -coCartesian if and only if f is p-coCartesian. It follows that p0
restricts to a coCartesian fibration C0/C → E/E . We have a pullback diagramof simplicial sets
(C0E )/C

{idE }

G/ C0/CG0/ E/E .

The right vertical map is smooth (Proposition 4.1.2.15) and G0 is rightanodyne, so that G is right anodyne, as desired.
Proposition 4.3.3.10. Let
/Y

δ

X
p



S



q

be a commutative diagram of simplicial sets, where p and q are coCartesianfibrations and δ carries p-coCartesian edges to q-coCartesian edges. Let f0 :X → C be a diagram in an ∞-category C and let f1 : Y → C, α : f0 → f1 ◦ δ
be a left extension of f0 . The following conditions are equivalent:(1) The transformation α exhibits f1 as a left Kan extension of f0 alongδ.

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LIMITS AND COLIMITS(2) For each vertex s ∈ S, the restriction αs : f0 |Xs → (f1 ◦ δ)|Xs exhibits
f1 |Ys as a left Kan extension of f0 |Xs along δs : Xs → Ys .
Proof. Choose an equivalence of simplicial categories C(S) → E, where E
is fibrant, and let [1] denote the linearly ordered set {0, 1} regarded as acategory. Let φ0 denote the induced map C(S × ∆1 ) → E ×[1]. Let M denotethe marked simplicial seta((X op )\ × (∆1 )] )(Y op )\ .
(X op )\ ×{0}++ E ×[1]Let St+denote the straightening functor
φ : (Set∆ )(S×∆1 )op → (Set∆ )defined in §3.2.1 and choose a fibrant replacementSt+
φM → Z
E ×[1]in (Set+. Let S 0 = N(E), so that S 0 × ∆1 ' N(E ×[1]), and let ψ :∆)C(S 0 × ∆1 ) → E ×[1] be the counit map. Then

Un+
ψ (Z)
is a fibrant object of (Set+∆ )/(S 0 ×∆1 )op , which we may identify with a coCartesian fibration of simplicial sets M → S 0 × ∆1 .
We may regard M as a correspondence from M0 = M ×∆1 {0} to M1 =M ×∆1 {1}. By construction, we have a unit map
u : M op (δ) → M ×S 0 S.
Theorem 3.2.0.1 implies that the induced maps u0 : X → M0 ×S 0 S, u1 :Y → M1 ×S 0 S are equivalences of coCartesian fibrations. Proposition 3.3.1.3
implies that the maps M0 ×S 0 S → M0 , M1 ×S 0 S → M1 are categoricalequivalences.
Let u0 denote the compositionu

M op (δ) → M ×S 0 S → M
and let u00 : X → M0 , u01 : Y → M1 be defined similarly. The above argumentshows that u00 and u01 are categorical equivalences. Consequently, the map
u0 is a quasi-equivalence of coCartesian fibrations over ∆1 and therefore acategorical equivalence (Proposition 3.2.2.7). Replacing M by the product
M ×K if necessary, where K is a contractible Kan complex, we may supposethat u0 is a cofibration of simplicial sets. Since D is an ∞-category, there
exists a functor F : M → D as indicated in the diagram below:
M op (δ)(f0 ,f1 ,α)

7/ D

F


M.
Consequently, we may reformulate condition (1) as follows:(10 ) The functor F is a left Kan extension of F | M0 .

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Proposition 3.3.1.5 now implies that, for each vertex s of S, the map
Xs → M0s is a categorical equivalence. Similarly, for each vertex s of S, theinclusion Ys → M1s is a categorical equivalence. It follows that the inclusion
M op (δ)s → Ms is a quasi-equivalence and therefore a categorical equivalence(Proposition 3.2.2.7). Consequently, we may reformulate condition (2) as
follows:(20 ) For each vertex s ∈ S, the functor F | Ms is a left Kan extension of
F | M0s .
Using Lemma 4.3.2.6, it is easy to see that the collection of objects s ∈ S 0such that F | Ms is a left Kan extension of F | M0s is stable under equivalence.
Since the inclusion S ⊆ S 0 is a categorical equivalence, we conclude that (20 )is equivalent to the following apparently stronger condition:(200 ) For every object s ∈ S 0 , the functor F | Ms is a left Kan extension of
F | M0s .
The equivalence of (10 ) and (200 ) follows from Lemma 4.3.3.9.

4.4 EXAMPLES OF COLIMITSIn this section, we will analyze in detail the colimits of some very simple diagrams. Our first three examples are familiar from classical category theory:
coproducts (§4.4.1), pushouts (§4.4.2), and coequalizers (§4.4.3).
Our fourth example is slightly more unfamiliar. Let C be an ordinary
category which admits coproducts. Then C is naturally tensored over the
category of sets. Namely, for each C ∈ C and S ∈ Set, we can define C ⊗ S
to be the coproduct of a collection of copies of C indexed by the set S. The
object C ⊗ S is characterized by the following universal mapping property:
HomC (C ⊗ S, D) ' HomSet (S, HomC (C, D)).
In the ∞-categorical setting, it is natural to try to generalize this definition
by allowing S to be an object of S. In this case, C ⊗ S can again be viewed
as a kind of colimit but cannot be written as a coproduct unless S is discrete.
We will study the situation in §4.4.4.
Our final objective in this section is to study the theory of retracts in
an ∞-category C. In §4.4.5, we will see that there is a close relationshipbetween retracts in C and idempotent endomorphisms, just as in classicalhomotopy theory. Namely, any retract of an object C ∈ C determines an
idempotent endomorphism of C; conversely, if C is idempotent complete,then every idempotent endomorphism of C determines a retract of C. We
will return to this idea in §5.1.4.

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4.4.1 Coproducts
In this section, we discuss the simplest type of colimit: namely, coproducts.
Let A be a set; we may regard A as a category with(∗ if I = J
HomA (I, J) =∅ if I 6= J.
We will also identify A with the (constant) simplicial set which is the nerveof this category. We note that a functor G : A → Set∆ is injectively fibrant
if and only if it takes values in the category
Q Kan of Kan complexes. If thiscondition is satisfied, then the product α∈A G(α) is a homotopy limit forG.
Let F : A → C be a functor from A to a fibrant simplicial category; inother words, F specifies a collection {Xα }α∈A of objects in C. A homotopy
colimit for F will be referred to as a homotopy coproduct of the objects{Xα }α∈A . Unwinding the definition, we see that a homotopy coproduct isan object X ∈ C equipped with morphisms φα : Xα → X such that theinduced map
Y
MapC (X, Y ) →MapC (Xα , Y )α∈A

is a homotopy equivalence for every object Y ∈ C. Consequently, we recover
the description given in Example 1.2.13.1. As we noted earlier, this characterization can be stated entirely in terms of the enriched homotopy category hC:
the maps {φα } exhibit X as a homotopy coproduct of the family {Xα }α∈A
if and only if the induced map
Y
MapC (X, Y ) →MapC (Xα , Y )α∈A

is an isomorphism in the homotopy category H of spaces for each Y ∈ C.
Now suppose that C is an ∞-category and let p : A → C be a map. As
above, we may identify this with a collection of objects {Xα }α∈A of C. To
specify an object of Cp/ is to give an object X ∈ C together with morphisms
φα : Xα → X for each α ∈ A. Using Theorem 4.2.4.1, we deduce that X isa colimit of the diagram p if and only if the induced map
Y
MapC (X, Y ) →MapC (Xα , Y )α∈A

is an isomorphism in H for each object Y ∈ C. In this case, we say that X
is a coproduct of the family {Xα }α∈A .
In either setting, we will denote the (homotopy) coproduct of a family ofobjects {Xα }α∈A by
a
Xα .
α∈A

It is well-defined up to (essentially unique) equivalence.
Using Corollary 4.2.3.10, we deduce the following:highertopoi

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Proposition 4.4.1.1. Let C be an ∞-category and let {pα : Kα → C}α∈A
be a family of diagrams in C indexed
by a set A. Suppose that each pα has`
a colimit Xα in C. Let K =Kα and let p : K → C be the result ofamalgamating the maps pα . Then p has a colimit in C if and only if thefamily {Xα }α∈A has `a coproduct in C; in this case, one may identify colimitsof p with coproducts α∈A Xα .
4.4.2 Pushouts
Let C be an ∞-category. A square in C is a map ∆1 × ∆1 → C. We will
typically denote squares in C by diagrams
X0

p0

q0


Y0

/X
q

p


/ Y,

with the “diagonal” morphism r : X 0 → Y and homotopies r ' q ◦ p0 ,r ' p ◦ q 0 being implicit.
We have isomorphisms of simplicial sets(Λ20 ). ' ∆1 × ∆1 ' (Λ22 )/ .
Consequently, given a square σ : ∆1 ×∆1 → C, it makes sense to ask whether
or not σ is a limit or colimit diagram. If σ is a limit diagram, we will also
say that σ is a pullback square or a Cartesian square and we will informally
write X 0 = X ×Y Y 0 . Dually, if σ is a colimit diagram, we will
` say that σ isa pushout square or a coCartesian square, and write Y = X X 0 Y 0 .
Now suppose that C is a (fibrant) simplicial category. By definition, acommutative diagram
X0

p0

q0


Y0

p

/X

/Y

q

is a homotopy pushout square if, for every object Z ∈ C, the diagram
MapC (Y, Z)/ MapC (Y 0 , Z)
MapC (X, Z)/ MapC (X 0 , Z)

is a homotopy pullback square in Kan. Using Theorem 4.2.4.1, we can reduce
questions about pushout diagrams in an arbitrary ∞-category to questions
about homotopy pullback squares in Kan.
The following basic transitivity property for pushout squares will be used
repeatedly throughout this book:highertopoi

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Lemma 4.4.2.1. Let C be an ∞-category and suppose we are given a map
σ : ∆2 × ∆1 → C which we will depict as a diagram
X

/Y

/Z


X0


/ Y0


/ Z 0.

Suppose that the left square is a pushout in C. Then the right square is apushout if and only if the outer square is a pushout.
Proof. For every subset A of {x, y, z, x0 , y 0 , z 0 }, let D(A) denote the corresponding full subcategory of ∆2 × ∆1 and let σ(A) denote the restriction of
σ to D(A). We may regard σ as determining an object σ
e ∈ Cσ({x,y,z,x0 ,y0 })/ .
Consider the mapsφ

ψCσ({x,z,x0 })/ ← Cσ({x,y,z,x0 ,y0 })/ → Cσ({x,y,x0 })/ .
The map φ is the composition of the trivial fibration
Cσ({x,y,z,x0 })/ → Cσ({x,z,x0 })/with a pullback ofCσ({x,y,x0 ,y0 })/ → Cσ({x,y,x0 })/ ,which is also a trivial fibration by virtue of our assumption that the square
X

/Y


X0


/ Y0

is a pushout in C. The map ψ is a trivial fibration because the inclusion
D({x, y, x0 }) ⊆ D({x, y, z, x0 , y 0 }) is left anodyne. It follows that φ(eσ ) isan initial object of Cσ({x,z,x0 })/ if and only if ψ(eσ ) is an initial object ofCσ({y,z,y0 })/ , as desired.
Our next objective is to apply Proposition 4.2.3.8 to show that in manycases complicated colimits may be decomposed as pushouts of simpler colimits. Suppose we are given a pushout diagram of simplicial sets
L0

K0

i

/L

/K

and a diagram p : K → C, where C is an ∞-category. Suppose furthermorethat p|K 0 , p|L0 , and p|L admit colimits in C, which we will denote by X, Y ,and Z, respectively. If we suppose further that the map i is a cofibration of
simplicial sets, then the hypotheses of Proposition 4.2.3.4 are satisfied andwe can deduce the following:highertopoi

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Proposition 4.4.2.2. Let C be an ∞-category and let p : K → C be `a mapof simplicial sets. Suppose we are given a decomposition K = K 0 L0 L,
where L0 → L is a monomorphism of simplicial sets. Suppose further thatp|K 0 has a colimit X ∈ C, p|L0 has a colimit Y ∈ C, and p|L `has a colimit
Z ∈ C. Then one may identify colimits for p with pushouts X Y Z.
Remark 4.4.2.3. The statement of Proposition 4.4.2.2 is slightly vague.
Implicit in the discussion is that identifications of X with the colimit ofp|K 0 and Y with the colimit of p|L0 induce a morphism Y → X in C (andsimilarly for Y and Z). This morphism is not uniquely determined, but it isdetermined up to a contractible space of choices: see the proof of Proposition
4.2.3.4.
It follows from Proposition 4.4.2.2 that any finite colimit can be built using
initial objects and pushout squares. For example, we have the following:
Corollary 4.4.2.4. Let C be an ∞-category. Then C admits all finite colimits if and only if C admits pushouts and has an initial object.
Proof. The “only if” direction is clear. For the converse, let us suppose that
C has pushouts and an initial object. Let p : K → C be any diagram, where
K is a finite simplicial set: that is, K has only finitely many nondegenerate
simplices. We will prove that p has a colimit. The proof proceeds by induction: first on the dimension of K, then on the number of simplices of K
having the maximal dimension.
If K is empty, then an initial object of C is a colimit for p. Otherwise, we
may fix a nondegenerate simplex` of K having the maximal dimension andthereby decompose K ' K0 ∂ ∆n ∆n . By the inductive hypothesis, p|K0
has a colimit X and p| ∂ ∆n has a colimit Y . The ∞-category ∆n has a final
object, so p|∆n has a colimit Z (which we may take to be p(v), where v isthe final`vertex of ∆n ). Now we simply apply Proposition 4.4.2.2 to deducethat X Y Z is a colimit for p.
Using the same argument, one can show:
Corollary 4.4.2.5. Let f : C → C0 be a functor between ∞-categories.
Assume that C has all finite colimits. Then f preserves all finite colimits if
and only if f preserves initial objects and pushouts.
We conclude by showing how all colimits can be constructed out of simple
ones.
Proposition 4.4.2.6. Let C be an ∞-category which admits pushouts andκ-small coproducts. Then C admits colimits for all κ-small diagrams.
Proof. If κ = ω, we have already shown this as Corollary 4.4.2.4. Let us
therefore suppose that κ > ω, and that C has pushouts and κ-small sums.
Let p : K → C be a diagram, where K is κ-small. We first suppose thatthe dimension n of K is finite: that is, K has no nondegenerate simplices ofhighertopoi

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dimension larger than n. We prove that p has a colimit, working by induction
on n.
If n = 0, then K consists of a finite disjoint union of fewer than κ vertices.
The colimit of p exists by the assumption that C has κ-small sums.
Now suppose that every diagram indexed by a κ-small simplicial set of
dimension n has a colimit. Let p : K → C be a diagram, with the dimension
0of K equal to n + 1. Let K n denote the n-skeleton of K and Kn+1
⊆ Kn+1the set of all nondegenerate (n+1)-simplices of K, so that there is a pushout
diagram of simplicial sets
a
0
(Kn+1
× ∆n+1 ) ' K.
Kn
0
Kn+1
×∂ ∆n+1

By Proposition 4.4.2.2, we can construct a colimit of p as a pushout using
00colimits for p|K n , p|(Kn+1× ∂ ∆n+1 ), and p|(Kn+1× ∆n+1 ). The first two
exist by the inductive hypothesis and the last because it is a sum of diagrams
which possess colimits.
Now let us suppose that K is not necessarily finite dimensional. In thiscase, we can filter K by its skeleta {K n }. This is a family of simplicial subsets
of K indexed by the set Z≥0 of nonnegative integers. By what we have shown
above, each p|K n has a colimit xn in C. Since this family is directed andcovers K, Corollary 4.2.3.10 shows that we may identify colimits of p withcolimits of a diagram N(Z≥0 ) → C which we may write informally as
x0 → x1 → · · · .
Let L be the simplicial subset of N(Z≥0 ) which consists of all vertices together with the edges which join consecutive integers. A simple computationshows that the inclusion L ⊆ N(Z≥0 ) is a categorical equivalence and therefore cofinal. Consequently, it suffices to construct the colimit of a diagram
L → C. But L is 1-dimensional and is κ-small since κ > ω.
The same argument also proves the following:
Proposition 4.4.2.7. Let κ be a regular cardinal and let f : C → D bea functor between ∞-categories, where C admits κ-small colimits. Then fpreserves κ-small colimits if and only if f preserves pushout squares andκ-small coproducts.
Let D be an ∞-category containing an object X and suppose that D
admits pushouts. Then DX/ admits pushouts, and these pushouts may becomputed in D. In other words, the projection f : DX/ → D preserves
pushouts. In fact, this is a special case of a very general result; it requires
only that f be a left fibration and that the simplicial set Λ20 be weakly
contractible.
Lemma 4.4.2.8. Let f : C → D be a left fibration of ∞-categories and let
K be a weakly contractible simplicial set. Then any map p : K . → C is an
f -colimit diagram.

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Proof. Let p = p|K. We must show that the map
φ : Cp/ → Cp/ ×Df ◦p/ Df ◦p/
is a trivial fibration of simplicial sets. In other words, we must show that we
can solve any lifting problem of the form`.
/6 C
(K ? A) K?A
 _ 0 (K ? A)
f


/ D.



K. ? A

Since f is a left fibration, it will suffice to prove that the left vertical map isleft anodyne, which follows immediately from Lemma 4.2.3.5.
Proposition 4.4.2.9. Let f : C → D be a left fibration of ∞-categories andlet p : K → C be a diagram. Suppose that K is weakly contractible. Then
(1) Let p : K . → C be an extension of p. Then p is a colimit of p if andonly if f ◦ p is a colimit of f ◦ p.
(2) Let q : K . → D be a colimit of f ◦ p. Then q = f ◦ p, where p is anextension (automatically a colimit by virtue of (1)) of p.
Proof. To prove (1), fix an extension p : K . → C. We have a commutative
diagramCp/

φψ0/ Cp/ ×D

Df p/
f p/


Df p/

ψ

/ Cp/


θ

/ Df p/ .

Lemma 4.4.2.8 implies that φ is a trivial Kan fibration. If f ◦ p is a colimit
diagram, the map ψ is a trivial fibration. Since ψ 0 is a pullback of ψ, we
conclude that ψ 0 is a trivial fibration. It follows that ψ 0 ◦φ is a trivial fibration
so that p is a colimit diagram. This proves the “if” direction of (1).
To prove the converse, let us suppose that p is a colimit diagram. The
maps φ and ψ 0 ◦ φ are both trivial fibrations. It follows that the fibers ofψ 0 are contractible. Using Lemma 4.2.3.6, we conclude that the map θ is atrivial fibration, and therefore surjective on vertices. It follows that the fibersof ψ are contractible. Since ψ is a left fibration with contractible fibers, it isa trivial fibration (Lemma 2.1.3.4). Thus f ◦ p is a colimit diagram, and theproof is complete.
To prove (2), it suffices to show that f has the right lifting property
with respect to the inclusion i : K ⊆ K . . Since f is a left fibration, it will
suffice to show that i is left anodyne, which follows immediately from Lemma
4.2.3.6.

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4.4.3 Coequalizers
Let I denote the category depicted by the diagram
X

F
G

// Y.

In other words, I has two objects X and Y satisfying the conditions
HomI (X, X) = HomI (Y, Y ) = ∗HomI (Y, X) = ∅HomI (X, Y ) = {F, G}.
To give a diagram p : N(I) → C in an ∞-category C, one must give a pairof morphisms f = p(F ), g = p(G) in C having the same domain x = p(X)and the same codomain y = p(Y ). A colimit for the diagram p is said to bea coequalizer of f and g.
Applying Corollary 4.2.3.10, we deduce the following:
Proposition 4.4.3.1. Let K and A be simplicial sets and let i0 , i1 : A → K
be embeddings having disjoint images in K. Let K 0 denote the coequalizer ofi0 and i1 : in other words, the simplicial set obtained from K by identifying
the image of i0 with the image of i1 . Let p : K 0 → C be a diagram in an∞-category C and let q : K → C be the compositionp

K → K0 → C .
Suppose that the diagrams q ◦ i0 = q ◦ i1 and q possess colimits x and y in
C. Then i0 and i1 induce maps j0 , j1 : x → y (well-defined up to homotopy);
colimits for p may be identified with coequalizers of j0 and j1 .
Like pushouts, coequalizers are a basic construction out of which othercolimits can be built. More specifically, we have the following:
Proposition 4.4.3.2. Let C be an ∞-category and κ a regular cardinal.
Then C has all κ-small colimits if and only if C has coequalizers and κ-small
coproducts.
Proof. The “only if” direction is obvious. For the converse, suppose that C
has coequalizers and κ-small coproducts. In view of Proposition 4.4.2.6, it
suffices to show that C has pushouts. Let p`: Λ20 be a pushout diagram in C.
We note that Λ20 is the quotient of ∆{0,1} ∆{0,2} obtained by identifying
the initial vertex of ∆{0,1} with the initial vertex`of ∆{0,2} . In view of Proposition 4.4.3.1, it suffices to show that p|(∆{0,1} ∆{0,2} ) and p|{0} possess
colimits in C. The second assertion is obvious.
` Since C has finite sums, toprove that there exists a colimit for p|(∆{0,1} ∆{0,2} ), it suffices to provethat p|∆{0,1} and p|∆{0,2} possess colimits in C. This is immediate because
both ∆{0,1} and ∆{0,2} have final objects.
Using the same argument, we deduce:
Proposition 4.4.3.3. Let κ be a regular cardinal and C be an ∞-category
which admits κ-small colimits. A full subcategory D ⊆ C is stable under κsmall colimits in C if and only if D is stable under coequalizers and underκ-small coproducts.

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4.4.4 Tensoring with Spaces
Every ordinary category C can be regarded as a category enriched over Set.
Moreover, if C admits coproducts, then C can be regarded as tensored overSet in an essentially unique way. In the ∞-categorical setting, one has asimilar situation: if C is an ∞-category which admits all small colimits, thenC may be regarded as tensored over the ∞-category S of spaces. To make thisidea precise, we would need a good theory of enriched ∞-categories, whichlies outside the scope of this book. We will instead settle for a slightly ad hoc
point of view which nevertheless allows us to construct the relevant tensor
products. We begin with a few remarks concerning representable functors in
the ∞-categorical setting.
Definition 4.4.4.1. Let D be a closed monoidal category and let C be acategory enriched over D. We will say that a D-enriched functor G : Cop → D
is representable if there exists an object C ∈ C and a map η : 1D → G(C)such that the induced map
MapC (X, C) ' MapC (X, C) ⊗ 1D → MapC (X, C) ⊗ G(C) → G(X)
is an isomorphism for every object X ∈ C. In this case, we will say that(C, η) represents the functor G.
Remark 4.4.4.2. In the situation of Definition 4.4.4.1, we will sometimes
abuse terminology and simply say that the functor F is represented by theobject C.
Remark 4.4.4.3. The dual notion of a corepresentable functor can be defined in an obvious way.
Definition 4.4.4.4. Let C be an ∞-category and let S denote the ∞category of spaces. We will say that a functor F : Cop → S is representable
if the underlying functor
hF : hCop → hS ' H
of (H-enriched) homotopy categories is representable. We will say that a pairC ∈ C, η ∈ π0 F (C) represents F if the pair (C, η) represents hF .
e → C be a right fibration of ∞-categories,
Proposition 4.4.4.5. Let f : Ceee ∈ C, and let F : Cop → S be a functor
let C be an object of C, let C = f (C)which classifies f (§3.3.2). The following conditions are equivalent:e C {C}) be the connected component containing(1) Let η ∈ π0 F (C) ' π0 (C×e
C. Then the pair (C, η) represents the functor F .
e is final.
e∈C(2) The object C
e is a contravariant equivalence in (Set∆ )/ C .
e ⊆C(3) The inclusion {C}highertopoi

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301

LIMITS AND COLIMITSProof. We have a commutative diagram of right fibrations
e e
C
/C

φ


C/C

e
/C

/ C.

Observe that the left vertical map is actually a trivial fibration. Fix an object
D ∈ C. The fiber of the upper horizontal map
e e ×C {D} → C
e ×C {D}φD : C/C

can be identified, in the homotopy category H, with the map MapC (D, C) →F (C). The map φD is a right fibration of Kan complexes and therefore a Kan
fibration. If (1) is satisfied, then φD is a homotopy equivalence and therefore
a trivial fibration. It follows that the fibers of φ are contractible. Since φ is aeright fibration, it is a trivial fibration (Lemma 2.1.3.4). This proves that Ceis a final object of C. Conversely, if (2) is satisfied, then φD is a trivial Kan
fibration and therefore a weak homotopy equivalence. Thus (1) ⇔ (2).
e is right anodyne and theree ⊆C
If (2) is satisfied, then the inclusion {C}fore a contravariant equivalence by Proposition 4.1.2.1. Thus (2) ⇒ (3).
Conversely, suppose that (3) is satisfied. The inclusion {idC } ⊆ C/C is rightanodyne and therefore a contravariant equivalence. It follows that the lifting
problem{idC }
_

e
C
e



C/C

/e
>C
f


/C

has a solution. We observe that e is a contravariant equivalence of rightfibrations over C and therefore a categorical equivalence. By construction, eee so that C
e is a final object of C.
carries a final object of C/C to C,
e → C is representable if C
e has a final
We will say that a right fibration C
object.
Remark 4.4.4.6. Let C be an ∞-category and let p : K → C be a diagram.
Then the right fibration C/p → C is representable if and only if p has a limit
in C.
Remark 4.4.4.7. All of the above ideas dualize in an evident way, so that
we may speak of corepresentable functors and corepresentable left fibrationsin the setting of ∞-categories.
Notation 4.4.4.8. For each diagram p : K → C in an ∞-category C, we
let Fp : hC → H denote the H-enriched functor corresponding to the left
fibration Cp/ → C.

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If p : ∗ → C is the inclusion of an object X of C, then we write FX for Fp .
We note that FX is the functor corepresented by X:
FX (Y ) = MapC (X, Y ).
Now suppose that X is an object in an ∞-category C and let p : K → C
be a constant map taking the value X. For every object Y of C, we have an
isomorphism of simplicial sets (Cp/ ) ×C {Y } ' (CX/ ×C {Y })K . This identification is functorial up to homotopy, so we actually obtain an equivalenceFp (Y ) ' MapC (X, Y )[K]in the homotopy category H of spaces, where [K] denotes the simplicial set
K regarded as an object of H. Applying Proposition 4.4.4.5, we deduce thefollowing:
Corollary 4.4.4.9. Let C be an ∞-category, X an object of C, and K a
simplicial set. Let p : K → C be the constant map taking the value X.
The objects of the fiber Cp/ ×C {Y } are classified (up to equivalence) by maps
ψ : [K] → MapC (X, Y ) in the homotopy category H. Such a map ψ classifies
a colimit for p if and only if it induces isomorphisms
MapC (Y, Z) ' MapC (X, Z)[K]in the homotopy category H for every object Z of C.
In the situation of Corollary 4.4.4.9, we will denote a colimit for p by
X ⊗ K if such a colimit exists. We note that X ⊗ K is well-defined up to(essentially unique) equivalence and that it depends (up to equivalence) only
on the weak homotopy type of the simplicial set K.
Corollary 4.4.4.10. Let C be an ∞-category, let K be a weakly contractible
simplicial set, and let p : K → C be a diagram which carries each edge of K
to an equivalence in C. Then
(1) The diagram p has a colimit in C.
(2) An arbitrary extension p : K . → C is a colimit for C if and only if pcarries each edge of K . → C to an equivalence in C.
Proof. Let C0 ⊆ C be the largest Kan complex contained in C. By assumption,p factors through C0 . Since K is weakly contractible, we conclude that p :K → C0 is homotopic to a constant map p0 : K → C0 . Replacing p by p0
if necessary, we may reduce to the case where p is constant, taking values
equal to some fixed object C ∈ C.
Let p : K . → C be the constant map with value C. Using the characterization of colimits in Corollary 4.4.4.9, we deduce that p is a colimit diagramin C. This proves (1) and (in view of the uniqueness of colimits up to equivalence) the “only if” direction of (2). To prove the converse, we suppose thatp0 is an arbitrary extension of p which carries each edge of K . to an equivalence in C. Then p0 factors through C0 . Since K . is weakly contractible, we
conclude as above that p0 is homotopic to a constant map and is therefore acolimit diagram.

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4.4.5 Retracts and Idempotents
Let C be a category. An object Y ∈ C is said to be a retract of an object
X ∈ C if there is a commutative diagram
>X
i

r

/Y

idY

Y

in C. In this case we can identify Y with a subobject of X via the monomorphism i and think of r as a retraction from X onto Y ⊆ X. We observe alsothat the map i ◦ r : X → X is idempotent. Moreover, this idempotent determines Y up to canonical isomorphism: we can recover Y as the equalizerof the pair of maps (idX , i ◦ r) : X → X (or, dually, as the coequalizer ofthe same pair of maps). Consequently, we obtain an injective map from thecollection of isomorphism classes of retracts of X to the set of idempotent
maps f : X → X. We will say that C is idempotent complete if this correspondence is bijective for every X ∈ C: that is, if every idempotent map
f : X → X comes from a (uniquely determined) retract of X. If C admits
equalizers (or coequalizers), then C is idempotent complete.
These ideas can be adapted to the ∞-categorical setting in a straightforward way. If X and Y are objects of an ∞-category C, then we say that Y
is a retract of X if it is a retract of X in the homotopy category hC. Equivalently, Y is a retract of X if there exists a 2-simplex ∆2 → C corresponding
to a diagram
>X
i

Y

r
idY

/ Y.

As in the classical case, there is a correspondence between retracts Y of X
and idempotent maps f : X → X. However, there is an important difference:in the ∞-categorical case one cannot recover a retract Y of X as the limit(or colimit) of a finite diagram involving X.
Example 4.4.5.1. Let R be a commutative ring and let C• (R) be thecategory of complexes of finite free R-modules, so that an object of C• (R)is a chain complex· · · → M1 → M0 → M−1 → · · ·such that each Mi is a finite free R-module and Mi = 0 for |i|  0; morphisms in C• (R) are given by morphisms of chain complexes. There is anatural simplicial enrichment of the category C• (R) for which the mapping
spaces are Kan complexes; let C = N(C• (R)) be the associated ∞-category.
Then C admits all finite limits and colimits (C is an example of a stable∞-category; see [50]). However, the ∞-category C is idempotent complete if
and only if every finitely generated projective R-module is stably free.

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The purpose of this section is to define the notion of an idempotent in
an ∞-category C and to obtain a correspondence between idempotents andretracts in C.
Definition 4.4.5.2. Let R denote the category consisting of two objects X
and Y , with morphism sets given by
HomR (X, X) = {idX , e}
HomR (X, Y ) = {r}HomR (Y, X) = {i}HomR (Y, Y ) = {idY }and composition law determined by r ◦ i = idY and i ◦ r = e (from which
it follows that r ◦ e = r, e ◦ i = i, and e ◦ e = e). We let Idem+ denote thenerve of the category J, and we let Idem denote the full subcategory of Idem
spanned by the object X.
Note that the commutative diagram
>X
i

Y

r
idY

/Y

in the category R determines a map of simplicial sets σ : ∆2 → Idem+ . We
let Ret ⊆ Idem+ denote the image of σ (so that Ret is isomorphic to thequotient of ∆2 obtained by collapsing the “long edge” ∆{0,2} ⊆ ∆2 to apoint).
Remark 4.4.5.3. The ∞-category Idem can be described as the classifying
space of the two-element monoid {id, e}, whose multiplication is given by
e2 = e. It contains exactly one nondegenerate simplex of each dimension,and the collection of nondegenerate simplices of Idem is closed under facemaps. In fact, Idem is characterized up to (unique) isomorphism by theseproperties.
Definition 4.4.5.4. Let C be an ∞-category.
(1) An idempotent in C is a functor of ∞-categories Idem → C. We willrefer to Fun(Idem, C) as the ∞-category of idempotents in C.
(2) A weak retraction diagram in C is a map of simplicial sets Ret → C. We
will refer to Fun(Ret, C) as the ∞-category of weak retraction diagramsin C.
(3) A strong retraction diagram in C is a functor of ∞-categories Idem+ →C. We will refer to Fun(Idem+ , C) as the ∞-category of strong retraction
diagrams in C.
Remark 4.4.5.5. Let C be an ∞-category. Then the datum of a weakretraction diagram in C is equivalent to the datum of a 2-simplex
>X
Y

idY

/Y

highertopoi

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in the ∞-category C (here we abuse notation by identifying the objects
X, Y ∈ R with their images in C). Note that this is exactly the data required
to exhibit Y as a retract of X in the homotopy category hC.
To give an idempotent F : Idem → C in C, it suffices to specify the image
under F of each nondegenerate simplex of Idem in each dimension n ≥ 0.
Taking n = 0, we obtain an object X ∈ C. Taking n = 1, we get a morphism
e : X → X. Taking n = 2, we get a 2-simplex of C corresponding to a
diagram
>X
e

e

/X

e

X

which verifies the equation e = e ◦ e in the homotopy category hC. Takingn > 2, we get higher-dimensional diagrams which express the idea that e is
not only idempotent “up to homotopy,” but “up to coherent homotopy.”
Let C be an ∞-category and suppose we are given a weak retraction diagram in C, which we indicate by
>X
i

Y

r
idY

/ Y.

Then e = r ◦ i is a morphism from X to itself, which is well-defined up
to homotopy. It follows immediately that e is idempotent in the homotopy
category hC: that is, e is homotopic to e2 . In fact, we can say more:
Proposition 4.4.5.6. The inclusion Ret ⊆ Idem+ is an inner anodyne mapof simplicial sets.
Corollary 4.4.5.7. Let C be an ∞-category. Then the restriction map
Fun(Idem+ , C) → Fun(Ret, C)from strong retraction diagrams to weak retraction diagrams is a trivial fibration of simplicial sets. In particular, every weak retraction diagram in C
can be extended to a strong retraction diagram.
The proof of Proposition 4.4.5.6 will require the following:
Lemma 4.4.5.8. Let J ⊆ {0, . . . , n} and let K ⊆ ∆n be the simplicial
subset spanned by the nondegenerate simplices of ∆n which do not contain∆J . Suppose that there exist 0 ≤ i < j < k ≤ n such that i, k ∈ J, j ∈/ J.
Then the inclusion K ⊆ ∆n is inner anodyne.
Proof. Let P denote the collection of all subsets J 0 ⊆ {0, . . . , n} which contain J ∪ {j}. Choose a linear ordering{J(1) ≤ · · · ≤ J(m)}highertopoi

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of P with the property that if J(i) ⊆ J(j), then i ≤ j. Let[
K(k) = K ∪∆J(i) .
1≤i≤kNote that there are pushout diagrams
Λj

J(i)

/ ∆J(i)K(i − 1)/ K(i).

It follows that the inclusions K(i − 1) ⊆ K(i) are inner anodyne. Therefore
the composite inclusion K = K(0) ⊆ K(m) = ∆n is also inner anodyne.
Proof of Proposition 4.4.5.6. For each positive integer m, let Ret(m) denotethe simplicial subset of Idem+ consisting of those simplices ∆k → Idem+
for which the projection map ∆k ×Idem+ Idem → Idem factors through the(m − 1)-skeleton of Idem. Then Ret(1) = Ret. Moreover, for m ≥ 2, there isa pushout diagram of simplicial sets
K

/ ∆2m


Ret(m − 1)/ Ret(m),τ

where the simplex τ classifies the diagram
i

r

i

r

i

r

Y →
− X−
→Y →
− X−
→Y →
− ··· −
→Yin the category J, and K ⊆ ∆2m denote the simplicial subset spanned bythose faces which do not contain ∆{1,3,...,2m−1} . Lemma 4.4.5.8 implies thatthe upper horizontal arrow is inner anodyne, so that the inclusion Ret(m −1) ⊆ Ret(m) is also inner anodyne. The inclusion Ret ⊆ Idem+ can be
identified with an infinite composition
Ret = Ret(1) ⊆ Ret(2) ⊆ · · ·of inner anodyne maps, and is therefore also inner anodyne.
According to Corollary 4.4.5.7, every weak retraction diagram
>X
Y

idY

/Y

in an ∞-category C can be extended to a strong retraction diagram F :Idem+ → C, which restricts to give an idempotent in C. Our next goal is toshow that F is canonically determined by the restriction F | Idem. First, weneed some elementary observations.

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Lemma 4.4.5.9. The ∞-category Idem is a retract of the cone Idem. . In
other words, the inclusion map Idem ,→ Idem. admits a left inverse F :Idem. → Idem.
Proof. Unwinding the definitions, we can identify Idem. with the nerve of
the category I having objects X and Z, with morphisms given by
HomI (X, X) = {idX , e}
HomI (Z, X) = ∅HomI (X, Z) = {f }HomI (Z, Z) = {idZ },and composition law given by e ◦ e = e, f ◦ e = f . The functor F is thengiven by F (Z) = X and F (f ) = e.
Lemma 4.4.5.10. The ∞-category Idem is weakly contractible.
Proof. By virtue of Lemma 4.4.5.9, it will suffice to show that the ∞-category
Idem. is weakly contractible. This is clear, since Idem. has a final object.
Lemma 4.4.5.11. The inclusion Idem ⊆ Idem+ is a cofinal map of simplicial sets.
Proof. According to Theorem 4.1.3.1, it will suffice to prove that the simplicial sets IdemX/ and IdemY / are weakly contractible. The simplicial set
IdemX/ is an ∞-category with an initial object and therefore weakly contractible. The projection map IdemY / → Idem is an isomorphism, so that
IdemY / is weakly contractible by Lemma 4.4.5.10.
Proposition 4.4.5.12. Let C be an ∞-category and let F : Idem+ → C bea strong retraction diagram. Then F is a left Kan extension of F | Idem.
Remark 4.4.5.13. Passing to opposite ∞-categories, it follows that a strong
retraction diagram F : Idem+ → C is also a right Kan extension of F | Idem.
Proof of Proposition 4.4.5.12. We must show that the induced map
G

F

+
.
(Idem/Y ). → (Idem+/Y ) → Idem → C

is a colimit diagram. Consider the commutative diagram
Idem/Y/ Idem+Idem

/ Idem+ .

/Y

The lower horizontal map is cofinal by Lemma 4.4.5.11, and the verticalmaps are isomorphisms: therefore the upper horizontal map is also cofinal.
Consequently, it will suffice to prove that F ◦ G is a colimit diagram, which
is obvious.
Corollary 4.4.5.14. Let C be an ∞-category and let F : Idem → C be an
idempotent in C. The following conditions are equivalent:highertopoi

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CHAPTER 4(a) The idempotent F : Idem → C extends to a map F + : Idem+ → C.
(b) The functor F admits a colimit in C.
(c) The functor F admits a limit in C.
Proof. The equivalence of (a) and (b) follows by combining Proposition
4.4.5.12 with Lemma 4.3.2.13. The equivalence of (a) and (c) follows by
applying the same argument to Cop .
We will say that an idempotent F : Idem → C in an ∞-category C iseffective if it satisfies the equivalent conditions of Corollary 4.4.5.14.
Corollary 4.4.5.15. Let C be an ∞-category. The following conditions areequivalent:(a) The ∞-category C is idempotent complete.
(b) Every diagram F : Idem → C admits a colimit.
(c) Every diagram F : Idem → C admits a limit.
Corollary 4.4.5.16. Let C be an ∞-category and let D ⊆ Fun(Idem, C) be
the full subcategory spanned by the effective idempotents in C. The restriction
map Fun(Idem+ , C) → D is a trivial Kan fibration. In particular, if C is
idempotent complete, then we have a diagram of trivial Kan fibrations
Fun(Ret, C) ← Fun(Idem+ , C) → Fun(Idem, C).
Proof. Combine Proposition 4.4.5.12 with Proposition 4.3.2.15.
The following result is useful for approximating the ∞-category Idem:
Proposition 4.4.5.17. Let A be a linearly ordered set and let p : N(A) →Idem be the unique map which carries nondegenerate simplices to nondegenerate simplices. If A has no largest element, then p is cofinal.
Remark 4.4.5.18. Let C be an ∞-category equipped with an idempotent
F : Idem → C, which determines an object X ∈ C and a map e : X → X.
Applying Proposition 4.4.5.17 in the special case A = Z≥0 , we deduce thatthe associated retract of X (if it exists) can be identified with a colimit ofthe diagram
e

e

e

e

X−
→X−
→X−
→X−
→ ···
Proof of Proposition 4.4.5.17. According to Theorem 4.1.3.1, it will suffice
to show that the fiber product E = N(A)×Idem IdemX/ is weakly contractible.
We observe that E is isomorphic to the nerve N(A0 ), where A0 denotes theset A × {0, 1} equipped with the partial ordering(α, i) < (α0 , j) ⇔ (j = 1) ∧ (α < α0 ).

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For each α ∈ A, let A<α = {α0 ∈ A : α0 < α} and let
A0α = {(α0 , i) ∈ A0 : (α0 < α) ∨ ((α0 , i) = (α, 1))}.
S
By hypothesis, we can write A as a filtered union α∈A A<α . It therefore
suffices to prove that for each α ∈ A the map
f : N(A<α ) ×Idem IdemX/ → N(A) ×Idem IdemX/
has a nullhomotopic geometric realization |f |. But this map factors throughN(A0α ), and | N(A0α )| is contractible because A0α has a largest element.
Let C be an ∞-category and let F : Idem → C be an idempotent in C.
Then F determines an idempotent in the homotopy category hC, which we
can identify with an object X ∈ C and a morphism e : X → X such that e2 ishomotopic to e. Beware that the converse is false: in general, an idempotent
in the homotopy category hC cannot necessarily be lifted to an idempotent
in C:
Counterexample 4.4.5.19. For example, let G denote the group of homeomorphism of the unit interval [0, 1] which fix its endpoints (which we regard
as a discrete group), and let λ : G → G denote the group homomorphism
given by the formula(
1g(2t) if 0 ≤ t ≤ 12
λ(g)(t) = 2
t
if 12 ≤ t ≤ 1.
Choose an element h ∈ G such that h(t) = 2t for 0 ≤ t ≤ 41 . Then λ(g) ◦ h =h ◦ λ(λ(g)) for each g ∈ G, so that the group homomorphisms λ, λ2 : G → G
are conjugate to one another. It follows that the induced map of classifying
spaces e : BG → BG is homotopic to e2 , and is therefore idempotent in
the homotopy category hS. However, we claim that e cannot be lifted to an
idempotent in the ∞-category of spaces. Otherwise, the natural map from
BG to the colimit of the sequenceeee
BG → BG → BG → BG → · · ·would admit a left homotopy inverse. Passing to fundamental groups, itwould follow that G surjects onto the colimit of the sequenceλλG → G → ··· .
This is clearly impossible, since the homomorphism λ is injective but not
bijective.
In the situation of Counterexample 4.4.5.19, the element h ∈ G determinesa homotopy from e to e2 , from which we can construct two homotopies frome to e3 (as maps from BG to itself). Concretely, these homotopies are givenby the elements h2 and hλ(h), respectively: note that for each g ∈ G, wehave
h−2 λ(g)h2 = h−1 λ(λ(g))h= λ(λ(λ(g)))= λ(h−1 λ(g)h)= λ(h)−1 λ(λ(g))λ(h)= λ(h)−1 h−1 λ(g)hλ(h).

highertopoi

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The failure of e to lift to an idempotent in the ∞-category S results fromthe fact that these two homotopies are not homotopic to one another (sinceh2 6= hλ(h) in the group G). In fact, this is the only possible obstruction:
Proposition 4.4.5.20. Let C be an ∞-category, let X be an object of C,
and let e : X → X be a morphism. The following conditions are equivalent:(1) The morphism e can be lifted to an idempotent in the ∞-category C.
That is, the map ∆1 → C determined by e extends to a map Idem → C,
where Idem is the ∞-category of Definition 4.4.5.2.
(2) Let E denote the fundamental groupoid of the space MapC (X, X), sothat composition of morphisms determines a composition product ◦ :E × E → E. Let [e] denote the object of E corresponding to the morphism
e. Then there exists an isomorphism h : [e] → [e] ◦ [e] in E such thatthe diagram σ :[e] ◦ [e]id[e] ◦h&/ [e] ◦ ([e] ◦ [e])h◦id[e]x([e] ◦ [e]) ◦ [e]commutes up to homotopy.
Before giving the proof of Proposition 4.4.5.20, let us note the following
consequence:
Corollary 4.4.5.21. Let {Cα } be a filtered diagram of idempotent complete∞-categories. Then the colimit C = lim Cα is idempotent complete.
−→
Proof. For each index α, let Uα : Cα → C be the canonical map. Let Idem
be as in Definition 4.4.5.2, and let f : Idem → C be a diagram; we wish toshow that f has a colimit in C. Let K ⊆ Idem be the 3-skeleton of Idem, sothat K is a finite simplicial set. Consequently, there exists an index α suchthat f |K is homotopic to a compositionfαUK −→ Cα →α C .
The map fα determines an object X ∈ Cα and a map e : X → X which
satisfies condition (2) of Proposition 4.4.5.20. Using Proposition 4.4.5.20, we
deduce that X and e can be extended to a map g : Idem → Cα . Since Cα is
idempotent complete, this diagram admits a colimit Y in Cα . It follows that
Uα (Y ) is a colimit of the diagram Uα ◦ g, so that Uα (Y ) is the colimit of the
diagramUα (e)Uα (e)Uα (X) −→ Uα (X) −→ · · · ,which is also a colimit of f (see Remark 4.4.5.18).

highertopoi

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LIMITS AND COLIMITSProof of Proposition 4.4.5.20. The implication (1) ⇒ (2) is obvious (if eb :Idem → C is a map extending e, then the value of eb on the nondegenerate
2-simplex of Idem determines an isomorphism [e] ' [e] ◦ [e], and the valueof eb on the nondegenerate 3-simplex of Idem witnesses the commutativity ofthe diagram σ). We will prove that (2) ⇒ (1). Enlarging C if necessary, we
can arrange that C admits sequential colimits (see Remark 5.3.5.9). Regard
the set Z≥0 of nonnegative integers as linearly ordered, and let K denote thesimplicial subset of N(Z≥0 ) consisting of all vertices of N(Z≥0 ), together withthose edges given by pairs i ≤ j where j ≤ i + 1. Let f : K → C be the map
which carries each vertex of K to the object X ∈ C, and each nondegenerateedge of K to the morphism e : X → X. Since C admits sequential colimits,
we can extend f to a colimit diagram f : K . → C. Let Y be the image under
f of the cone point of K . . For each integer n ≥ 0, the restriction of f to{n}. determines a morphism φn : X → Y , so that f determines a sequenceof 2-simplices σn of C which witness the commutativity of the diagramsX/Xe
φn φn+1Y.

~

Let σ 0 be a 2-simplex of C representing the homotopy h (so that the
restriction of σ to each nondegenerate 1-simplex of ∆2 coincides with e),
and let f : K / → C be the map which carries each vertex of K . to X, each
nondegenerate edge of K / to e, and each nondegenerate 2-simplex of K / to0
σ. Then f determines a morphism i : Y = lim(f ) → X. We will prove that−→the map φ0 ◦ i : Y → Y is homotopic to the identity, so that φ0 and i exhibit
Y as a retract of X. The idempotence of e will then follow from Corollary
4.4.5.7, since the composition i ◦ φ0 is homotopic to e.
Let us abuse notation by identifying X and Y with objects of CY / (via
the morphisms i : Y → X and idY : Y → Y ). The statement that φ0 ◦ i is
homotopic to the identity is equivalent to the requirement that φ0 can belifted to a morphism from X to Y in CY / . Since Y is a colimit of f , this is0
equivalent to the assertion that φ0 can be lifted to a morphism from f to f in
Cf / . Equivalently, we must show that there exists a map g : K ?{u}?{v} → C
0whose restriction to K ? {u} is given by f , whose restriction to K ? {v} isgiven by f , and whose restriction to {u} ? {v} is given by φ0 .
Let D denote the fundamental groupoid of the mapping space MapC (X, Y ).
For each morphism s : X → Y , we will denote the corresponding object of D
by [s]. The composition product MapC (X, X)×MapC (X, Y ) → MapC (X, Y )determines a right action of E on D, which we will denote by ◦ : D × E → E.
Note that each of the 2-simplices σn determines an isomorphism αn : [φn ] →[φn+1 ] ◦ [e] in the category D.
Suppose we are given a sequence of isomorphisms ρn : [φn ] → [φ0 ] ◦ [e] inhighertopoi

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CHAPTER 4

the category D. Each of these isomorphisms is witnessed by a 2-simplex τn :
/X

e

X
φn

Y.

~

φ0

in the category C. We then have a unique map g0 from the 2-skeleton of
K ? {u} ? {v} to C satisfying
g0 |K?{u} = f0

g0 |K?{v} = fg0 |{n}?{u}?{v} = τn .

Unwinding the definitions, we see that g0 can be extended to the 3-skeleton
of K ? {u} ? {v} (which coincides with K ? {u} ? {v}) if and only if, for eachn ≥ 0, the diagram/ [φ0 ] ◦ [e]

ρn

[φn ]
αn

h

[φn+1 ] ◦ [e]/ [φ0 ] ◦ ([e] ◦ [e])ρn+1

/ ([φ0 ] ◦ [e]) ◦ [e]commutes.
We will construct a sequence of isomorphisms βn : [φn+1 ] ◦ [e] → [φ0 ] ◦ [e]in the category D for which the diagrams/ [φ0 ] ◦ [e]βn[φn+1 ] ◦ [e]
αn+1

h


βn+1
/ ([φ0 ] ◦ [e]) ◦ [e]([φn+2 ] ◦ [e]) ◦ [e]/ [φ0 ] ◦ ([e] ◦ [e])commute. Assuming this has been done, we can complete the proof by taking
ρn = βn ◦ αn .
For every pair of morphisms s, s0 : X → Y in C, let H(s, s0 ) denote theset HomD ([s] ◦ [e], [s0 ] ◦ [e]). Let us say that an element β ∈ H(s, s0 ) is good
if the diagram[s] ◦ [e]h[s] ◦ ([e] ◦ [e])β/ [s0 ] ◦ [e]h
[s0 ] ◦ ([e] ◦ [e])



β×id
/ ([s0 ] ◦ [e]) ◦ [e]([s] ◦ [e]) ◦ [e]commutes. The collection of good morphisms has the following evident properties:(i) For each morphism s : X → Y , the composite maph[s] ◦ [e] → [s] ◦ ([e] ◦ [e]) ' ([s] ◦ [e]) ◦ [e]highertopoi

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313

LIMITS AND COLIMITS

is a good element of H(s, s ◦ e). This follows from the commutativityof the diagram[e] ◦ [e]id[e] ◦hx([e] ◦ [e]) ◦ [e]&/ [e] ◦ ([e] ◦ [e])h◦id[e]in E.
(ii) For each morphism β : [s] → [s0 ] in D, the induced map [s]◦[e] → [s]◦[e]is a good element of H(s, s0 ).
(iii) If β : [s] ◦ [e] → [s0 ] ◦ [e] and β 0 : [s0 ] ◦ [e] → [s00 ] ◦ [e] are good elementsof H(s, s0 ) and H(s0 , s00 ), respectively, then the composition β 0 ◦ β is agood element of H(s, s00 ).
We will complete the proof by establishing the following:(a) There exists a good element β0 ∈ H(φ1 , φ0 ).
(b) Let βn be a good element of H(φn+1 , φ0 ). Then there exists a goodelement βn+1 ∈ H(φn+2 , φ0 ) for which the diagram[φn+1 ] ◦ [e]/ [φ0 ] ◦ [e]

βn

αn+1

h


βn+1
/ ([φ0 ] ◦ [e]) ◦ [e]([φn+2 ] ◦ [e]) ◦ [e]commutes.

/ [φ0 ] ◦ ([e] ◦ [e])To prove (a), it will suffice (by virtue of (ii)) to show that [φ0 ] and [φ1 ]are isomorphic. This is clear, since there exists a chain of isomorphisms[φ1 ] ' [φ2 ] ◦ [e] ' [φ2 ] ◦ ([e] ◦ [e]) ' ([φ2 ] ◦ [e]) ◦ [e] ' [φ1 ] ◦ [e] ' [φ0 ].
We now prove (b). Suppose that we are given a good element βn ∈H(φn+1 , φ0 ). Let βn+1 ∈ H(φn+2 , φ0 ) be the map given by the composition
α−1
n+1

h

βn

[φn+2 ]◦[e] → [φn+2 ]◦([e]◦[e]) ' ([φn+2 ]◦[e])◦[e] → [φn+1 ]◦[e] → [φ0 ]◦[e].
Using (i), (ii), and (iii), we see that βn+1 is good. It follows that the composite map
h−1

βn+1

([φn+2 ] ◦ [e]) ◦ [e] → ([φ0 ] ◦ [e]) ◦ [e] ' [φ0 ] ◦ ([e] ◦ [e]) → [φ0 ] ◦ [e]is given by
h−1

βn+1

([φn+2 ] ◦ [e]) ◦ [e] ' [φn+2 ] ◦ ([e] ◦ [e]) −−→ [φn+2 ] ◦ [e] −−−→→ [φ0 ] ◦ [e].
Consequently, the commutativity of the diagram appearing in (b) is equivalent to the assertion that βn facts as a composition
αn+1

h−1

βn+1

[φn+1 ]◦[e] −−−→ ([φn+2 ]◦[e])◦[e] ' [φn+2 ]◦([e]◦[e]) −−→ [φn+2 ]◦[e] −−−→ [φ0 ]◦[e].
This follows immediately from the definition of βn+1 .

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Chapter Five
Presentable and Accessible ∞-Categories
Many categories which arise naturally, such as the category A of abeliangroups, are large: they have a proper class of objects even when the objects are considered only up to isomorphism. However, though A itself islarge, it is in some sense determined by the much smaller category A0 of
finitely generated abelian groups: A is naturally equivalent to the categoryof Ind-objects of A0 . This remark carries more than simply philosophicalsignificance. When properly exploited, it can be used to prove statements
such as the following:
Proposition 5.0.0.1. Let F : A → Set be a contravariant functor from A
to the category of sets. Then F is representable by an object of A if and only
if it carries colimits in A to limits in Set.
Proposition 5.0.0.1 is valid not only for the category A of abelian groups
but for any presentable category: that is, any category which possesses all(small) colimits and satisfies mild set-theoretic assumptions (such categories
are referred to as locally presentable in [1]). Our goal in this chapter is todevelop an ∞-categorical generalization of the theory of presentable categories and to obtain higher-categorical analogues of Proposition 5.0.0.1 andrelated results (such as the adjoint functor theorem).
The most basic example of a presentable ∞-category is the ∞-category S
of spaces. More generally, we can define an ∞-category P(C) of presheaves(of spaces) on an arbitrary small ∞-category C. We will study the properties of P(C) in §5.1; in particular, we will see that there exists a Yoneda
embedding j : C → P(C) which is fully faithful, just as in ordinary category
theory. Moreover, we give a characterization of P(C) in terms of C: it is freely
generated by the essential image of j under (small) colimits.
For every small ∞-category C, the ∞-category P(C) is presentable. Conversely, any presentable ∞-category can be obtained as a localization of somepresheaf ∞-category P(C) (Proposition 5.5.1.1). To make sense of this statement, we need a theory of localizations of ∞-categories. We will develop sucha theory in §5.2 as part of a more general theory of adjoint functors between∞-categories.
In §5.3, we will introduce, for every small ∞-category C, an ∞-category
Ind(C) of Ind-objects of C. Roughly speaking, this is an ∞-category which
is obtained from C by freely adjoining colimits for all filtered diagrams. It
is characterized up to equivalence by the fact that Ind(C) contains a fullsubcategory equivalent to C, which generates Ind(C) under filtered colimits

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315

and consists of compact objects.
The construction of Ind-categories will be applied in §5.4 to the study ofaccessible ∞-categories. Roughly speaking, an ∞-category C is accessible if
it is generated under (sufficiently) filtered colimits by a small subcategory
C0 ⊆ C. We will prove that the class of accessible ∞-categories is stable undervarious categorical constructions. Results of this type will play an important
technical role later in this book: they generally allow us to dispense with the
set-theoretic aspects of an argument (such as cardinality estimation) and to
focus instead on the more conceptual aspects.
We will say that an ∞-category C is presentable if C is accessible andadmits (small) colimits. In §5.5, we will describe the theory of presentable ∞categories in detail. In particular, we will generalize Proposition 5.0.0.1 to the∞-categorical setting and prove an analogue of the adjoint functor theorem.
We will also study localizations of presentable ∞-categories following ideas
of Bousfield. The theory of presentable ∞-categories will play a vital role inthe study of ∞-topoi, which is the subject of the next chapter.

5.1 ∞-CATEGORIES OF PRESHEAVESThe category of sets plays a central role in classical category theory. The primary reason for this is Yoneda’s lemma, which asserts that for any category
C, the Yoneda embeddingj : C → SetC

op

C 7→ HomC (•, C)
is fully faithful. Consequently, objects in C can be thought of as a kindof “generalized sets,” and various questions about the category C can be
reduced to questions about the category of sets.
If C is an ∞-category, then the mapping sets of the above discussion should
be replaced by mapping spaces. Consequently, one should expect the Yoneda
embedding to take values in presheaves of spaces rather than in presheavesof sets. To formalize this, we introduce the following notation:
Definition 5.1.0.1. Let S be a simplicial set. We let P(S) denote the simplicial set Fun(S op , S); here S denotes the ∞-category of spaces defined in§1.2.16. We will refer to P(S) as the ∞-category of presheaves on S.
Remark 5.1.0.2. More generally, for any ∞-category C, we might refer toFun(S op , C) as the ∞-category of C-valued presheaves on S. Unless otherwise
specified, the word “presheaf” will always refer to a S-valued presheaf. Thisis somewhat nonstandard terminology: one usually understands the term“presheaf” to refer to a presheaf of sets rather than to a presheaf of spaces.
The shift in terminology is justified by the fact that the important role of Set
in ordinary category theory is taken on by S in the ∞-categorical setting.

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Our goal in this section is to establish the basic properties of P(S). We
begin in §5.1.1 by reviewing two other possible definitions of P(S): one via
the theory of right fibrations over S, another via simplicial presheaves on
the category C[S]. Using the “straightening” results of §2.2.3 and §4.2.4, we
will show that all three of these definitions are equivalent.
The presheaf ∞-categories P(S) are examples of presentable ∞-categories(see §5.5). In particular, each P(S) admits small limits and colimits. We will
give a proof of this assertion in §5.1.2 by reducing to the case where S is apoint.
The main question regarding the ∞-category P(S) is how it relates to theoriginal simplicial set S. In §5.1.3, we will construct a map j : S → P(S),which is an ∞-categorical analogue of the usual Yoneda embedding. Just
as in classical category theory, the Yoneda embedding is fully faithful. In
particular, we note that any ∞-category C can be embedded in a larger∞-category which admits limits and colimits; this observation allows us toconstruct an idempotent completion of C, which we will study in §5.1.4.
In §5.1.5, we will characterize the ∞-category P(S) in terms of the Yoneda
embedding j : S → P(S). Roughly speaking, we will show that P(S) is freely
generated by S under colimits (Theorem 5.1.5.6). In particular, if C is an ∞category which admits small colimits, then any diagram f : S → C extends
uniquely (up to homotopy) to a functor F : P(S) → C. In §5.1.6, we will givea criterion for determining whether or not F is an equivalence.
5.1.1 Other Models for P(S)Let S be a simplicial set. We have defined the ∞-category P(S) of presheaves
on S to be the mapping space Fun(S op , S). However, there are several equivalent models which would serve equally well; we discuss two of them in thissection.
Let P0∆ (S) denote the full subcategory of (Set∆ )/S spanned by the rightfibrations X → S. We define P0 (S) to be the simplicial nerve N(P0∆ (S)).
Because P0∆ (S) is a fibrant simplicial category, P0 (S) is an ∞-category. We
will see in a moment that P0 (S) is (naturally) equivalent to P(S). In orderto do this, we need to introduce a third model.
Let φ : C[S]op → C be an equivalence of simplicial categories. Let SetC∆
denote the category of simplicial functors C → Set∆ (which we may view as
simplicial presheaves on Cop ). We regard SetC
∆ as being endowed with the
projective model structure defined in §A.3.3. With respect to this structure,
00
C ◦
SetC
∆ is a simplicial model category; we let P∆ (φ) = (Set∆ ) denote the full
simplicial subcategory consisting of fibrant-cofibrant objects, and we define
P00 (φ) to be the simplicial nerve N(P00∆ (φ)).
We are now ready to describe the relationship between these different
models:
Proposition 5.1.1.1. Let S be a simplicial set and let φ : C[S]op → C be an
equivalence of simplicial categories. Then there are (canonical) equivalenceshighertopoi

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of ∞-categoriesfg

P(S) ← P00 (φ) → P0 (S).
Proof. The map f was constructed in Proposition 4.2.4.4; it therefore suffices
to give a construction of g. We will regard the category (Set∆ )/S of simplicial
sets over S as endowed with the contravariant model structure defined in§2.1.4. This model structure is simplicial (Proposition 2.1.4.8) and the fibrantobjects are precisely the right fibrations over S (Corollary 2.2.3.12). Thus, we
may identify P0∆ (S) with the simplicial category (Set∆ )◦/S of fibrant-cofibrantobjects of (Set∆ )/S .
According to Theorem 2.2.1.2, the straightening and unstraightening functors (Stφ , Unφ ) determine a Quillen equivalence between the model categories (Set∆ )C and (Set∆ )/S . Moreover, for any X ∈ (Set∆ )/S and any
simplicial set K, there is a natural chain of equivalences
Stφ (X × K) → (Stφ X) ⊗ |K|Q• → (Stφ X) ⊗ K.
(The fact that the first map is an equivalence follows easily from Proposition
3.2.1.13.) It follows from Proposition A.3.1.10 that Unφ is endowed with thestructure of a simplicial functor and induces an equivalence of simplicial
categories
◦
◦
(SetC
∆ ) → (Set∆ )/S .

We obtain the desired equivalence g by passing to the simplicial nerve.
5.1.2 Colimits in ∞-Categories of Functors
Let S be an arbitrary simplicial set. Our goal in this section is to prove thatthe ∞-category P(S) of presheaves on S admits small limits and colimits.
There are (at least) three approaches to proving this:(1) According to Proposition 5.1.1.1, we may identify P(S) with the ∞C[S]op
category underlying the simplicial model category Set∆
. We canthen deduce the existence of limits and colimits in P(S) by invoking
Corollary 4.2.4.8.
(2) Since the ∞-category S classifies left fibrations, the ∞-category P(S)classifies left fibrations over S op : in other words, homotopy classes ofmaps K → P(S) can be identified with equivalence classes of left fibrations X → K × S op . It is possible to generalize Proposition 3.3.4.5
and Corollary 3.3.3.3 to describe limits and colimits in P(S) entirelyin the language of left fibrations. The existence problem can then be
solved by exhibiting explicit constructions of left fibrations.
(3) Applying either (1) or (2) in the case where S is a point, we can deduce that the ∞-category S ' P(∗) admits limits and colimits. We canthen attempt to deduce the same result for P(S) = Fun(S op , S) using a general result about (co)limits in functor categories (Proposition
5.1.2.2).

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Although approach (1) is probably the quickest, we will adopt approach(3) because it gives additional information: our proof will show that limits
and colimits in P(S) can be computed pointwise. The same proof will also
apply to ∞-categories of C-valued presheaves in the case where C is notnecessarily the ∞-category S of spaces.
Lemma 5.1.2.1. Let q : Y → S be a Cartesian fibration of simplicial
sets and let C = MapS (S, Y ) denote the ∞-category of sections of q. Let
p : S → Y be an object of C having the property that p(s) is an initial objectof the fiber Ys for each vertex s of S. Then p is an initial object of C.
Proof. By Proposition 4.2.2.4, the map Y pS / → S is a Cartesian fibration.
By hypothesis, for each vertex s of S, the map Y pS / ×S {s} → Ys is atrivial fibration. It follows that the projection Y pS / → Y is an equivalenceof Cartesian fibrations over S and therefore a categorical equivalence; taking
sections over S, we obtain another categorical equivalence
MapS (S, Y pS / ) → MapS (Y, S).
But this map is just the left fibration j : Cp/ → C; it follows that j is a
categorical equivalence. Applying Propostion 3.3.1.5 to the diagram
j

Cp/

/C

j
idC

C,
we deduce that j induces categorical equivalences Cp/ ×C {t} → {t} for eachvertex t of Q. Thus the fibers of j are contractible Kan complexes, so thatj is a trivial fibration (by Lemma 2.1.3.4) and p is an initial object of C, as
desired.
Proposition 5.1.2.2. Let K be a simplicial set, q : X → S a Cartesianfibration, and p : K → MapS (S, X) a diagram. For each vertex s of S, we
let ps : K → Xs be the induced map. Suppose, furthermore, that each ps has
a colimit in the ∞-category Xs . Then
(1) There exists a map p : K  ∆0 → MapS (S, X) which extends p andinduces a colimit diagram p : K  ∆0 → Xs for each vertex s ∈ S.
(2) An arbitrary extension p : K  ∆0 → MapS (S, X) of p is a colimit for
p if and only if each ps : K  ∆0 → Xs is a colimit for ps .
Proof. Choose a factorization K → K 0 → MapS (S, X) of p, where K → K 0
is inner anodyne (and therefore a categorical equivalence) and K 0 → CS is
an inner fibration (so that K 0 is an ∞-category). The map K → K 0 is a
categorical equivalence and therefore cofinal. We are free to replace K by K 0
and may thereby assume that K is an ∞-category.

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We apply Proposition 4.2.2.7 to the Cartesian fibration X → S and the
diagram pS : K × S → X determined by the map p. We deduce that there
exists a map
pS : (K × S) S S = (K  ∆0 ) × S → X
having the property that its restriction to the fiber over each s ∈ S is acolimit of ps ; this proves (1).
The “if” direction of (2) follows immediately from Lemma 5.1.2.1. The“only if” direction follows from (1) and the fact that colimits, when they
exist, are unique up to equivalence.
Corollary 5.1.2.3. Let K and S be simplicial sets and let C be an ∞category which admits K-indexed colimits. Then
(1) The ∞-category Fun(S, C) admits K-indexed colimits.
(2) A map K . → Fun(S, C) is a colimit diagram if and only if, for eachvertex s ∈ S, the induced map K . → C is a colimit diagram.
Proof. Apply Proposition 5.1.2.2 to the projection C ×S → S.
Corollary 5.1.2.4. Let S be a simplicial set. The ∞-category P(S) of presheaves on S admits all small limits and colimits.
5.1.3 Yoneda’s Lemma
In this section, we will construct the ∞-categorical analogue of the Yoneda
embedding and prove that it is fully faithful. We begin with a somewhat
naive approach based on the formalism of simplicial categories. We notethat an analogue of Yoneda’s Lemma is valid in enriched category theory(with the usual proof). Namely, suppose that C is a category enriched overanother category E. Then there is an enriched Yoneda embeddingi : C → EC

op

X 7→ MapC (•, X).
Consequently, for any simplicial category C, one obtains a fully faithfulembedding i of C into the simplicial category MapCat∆ (Cop , Set∆ ) of simplicial functors from Cop into Set∆ . In fact, i is fully faithful in the strong sensethat it induces isomorphisms of simplicial sets
MapC (X, Y ) → MapSetC∆op (i(X), i(Y ))rather than merely weak homotopy equivalences. Unfortunately, this assertion does not necessarily
have any ∞-categorical content because the simop
plicial category SetC
does
not generally represent the correct ∞-category∆of functors from Cop to Set∆ .

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Let us describe an analogous construction in the setting of ∞-categories.
Let K be a simplicial set and set C = C[K]. Then C is a simplicial category,so(X, Y ) 7→ Sing | HomC (X, Y )|determines a simplicial functor from Cop × C to the category Kan. The functor C does not commute with products, but there exists a natural map
C[K op × K] → Cop × C. Composing with this map, we obtain a simplicial
functor C[K op × K] → Kan . Passing to the adjoint, we get a map of simplicial sets K op × K → S, which we can identify withj : K → Fun(K op , S) = P(K).
We shall refer to j (or more generally, to any functor equivalent to j) as the
Yoneda embedding.
Proposition 5.1.3.1 (∞-Categorical Yoneda Lemma). Let K be a simplicial set. Then the Yoneda embedding j : K → P(K) is fully faithful.
Proof. Let C0 = Sing | C[K op ]| be the “fibrant replacement” for C = C[K op ].
0
We endow SetC
∆ with the projective model structure described in §A.3.3.
We note that the Yoneda embedding factors as a composition
j0

0

j 00

◦
op
K → N((SetC∆ ) ) → Fun(K , S),

where j 00 is the map of Proposition 4.2.4.4 and consequently a categoricalequivalence. It therefore suffices to prove that j 0 is fully faithful. For this,we need only show that the adjoint map
0

J : C[K] → SetC∆

is a fully faithful functor between simplicial categories. We now observe that
J is the composition of an equivalence C[K] → (C0 )op with the (simplicially0enriched) Yoneda embedding (C0 )op → SetC
∆ , which is fully faithful by virtueof the classical (enriched) version of Yoneda’s Lemma.
We conclude by establishing another pleasant property of the Yoneda embedding:
Proposition 5.1.3.2. Let C be a small ∞-category and j : C → P(C) the
Yoneda embedding. Then j preserves all small limits which exist in C.
Proof. Let p : K → C be a small diagram having a limit in C. We wish toshow that j carries any limit for p to a limit of j ◦ p. Choose a category I
and a cofinal map N(Iop ) → K op (the existence of which is guaranteed by
Proposition 4.2.3.14). Replacing K by N(I), we may suppose that K is thenerve of a category. Let p : N(I)/ → C be a limit for p.
We recall the definition of the Yoneda embedding. It involves the choice ofan equivalence C[C] → D, where D is a fibrant simplicial category. For definiteness, we took D to be Sing | C[C]|. However, we could just as well choosehighertopoi

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some other fibrant simplicial category D0 equivalent to C[C] and obtain a“modified” Yoneda embedding j 0 : C → P(C); it is easy to see that j 0 and j
are equivalent functors, so it suffices to show that j 0 preserves the limit of p.
Using Corollary 4.2.4.7, we may suppose that p is obtained from a functor
between simplicial categories q : {x} ? I → D by passing to the nerve. According to Theorem 4.2.4.1, q is a homotopy limit of q = q| I. Consequently,
for each object Z ∈ D, the induced functorq Z : I 7→ HomD (Z, q(I))
is a homotopy limit of qZ = q Z | I. Taking Z to be the image of an object C
of C, we deduce thatj0

N(I)/ → C → P(C) → S
is a limit for its restriction to N(I), where the map on the right is given byevaluation at C. Proposition 5.1.2.2 now implies that j 0 ◦p is a limit for j 0 ◦p,
as desired.
5.1.4 Idempotent Completions
Recall that an ∞-category C is said to be idempotent complete if every functor
Idem → C admits a colimit in C (see §4.4.5). If an ∞-category C is not
idempotent complete, then we can attempt to correct the situation by passing
to a larger ∞-category.
Definition 5.1.4.1. Let f : C → D be a functor between ∞-categories.
We will say that f exhibits D as an idempotent completion of C if D is
idempotent complete, f is fully faithful, and every object of D is a retractof f (C) for some object C ∈ C.
Our goal in this section is to show that ∞-category C has an idempotent
completion D which is unique up to equivalence. The uniqueness is a consequence of Proposition 5.1.4.9 below. The existence question is much easier
to address.
Proposition 5.1.4.2. Let C be an ∞-category. Then C admits an idempotent completion.
Proof. Enlarging the universe if necessary, we may suppose that C is small.
Let C0 denote the full subcategory of P(C) spanned by those objects which
are retracts of objects which belong to the image of the Yoneda embeddingj : C → P(C). Then C0 is stable under retracts in P(C). Since P(C) admits
all small colimits, it is idempotent complete (see Corollary 4.4.5.15). It follows that C0 is idempotent complete. Proposition 5.1.3.1 implies that the
Yoneda embedding j : C → C0 is fully faithful and therefore exhibits C0 as
an idempotent completion of C.
We now address the question of uniqueness for idempotent completions.
First, we need a few preliminary results.

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Lemma 5.1.4.3. Let C be an ∞-category which is idempotent complete andlet p : K → C be a diagram. Then C/p and Cp/ are also idempotent complete.
Proof. By symmetry, it will suffice to prove that C/p is idempotent complete.
Let q : C/p → C be the associated right fibration and let F : Idem → C/p
be an idempotent. We will show that F has a limit. Since C is idempotent
complete, q ◦ F has a limit q ◦ F : Idem/ → C. Consider the lifting problem
Idem

/ C/p

F

;
F


q◦F
Idem/

q


/ C.

The right vertical map is a right fibration, and the left vertical map is rightanodyne (Lemma 4.2.3.6), so that there exists a dotted arrow F as indicated.
Using Proposition 4.4.2.9, we deduce that F is a limit of F .
Lemma 5.1.4.4. Let f : C → D be a functor between ∞-categories which
exhibits D as an idempotent completion of C and let p : K → D be a diagram.
Then the induced map f/p : C ×D D/p → D/p exhibits D/p as an idempotent
completion of C ×D D/p .
Proof. Lemma 5.1.4.3 asserts that D/p is idempotent complete. We mustshow that every object D ∈ D/p is a retract of f/p (C) for some C ∈C ×D D/p . Let q : D/p → D be the projection and set D = q(D). Since
f exhibits D as an idempotent completion of C, there is a diagram
f (C)
<
D0

g

!
/D

in D, where g is an equivalence. Since q is a right fibration, we can lift thisto a diagram
f (C)
=

D

0

g

/D

in D/q . Since g is q-Cartesian and g is an equivalence, g is an equivalence.
It follows that D is a retract of f (C). By construction, f (C) = f/p (C) for
an appropriately chosen object C ∈ C ×D D/p .
Lemma 5.1.4.5. Let f : C → D be a functor between ∞-categories which
exhibits D as an idempotent completion of C. Suppose that D has an initial
object ∅. Then C is weakly contractible as a simplicial set.

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIESProof. Without loss of generality, we may suppose that C is a full subcategory of D and that f is the inclusion. Since f exhibits D as an idempotent
completion of C, the initial object ∅ of D admits a map f : C → ∅, whereC ∈ C. The ∞-category CC/ has an initial object and is therefore weakly
contractible. Since composition
Cf / → CC/ → C
is both a weak homotopy equivalence (in fact, a trivial fibration) and weakly
nullhomotopic, we conclude that C is weakly contractible.
Lemma 5.1.4.6. Let f : C → D be a functor between ∞-categories which
exhibits D as an idempotent completion of C. Then f is cofinal.
Proof. According to Theorem 4.1.3.1, it suffices to prove that for every object
D ∈ D, the simplicial set C ×D DD/ is weakly contractible. Lemma 5.1.4.4
asserts that fD/ is also an idempotent completion, and Lemma 5.1.4.5 completes the proof.
Lemma 5.1.4.7. Let F : C → D be a functor between ∞-categories andlet C0 ⊆ C be a full subcategory such that the inclusion exhibits C as an
idempotent completion of C0 . Then F is a left Kan extension of F | C0 .
Proof. We must show that for every object C ∈ C, the composite map
GF(C0/C ). → (C/C ). → C → D
is a colimit diagram in D. Lemma 5.1.4.4 guarantees that C0/C ⊆ C/C is
an idempotent completion and therefore cofinal by Lemma 5.1.4.6. Consequently, it suffices to prove that F ◦ G is a colimit diagram, which is obvious.
Lemma 5.1.4.8. Let C and D be ∞-categories which are idempotent complete and let C0 ⊆ C be a full subcategory such that the inclusion exhibits C
as an idempotent completion of C0 . Then any functor F0 : C0 → D has anextension F : C → D.
Proof. We will suppose that the ∞-categories C and D are small. Let P(D) be
the ∞-category of presheaves on D (see §5.1), j : D → P(D) the Yoneda embedding, and D0 the essential image of j. According to Proposition A.2.3.1,
it will suffice to prove that j ◦ F0 can be extended to a functor F 0 : C → D0 .
Since P(D) admits small colimits, we can choose F 0 : C → P(D) to be a left
Kan extension of j ◦ F0 . Every object of C is a retract of an object of C0 ,so that every object in the essential image of F 0 is a retract of the Yoneda
image of an object of D. Since D is idempotent complete, it follows that theF 0 factors through D0 .
Proposition 5.1.4.9. Let f : C → D be a functor which exhibits D as theidempotent completion of C and let E be an ∞-category which is idempotent
complete. Then composition with f induces an equivalence of ∞-categoriesf ∗ : Fun(D, E) → Fun(C, E).

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Proof. Without loss of generality, we may suppose that f is the inclusion ofa full subcategory. In this case, we combine Lemma 5.1.4.7, Lemma 5.1.4.8,
and Proposition 4.3.2.15 to deduce that f ∗ is a trivial fibration.
Remark 5.1.4.10. Let C be a small ∞-category and let f : C → C0 be an
idempotent completion of C. The proof of Proposition 5.1.4.2 shows that C0
is equivalent to a full subcategory of P(C) and therefore locally small (see§5.4.1). Moreover, every object of hC0 is the image of some retraction map in
hC; it follows that the set of equivalence classes of objects in C0 is boundedin size. It follows that C0 is essentially small.
5.1.5 The Universal Property of P(S)Let S be a (small) simplicial set. We have defined P(S) to be the ∞-categoryof maps from S op into the ∞-category S of spaces. Informally, we may view
P(S) as the limit of a diagram in the ∞-bicategory of (large) ∞-categories:
namely, the constant diagram carrying S op to S. In more concrete terms,
our definition of P(S) leads immediately to a characterization of P(S) by auniversal mapping property: for every ∞-category C, there is an equivalenceof ∞-categories (in fact, an isomorphism of simplicial sets)
Fun(C, P(S)) ' Fun(C ×S op , S).
The goal of this section is to give a dual characterization of P(S): it may also
be viewed as a colimit of copies of S indexed by S. However, this colimit needs
to be understood in an appropriate ∞-bicategory of ∞-categories where themorphisms are given by colimit-preserving functors. In other words, we willshow that P(S) is in some sense “freely generated” by S under small colimits(Theorem 5.1.5.6). First, we need to introduce a bit of notation.
Notation 5.1.5.1. Let C be an ∞-category and S a simplicial set. We will
let FunL (P(S), C) denote the full subcategory of Fun(P(S), C) spanned bythose functors P(S) → C which preserve small colimits.
The motivation for this notation is as follows: in §5.2.6, we will use thenotation FunL (D, C) to denote the full subcategory of Fun(D, C) spannedby those functors which are left adjoints. In §5.5.2, we will see that when
D = P(S) (or more generally, when D is presentable), then a functor D → C
is a left adjoint if and only if it preserves small colimits (see Corollary 5.5.2.9
and Remark 5.5.2.10).
We wish to prove that if C is an ∞-category which admits small colimits,then any map S → C extends in an essentially unique fashion to a colimitpreserving functor P(S) → C. To prove this, we need a second characterization of the colimit-preserving functors f : P(S) → C: they are precisely
those functors which are left Kan extensions of their restriction to the essential image of the Yoneda embedding.
Lemma 5.1.5.2. Let S be a small simplicial set, let s be a vertex of S, lete : P(S) → S be the map given by evaluation at s, and let f : C → P(S)highertopoi

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be the associated left fibration (see §3.3.2). Then f is corepresentable by theobject j(s) ∈ P(S), where j : S → P(S) denotes the Yoneda embedding.
Proof. Without loss of generality, we may suppose that S is an ∞-category.
We make use of the equivalent model P0 (S) of §5.1.1. Observe that thefunctor f : P(S) → S is equivalent to f 0 : P0 (S) → S, where f 0 is the nerve of
the simplicial functor P0∆ (S) → Kan which associates to each left fibration
Y → S the fiber Ys = Y ×S {s}. Furthermore, under the equivalence of P(S)with P0 (S), the object j(s) corresponds to a left fibration X(s) → S which is
corepresented by s. Then X(s) contains an initial object x lying over s. The
choice of x determines a point η ∈ π0 f 0 (X(s)). According to Proposition
4.4.4.5, to show that X(s) corepresents f 0 , it suffices to show that for every
left fibration X → S, the map
MapS (X(s), Y ) → Ys ,given by evaluation at x, is a homotopy equivalence of Kan complexes. We
may rewrite the space on the right hand side as MapS ({x}, Y ). According to
Proposition 2.1.4.8, the covariant model structure on (Set∆ )/S is compatible
with the simplicial structure. It therefore suffices to prove that the inclusioni : {x} ⊆ X(s) is a covariant equivalence. But this is clear since i is theinclusion of an initial object and therefore left anodyne.
Lemma 5.1.5.3. Let S be a small simplicial set and let j : S → P(S) denote
the Yoneda embedding. Then idP(S) is a left Kan extension of j along itself.
Proof. Let C ⊆ P(S) denote the essential image of j. According to Proposition 5.1.3.1, j induces an equivalence S → C. It therefore suffices to provethat idP(S) is a left Kan extension of its restriction to C. Let X be an objectof P(S); we must show that the natural map
φ : C./X ⊆ P(S)./X → P(S)is a colimit diagram.
According to Proposition 5.1.2.2, it will suffice to prove that, for eachvertex s of S, the map
φs : C./X → S
given by composing φ with the evaluation map is a colimit diagram in S.
Let D → C./X be the pullback of the universal left fibration along φs andlet D0 ⊆ D be the preimage in D of C/X ⊆ C./X . According to Proposition
3.3.4.5, it will suffice to prove that the inclusion D0 ⊆ D is a weak homotopy
equivalence of simplicial sets.
Let C = j(s). Let E = C./X ×P(S) P(S)C/ , let E0 = C/X ×C CC/ ⊆ E, andlet E1 = C/X ×C {idC } ⊆ E0 . Lemma 5.1.5.2 implies that the left fibrations
D → C./X ← E
are equivalent. It therefore suffices to show that the inclusion E0 ⊆ E is aweak homotopy equivalence. To prove this, we observe that both E and E0

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contain E1 as a deformation retract (that is, there is a retraction r : E → E1
and a homotopy E ×∆1 → E from r to idE , so that the inclusion E1 ⊆ E isa homotopy equivalence; the situation for E0 is similar).
Lemma 5.1.5.4. Let
/B

f

A




g

S
be a diagram of simplicial sets. The following conditions are equivalent:(1) The map f is a covariant equivalence in (Set∆ )/S .
(2) For every diagram p : S → C taking values in an ∞-category C andevery limit p ◦ g : B / → C of p ◦ g, the composition p ◦ g ◦ f / : A/ → C
is a limit diagram.
(3) For every diagram p : S → S taking values in the ∞-category S ofspaces and every limit p ◦ g : B / → S of p◦g, the composition p ◦ g◦f / :A/ → S is a limit diagram.
Proof. The equivalence of (1) and (3) follows from Corollary 3.3.3.4 (andthe definition of a contravariant equivalence). The implication (2) ⇒ (3) isobvious. We show that (3) ⇒ (2). Let p : S → C and p ◦ g be as in (2).
Passing to a larger universe if necessary, we may suppose that C is small.
For each object C ∈ C, let jC : C → S denote the composition of the Yoneda
embedding j : C → P(C) with the map P(C) → S given by evaluation at C.
Combining Proposition 5.1.3.2 with Proposition 5.1.2.2, we deduce that eachjC ◦p ◦ g is a limit diagram. Applying (3), we conclude that each jC ◦p ◦ g◦f /is a limit diagram. We now apply Propositions 5.1.3.2 and 5.1.2.2 to conclude
that p ◦ g ◦ f / is a limit diagram, as desired.
Lemma 5.1.5.5. Let S be a small simplicial set, let j : S → P(S) be theYoneda embedding, and let C denote the full subcategory of P(S) spanned by
the objects j(s), where s is a vertex of S. Let D be an arbitrary ∞-category.
(1) Let f : P(S) → D be a functor. Then f is a left Kan extension of f | C
if and only if f preserves small colimits.
(2) Suppose that D admits small colimits and let f0 : C → D be an arbitrary
functor. There exists an extension f : P(S) → D which is a left Kan
extension of f0 = f | C.
Proof. Assertion (2) follows from Lemma 4.3.2.13 since the ∞-category C/X
is small for each object X ∈ P(S). We will prove (1). Suppose first thatf preserves small colimits. We must show that for each X ∈ P(S), thecomposition
δfC./X → P(S) → D

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is a colimit diagram. Lemma 5.1.5.3 implies that δ is a colimit diagram; if fpreserves small colimits, then f ◦ δ is also a colimit diagram.
Now suppose that f is a left Kan extension of f0 = f | C. We wish toprove that f preserves small colimits. Let K be a small simplicial set andlet p : K . → P(S) be a colimit diagram. We must show that f ◦ p is also acolimit diagram.
Let
E = C ×Fun({0},P(S)) Fun(∆1 , P(S)) ×Fun({1},P(S)) K .
and let E = E ×K . K ⊆ E. We have a commutative diagram
E

/E


K


/ K .,

where the vertical arrows are coCartesian fibrations (Corollary 2.4.7.12). Letη : E K . K . → P(S) be the natural map and set η = η| E K K. Proposition
4.3.3.10 implies that f ◦ η exhibits f ◦ p as a left Kan extension of f ◦ (η| E)
along q| E. Similarly, f ◦ η exhibits f ◦ p as a left Kan extension of f ◦ (η|E). It
will therefore suffice to prove that every colimit of f ◦(η|E) is also a colimit off ◦ (η| E). According to Lemma 5.1.5.4, it suffices to show that the inclusion
E ⊆ E is a contravariant equivalence in (Set∆ )/ C .
Since the map E → K . × C is a bivariant fibration, we can apply Propoop
sition 4.1.2.16 to deduce that the map E → Cop is smooth. Similarly,op
op
E → C is smooth. According to Proposition 4.1.2.18, the inclusion E ⊆ E
is a contravariant equivalence if and only if, for every object C ∈ C, theinclusion of fibers EC ⊆ EC is a weak homotopy equivalence. Lemma 5.1.5.2
implies that EC → K . is equivalent to the left fibration given by the pullbackof the universal left fibration along the mapps

K . → P(S) → S .
We now conclude by applying Proposition 3.3.4.5, noting that p is a colimit
diagram by assumption and that s preserves colimits by Proposition 5.1.2.2.
Theorem 5.1.5.6. Let S be a small simplicial set and let C be an ∞-category
which admits small colimits. Composition with the Yoneda embedding j :S → P(S) induces an equivalence of ∞-categories
FunL (P(S), C) → Fun(S, C).
Proof. Combine Corollary 4.3.2.16 with Lemma 5.1.5.5.
Definition 5.1.5.7. Let C be an ∞-category. A full subcategory C0 ⊆ C
is stable under colimits if, for any small diagram p : K → C0 which has acolimit p : K . → C in C, the map p factors through C0 .

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Let C be an ∞-category which admits all small colimits. Let A be a collection of objects of C. We will say that A generates C under colimits if thefollowing condition is satisfied: for any full subcategory C0 ⊆ C containing
every element of A, if C0 is stable under colimits, then C = C0 .
We say that a map f : S → C generates C under colimits if the image
f (S0 ) generates C under colimits.
Corollary 5.1.5.8. Let S be a small simplicial set. Then the Yoneda embedding j : S → P(S) generates P(S) under small colimits.
Proof. Let C be the smallest full subcategory of P(S) which contains theessential image of j and is stable under small colimits. Applying Theorem
5.1.5.6, we deduce that the diagram j : S → C is equivalent to F ◦ j for somecolimit-preserving functor F : P(S) → C. We may regard F as a colimitpreserving functor from P(S) to itself. Applying Theorem 5.1.5.6 again, we
deduce that F is equivalent to the identity functor from P(S) to itself. It
follows that every object of P(S) is equivalent to an object which lies in C,
so that C = P(S), as desired.
5.1.6 Complete Compactness
Let S be a small simplicial set and f : S → C a diagram in an ∞-category
C. Our goal in this section is to analyze the following question: when is thediagram f : S → C equivalent to the Yoneda embedding j : S → P(S)? An
obvious necessary condition is that C admit small colimits (Corollary 5.1.2.4).
Conversely, if C admits small colimits, then Theorem 5.1.5.6 implies that f
is equivalent to F ◦ j, where F : P(S) → C is a colimit-preserving functor.
We are now reduced to the question of deciding whether or not the functor
F is an equivalence. There are two obvious necessary conditions for this tobe so: f must be fully faithful (Proposition 5.1.3.1), and f must generate
C under colimits (Corollary 5.1.5.8). We will show that the converse holds
provided that the essential image of f consists of completely compact objectsof C (see Definition 5.1.6.2 below).
We begin by considering an arbitrary simplicial set S and a vertex s of S.
Composing the Yoneda embedding j : S → P(S) with the evaluation map
P(S) = Fun(S op , S) → Fun({s}, S) ' S,
we obtain a map js : S → S. We will refer to js as the functor corepresented
by s.
Remark 5.1.6.1. The above definition makes sense even when the simplicial
set S is not small. However, in this case we need to replace S (the simplicial
nerve of the category of small Kan complexes) by the (very large) ∞-categoryb
S, where b
S is the simplicial nerve of the category of all Kan complexes (notnecessarily small).
Definition 5.1.6.2. Let C be an ∞-category which admits small colimits.
We will say that an object C ∈ C is completely compact if the functor jC :C→b
S corepresented by C preserves small colimits.

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIESThe requirement that an object C of an ∞-category C be completely
compact is very restrictive (see Example 5.1.6.9 below). We introduce thisnotion not because it is a generally useful one but because it is relevant forthe purpose of characterizing ∞-categories of presheaves.
Our first goal is to establish that the class of completely compact objectsof C is stable under retracts.
Lemma 5.1.6.3. Let C be an ∞-category, K a simplicial set, and p, q :K . → C a pair of diagrams. Suppose that q is a colimit diagram and that p
is a retract of q in the ∞-category Fun(K . , C). Then p is a colimit diagram.
Proof. Choose a map σ : ∆2 × K . → C such that σ|{1} × K . = q andσ|∆{0,2} × K . = p ◦ πK . . We have a commutative diagram of simplicial sets:/ Cσ|∆2 ×K/

Cσ/
Cσ|∆{1,2} ×K . /f/ Cσ|∆{1,2} ×K/
Cσ|{2}×K . /f0


/ Cσ|{2}×K/ .

We first claim that both vertical compositions are categorical equivalences.
We give the argument for the right vertical composition; the other case issimilar. We have a factorization
g0

g 00

Cσ|∆2 ×K/ → Cσ|∆{0,2} ×K/ → Cσ|{2}×K/ ,where g 0 is a trivial fibration and g 00 admits a section s. The map s is also
a section of the trivial fibration Cσ|∆{0,2} ×K/ → Cσ|{0}×K/ . Consequently, s
and g 00 are categorical equivalences. It follows that the map f 0 is a retract off in the homotopy category of Set∆ (taken with respect to the Joyal model
structure).
The map f sits in a commutative diagram
Cσ|∆{1,2} ×K . /

Cq/

f

/ Cσ|∆{1,2}/ ×K
/ Cq/ ,

where the vertical maps and the lower horizontal map are trivial fibrations.
It follows that f is a categorical equivalence. Since f 0 is a retract of f , f 0
is also a categorical equivalence. Since f 0 is a left fibration, we deduce thatf 0 is a trivial fibration (Corollary 2.4.4.6), so that p is a colimit diagram as
desired.
Lemma 5.1.6.4. Let C be an ∞-category which admits small colimits. Let
C and D be objects of C. Suppose that C is completely compact and that D

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is a retract of C (that is, there exist maps f : D → C and r : C → D withr ◦ f ' idD . Then D is completely compact. In particular, if C and D areequivalent, then D is completely compact.
Proof. Let j : Cop → SC denote the Yoneda embedding (for Cop ). Since D
is a retract of C, j(D) is a retract of j(C). Let p : K . → C be a diagram.
Then j(D) ◦ p : K . → S is a retract of j(C) ◦ p : K . → S in the ∞-category
Fun(K . , S). If p is a colimit diagram, then j(C) ◦ p is a colimit diagram(since C is completely compact). Lemma 5.1.6.3 now implies that j(D) ◦ p
is a colimit diagram as well.
In order to study the condition of complete compactness in more detail,it is convenient to introduce a slightly more general notion.
Definition 5.1.6.5. Let C be an ∞-category which admits small colimits
e → C be a left fibration. We will say that φ is completely compact
and let φ : C
if it is classified by a functor C → b
S that preserves small colimits.
Lemma 5.1.6.6. Let C be an ∞-category which admits small colimits, let
f : X 0 → X be a map of Kan complexes, and let
/F
F0


f ×idC/ X ×C
X0 × Cbe a diagram of left fibrations over C which is a homotopy pullback square(with respect to the covariant model structure on (Set∆ )/ C ). If F → C iscompletely compact, then F0 → C is completely compact.
Proof. Replacing the diagram by an equivalent one if necessary, we may
suppose that it is Cartesian and that f is a Kan fibration. Let p : K . → C
be a colimit diagram and let F : C → b
S be a functor which classifies the left
fibration F0 . We wish to show that F ◦ p is a colimit diagram in b
S.
We have a pullback diagram/ K ×C F
K ×C F 0
ψ0
K . ×C F 0ψ/ K . ×C F

of simplicial sets which is homotopy Cartesian (with respect to the usualmodel structure on Set∆ ) since the horizontal maps are pullbacks of f . Since
F is completely compact, Proposition 3.3.4.5 implies that the inclusion ψ isa weak homotopy equivalence. It follows that ψ 0 is also a weak homotopy
equivalence. Applying Proposition 3.3.4.5 again, we deduce that F ◦ p is acolimit diagram, as desired.
Lemma 5.1.6.7. Let C be a presentable ∞-category, let p : K → C be asmall diagram, and let X ∈ C/p be an object whose image in C is completely
compact. Then X is completely compact.

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIESProof. Let p : K / → C be a limit of p carrying the cone point to an object
Z ∈ C. Then we have trivial fibrations
C/Z ← C/p → C/p .
Consequently, we may replace the diagram p : K → C with the inclusion{Z} → C.
We may identify the object X ∈ C/Z with a morphism f : Y → Z in C.
We have a commutative diagram of simplicial sets
(C/Z )f /θθ0/ (C/Y )f // (C/Y )f /

ψ

$


C/Z

θ00



ψ0

/ C/Z ,

where θ is an isomorphism, the maps θ0 and θ00 are categorical equivalences(see §4.2.1), and the vertical maps are left fibrations. We wish to prove that
ψ is a completely compact left fibration. It will therefore suffice to prove thatψ 0 is completely compact. We have a (homotopy) pullback diagram
CY /

/f

/ C∆
Y / × {1} {Z}


C/Z


/ (CY / ×C {Z}) × C/Z

1

C

of left fibrations over C/Z . We observe that the left fibrations in the lower
part of the diagram are constant. According to Lemma 5.1.6.6, to provethat ψ 0 is completely compact, it will suffice to prove that the left fibration
1

ψ 00

/Z
C∆
is completely compact. We observe that ψ 00 admits a
Y / ×C{1} {Z} → C
factorization
1

φ

φ0

/Z
C∆
→ C/Z ,
Y / ×C{1} {Z} → CY / ×C{0} C

where φ is a trivial fibration and φ0 is a pullback of the left fibration φ00 :CY / → C. Since Y is completely compact, φ00 is completely compact. The
projection C/Z → C is equivalent to C/Z → C and therefore commutes withcolimits by Proposition 1.2.13.8. It follows that φ0 is completely compact,which completes the proof.
Proposition 5.1.6.8. Let S be a small simplicial set and let j : S → P(S)denote the Yoneda embedding. Let C be an object of P(S). The following
conditions are equivalent:(1) The object C ∈ P(S) is completely compact.
(2) There exists a vertex s of S such that C is a retract of j(s).

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Proof. Suppose first that (1) is satisfied. Let S/C = S ×P(S) P(S)/C . According to Lemma 5.1.5.3, the natural map
j0

.
S/C
→ P(S)./C → P(S)is a colimit diagram. Let f : P(S) → S be the functor corepresented by C.
Since C is completely compact. f (C) can be identified with a colimit of the
diagram f |S/C . The space f (C) is homotopy equivalent to MapP(S) (C, C)
and therefore contains a point corresponding to idC . It follows that idClies in the image of MapP(S) (C, j 0 (es)) → MapP(S) (C, C) for some vertex seof S/C . The vertex se classifies a vertex s ∈ S equipped with a morphism
α : j(s) → C. It follows that there is a commutative triangle
=

j(s)

C

α
idC

!
/C

in the ∞-category P(S), so that C is a retract of j(s).
Now suppose that (2) is satisfied. According to Lemma 5.1.6.4, it suffices
to prove that j(s) is completely compact. Using Lemma 5.1.5.2, we may
identify the functor P(S) → S corepresented by j(s) with the functor givenby evaluation at s. Proposition 5.1.2.2 implies that this functor preserves all
limits and colimits that exist in P(S).
Example 5.1.6.9. Let C be the ∞-category S of spaces. Then an object
C ∈ S is completely compact if and only if it is equivalent to ∗, the final
object of S.
We now use the theory of completely compact objects to give a characterization of presheaf ∞-categories.
Proposition 5.1.6.10. Let S be a small simplicial set and C an ∞-category
which admits small colimits. Let F : P(S) → C be a functor which preserves
small colimits and let f = F ◦j be its composition with the Yoneda embeddingj : S → P(S). Suppose further that
(1) The functor f is fully faithful.
(2) For every vertex s of S, the object f (s) ∈ C is completely compact.
Then F is fully faithful.
Proof. Let C and D be objects of P(S). We wish to prove that the naturalmap
ηC,D : MapP(S) (C, D) → MapC (F (C), F (D))
is an isomorphism in the homotopy category H. Suppose first that C belongs
to the essential image of j. Let G : P(S) → S be a functor corepresented by
C and let G0 : C → S be a functor corepresented by F (C). Then we havehighertopoi

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a natural transformation of functors G → G0 ◦ F . Assumption (2) implies
that G0 preserves small colimits, so that G0 ◦ F preserves small colimits.
Proposition 5.1.6.8 implies that G preserves small colimits. It follows thatthe collection of objects D ∈ P(S) such that ηC,D is an equivalence is stable
under small colimits. If D belongs to the essential image of j, then assumption (1) implies that ηC,D is an equivalence. It follows from Lemma 5.1.5.3
that the essential image of j generates P(S) under small colimits; thus ηC,D
is an isomorphism in H for every object D ∈ P(S).
We now prove the result in general. Fix D ∈ P(S). Let H : P(S)op → S bea functor represented by D and let H 0 : Cop → S be a functor represented
by F D. Then we have a natural transformation of functors H → H 0 ◦ F op ,which we wish to prove is an equivalence. By assumption, F op preserves
small limits. Proposition 5.1.3.2 implies that H and H 0 preserve small limits.
It follows that the collection P of objects C ∈ P(S) such that ηC,D is anequivalence is stable under small colimits. The special case above establishedthat P contains the essential image of the Yoneda embedding. We once again
invoke Lemma 5.1.5.3 to deduce that every object of P(S) belongs to P , asdesired.
Corollary 5.1.6.11. Let C be an ∞-category which admits small colimits.
Let S be a small simplicial set and F : P(S) → C a colimit-preserving functor. Then F is an equivalence if and only if the following conditions are
satisfied:(1) The composition f = F ◦ j : S → C is fully faithful.
(2) For every vertex s ∈ S, the object f (s) ∈ C is completely compact.
(3) The set of objects {f (s) : s ∈ S0 } generates C under colimits.
Proof. If (1), (2), and (3) are satisfied, then F is fully faithful (Proposition
5.1.6.10). Since P(S) admits small colimits and F preserves small colimits,
the essential image of F is stable under small colimits. Using (3), we conclude
that F is essentially surjective and therefore an equivalence of ∞-categories.
For the converse, it suffices to check that idP(S) : P(S) → P(S) satisfies (1),(2), and (3). For this, we invoke Propsition 5.1.3.1, Proposition 5.1.6.8, andLemma 5.1.5.3, respectively.
Corollary 5.1.6.12. Let C be a small ∞-category, let p : K → C be adiagram, let p0 : K → P(C) be the composition of p with the Yoneda embedding j : C → P(C), and let f : C/p → P(C)/p0 be the induced map. Let
F : P(C/p ) → P(C)/p0 be a colimit-preserving functor such that F ◦ j 0 is
equivalent to f , where j 0 : C/p → P(C/p ) denotes the Yoneda embedding for
C/p (according to Theorem 5.1.5.6, F exists and is unique up to equivalence).
Then F is an equivalence of ∞-categories.
Proof. We will show that f satisfies conditions (1) through (3) of Corollary
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assertion that j is fully faithful (Proposition 5.1.3.1). To prove that the essential image of f consists of completely compact objects, we use Lemma
5.1.6.7 to reduce to proving that the essential image of j consists of completely compact objects of P(C), which follows from Proposition 5.1.6.8. It
remains to prove that P(C)/p0 is generated under colimits by f . Let X be
an object of P(C)/p0 and X its image in P(C). Let D ⊆ P(C) be the essential image of j and D the inverse image of D in P(C)/p0 , so that D is theessential image of f . Using Lemma 5.1.5.3, we can choose a colimit diagramq : L. → P(C) which carries the cone point to X such that q = q|L factors
through D. Since the inclusion of the cone point into L. is right anodyne,
there exists a map q 0 : L. → P(C)/p0 lifting q, which carries the cone pointof L. to X. Proposition 1.2.13.8 implies that q 0 is a colimit diagram, so that
X can be written as the colimit of a diagram L → D.

5.2 ADJOINT FUNCTORSLet C and D be (ordinary) categories. Two functors
Co

F
G

/D

are said to be adjoint to one another if there is a functorial bijection
HomD (F (C), D) ' HomC (C, G(D))
defined for C ∈ C, D ∈ D. Our goal in this section is to extend the theoryof adjoint functors to the ∞-categorical setting.
By definition, a pair of functors F and G (as above) are adjoint if andonly if they determine the same correspondence
Cop × D → Set .
In §2.3.1, we introduced an ∞-categorical generalization of the notion of acorrespondence. In certain cases, a correspondence M from an ∞-category
C to an ∞-category D determines a functor F : C → D, which we say
is a functor associated to M. We will study these associated functors in§5.2.1. The notion of a correspondence is self-dual, so it is possible that thecorrespondence M also determines an associated functor G : D → C. In thiscase, we will say that F and G are adjoint. We will study the basic propertiesof adjoint functors in §5.2.2.
One of the most important features of adjoint functors is their behavior
with respect to limits and colimits: left adjoints preserve colimits, whileright adjoints preserve limits. We will prove an ∞-categorical analogue of
this statement in §5.2.3. In certain situations, the adjoint functor theorem
provides a converse to this statement: see §5.5.2.
The theory of model categories provides a host of examples of adjoint functors between ∞-categories. In §5.2.4, we will show that a simplicial Quillen
adjunction between a pair of model categories (A, A0 ) determines an ad◦junction between the associated ∞-categories (N(A◦ ), N(A0 )). We will alsohighertopoi

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consider some other examples of situations which give rise to adjoint functors.
In §5.2.5, we study the behavior of adjoint functors when restricted toovercategories. Our main result (Proposition 5.2.5.1) can be summarized asfollows: suppose that F : C → D is a functor between ∞-categories which
admits a right adjoint G. Assume further that the ∞-category C admits
pullbacks. Then for every object C, the induced functor C/C → D/F C admits
a right adjoint given by the formula(D → F C) 7→ (GD ×GF C C → C).
If a functor F : C → D has a right adjoint G, then G is uniquely determined
up to equivalence. In §5.2.6, we will prove a strong version of this statement(which we phrase as an (anti)equivalence of functor ∞-categories).
In §5.2.7, we will restrict the theory of adjoint functors to the special casein which one of the functors is the inclusion of a full subcategory. In thiscase, we obtain the theory of localizations of ∞-categories. This theory willplay a central role in our study of presentable ∞-categories (§5.5) and laterin the study of ∞-topoi (§6). It is also useful in the study of factorization
systems on ∞-categories, which we will discuss in §5.2.8.
5.2.1 Correspondences and Associated Functors
Let p : X → S be a Cartesian fibration of simplicial sets. In §3.3.2, we sawthat p is classified by a functor S op → Cat∞ . In particular, if S = ∆1 , thenp determines a diagram
G:D→C
in the ∞-category Cat∞ , which is well-defined up to equivalence. We canobtain this diagram by applying the straightening functor St+
S to the marked
simplicial set X \ and then taking a fibrant replacement. In general, thisconstruction is rather complicated. However, in the special case where S =∆1 , it is possible to give a direct construction of G; that is our goal in thissection.
Definition 5.2.1.1. Let p : M → ∆1 be a Cartesian fibration and suppose
we are given equivalences of ∞-categories h0 : C → p−1 {0} and h1 : D →p−1 {1}. We will say that a functor g : D → C is associated to M if there is
a commutative diagram
/M

s

D ×∆1
#

∆1

~

such that s| D ×{1} = h1 , s| D ×{0} = h0 ◦g, and s|{x}×∆1 is a p-Cartesianedge of M for every object x of D.

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Remark 5.2.1.2. The terminology of Definition 5.2.1.1 is slightly abusive:
it would be more accurate to say that g is associated to the triple (p : M →∆1 , h0 : C → p−1 {0}, h1 : D → p−1 {1}).
Proposition 5.2.1.3. Let C and D be ∞-categories and let g : D → C be afunctor.
(1) There exists a diagram
C

/Mo


{0}


/ ∆1 o

p

D

{1},

where p is a Cartesian fibration, the associated maps C → p−1 {0} andD → p−1 {1} are isomorphisms, and g is associated to M.
(2) Suppose we are given a commutative diagram/ M0 o
D
C
s


M

{0}

p


/ ∆1 o{1}where s is a categorical equivalence, p and p0 = p ◦ s are Cartesianfibrations, and the maps C → p−1 {0}, D → p−1 {1} are categoricalequivalences. The functor g is associated to M if and only if it is associated to M0 .
(3) Suppose we are given diagramsC/ M0 o{0}/ ∆1 o{1}C/ M00 o

D


{0}


/ ∆1 o

p0

p00

D


{1}

as above, such that g is associated to both M0 and M00 . Then there
exists a third such diagram
/Mo
C
D

{0}

p


/ ∆1 o{1}highertopoi

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIES

and a diagramM0 ← M → M00
of categorical equivalences in (Set∆ )C ` D / /∆1 .
Proof. We begin with (1). Let C\ and D\ denote the simplicial sets C and D
considered as marked simplicial sets, where the marked edges are precisely
the equivalences. We seta
N = (D\ ×(∆1 )] )C\ .
D\ ×{0}]The small object argument implies the existence of a factorization
N → N (∞) → (∆1 )] ,where the left map is marked anodyne and the right map has the right lifting
property with respect to all marked anodyne morphisms. We remark that
we can obtain N (∞) as the colimit of a transfinite sequence of simplicial
sets N (α), where N (0) = N , N (α) is the colimit of the sequence {N (β)}β<αwhen α is a limit ordinal and each N (α + 1) fits into a pushout diagramA/ N (α);B/ N (α + 1)%
/ (∆1 )] ,where the left vertical map is one of the generators for the class of marked
anodyne maps given in Definition 3.1.1.1. We may furthermore assume that
there does not exist a dotted arrow as indicated in the diagram. It follows by
induction on α that N (α) ×∆1 {0} ' C\ and N (α) ×∆1 {1} ' D\ . According
to Proposition 3.1.1.6, N (∞) ' M\ for some Cartesian fibration M → ∆1 . It
follows immediately that C ' M{0} , that D ' M{1} , and that g is associatedto M.
We now prove (2). The “if” direction is immediate from the definition.
Conversely, suppose that g is associated to M. To show that g is associatedto M0 , we need to produce the dotted arrow indicated in the diagramD × ∂ _ ∆1

/ M0
:


D ×∆1 .
According to Proposition A.2.3.1, we may replace M0 by the equivalent ∞category M; the desired result then follows from the assumption that g isassociated to M.
To prove (3), we take M to be the correspondence constructed in thecourse of proving (1). It will suffice to construct an appropriate categoricalhighertopoi

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equivalence M → M0 ; the same argument will construct the desired map
M → M00 . Consider the diagram

s0

/ M0
=

s

N _
s00


/ ∆1 .

M(Here we identify N with its underlying simplicial set by forgetting the classof marked edges, and the top horizontal map exhibits g as associated to M0 .)In the terminology of §3.2.2, the maps s and s0 are both quasi-equivalences.
By Proposition 3.2.2.10, they are categorical equivalences. The projection
M0 → ∆1 is a categorical fibration and s0 is a trivial cofibration, which
ensures the existence of the arrow s00 . The factorization s = s00 ◦ s0 showsthat s00 is a categorical equivalence and completes the proof.
Proposition 5.2.1.3 may be informally summarized by saying that every
functor g : D → C is associated to some Cartesian fibration p : M → ∆1 andthat M is determined up to equivalence. Conversely, the Cartesian fibration
also determines g:
Proposition 5.2.1.4. Let p : M → ∆1 be a Cartesian fibration and let
h0 : C → p−1 {0} and h1 : D → p−1 {1} be categorical equivalences. Thereexists a functor g : D → C associated to M. Any other functor g 0 : D → C
is associated to p if and only if g is equivalent to g 0 as an object of the∞-category CD .
Proof. Consider the diagram/ M\
9

D[ ×{1}sD[ ×(∆1 )]/ (∆1 )] .

By Proposition 3.1.2.3, the left vertical map is marked anodyne, so the dottedarrow exists. Consider the map s0 : s| D ×{0} : D → p−1 {0}. Since h0 is a
categorical equivalence, there exists a map g : D → C such that the functionsh0 ◦ g and s0 are equivalent. Let e : D ×∆1 → M be an equivalence fromh0 ◦ g to s0 . Let e0 : D ×Λ21 → M be the result of amalgamating e with s.
Then we have a commutative diagram of marked simplicial sets2 ]D[ ×(Λ
 _ 1)

D[ ×(∆2 )]

e0

/ M\
9

e00


/ (∆1 )] .

Because the left vertical map is marked anodyne, there exists a morphism
e00 as indicated which renders the diagram commutative. The restriction
e00 | D ×∆{0,2} exhibits g as associated to M.

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339

Now suppose that g 0 is another functor associated to p. Then there exists
a commutative diagram of marked simplicial sets
D[ ×{1}
_

/ M\
9
s0


D[ ×(∆1 )]/ (∆1 )] ,with g 0 = s0 | D ×{0}. Let s00 be the map obtained by amalgamating s ands0 . Consider the diagram
2 ]D[ ×(Λ
 _ 2)

s00

/ M\
9

s000


D[ ×(∆2 )]/ (∆1 )] .

Since the left vertical map is marked anodyne, the indicated dotted arrows00 exists. The restriction s00 | D ×∆{0,1} is an equivalence between h0 ◦ g
and h0 ◦ g 0 . Since h0 is a categorical equivalence, g and g 0 are themselves
homotopic.
Conversely, suppose that f : D ×∆1 → C is an equivalence from g 0 to g.
The maps s and h0 ◦ f amalgamate to give a map f 0 : D ×Λ21 → C which fits
into a commutative diagram of marked simplicial sets:2 ]D[ ×(Λ
 _ 1)

f0

/ M\
9

f 00


D ×(∆2 )]/ (∆1 )] .

[The left vertical map is marked anodyne, so there exists a dotted arrow f 00
as indicated; then the map f 00 | D ×∆{0,2} exhibits that g 0 is associated top.
Proposition 5.2.1.5. Let p : M → ∆2 be a Cartesian fibration and suppose we are given equivalences of ∞-categories C → p−1 {0}, D → p−1 {1},and E → p−1 {2}. Suppose that M ×∆2 ∆{0,1} is associated to a functor
f : D → C and that M ×∆2 ∆{1,2} is associated to a functor g : E → D.
Then M ×∆2 ∆{0,2} is associated to the composite functor f ◦ g.
Proof. Let X be the mapping simplex of the sequence of functors
gf

E → D → C.
Since f and g are associated to restrictions of M, we obtain a commutative
diagram/
X ×∆ 2 Λ21
:M
_
s


X


/ ∆2 .

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CHAPTER 5

The left vertical inclusion is a pushout of E ×Λ21 ⊆ E ×∆2 , which is inner
anodyne. Since p is inner anodyne, there exists a dotted arrow s as indicatedin the diagram. The restriction s|X ×∆2 ∆{0,2} exhibits that the functor f ◦g
is associated to the correspondence M ×∆2 ∆{0,2} .
Remark 5.2.1.6. Taken together, Propositions 5.2.1.3 and 5.2.1.4 assertthat there is a bijective correspondence between equivalence classes of functors D → C and equivalence classes of Cartesian fibrations p : M → ∆1equipped with equivalences C → p−1 {0}, D → p−1 {1}.
5.2.2 Adjunctions
In §5.2.1, we established a dictionary that allows us to pass back and forth
between functors g : D → C and Cartesian fibrations p : M → ∆1 . The
dual argument shows that if p is a coCartesian fibration, it also determines afunctor f : C → D. In this case, we will say that f and g are adjoint functors.
Definition 5.2.2.1. Let C and D be ∞-categories. An adjunction between
C and D is a map q : M → ∆1 which is both a Cartesian fibration and a
coCartesian fibration together with equivalences C → M{0} and D → M{1} .
Let M be an adjunction between C and D and let f : C → D and g : D → C
be functors associated to M. In this case, we will say that f is left adjoint
to g and g is right adjoint to f .
Remark 5.2.2.2. Propositions 5.2.1.3 and 5.2.1.4 imply that if a functor
f : C → D has a right adjoint g : D → C, then g is uniquely determined up tohomotopy. In fact, we will later see that g is determined up to a contractible
ambiguity.
We now verify a few basic properties of adjunctions:Lemma 5.2.2.3. Let p : X → S be a locally Cartesian fibration of simplicial
sets. Let e : s → s0 be an edge of S with the following property:(∗) For every 2-simplex
0

?x

e0

e
00

e
/ x00
x
in X such that p(e) = e, if e and e0 are locally p-Cartesian, then e00 is
locally p-Cartesian.

Let e : x → y be a locally p-coCartesian edge such that p(e) = e. Then e is
p-coCartesian.
Proof. We must show that for any n ≥ 2 and any diagram
Λn0
_

∆n

f

/X
>

/S

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIES

such that f |∆{0,1} = e, there exists a dotted arrow as indicated. Pullingback along the bottom horizontal map, we may reduce to the case S = ∆n ;in particular, X and S are both ∞-categories.
According to (the dual of) Proposition 2.4.4.3, it suffices to show thatcomposition with e gives a homotopy Cartesian diagram
MapX (y, z)/ MapX (x, z)MapS (p(y), p(z))

/ MapS (p(x), p(z)).

There are two cases to consider: if MapS (p(y), p(z)) = ∅, there is nothing toprove. Otherwise, we must show that composition with f induces a homotopy
equivalence MapX (y, z) → MapX (x, z).
In view of the assumption that S = ∆n , there is a unique morphism
g0 : p(y) → p(z). Let g : y 0 → z be a locally p-Cartesian edge lifting g0 . We
have a commutative diagram
MapX (y, y 0 )/ MapX (x, y 0 )MapX (y, z)/ MapX (x, z).

Since g is locally p-Cartesian, the left vertical arrow is a homotopy equivalence. Since e is locally p-coCartesian, the top horizontal arrow is a homotopy
equivalence. It will therefore suffice to show that the map MapX (x, y 0 ) →MapX (x, z) is a homotopy equivalence.
Choose a locally p-Cartesian edge e0 : x0 → y 0 in X with p(e0 ) = e, so that
we have another commutative diagram
MapX (x, x0 )w
MapX (x, y 0 )'/ MapX (x, z).

Using the two-out-of-three property, we are reduced to proving that bothof the diagonal arrows are homotopy equivalences. For the diagonal arrowon the left, this follows from our assumption that e0 is locally p-Cartesian.
For the arrow on the right, it suffices to show that the composition g ◦ e0 islocally p-coCartesian, which follows from assumption (∗).
Corollary 5.2.2.4. Let p : X → S be a Cartesian fibration of simplicial
sets. An edge e : x → y of X is p-coCartesian if and only if it is locallyp-coCartesian (see the discussion preceding Proposition 2.4.2.8).
Corollary 5.2.2.5. Let p : X → S be a Cartesian fibration of simplicial
sets. The following conditions are equivalent:(1) The map p is a coCartesian fibration.

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CHAPTER 5(2) For every edge f : s → s0 of S, the induced functor f ∗ : Xs0 → Xs has
a left adjoint.
Proof. By definition, the functor corresponding to an edge f : ∆1 → S has
a left adjoint if and only if the pullback X ×S ∆1 → ∆1 is a coCartesianfibration. In other words, condition (2) is equivalent to the assertion that forevery edge f : s → s0 and every vertex se of X lifting s, there exists a locallyp-coCartesian edge fe : se → se0 lifting f . Using Corollary 5.2.2.4, we conclude
that fe is automatically p-coCartesian, so that (2) is equivalent to (1).
Proposition 5.2.2.6. Let f : C → D and f 0 : D → E be functors between∞-categories. Suppose that f has a right adjoint g and that f 0 has a rightadjoint g 0 . Then g ◦ g 0 is right adjoint to f 0 ◦ f .
Proof. Let φ denote the composable sequence of morphisms
g0

gC ← D ← E.
Let M (φ) denote the mapping simplex and choose a factorizationqs

M (φ) → X → ∆2 ,where s is a quasi-equivalence and X → ∆2 is a Cartesian fibration (using
Proposition 3.2.2.11). We first show that q is a coCartesian fibration. In
other words, we must show that, for every object x ∈ C and every morphisme : q(x) → y, there is a q-Cartesian edge e : x → y lifting e. This is clear if
e is degenerate. If e = ∆{0,1} ⊆ ∆2 , then the existence of a left adjoint to g
implies that e has a locally q-coCartesian lift e. Lemma 5.2.2.3 implies that e
is q-coCartesian. Similarly, if e = ∆{1,2} , then we can find a q-coCartesian liftof e. Finally, if e is the long edge ∆{0,2} , then we may write e as a composite
e0 ◦ e00 ; the existence of a q-coCartesian lift of e follows from the existenceof q-coCartesian lifts of e0 and e00 . We now apply Proposition 5.2.1.5 anddeduce that the adjunction X ×∆2 ∆{0,2} is associated to both g ◦ g 0 and
f0 ◦ f.
In classical category theory, one can spell out the relationship between apair of adjoint functors f : C → D and g : D → C by specifying a unittransformation idC → g ◦ f (or, dually, a counit f ◦ g → idD ). This concept
generalizes to the ∞-categorical setting as follows:
Definition 5.2.2.7. Suppose we are given a pair of functors
Co

f
g

/D

between ∞-categories. A unit transformation for (f, g) is a morphism u :idC → g ◦f in Fun(C, C) with the following property: for every pair of objects
C ∈ C, D ∈ D, the compositionu(C)MapD (f (C), D) → MapC (g(f (C)), g(D)) → MapC (C, g(D))
is an isomorphism in the homotopy category H.

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Proposition 5.2.2.8. Let f : C → D and g : D → C be a pair of functors
between ∞-categories C and D. The following conditions are equivalent:(1) The functor f is a left adjoint to g.
(2) There exists a unit transformation u : idC → g ◦ f .
Proof. Suppose first that (1) is satisfied. Choose an adjunction p : M → ∆1
which is associated to f and g; according to (1) of Proposition 5.2.1.3 we may
identify M{0} with C and M{1} with D. Since f is associated to M , there isa map F : C ×∆1 → M such that F | C ×{0} = idC and F | C ×{1} = f , witheach edge F |{c} × ∆1 p-coCartesian. Similarly, there is a map G : D ×∆1 →M with G| D ×{1} = idD , G| D ×{0} = g, and such that G|{d} × ∆1 is
p-Cartesian for each object d ∈ D. Let F 0 : Λ22 × C → M be such thatF 0 |∆{0,2} × C = F and F 0 |∆{1,2} × C = G ◦ (f × id∆1 ). Consider the diagramΛ22 × C_F

∆2 × C

F0

/M
;

00


/ ∆1 .

Using the fact that F 0 |{c} × ∆{1,2} is p-Cartesian for every object c ∈ C,
we deduce the existence of the indicated dotted arrow F 00 . We now defineu = F 00 | C ×∆{0,1} . We may regard u as a natural transformation idC → g◦f .
We claim that u is a unit transformation. In other words, we must show thatfor any objects C ∈ C, D ∈ D, the composite map
u

MapD (f C, D) → MapC (gf C, gD) → MapC (C, gD)
is an isomorphism in the homotopy category H of spaces. This composite
map fits into a commutative diagram/ MapD (g(f (C)), g(D))

MapD (f (C), D)/ MapD (C, g(D))
/ MapM (C, D).


MapM (C, D)

The left and right vertical arrows in this diagram are given by composition with a p-coCartesian and a p-Cartesian morphism in M , respectively.
Proposition 2.4.4.2 implies that these maps are homotopy equivalences.
We now prove that (2) ⇒ (1). Choose a correspondence p : M → ∆1 fromC to D which is associated to the functor g via a map G : D ×∆1 → M as
above. We have natural transformations
idC

u

/ g◦f

G◦(f ×id∆1 )/ f.

Let F : C ×∆1 → M be a composition of these transformations. We willcomplete the proof by showing that F exhibits M as a correspondence associated to the functor f . It will suffice to show that for each object C ∈ C,

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F (C) : C → f C is p-coCartesian. According to Proposition 2.4.4.3, it will
suffice to show that for each object D ∈ D, composition with F (C) induces ahomotopy equivalence MapD (f (C), D) → MapM (C, D). As above, this map
fits into a commutative diagram
MapD (f (C), D)/ MapD (g(f (C)), g(D))


MapM (C, D)/ MapD (C, g(D))
/ MapM (C, D)

where the upper horizontal composition is an equivalence (since u is a unittransformation) and the right vertical arrow is an equivalence (since it isgiven by composition with a p-Cartesian morphism). It follows that the left
vertical arrow is also a homotopy equivalence, as desired.
Proposition 5.2.2.9. Let C and D be ∞-categories and let f : C → D
and g : D → C be adjoint functors. Then f and g induce adjoint functors
hf : hC → hD and hg : hD → hC between (H-enriched) homotopy categories.
Proof. This follows immediately from Proposition 5.2.2.8 because a unittransformation idC → g ◦ f induces a unit transformation idhC → (hg) ◦(hf ).
The converse to Proposition 5.2.2.9 is false. If f : C → D and g : D → C
are functors such that hf and hg are adjoint to one another, then f andg are not necessarily adjoint. Nevertheless, the existence of adjoints can betested at the level of (enriched) homotopy categories.
Lemma 5.2.2.10. Let p : M → ∆1 be an inner fibration of simplicial sets
giving a correspondence between the ∞-categories C = M{0} and D = M{1} .
Let c be an object of C, d an object of D, and f : c → d a morphism. The
following are equivalent:(1) The morphism f is p-Cartesian.
(2) The morphism f gives rise to a Cartesian morphism in the enrichedhomotopy category hM; in other words, composition with f induces
homotopy equivalences
MapC (c0 , c) → MapM (c0 , d)for every object c0 ∈ C.
Proof. This follows immediately from Proposition 2.4.4.3.
Lemma 5.2.2.11. Let p : M → ∆1 be an inner fibration, so that M canbe identified with a correspondence from C = p−1 {0} to D = p−1 {1}. The
following conditions are equivalent:(1) The map p is a Cartesian fibration.

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIES(2) There exists a H-enriched functor g : hD → hC and a functorial identification
MapM (c, d) ' MapC (c, g(d)).
Proof. If p is a Cartesian fibration, then there is a functor D → C associated to M; we can then take g to be the associated functor on enrichedhomotopy categories. Conversely, suppose that there exists a functor g as
above. We wish to show that p is a Cartesian fibration. In other words, wemust show that for every object d ∈ D, there is an object c ∈ C and a pCartesian morphism f : c → d. We take c = g(d); in view of the identification
MapM (c, d) ' MapC (c, c), there exists a morphism f : c → d corresponding to the identity idc . Lemma 5.2.2.10 implies that f is p-Cartesian, as
desired.
Proposition 5.2.2.12. Let f : C → D be a functor between ∞-categories.
Suppose that the induced functor of H-enriched categories hf : hC → hD
admits a right adjoint. Then f admits a right adjoint.
Proof. According to (1) of Proposition 5.2.1.3, there is a coCartesian fibration p : M → ∆1 associated to f . Let hg be the right adjoint of hf . Applying
Lemma 5.2.2.11, we deduce that p is a Cartesian fibration. Thus p is an
adjunction, so that f has a right adjoint, as desired.
5.2.3 Preservation of Limits and Colimits
Let C and D be ordinary categories and let F : C → D be a functor. If F
has a right adjoint G, then F preserves colimits; we have a chain of natural
isomorphisms
HomD (F (lim Cα ), D) ' HomC (lim Cα , G(D))−→−→' lim HomC (Cα , G(D))←−' lim HomD (F (Cα ), D)←−' HomD (lim F (Cα ), D).
−→In fact, this is in some sense the defining feature of left adjoints: undersuitable set-theoretic assumptions, the adjoint functor theorem asserts thatany colimit-preserving functor admits a right adjoint. We will prove an ∞categorical version of the adjoint functor theorem in §5.5.2. Our goal inthis section is to lay the groundwork by showing that left adjoints preserve
colimits in the ∞-categorical setting. We will first need to establish several
lemmas.
Lemma 5.2.3.1. Suppose we are given a diagram/M

P

K × ∆1
#

∆

1

~

q

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of simplicial sets, where M is an ∞-category and P |{k}×∆1 is q-coCartesianfor every vertex k of K. Let p = P |K × {0}. Then the induced map
ψ : MP/ → Mp/
induces a trivial fibration
ψ1 : MP/ ×∆1 {1} → Mp/ ×∆1 {1}.
Proof. If K is a point, then the assertion reduces immediately to the definition of a coCartesian edge. In the general case, we note that ψ and ψ1 areboth left fibrations between ∞-categories. Consequently, it suffices to showthat ψ1 is a categorical equivalence. In doing so, we are free to replace ψ bythe equivalent map ψ 0 : MP/ → Mp/ . To prove that ψ10 : MP/ ×∆1 {1} →Mp/ ×∆1 {1} is a trivial fibration, we must show that for every inclusionA ⊆ B of simplicial sets and any mapak0 : ((K × ∆1 )  A)((K × {0})  B) → M(K×{0})A

with k0 |K × ∆ = P and k0 (B) ⊆ q −1 {1}, there exists an extension of k0 to
a map k : (K × ∆1 )  B → M. Leta
X = (K × ∆1 )(K × ∆1 × B × ∆1 )1

K×∆1 ×B×{0}and let h : X → K  B be the natural map. Leta
X 0 = h−1 ((K × ∆1 )  A)((K × {0})  B) ⊆ X(K×{0})A

and let e
k0 : X 0 → M be the composition k0 ◦ h. It suffices to prove that
there exists an extension of e
k0 to a map ek : X → M. Replacing M by
Map∆1 (K, M), we may reduce to the case where K is a point, which wealready treated above.
Lemma 5.2.3.2. Let q : M → ∆1 be a correspondence between ∞-categories
C = q −1 {0} and D = q −1 {1} and let p : K → C be a diagram in C. Letf : c → d be a q-Cartesian morphism in M from c ∈ C to d ∈ D. Letr : Mp/ → M be the projection and let d be an object of Mp/ with r(d) = d.
Then(1) There exists a morphism f : c → d in Mp/ satisfying f = r(f ).
(2) Any morphism f : c → d which satisfies r(f ) = f is r-Cartesian.
Proof. We may identify d with a map d : K → M/d . Consider the set ofpairs (L, s), where L ⊆ K and s : L → M/f sits in a commutative diagram
L _

/ M/f


K


/ M/d .

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We order these pairs by setting (L, s) ≤ (L0 , s0 ) if L ⊆ L0 and s = s0 |L. By
Zorn’s lemma, there exists a pair (L, s) which is maximal with respect tothis ordering. To prove (1), it suffices to show`that L = K. Otherwise, we
may obtain a larger simplicial subset L0 = L ∂ ∆n ∆n ⊆ K by adjoining
a single nondegenerate simplex. By maximality, there is no solution to theassociated lifting problem/ M/f∂ ∆ _n
<

/ M/d


∆n
nor to the associated lifting problem

/M
<

s

Λn+2
n+2
_

q




/ ∆1 ,∆n+2

which contradicts the fact that s carries ∆{n+1,n+2} to the q-Cartesian morphism f in M.
Now suppose that f is a lift of f . To prove that f is r-Cartesian, it suffices
to show that for every m ≥ 2 and every diagram
Λm
m
_

g0
g

/ Mp/
<

/M


∆m

such that g0 |∆{m−1,m} = fe, there exists a dotted arrow g as indicated,
rendering the diagram commutative. We can identify the diagram with amap
at0 : (K ? Λm∆m → M .
m)Λmm

Consider the set of all pairs (L, t), where L ⊆ K andat : (K ? Λm
(L ? ∆m ) → M
m)
L?Λm
m

is an extension of t0 . As above, we order the set of such pairs by declaring(L, t) ≤ (L0 , t0 ) if L ⊆ L0 and t = t0 |L. Zorn’s lemma guarantees the existence of a maximal pair (L, t). If L = K, we are done; otherwise let L0 be
obtained from L by adjoining a single nondegenerate n-simplex of K. By
maximality, the map t does not extend to L0 ; consequently, the associatedmapping problem`(∆n ? Λm(∂ ∆n ? ∆m )5/ Mm)
∂ ∆n ?Λ
m
_ m
∆n ? ∆m/ ∆1

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has no solution. But this contradicts the assumption that r(fe) = f is aq-Cartesian edge of M.
Lemma 5.2.3.3. Let q : M → ∆1 be a correspondence between the ∞categories C = q −1 {0} and D = q −1 {1}. Let f : c → d be a morphism in M
between objects c ∈ C and d ∈ D. Let p : K → C be a diagram and consider
an associated map
k : Mp/ ×M {c} → Mp/ ×M {d}(the map k is well-defined up to homotopy according to Lemma 2.1.1.4). If
f is q-Cartesian, then k is a homotopy equivalence.
1

Proof. Let X = (Mp/ )∆ ×M∆1 {f } and consider the diagramXuy
Mp/ ×M {c}v%
Mp/ ×M {d}.

The map u is a homotopy equivalence, and k is defined as the composition ofv with a homotopy inverse to u. Consequently, it will suffice to show that v
is a trivial fibration. To prove this, we must show that v has the right lifting
property with respect to ∂ ∆n ⊆ ∆n , which is equivalent to solving a lifting
problem`(∂ ∆n × ∆1 ) ∂ ∆n ×{1} (∆n × {1})5/ Mp/
_
∆n × ∆1r

/ M.

If n = 0, we invoke (1) of Lemma 5.2.3.2. If n > 0, then Proposition 2.4.1.8
implies that it suffices to show that the upper horizontal map carries {n} ×∆1 to an r-Cartesian edge of Mp/ , which also follows from assertion (2) ofLemma 5.2.3.2.
Lemma 5.2.3.4. Let q : M → ∆1 be a Cartesian fibration and let C =q −1 {0}. The inclusion C ⊆ M preserves all colimits which exist in C.
Proof. Let p : K . → C be a colimit of p = p|K. We wish to show thatMp/ → Mp/ is a trivial fibration. Since we have a diagram
Mp/ → Mp/ → M
of left fibrations, it will suffice to show that the induced map
Mp/ ×M {d} → Mp/ ×M {d}is a homotopy equivalence of Kan complexes for each object d of M. If
d belongs to C, this is obvious. In general, we may choose a q-Cartesianhighertopoi

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morphism f : c → d in M. Composition with f gives a commutative diagram[Mp/ ×M {c}]/ [Mp/ ×M {c}]
[Mp/ ×M {d}]
/ [Mp/ ×M {d}]in the homotopy category H of spaces. The upper horizontal map is a homotopy equivalence since p is a colimit of p in C. The vertical maps are homotopy
equivalences by Lemma 5.2.3.3. Consequently, the bottom horizontal map isalso a homotopy equivalence, as desired.
Proposition 5.2.3.5. Let f : C → D be a functor between ∞-categories
which has a right adjoint g : D → C. Then f preserves all colimits which
exist in C, and g preserves all limits which exist in D.
Proof. We will show that f preserves colimits; the analogous statement forg follows by a dual argument. Let p : K . → C be a colimit for p = p|K. We
must show that f ◦ p is a colimit of f ◦ p.
Let q : M → ∆1 be an adjunction between C = M{0} and D = M{1} whichis associated to f and g. We wish to show that
φ1 : Df p/ → Df p/
is a trivial fibration. Since φ1 is a left fibration, it suffices to show that φ1 isa categorical equivalence.
Since M is associated to f , there is a map F : C ×∆1 → M such that
F | C ×{0} = idC , F | C ×{1} = f , and F |{c} × ∆1 a q-coCartesian morphismof M for every object c ∈ C. Let P = F ◦ (p × id∆1 ) be the induced map
K . × ∆1 → M and let P = P |K × ∆1 .
Consider the diagram
Mp/
O

φ0

/ Mp/
O
v

v

/ MP/

MP /


u

u

Mf p/

φ



/ Mf p/ .

We note that every object in this diagram is an ∞-category with a map to∆1 ; moreover, the map φ1 is obtained from φ by passage to the fiber over{1} ⊆ ∆1 . Consequently, to prove that φ1 is a categorical equivalence, itsuffices to verify three things:(1) The bottom vertical maps u and u are trivial fibrations. This follows
from the fact that K × {1} ⊆ K × ∆1 and K . × {1} ⊆ K . × ∆1 areright anodyne inclusions (Proposition 2.1.2.5).

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CHAPTER 5(2) The upper vertical maps v and v are trivial fibrations when restricted to
D ⊆ M. This follows from Lemma 5.2.3.1 since F carries each {c} × ∆1
to a q-coCartesian edge of M.
(3) The map φ0 is a trivial fibration since p is a colimit of p in M according
to Lemma 5.2.3.4.

Remark 5.2.3.6. Under appropriate set-theoretic hypotheses, one can establish a converse to Proposition 5.2.3.5. See Corollary 5.5.2.9.
5.2.4 Examples of Adjoint Functors
In this section, we describe a few simple criteria for establishing the existenceof adjoint functors.
Lemma 5.2.4.1. Let q : M → ∆1 be a coCartesian fibration associated to afunctor f : C → D, where C = q −1 {0} and D = q −1 {1}. Let D be an objectof D. The following are equivalent:(1) There exists a q-Cartesian morphism g : C → D in M, where C ∈ C.
(2) The right fibration C ×D D/D → C is representable.
Proof. Let F : C ×∆1 → M be a p-coCartesian natural transformationfrom idC to f . Define a simplicial set X so that for every simplicial set
K, HomSet∆ (K, X) parametrizes maps H : K × ∆2 → M such that h =H|K × {0} factors through C, H|K × ∆{0,1} = F ◦ (h|(K × {0}) × id∆1 ), and
H|K × {2} is the constant map at the vertex D. We have restriction maps
X

C ×M Mz/D$
C ×D D/D .

which are both trivial fibrations (the map on the right because M is an ∞category, and the map on the left because F is p-coCartesian). Consequently,(2) is equivalent to the assertion that the ∞-category C ×M M/D has a final
object. It now suffices to observe that a final object of C ×M M/D is precisely
a q-Cartesian morphism C → D, where C ∈ C.
Proposition 5.2.4.2. Let F : C → D be a functor between ∞-categories.
The following are equivalent:(1) The functor F has a right adjoint.

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIES(2) For every pullback diagram
/D

C
p0


C

p
F


/ D,

if p is a representable right fibration, then p0 is also a representableright fibration.
Proof. Let M be a correspondence from C to D associated to F and apply
Lemma 5.2.4.1 to each object of D.
Proposition 5.2.4.3. Let p : C → D be a Cartesian fibration of ∞categories and let s : D → C be a section of p such that s(D) is an initial
object of CD = C ×D {D} for every object D ∈ D. Then s is a left adjoint ofp.
Proof. Let C0 ⊆ C denote the full subcategory of C spanned by those objects C ∈ C such that C is initial in the ∞-category Cp(C) . According to
Proposition 2.4.4.9, the restriction p| C0 is a trivial fibration from C0 to D.
Consequently, it will suffice to show that the inclusion C0 ⊆ C is left adjoint
to the composition s ◦ p : C → C0 . Let M ⊆ C ×∆1 be the full subcategory
spanned by the vertices (C, {i}), where i = 1 or C ∈ C0 . Let q : M → ∆1 be
the projection. It is clear that q is a coCartesian fibration associated to theinclusion C0 ⊆ C. To complete the proof, it will suffice to show that q is alsoa Cartesian fibration associated to s ◦ p.
We first show that q is a Cartesian fibration. It will suffice to show thatfor any object C ∈ C, there is a q-Cartesian edge (C 0 , 0) → (C, 1) in M. By
assumption, C 0 = (s ◦ p)(C) is an initial object of Cp(C) . Consequently, there
exists a morphism f : C 0 → C in Cp(C) ; we will show that f × id∆1 is aq-Cartesian edge of M. To prove this, it suffices to show that for every n ≥ 2and every diagram
Λnn
_

G0

/M
=

G


/ ∆1


∆n

such that F0 |∆{n−1,n} = f × id∆1 , there exists a dotted arrow F : ∆n → M
as indicated, rendering the diagram commutative. We may identify G0 with amap g0 : Λnn → C. The composite map p◦g0 carries ∆{n−1,n} to a degenerateedge of D and therefore admits an extension g : ∆n → D. Consider the
diagram
Λnn
_

∆n

g0
g

/C
=
p

g


/ D.

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Since g0 carries the initial vertex v of ∆n to an initial object of the fiberCg(v) , Lemma 2.4.4.8 implies the existence of the indicated map g rendering
the diagram commutative. This gives rise to a map G : ∆n → M with thedesired properties and completes the proof that q is a Cartesian fibration.
We now wish to show that s ◦ p is associated to q. To prove this, it suffices
to prove the existence of a map H : C ×∆1 → C such that p ◦ H = p ◦ πC ,H| C ×{1} = idC , and H|C×{0} = s◦p. We construct the map H inductively,
working simplex by simplex on C. Suppose that we have a nondegenerate
simplex σ : ∆n → C and that H has already been defined on skn−1 C ×∆1 .
To define H ◦(σ×id∆1 ), we must solve a lifting problem that may be depicted
as follows:
`
h0
/2 C
(∂ ∆n × ∆1 ) ∂ ∆n ×∂ ∆1 (∆n × ∂ ∆1 )_h∆n × ∆1p/ D.

We now consider the filtration
X(n + 1) ⊆ X(n) ⊆ · · · ⊆ X(0) = ∆n × ∆1`
defined in the proof of Proposition 2.1.2.6. Let Y (i) = X(i) ∂ ∆n ×{0} (∆n ×{1}). For i > 0, the inclusion Y (i + 1) ⊆ Y (i) is a pushout of the inclusion
X(i + 1) ⊆ X(i) and therefore inner anodyne. Consequently, we may use theassumption that p is an inner fibration to extend h0 to a map defined on
Y (1). The inclusion Y (1) ⊆ ∆n × ∆1 is a pushout of ∂ ∆n+1 ⊆ ∆n+1 ; wethen obtain the desired extension h by applying Lemma 2.4.4.8.
Proposition 5.2.4.4. Let M be a fibrant simplicial category equipped witha functor p : M → ∆1 (here we identify ∆1 with the two-object category
whose nerve is ∆1 ), so that we may view M as a correspondence between
the simplicial categories C = p−1 {0} and D = p−1 {1}. The following are
equivalent(1) The map p is a Cartesian fibration.
(2) For every object D ∈ D, there exists a morphism f : C → D in M
which induces homotopy equivalences
MapC (C 0 , C) → MapM (C 0 , D)for every C 0 ∈ C.
Proof. This follows immediately from Proposition 2.4.1.10 since nonempty
morphism spaces in ∆1 are contractible.
Corollary 5.2.4.5. Let C and D be fibrant simplicial categories and let
Co

F
G

/D

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353

be a pair of adjoint functors F : C → D (in the sense of enriched category
theory, so that there is a natural isomorphism of simplicial sets
MapD (F (C), D) ' MapC (C, G(D))
for C ∈ C, D ∈ D). Then the induced functors
N(C) ofg/ N(D)

are also adjoint to one another.
Proof. Let M be the correspondence associated to the adjunction (F, G). In
other words, M is a simplicial category containing C and D as full (simplicial)subcategories, with
MapM (C, D) = MapC (C, G(D)) = MapD (F (C), D)
MapM (D, C) = ∅for every pair of objects C ∈ C, D ∈ D. Let M = N(M). Then M is acorrespondence between N(C) and N(D). By Proposition 5.2.4.4, it is an
adjunction. It is easy to see that this adjunction is associated to both f andg.
The following variant on the situation of Corollary 5.2.4.5 arises very often
in practice:
Proposition 5.2.4.6. Let A and A0 be simplicial model categories and let
Ao

F

/ A0

G

be a (simplicial) Quillen adjunction. Let M be the simplicial category defined
as in the proof of Corollary 5.2.4.5 and let M◦ be the full subcategory of M
consisting of those objects which are fibrant-cofibrant (either as objects of A
or as objects of A0 ). Then N(M◦ ) determines an adjunction between N(A◦ )◦and N(A0 ).
Proof. We need to show that N(M◦ ) → ∆1 is both a Cartesian fibration anda coCartesian fibration. We will argue the first point; the second follows froma dual argument. According to Proposition A.2.3.1, it suffices to show thatfor every fibrant-cofibrant object D of A0 , there is a fibrant-cofibrant object
C of A and a morphism f : C → D in M◦ which induces weak homotopy
equivalences
MapA (C 0 , C) → MapM (C 0 , D)for every fibrant-cofibrant object C 0 ∈ A. We define C to be a cofibrantreplacement for GD: in other words, we choose a cofibrant object C with atrivial fibration C → G(D) in the model category A. Then MapA (C 0 , C) →MapM (C 0 , D) = MapA (C 0 , G(D)) is a trivial fibration of simplicial sets,
whenever C 0 is a cofibrant object of A.

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Remark 5.2.4.7. Suppose that F : A → A0 and G : A0 → A are asin Proposition 5.2.4.6. We may associate to the adjunction N(M ◦ ) a pair◦◦of adjoint functors f : N(A◦ ) → N(A0 ) and g : N(A0 ) → N(A◦ ). In
this situation, f is often called a (nonabelian) left derived functor of F ,and g a (nonabelian) right derived functor of G. On the level of homotopy
categories, f and g reduce to the usual derived functors associated to the
Quillen adjunction (see §A.2.5).
5.2.5 Adjoint Functors and Overcategories
Our goal in this section is to prove the following result:
Proposition 5.2.5.1. Suppose we are given an adjunction of ∞-categories
F

Co

G

/D.

Assume that the ∞-category C admits pullbacks and let C be an object of C.
Then(1) The induced functor f : C/C → D/F C admits a right adjoint g.
(2) The functor g is equivalent to the composition
g0

g 00

D/F C → C/GF C → C/C ,where g 0 is induced by G and g 00 is induced by pullback along the unit
map C → GF C.
Proposition 5.2.5.1 is an immediate consequence of the following moregeneral result, which we will prove at the end of this section:
Lemma 5.2.5.2. Suppose we are given an adjunction between ∞-categories
F

Co

G

/D.

Let K be a simplicial set and suppose we are given a pair of diagrams p0 :K → C, p1 : K → D and a natural transformation h : F ◦ p0 → p1 . Assume
that C admits pullbacks and K-indexed limits. Then
(1) Let f : C/p0 → D/p1 denote the compositionh◦C/p0 → D/F p0 −→ D/p1 .
Then f admits a right adjoint g.
(2) The functor g is equivalent the composition
g0

g 00

D/p1 → C/Gp1 → C/p0 .
Here g 00 is induced by pullback along the natural transformation p0 →Gp1 adjoint to h (see below).

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355

PRESENTABLE AND ACCESSIBLE ∞-CATEGORIESWe begin by recalling a bit of notation which will be needed in the proof.
Suppose that q : X → S is an inner fibration of simplicial sets and pS :K → X is an arbitrary map, then we have defined a map of simplicial sets
X /pS → S, which is characterized by the following universal property: forevery simplicial set Y equipped with a map to S, there is a pullback diagram
HomS (Y, X /pS )/ HomS (Y S S, X)


{p}


/ HomS (S, X).

We refer the reader to §4.2.2 for a more detailed discussion.
Lemma 5.2.5.3. Let q : M → ∆1 be a coCartesian fibration of simplicial
sets classifying a functor F from C = M ×∆1 {0} to D = M ×∆1 {1}. Let K
be a simplicial set and suppose we are given a commutative diagramg∆1

K × ∆1

#

∆1 ,

/M
~

which restricts to give a pair of diagrams
g0

g1

C ← K → D.
Then(1) The projection q 0 : M/g∆1 → ∆1 is a coCartesian fibration of simplicial
sets classifying a functor F 0 : C/g0 → D/g1 . Moreover, an edge ofM/g∆1 is q 0 -coCartesian if and only if its image in M is q-coCartesian.
(2) Suppose that for every vertex k in K, the map g∆ carries {k}×∆1 to a
q-coCartesian morphism in M, so that g∆1 determines an equivalence
g1 ' F ◦ g0 . Then F 0 is homotopic to the composite functor
C/g0 → D/F g0 ' D/g1 .
(3) Suppose that M = D ×∆1 and that q is the projection onto the second
factor, so that we can identify F with the identity functor from D
to itself. Let g : K × ∆1 → D denote the composition g∆1 with the
projection map M → D, so that we can regard g as a morphism fromg0 to g1 in Fun(K, D). Then the functor F 0 : D/g0 → D/g1 is inducedby composition with g.
Proof. Assertion (1) follows immediately from Proposition 4.2.2.4.
We now prove (2). Since F is associated to the correspondence M, there
exists a natural transformation α : C ×∆1 → M from idC to F , such that foreach C 0 ∈ C the induced map αC : C 0 → F C 0 is q-coCartesian. Without lossof generality, we may assume that g∆1 is given by the compositiong0α

K × ∆1 → C ×∆1 → M .

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In this case, α induces a map α0 : C/g0 ×∆1 → M/g∆1 , which we may identify
with a natural transformation from idC/g0 to the functor C/g0 → D/F g0
determined by F . To show that this functor coincides with F 0 , it will suffice
to show that α0 carries each object of C/g0 to a q 0 -coCartesian morphism inM/g∆1 . This follows immediately from the description of the q 0 -coCartesianedges given in assertion (1).
We next prove (3). Consider the diagrampp0

D/g0 ← D/g → D/g1 .
By definition, “composition with g” refers to a functor from D/g0 to D/g1
obtained by composing p0 with a section to the trivial fibration p. To provethat this functor is homotopic to F 0 , it will suffice to show that F 0 ◦ p ishomotopic to p0 . For this, we must produce a map β : D/g ×∆1 → M/g∆1
from p to p0 , such that β carries each object of D/g to a q 0 -coCartesian edgeof M/g∆1 . We observe that D/g ×∆1 can be identified with M/h∆1 , whereh : ∆1 × ∆1 → M ' D ×∆1 is the product of g with the identity map.
We now take β to be the restriction map M/h∆1 → M/g∆1 induced by the
diagonal inclusion ∆1 ⊆ ∆1 × ∆1 . Using (1), we readily deduce that β has
the desired properties.
We will also need the following counterpart to Proposition 4.2.2.4:
Lemma 5.2.5.4. Suppose we are given a commutative diagram of simplicial
sets
K ×S

pS

/X

q

/Y

"  
S,
where the left diagonal arrow is projection onto the second factor and q is a
Cartesian fibration. Assume further that the following condition holds:(∗) For every vertex k ∈ K, the map pS carries each edge of {k} × S to aq-Cartsian edge in X.
0

Let p0S = q ◦ pS . Then the map q 0 : X /pS → Y /pS is a Cartesian fibration.
Moreover, an edge of X /pS is q 0 -Cartesian if and only if its image in X is
q-Cartesian.
Proof. To give the proof, it is convenient to use the language of marked
simplicial sets (see §3.1). Let X \ denote the marked simplicial set whose
underlying simplicial set is X, where we consider an edge of X \ to be marked\
if it is q-Cartesian. Let X denote the marked simplicial set whose underlying
simplicial set is X /pS , where we consider an edge to be marked if and only if
its image in X is marked. According to Proposition 3.1.1.6, it will suffice to0\show that the map X → (Y /pS )] has the right lifting property with respecthighertopoi

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIES

to every marked anodyne map i : A → B. Let A and B denote the simplicial
sets underlying A and B, respectively. Suppose we are given a diagram ofmarked simplicial sets
/ X\
<

A _
i


/ (Y /p0S )] .

BWe wish to show that there exists a dotted arrow rendering the diagram commutative. We begin by choosing a solution to the associated lifting problem
/ X\
A _
>


/ Y ],
B
which is possible in view of our assumption that q is a Cartesian fibration. To
extend this to a solution to the original problem, it suffices to solve anotherlifting problem`f
4/ X(A × K × ∆1 ) (A×K×∂ ∆1 ) (B × K × ∂ ∆1 )
_
q

j



/ Y.
B × K × ∆1
By construction, the map f induces a map of marked simplicial sets from
B × K [ × {0} to X \ . Using assumption (∗), we conclude that f also induces
a map of marked simplicial sets from B × K [ × {1} to X \ . Using Proposition
3.1.1.6 again (and our assumption that q is a Cartesian fibration), we are
reduced to proving that the map j induces a marked anodyne mapa(A × (K × ∆1 )[ )(B × (K × ∂ ∆1 )[ ) → B × (K × ∆1 )[ .
A×(K×∂ ∆1 )[Since i is marked anodyne by assumption, this follows immediately from
Proposition 3.1.2.3.
Lemma 5.2.5.5. Let q : M → ∆1 be a Cartesian fibration of simplicial sets
associated to a functor G from D = M ×∆1 {1} to C = M ×∆1 {0}. Suppose
we are given a simplicial set K and a commutative diagramg∆1

K × ∆1#∆1 ,so that g∆1 restricts to a pair of functors
g0

g1

/M
~

C ← K → D.
Suppose furthermore that, for every vertex k of K, the corresponding morphism g0 (k) → g1 (k) is q-Cartesian. Thenhighertopoi

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CHAPTER 5(1) The induced map q 0 : M/f∆1 → ∆1 is a Cartesian fibration. Moreover,an edge of M/f∆1 is q 0 -Cartesian if and only if its image in M is qCartesian.
(2) The associated functor D/g1 → C/g0 is homotopic to the compositionof the functor G0 : D/g1 → C/Gg1 induced by G and the equivalence
C/Gg1 ' C/g0 determined by the map g∆1 .
Proof. Assertion (1) follows immediately from Lemma 5.2.5.4. We will prove(2). Since the functor G is associated to q, there exists a map α : D ×∆1 → M
which is a natural transformation from G to idD , such that for every object
D ∈ D the induced map αD : {D} × ∆1 → M is a q-Cartesian edge of
M. Without loss of generality, we may assume that g coincides with thecomposition
g1

α

K × ∆1 → D ×∆1 → M .
In this case, α induces a map α0 : D/g1 ×∆1 → M/f∆1 , which is a naturaltransformation from G0 to the identity. Using (1), we deduce that α0 carrieseach object of D/g1 to a q 0 -Cartesian edge of M/f∆1 . It follows that α0 exhibits
G0 as the functor associated to the Cartesian fibration q 0 , as desired.

Proof of Lemma 5.2.5.2. Let q : M → ∆1 be a correspondence from C =M ×∆1 {0} to D = M ×∆1 {1}, which is associated to the pair of adjoint
functors F and G. The natural transformation h determines a map α :K × ∆1 → M, which is a natural transformation from p0 to p1 . Using thefact that q is both a Cartesian and a coCartesian fibration, we can form acommutative square σ
Gp
= 1
α

p0
ψ

φ

/ ! p1=!
F p0
in the ∞-category Fun(K, M), where the morphism φ is q-Cartesian and themorphism ψ is q-coCartesian.
Let N = M ×∆1 . We can identify σ with a map σ∆1 ×∆1 : K × ∆1 × ∆1 →M ×∆1 . Let N0 = N/σ∆1 ×∆1 . Proposition 4.2.2.4 implies that the projection
N0 → ∆1 × ∆1 is a coCartesian fibration associated to some diagram of

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIES∞-categories
f 00

C; /Gp1

f0

$

C/p0

/ D/p1:#
D/F p0 .
Lemma 5.2.5.3 allows us to identify the functors in the lower triangle, so
we see that the horizontal composition is homotopic to the functor f . To
complete the proof of (1), it will suffice to show that the functors f 0 and
f 00 admit right adjoints. To prove (2), it suffices to show that those rightadjoints are given by g 0 and g 00 , respectively. The adjointness of f 0 and g 0
follows from Lemma 5.2.5.5.
It follows from Lemma 5.2.5.3 that the functor f 00 : C/p0 → C/Gp1 is given
by composition with the transformation h0 : p0 → Gp1 which is adjoint toh. The pullback functor g 00 is right adjoint to f 00 by definition; the only
nontrivial point is to establish the existence of g 00 . Here we must use ourhypotheses on the ∞-category C. Let p0 : K / → C be a limit of p0 and let
Gp1 : K / → C be a limit of Gp1 . Let us identify h0 with a map K × ∆1 → C
0and choose an extension h : K / × ∆1 → C which is a natural transformationfrom p0 to Gp1 . Let C ∈ C denote the image under p0 of the cone point of
K / , let C 0 ∈ C denote the image under Gp1 of the cone point of K / , and0
let j : C → C 0 be the morphism induced by h . We have a commutative
diagram of ∞-categories:
C/pO 0 o
C/p0 o

f000

C/h
O

0

f100

0

C/h

/ C/Gp1
O/ C/Gp1


0/ C/C .
C/C o
C/j
In this diagram, the left horizontal arrows are trivial Kan fibrations, as are
all of the vertical arrows. The functor f 00 is obtained by composing f000 witha section to the trivial Kan fibration f100 . Utilizing the vertical equivalences,0
we can identify f 00 with the functor C/C → C/C given by composition with
j. But this functor admits a right adjoint in view of our assumption that C
admits pullbacks.
5.2.6 Uniqueness of Adjoint Functors
We have seen that if f : C → D is a functor which admits a right adjoint
g : D → C, then g is uniquely determined up to homotopy. Our next result

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is a slight refinement of this assertion.
Definition 5.2.6.1. Let C and D be ∞-categories. We let FunL (C, D) ⊆Fun(C, D) denote the full subcategory of Fun(C, D) spanned by those functors F : C → D which are left adjoints. Similarly, we define FunR (C, D) tobe the full subcategory of Fun(C, D) spanned by those functors which areright adjoints.
Proposition 5.2.6.2. Let C and D be ∞-categories. Then the ∞-categories
FunL (C, D) and FunR (D, C)op are (canonically) equivalent to one another.
Proof. Enlarging the universe if necessary, we may assume without loss ofgenerality that C and D are small. Let j : D → P(D) be the Yoneda embedding. Composition with j induces a fully faithful embeddingi : Fun(C, D)→ Fun(C, P(D)) ' Fun(C × Dop , S).
The essential image of i consists of those functors G : C × Dop → S withthe property that, for each C ∈ C, the induced functor GC : Dop → S isrepresentable by an object D ∈ D. The functor i induces a fully faithfulembeddingi0 : FunR (C, D) → Fun(C × Dop , S)
whose essential image consists of those functors G which belong to the essential image of i and furthermore satisfy the additional condition that foreach D ∈ D, the induced functor GD : C → S is corepresentable by an object C ∈ C (this follows from Proposition 5.2.4.2). Let E ⊆ Fun(C × Dop , S)
be the full subcategory spanned by those functors which satisfy these two
conditions, so that the Yoneda embedding induces an equivalence
FunR (C, D) → E .
We note that the above conditions are self-dual, so that the same reasoning
gives an equivalence of ∞-categories
FunR (Dop , Cop ) → E .
We now conclude by observing that there is a natural equivalence of ∞categories FunR (Dop , Cop ) ' FunL (D, C)op .
We will later need a slight refinement of Proposition 5.2.6.2, which exhibits
some functoriality in C. We begin with a few preliminary remarks concerning
the construction of presheaf ∞-categories.
Let f : C → C0 be a functor between small ∞-categories. Then composition
with f induces a restriction functor G : P(C0 ) → P(C). However, there isanother slightly less evident functoriality of the construction C 7→ P(C).
Namely, according to Theorem 5.1.5.6, there is a colimit-preserving functor
P(f ) : P(C) → P(C0 ), uniquely determined up to equivalence, such that the
diagram
C

P(C)

f

P(f )

/ C0

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commutes up to homotopy (here the vertical arrows are given by the Yoneda
embeddings).
The functor P(f ) has an alternative characterization in the language ofadjoint functors:
Proposition 5.2.6.3. Let f : C → C0 be a functor between small ∞categories and let G : P(C0 ) → P(C) be the functor given by composition
with f . Then G is right adjoint to P(f ).
Proof. We first prove that G admits a left adjoint. Let
e : P(C) → Fun(P(C)op , bS)denote the Yoneda embedding. According to Proposition 5.2.4.2, it will suffice to show that for each M ∈ P(C), the composite functor e(M ) ◦ G is
corepresentable. Let D denote the full subcategory of P(C) spanned by thoseobjects M such that G ◦ eM is corepresentable. Since P(C) admits small
colimits, Proposition 5.1.3.2 implies that the collection of corepresentablefunctors on P(C) is stable under small colimits. According to Propositions
5.1.3.2 and 5.1.2.2, the functor M 7→ e(M )◦G preserves small colimits. It follows that D is stable under small colimits in P(C). Since P(C) is generated
under small colimits by the Yoneda embedding jC : C → P(C) (Corollary
5.1.5.8), it will suffice to show that jC (C) ∈ D for each C ∈ C. According
to Lemma 5.1.5.2, e(jC (C)) is equivalent to the functor P(C) → b
S given byevaluation at C. Then e(jC (C)) ◦ G is equivalent to the functor given byevaluation at f (C) ∈ C0 , which is corepresentable (Lemma 5.1.5.2 again).
We conclude that G has a left adjoint F .
To complete the proof, we must show that F is equivalent to P(f ). To
prove this, it will suffice to show that F preserves small colimits and thatthe diagram
C

P(C)

f

F

/ C0

/ P(C0 )commutes up to homotopy. The first point is obvious: since F is a left adjoint,
it preserves all colimits which exist in P(C) (Proposition 5.2.3.5). For thesecond, choose a counit map v : F ◦ G → idP(C0 ) . By construction, thefunctor f induces a natural transformation u : jC → G ◦ jC0 ◦ f . To complete
the proof, it will suffice to show that the compositionuv

θ : F ◦ j C → F ◦ G ◦ j C0 ◦ f → j C0 ◦ f
is an equivalence of functors from C to P(C0 ). Fix objects C ∈ C, M ∈ P(C0 ).

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We have a commutative diagram
MapP(C0 ) (jC0 (f (C)), M )MapP(C0 ) (jC0 (f (C)), M )
MapP(C) (G(jC0 (f (C))), G(M ))/ MapP(C0 ) (F (G(jC0 (f (C)))), M )
MapP(C) (jC (C), G(M ))/ MapP(C0 ) (F (jC (C)), M )in the homotopy category H of spaces, where the vertical arrows are isomorphisms. Consequently, to prove that the lower horizontal composition is
an isomorphism, it suffices to prove that the upper horizontal composition
is an isomorphism. Using Lemma 5.1.5.2, we reduce to the assertion that
M (f (C)) → (G(M ))(C) is an isomorphism in H, which follows immediatelyfrom the definition of G.
Remark 5.2.6.4. Suppose we are given a functor f : D → D0 which admits
a right adjoint g. Let E ⊆ Fun(C × Dop , S) and E0 ⊆ Fun(C ×(D0 )op , S) be
defined as in the proof of Proposition 5.2.6.2 and consider the diagram
FunR (C, D)◦gFunR (C, D0 )/Eo

FunL (D, C)op
/ E0 o
FunL (D0 , C)op .

◦f

Here the middle vertical map is given by composition with idC ×f . The
square on the right is manifestly commutative, but the square on the left
commutes only up to homotopy. To verify the second point, we observe thatthe square in question is given by applying the functor Map(C, •) to thediagramDg
D0/ P(D)G/ P(D0 ),where G is given by composition with f and the horizontal arrows are givenby the Yoneda embedding. Let P0 (D) ⊆ P(D) and P0 (D0 ) denote the essential images of the Yoneda embeddings. Proposition 5.2.4.2 asserts that G
carries P0 (D0 ) into P0 (D), so that it will suffice to verify that the diagram
D
g


D0

/ P0 (D)G0


/ P0 (D0 )highertopoi

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is homotopy commutative. In view of Proposition 5.2.2.6, it will suffice toshow that G0 admits a left adjoint F 0 and that the diagram/ P0 (D)O

DO

F0/ P0 (D0 )D0is homotopy commutative. According to Proposition 5.2.6.3, the functor G
has a left adjoint P(f ) which fits into a commutative diagram/ P(D)
O

DO
f

P(f )

D0

/ P(D0 ).

In particular, P(f ) carries P0 (D) into P0 (D0 ) and therefore restricts to givea left adjoint F 0 : P0 (D) → P0 (D0 ) which verifies the desired commutativity.
We conclude this section by establishing the following consequence of
Proposition 5.2.6.3:
Corollary 5.2.6.5. Let C be a small ∞-category and D a locally small∞-category which admits small colimits. Let F : P(C) → D be a colimitpreserving functor, let f : C → D denote the composition of F with the
Yoneda embedding of C, and let G : D → P(C) be the functor given by thecomposition
j0◦f

D → Fun(Dop , S) → P(C).
Then G is a right adjoint to F . Moreover, the map
f = F ◦ j → (F ◦ (G ◦ F )) ◦ j = (F ◦ G) ◦ f
exhibits F ◦ G as a left Kan extension of f along itself.
The proof requires a few preliminaries:
Lemma 5.2.6.6. Suppose we are given a pair of adjoint functors
Co

f
g

/D

between ∞-categories. Let T : C → X be any functor. Then T ◦ g : D → X isa left Kan extension of T along f .
Proof. Let p : M → ∆1 be a correspondence associated to the pair of adjoint
functors f and g. Choose a p-Cartesian homotopy h from r to idM , where r
is a functor from M to C; thus r| D is homotopic to g. It will therefore suffice
to show that the compositionr

T

T :M→C→X

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is a left Kan extension of T | C ' T . For this, we must show that for each
D ∈ D, the functor T exhibits T (D) as a colimit of the diagram
T(C ×M M/D ) → M → X .
We observe that C ×M M/D has a final object given by any p-Cartesianmorphism e : C → D. It therefore suffices to show that T (e) is an equivalencein X, which follows immediately from the construction of T .
Lemma 5.2.6.7. Let f : C → C0 be a functor between small ∞-categories
and X an ∞-category which admits small colimits. Let H : P(C) → X be afunctor which preserves small colimits and h : C → X the composition of H
with the Yoneda embedding jC : C → P(C). Then the composition
j 0

◦f

H

C
C0 →
P(C0 ) → P(C) → X

is a left Kan extension of h along f .
Proof. Let G : P(C0 ) → P(C) be the functor given by composition with f . Inview of Proposition 4.3.2.8, it will suffice to show that H ◦ G is a left Kan
extension of h along jC ◦ f .
Theorem 5.1.5.6 implies the existence of a functor F : P(C) → P(C0 ) which
preserves small colimits, such that F ◦ jC ' jC0 ◦ f . Moreover, Lemma 5.1.5.5
ensures that F is a left Kan extension of f along the fully faithful Yoneda
embedding jC . Using Proposition 4.3.2.8 again, we are reduced to provingthat H ◦ G is a left Kan extension of H along F . This follows immediatelyfrom Proposition 5.2.6.3 and Lemma 5.2.6.6.
Proof of Corollary 5.2.6.5. The first claim follows from Proposition 5.2.6.3.
To prove the second, we may assume without loss of generality that D is
minimal, so that D is a union of small full subcategories {Dα }. It will suffice
to show that, for each index α such that f factors through Dα , the restrictedtransformation f → ((F ◦ G)| Dα ) ◦ f exhibits (F ◦ G)| Dα as a left Kan
extension of f along the induced map C → Dα , which follows from Lemma
5.2.6.7.
5.2.7 Localization Functors
Suppose we are given an ∞-category C and a collection S of morphisms of C
which we would like to invert. In other words, we wish to find an ∞-category
S −1 C equipped with a functor η : C → S −1 C which carries each morphismin S to an equivalence and is in some sense universal with respect to theseproperties. One can give a general construction of S −1 C using the formalismof §3.1.1. Without loss of generality, we may suppose that S contains all theidentity morphisms in C. Consequently, the pair (C, S) may be regarded asa marked simplicial set, and we can choose a marked anodyne map (C, S) →(S −1 C, S 0 ), where S −1 C is an ∞-category and S 0 is the collection of allequivalences in S −1 C. However, this construction is generally very difficult

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to analyze, and the properties of S −1 C are very difficult to control. For
example, it might be the case that C is locally small and S −1 C is not.
Under suitable hypotheses on S (see §5.5.4), there is a drastically simpler
approach: we can find the desired ∞-category S −1 C inside C as the fullsubcategory of S-local objects of C.
Example 5.2.7.1. Let C be the (ordinary) category of abelian groups, let
p be a prime number, and let S denote the collection of morphisms f whose
kernel and cokernel consist entirely of p-power torsion elements. A morphism
f lies in S if and only if it induces an isomorphism after inverting the prime
number p. In this case, we may identify S −1 C with the full subcategory of C
consisting of those abelian groups which are uniquely p-divisible. The functor
C → S −1 C is given by1
M 7→ M ⊗Z Z[ ].
p
In Example 5.2.7.1, the functor C → S −1 C is actually left adjoint to an
inclusion functor. We will take this as our starting point.
Definition 5.2.7.2. A functor f : C → D between ∞-categories is a localization if f has a fully faithful right adjoint.
Warning 5.2.7.3. Let f : C → D be a localization functor and let S denotethe collection of all morphisms α in C such that f (α) is an equivalence. Then,for any ∞-category E, composition with f induces a fully faithful functor◦f

Fun(D, E) → Fun(C, E)
whose essential image consists of those functors p : C → E which carry eachα ∈ S to an equivalence in E (Proposition 5.2.7.12). We may describe thesituation more informally by saying that D is obtained from C by inverting
the morphisms of S.
Some authors use the term “localization” in a more general sense to describe any functor f : C → D in which D is obtained by inverting somecollection S of morphisms in C. Such a morphism f need not be a localization in the sense of Definition 5.2.7.2; however, it is in many cases (seeProposition 5.5.4.15).
If f : C → D is a localization of ∞-categories, then we will also say that
D is a localization of C. In this case, a right adjoint g : D → C of f gives anequivalence between D and a full subcategory of C (the essential image of g).
We let L : C → C denote the composition g ◦ f . We will abuse terminology
by referring to L as a localization functor if it arises in this way.
The following result will allow us to recognize localization functors:
Proposition 5.2.7.4. Let C be an ∞-category and let L : C → C be a functor
with essential image L C ⊆ C. The following conditions are equivalent:(1) There exists a functor f : C → D with a fully faithful right adjoint
g : D → C and an equivalence between g ◦ f and L.

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CHAPTER 5(2) When regarded as a functor from C to L C, L is a left adjoint of the
inclusion L C ⊆ C.
(3) There exists a natural transformation α : C ×∆1 → C from idC to L
such that, for every object C of C, the morphisms L(α(C)), α(LC) :LC → LLC of C are equivalences.
Proof. It is obvious that (2) implies (1) (take D = L C, f = L, and g to
be the inclusion). The converse follows from the observation that, since g isfully faithful, we are free to replace D by the essential image of g (which isequal to the essential image of L).
We next prove that (2) implies (3). Let α : idC → L be a unit for the
adjunction. Then, for each pair of objects C ∈ C, D ∈ L C, composition with
α(C) induces a homotopy equivalence
MapC (LC, D) → MapC (C, D)
and, in particular, a bijection HomhC (LC, D) → HomhC (C, D). If C belongs
to L C, then Yoneda’s lemma implies that α(C) is an isomorphism in hC.
This proves that α(LC) is an equivalence for every C ∈ C. Since α is anatural transformation, we obtain a diagram
C

α(C)

α(C)

/ LC
Lα(C)

 α(LC) / LLC.
LC
Since composition with α(C) gives an injective map from HomhC (LC, LLC)
to HomhC (C, LLC), we conclude that α(LC) is homotopic to Lα(C); in
particular, α(LC) is also an equivalence. This proves (3).
Now suppose that (3) is satisfied; we will prove that α is the unit of an
adjunction between C and L C. In other words, we must show that for eachC ∈ C and D ∈ C, composition with α(C) induces a homotopy equivalence
φ : MapC (LC, LD) → MapC (C, LD).
By Yoneda’s lemma, it will suffice to show that for every Kan complex K,
the induced map
HomH (K, MapC (LC, LD)) → HomH (K, MapC (C, LD))is a bijection of sets, where H denotes the homotopy category of spaces.
Replacing C by Fun(K, C), we are reduced to proving the following:(∗) Suppose that α : idC → L satisfies (3). Then, for every pair of objects
C, D ∈ C, composition with α(C) induces a bijection of sets
φ : HomhC (LC, LD) → HomhC (C, LD).
We first show that φ is surjective. Let f be a morphism from C to LD.
We then have a commutative diagram
f

C
α(C)


LC

Lf

/ LD
α(LD)


/ LLD,

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so that f is homotopic to the composition (α(LD)−1 ◦Lf )◦α(C); this provesthat the homotopy class of f lies in the image of φ.
We now show that φ is injective. Let g : LC → LD be an arbitrary
morphism. We have a commutative diagram
LC

g

α(LC)


LLC

Lg

/ LD
α(LD)


/ LLD,

so that g is homotopic to the compositionα(LD)−1 ◦ Lg ◦ α(LC) ' α(LD)−1 ◦ Lg ◦ Lα(C) ◦ (Lα(C))−1 ◦ α(LC)' α(LD)−1 ◦ L(g ◦ α(C)) ◦ (Lα(C))−1 ◦ α(LC).
In particular, g is determined by g ◦ α(C) up to homotopy.
Remark 5.2.7.5. Let L : C → D be a localization functor and K a simplicial
set. Suppose that every diagram p : K → C admits a colimit in C. Then the∞-category D has the same property. Moreover, we can give an explicit
prescription for computing colimits in D. Let q : K → D be a diagram andlet p : K → C be the composition of q with a right adjoint to L. Choose acolimit p : K . → C. Since L is a left adjoint, L ◦ p is a colimit diagram in D,
and L ◦ p is equivalent to the diagram q.
We conclude this section by introducing a few ideas which will allow us torecognize localization functors when they exist.
Definition 5.2.7.6. Let C be an ∞-category and C0 ⊆ C a full subcategory.
We will say that a morphism f : C → D in C exhibits D as a C0 -localizationof C if D ∈ C0 , and composition with f induces an isomorphism
MapC0 (D, E) → MapC (C, E)
in the homotopy category H for each object E ∈ C0 .
Remark 5.2.7.7. In the situation of Definition 5.2.7.6, a morphism f : C →D exhibits D as a localization of C if and only if f is an initial object of the∞-category C0C/ = CC/ ×C C0 . In particular, f is uniquely determined up toequivalence.
Proposition 5.2.7.8. Let C be an ∞-category and C0 ⊆ C a full subcategory.
The following conditions are equivalent:(1) For every object C ∈ C, there exists a localization f : C → D relative
to C0 .
(2) The inclusion C0 ⊆ C admits a left adjoint.

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Proof. Let D be the full subcategory of C ×∆1 spanned by objects of theform (C, i), where C ∈ C0 if i = 1. Then the projection p : D → ∆1 isa correspondence from C to C0 which is associated to the inclusion functor
i : C0 ⊆ C. It follows that i admits a left adjoint if and only if p is a
coCartesian fibration. It now suffices to observe that if C is an object ofC, then we may identify p-coCartesian edges f : (C, 0) → (D, 1) of D withlocalizations C → D relative to C0 .
Remark 5.2.7.9. By analogy with classical category theory, we will saythat a full subcategory C0 of an ∞-category C is a reflective subcategory if
the hypotheses of Proposition 5.2.7.8 are satisfied by the inclusion C0 ⊆ C.
Example 5.2.7.10. Let C be an ∞-category which has a final object and
let C0 be the full subcategory of C spanned by the final objects. Then the
inclusion C0 ⊆ C admits a left adjoint.
Corollary 5.2.7.11. Let p : C → D be a coCartesian fibration between ∞categories, let D0 ⊆ D be a full subcategory and let C0 = C ×D D0 . If the
inclusion D0 ⊆ D admits a left adjoint, then the inclusion C0 ⊆ C admits aleft adjoint.
Proof. In view of Proposition 5.2.7.8, it will suffice to show that for every
object C ∈ C, there a morphism f : C → C0 which is a localization of C
relative to C0 . Let D = p(C), let f : D → D0 be a localization of D relative
to D0 , and let f : C → C0 be a p-coCartesian morphism in C lifting f . We
claim that f has the desired property. Choose any object C 0 ∈ C0 and let
D0 = p(C 0 ) ∈ D0 . We obtain a diagram of spaces
MapC (C0 , C 0 )MapD (D0 , D0 )φψ

/ MapC (C, C 0 )/ MapD (D, D0 )which commutes up to preferred homotopy. By assumption, the map ψ is a
homotopy equivalence. Since f is p-coCartesian, the map φ induces a homotopy equivalence after passing to the homotopy fibers over any pair of pointsη ∈ MapD (D0 , D0 ), ψ(η) ∈ MapD (D, D0 ). Using the long exact sequence of
homotopy groups associated to the vertical fibrations, we conclude that φ isa homotopy equivalence, as desired.
Proposition 5.2.7.12. Let C be an ∞-category and let L : C → C be a
localization functor with essential image L C. Let S denote the collection of
all morphisms f in C such that Lf is an equivalence. Then, for every ∞category D, composition with L induces a fully faithful functor
ψ : Fun(L C, D) → Fun(C, D).
Moreover, the essential image of ψ consists of those functors F : C → D
such that F (f ) is an equivalence in D for each f ∈ S.

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Proof. Let S0 be the collection of all morphisms C → D in C which exhibit
D as an L C-localization of C. We first claim that, for any functor F : C → D,
the following conditions are equivalent:(a) The functor F is a right Kan extension of F |L C.
(b) The functor F carries each morphism in S0 to an equivalence in D.
(c) The functor F carries each morphism in S to an equivalence in D.
The equivalence of (a) and (b) follows immediately from the definitions(since a morphism f : C → D exhibits D as an L C-localization of C if andonly if f is an initial object of (L C) ×C CC/ ), and the implication (c) ⇒ (b)is obvious. To prove that (b) ⇒ (c), let us consider any map f : C → D
which belongs to S. We have a commutative diagram
f

C

/ LC
Lf



/ LD.
D
Since f ∈ S, the map Lf is an equivalence in C. If F satisfies (b), then F
carries each of the horizontal maps to an equivalence in D. It follows fromthe two-out-of-three property that F f is an equivalence in D as well, so that
F satisfies (c).
Let Fun0 (C, D) denote the full subcategory of Fun(C, D) spanned by thosefunctors which satisfy (a), (b), and (c). Using Proposition 4.3.2.15, we deducethat the restriction functor φ : Fun0 (C, D) → Fun(L C, D) is fully faithful.
We now observe that ψ is a right homotopy inverse to φ. It follows that φis essentially surjective and therefore an equivalence. Being right homotopy
inverse to an equivalence, the functor ψ must itself be an equivalence.
5.2.8 Factorization Systems
Let f : X → Z be a map of sets. Then f can be written as a composition
f0

f 00

X → Y → Z,
000
where f is surjective and f is injective. This factorization is uniquely determined up to (unique) isomorphism: the set Y can be characterized either
as the image of the map f or as the quotient of X by the equivalence relation R = {(x, y) ∈ X 2 : f (x) = f (y)}. We can describe the situation
formally by saying that the collections of surjective and injective maps forma factorization system on the category Set of sets (see Definition 5.2.8.8).
In this section, we will describe a theory of factorization systems in the ∞categorical setting. These ideas are due to Joyal, and we refer the reader to[44] for further details.
Definition 5.2.8.1. Let f : A → B and g : X → Y be morphisms in an∞-category C. We will say that f is left orthogonal to g (or that g is rightorthogonal to f ) if the following condition is satisfied:highertopoi

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CHAPTER 5(∗) For every commutative diagram
/X

A
f


B


/Y

g

in C, the mapping space MapCA/ /Y (B, X) is contractible. (Here weabuse notation by identifying B and X with the corresponding objectsof CA/ /Y .)In this case, we will write f ⊥ g.
Remark 5.2.8.2. More informally, a morphism f : A → B in an ∞-category
C is left orthogonal to another morphism g : X → Y if, for every commutative
diagram
A
f


B

/X
?
g


/ Y,

the space of dotted arrows which render the diagram commutative is contractible.
Remark 5.2.8.3. Let f : A → B and g : X → Y be morphisms in an ∞category C. Fix a morphism A → Y , which we can identify with an objecte ∈ CA/ /Y is equivalent toY ∈ CA/ . Lifting g : X → Y to an object of X
lifting g to a morphism g : X → Y in CA/ . The map f : A → B determinese ∈ CA/ /Y is equivalent toan object B ∈ CA/ , and lifting f to an object B
giving a map h : B → Y in CA/ . We therefore have a fiber sequence of spacese X)
e → MapC (B, X) → MapC (B, Y ),MapCA/ /Y (B,A/A/
where the fiber is taken over the point h. Consequently, condition (∗) ofDefinition 5.2.8.1 can be reformulated as follows: for every morphism g :X → Y in CA/ lifting g, composition with g induces a homotopy equivalence
MapCA/ (B, X) → MapCA/ (B, Y ).
Notation 5.2.8.4. Let C be an ∞-category and let S be a collection ofmorphisms in C. We let S ⊥ denote the collection of all morphisms in C
which are right orthogonal to S, and ⊥ S the collection of all morphisms in
C which are left orthogonal to S.
Remark 5.2.8.5. Let C be an ordinary category containing a pair of morphisms f and g. If f ⊥ g, then f has the left lifting property with respect
to g, and g has the right lifting property with respect to f . It follows thatfor any collection S of morphisms in C, we have inclusions S ⊥ ⊆ S⊥ and⊥S ⊆⊥ S, where the latter classes of morphisms are defined in §A.1.2.

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Applying Remark 5.2.8.3 to an ∞-category C and its opposite, we obtainthe following result:
Proposition 5.2.8.6. Let C be an ∞-category and S a collection of morphisms in C.
(1) The sets of morphisms S ⊥ and ⊥ S contain every equivalence in C.
(2) The sets of morphisms S ⊥ and ⊥ S are closed under the formation ofretracts.
(3) Suppose we are given a commutative diagram
>Y
f

g


/Z

h

X

in C, where g ∈ S ⊥ . Then f ∈ S ⊥ if and only if h ∈ S ⊥ . In particular,S ⊥ is closed under composition.
(4) Suppose we are given a commutative diagram
>Y
f

g


/Z

h

X

in C, where f ∈ ⊥ S. Then g ∈ ⊥ S if and only if h ∈ ⊥ S. In particular,⊥
S is closed under composition.
(5) The set of morphisms S ⊥ is stable under pullbacks: that is, given apullback diagram
X0
g0


Y0

/X

/Y

g

in C, if g belongs to S ⊥ , then g 0 belongs to S ⊥ .
(6) The set of morphisms ⊥ S is stable under pushouts: that is, given a
pushout diagram
A
f


B

if f belongs to ⊥ S, then so does f 0 .

/ A0
f0


/ B0,

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CHAPTER 5(7) Let K be a simplicial set such that C admits K-indexed colimits. Thenthe full subcategory of Fun(∆1 , C) spanned by the elements of ⊥ S isclosed under K-indexed colimits.
(8) Let K be a simplicial set such that C admits K-indexed limits. Thenthe full subcategory of Fun(∆1 , C) spanned by the elements of S ⊥ isclosed under K-indexed limits.
Remark 5.2.8.7. Suppose we are given a pair of adjoint functors
F

Co

G

/D .

Let f be a morphism in C and g a morphism in D. Then f ⊥ G(g) if andonly if F (f ) ⊥ g.
Definition 5.2.8.8 (Joyal). Let C be an ∞-category. A factorization system
on C is a pair (SL , SR ), where SL and SR are collections of morphisms of C
which satisfy the following axioms:(1) The collections SL and SR are stable under the formation of retracts.
(2) Every morphism in SL is left orthogonal to every morphism in SR .
(3) For every morphism h : X → Z in C, there exists a commutative
triangle
?Y
f

g


/ Z,

h

X
where f ∈ SL and g ∈ SR .

We will call SL the left set of the factorization system and SR the right setof the factorization system.
Example 5.2.8.9. Let C be an ∞-category. Then C admits a factorization
system (SL , SR ), where SL is the collection of all equivalences in C and SR
consists of all morphisms of C.
Remark 5.2.8.10. Let (SL , SR ) be a factorization system on an ∞-category
C. Then (SR , SL ) is a factorization system on the opposite ∞-category Cop .
Proposition 5.2.8.11. Let C be an ∞-category and let (SL , SR ) be a factorization system on C. Then SL = ⊥ SR and SR = SL⊥ .
Proof. By symmetry, it will suffice to prove the first assertion. The inclusion
SL ⊆ ⊥ SR follows immediately from the definition. To prove the reverse
inclusion, let us suppose that h : X → Z is a morphism in C which is leftorthogonal to every morphism in SR . Choose a commutative triangle
>Y
f

X

g
h



/Z

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where f ∈ SL and g ∈ SR , and consider the associated diagram
X

f

/Y
>
g

h



id /
Z
Z.
Since h ⊥ g, we can complete this diagram to a 3-simplex of C as indicated.
This 3-simplex exhibits h as a retract of f , so that h ∈ SL , as desired.
Remark 5.2.8.12. It follows from Proposition 5.2.8.11 that a factorization
system (SL , SR ) on an ∞-category C is completely determined by either theleft set SL or the right set SR .
Corollary 5.2.8.13. Let C be an ∞-category and let (SL , SR ) be a factorization system on C. Then the collections of morphisms SL and SR containall equivalences and are stable under composition.
Proof. Combine Propositions 5.2.8.11 and 5.2.8.6.
Remark 5.2.8.14. It follows from Corollary 5.2.8.13 that a factorization
system (SL , SR ) on C determines a pair of subcategories CL , CR ⊆ C, eachcontaining all the objects of C: the morphisms of CL are the elements of SL ,and the morphisms of CR are the elements of SR .
Example 5.2.8.15. Let p : C → D be a coCartesian fibration of ∞categories. Then there is an associated factorization system (SL , SR ) on C,
where SL is the class of p-coCartesian morphisms of C and SR is the classof morphisms g of C such that p(g) is an equivalence in D. If D ' ∆0 , thisrecovers the factorization system of Example 5.2.8.9; if p is an isomorphism,
this recovers the opposite of the factorization system of Example 5.2.8.9.
Example 5.2.8.16. Let X be an ∞-topos and let n ≥ −2 be an integer.
Then there exists a factorization system (SL , SR ) on X, where SL denotes
the collection of (n + 1)-connective morphisms of X and SR denotes thecollection of n-truncated morphisms of X. See §6.5.1.
Let (SL , SR ) be a factorization system on an ∞-category C, so that any
morphism h : X → Z factors as a composition
?Y
f

X

g
h


/ Z,

where f ∈ SL and g ∈ SR . For many purposes, it is important to know that
this factorization is canonical. More precisely, we have the following result:
Proposition 5.2.8.17. Let C be an ∞-category and let SL and SR be collections of morphisms in C. Suppose that SL and SR are stable under equivalence in Fun(∆1 , C) and contain every equivalence in C. The following conditions are equivalent:highertopoi

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CHAPTER 5(1) The pair (SL , SR ) is a factorization system on C.
(2) The restriction map p : Fun0 (∆2 , C) → Fun(∆{0,2} , C) is a trivial Kan
fibration. Here Fun0 (∆2 , C) denotes the full subcategory of Fun(∆2 , C)spanned by those diagrams
>Y
f

g



/Z

h

X
such that f ∈ SL and g ∈ SR .

Corollary 5.2.8.18. Let C be an ∞-category equipped with a factorization system (SL , SR ) and let K be an arbitrary simplicial set. Then theK∞-category Fun(K, C) admits a factorization system (SLK , SR), where SLK
denotes the collection of all morphisms f in Fun(K, C) such that f (v) ∈ SLK
for each vertex v of K, and SR
is defined likewise.
The remainder of this section is devoted to the proof of Proposition
5.2.8.17. We begin with a few preliminary results.
Lemma 5.2.8.19. Let C be an ∞-category and let (SL , SR ) be a factorization system on C. Let D be the full subcategory of Fun(∆1 , C) spanned by theelements of SR . Then
(1) The ∞-category D is a localization of Fun(∆1 , C); in other words, theinclusion D ⊆ Fun(∆1 , C) admits a left adjoint.
(2) A morphism α : h → g in Fun(∆1 , C) corresponding to a commutative
diagram
/Y

f

X

g

h



e
/Z
Z0
exhibits g as a D-localization of h (see Definition 5.2.7.6) if and only
if g ∈ SR , f ∈ SL , and e is an equivalence.
Proof. We will prove the “if” direction of assertion (2). It follows from thedefinition of a factorization system that for every object h ∈ Fun(∆1 , C),
there exists a morphism α : h → g satisfying the condition stated in (2),which therefore exhibits g as a D-localization of h. Invoking Proposition
5.2.7.8, we will deduce (1). Because a D-localization of h is uniquely determined up to equivalence, we will also deduce the “only if” direction ofassertion (2).
Suppose we are given a commutative diagram
X

f

g

h


Z0

/Y

e


/ Z,

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where f ∈ SL , g ∈ SR , and e is an equivalence, and let g : Y → Z be anotherelement of SR . We have a diagram of spaces
MapFun(∆1 ,C) (g, g)
MapC (Z, Z)ψ/ MapFun(∆1 ,C) (h, g)/ MapC (Z 0 , Z)

ψ0which commutes up to canonical homotopy. We wish to prove that ψ is a
homotopy equivalence.
Since e is an equivalence in C, the map ψ0 is a homotopy equivalence. It will
therefore suffice to show that ψ induces a homotopy equivalence after passing
to the homotopy fibers over any point of MapC (Z, Z) ' MapC (Z 0 , Z). These
homotopy fibers can be identified with the homotopy fibers of the verticalarrows in the diagram
MapC (Y, Y )/ MapC (X, Y )
MapC (Y, Z)/ MapC (X, Z).

It will therefore suffice to show that this diagram (which commutes up to
specified homotopy) is a homotopy pullback. Unwinding the definition, thisis equivalent to the assertion that f is left orthogonal to g, which is part ofthe definition of a factorization system.
Lemma 5.2.8.20. Let K, A, and B be simplicial sets. Then the diagram
K ×B/ K × (A ? B)B/ (K × A) ? B

is a homotopy pushout square of simplicial sets (with respect to the Joyalmodel structure).
Proof. We consider the larger diagram
K ×B/ K × (A  B)

/ K × (A ? B)B/ (K × A)  B

/ (K × A) ? B.

The square on the left is a pushout square in which the horizontal maps
are monomorphisms of simplicial sets and therefore is a homotopy pushoutsquare (since the Joyal model structure is left proper). The square on theright is a homotopy pushout square since the horizontal arrows are both categorical equivalences (Proposition 4.2.1.2). It follows that the outer rectangle
is also a homotopy pushout as desired.

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Notation 5.2.8.21. In the arguments which follow, we let Q denote thesimplicial subset of ∆3 spanned by all simplices which do not contain ∆{1,2} .
Note that Q is isomorphic to the product ∆1 × ∆1 as a simplicial set.
Lemma 5.2.8.22. Let C be an ∞-category and let σ : Q → C be a diagram,which we depict as
/X
A

B


/ Y.

Then there is a canonical categorical equivalenceθ : Fun(∆3 , C) ×Fun(Q,C) {σ} → MapCA/ /Y (B, X).
In particular, Fun(∆3 , C) ×Fun(Q,C) {σ} is a Kan complex.
Proof. We will identify MapCA/ /Y (B, X) with the simplicial set Z definedby the following universal property: for every simplicial set K, we have apullback diagram of sets
HomSet∆ (K, Z)/ HomSet∆ (∆0 ? (K × ∆1 ) ? ∆0 , C)∆0
/ HomSet∆ (∆0 ? (K × ∂ ∆1 ) ? ∆0 , C).

The map θ is then induced by the natural transformation
K × ∆3 ' K × (∆0 ? ∆1 ? ∆0 ) → ∆0 ? (K × ∆1 ) ? ∆0 .
We wish to prove that θ is a categorical equivalence. Since C is an ∞category, it will suffice to show that for every simplicial set K, the bottom
square of the diagram``/ ∆{0} ∆{3}
K × (∆{0} ∆{3} )

K ×C


/ ∆0 ? (K × ∂ ∆1 ) ? ∆0K × ∆3/ ∆0 ? (K × ∆1 ) ? ∆0

is a homotopy pushout square (with respect to the Joyal model structure).
For this we need only verify that the top and outer squares are homotopy
pushout diagrams; this follows from repeated application of Lemma 5.2.8.20.
Proof of Proposition 5.2.8.17. We first show that (1) ⇒ (2). Assume that
(SL , SR ) is a factorization system on C. The restriction map
p : Fun0 (∆2 , C) → Fun(∆{0,2} , C)highertopoi

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIES

is obviously a categorical fibration. It will therefore suffice to show that p isa categorical equivalence.
Let D be the full subcategory of Fun(∆1 × ∆1 , C) spanned by those diagrams of the form
/Y

f

X

g

h


Z0


/ Z,

e

where f ∈ SL , g ∈ SR , and e is an equivalence in C. The map p factors as acomposition
p0

p00

Fun0 (∆2 , C) → D → Fun(∆1 , C),
where p0 carries a diagram
>Y
f

g


/Z

h

X
to the partially degenerate square
X
h


Z

/Y

f
h
id

g


/Z

and p00 is given by restriction to the left vertical edge of the diagram. To
complete the proof, it will suffice to show that p0 and p00 are categoricalequivalences.
We first show that p0 is a categorical equivalence. The map p0 admits a left
inverse q given by composition with an inclusion ∆2 ⊆ ∆1 × ∆1 . We notethat q is a pullback of the restriction map q 0 : Fun00 (∆2 , C) → Fun(∆{0,2} , C),
where Fun00 (∆2 , C) is the full subcategory spanned by diagrams of the form
X

Z0

e

/ Z,

where e is an equivalence. Since q 0 is a trivial Kan fibration (Proposition
4.3.2.15), q is a trivial Kan fibration, so that p0 is a categorical equivalence,
as desired.
We now complete the proof by showing that p00 is a trivial Kan fibration.
Let E denote the full subcategory of Fun(∆1 , C) × ∆1 spanned by those pairs(g, i) where either i = 0 or g ∈ SR . The projection map r : E → ∆1 is
a Cartesian fibration associated to the inclusion Fun0 (∆1 , C) ⊆ Fun(∆1 , C),highertopoi

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where Fun0 (∆1 , C) is the full subcategory spanned by the elements of SR .
Using Lemma 5.2.8.19, we conclude that r is also a coCartesian fibration.
Moreover, we can identify
D ⊆ Fun(∆1 × ∆1 , C) ' Map∆1 (∆1 , E)with the full subcategory spanned by the coCartesian sections of r. In termsof this identification, p00 is given by evaluation at the initial vertex {0} ⊆ ∆1
and is therefore a trivial Kan fibration, as desired. This completes the proof
that (1) ⇒ (2).
Now suppose that (2) is satisfied and choose a section s of the trivial Kan
fibration p. Let s carry each morphism f : X → Z to a commutative diagram>YsL (f )sR (f )f/ Z.
X
If sR (f ) is an equivalence, then f is equivalent to sL (f ) and therefore belongs
to SL . Conversely, if f belongs to SL , then the diagram
>Z
f

id


f
/Z
X
is a preimage of f under p and therefore equivalent to s(f ); this implies thatsL (f ) is an equivalence. We have proved the following:(∗) A morphism f of C belongs to SL if and only if sL (f ) is an equivalence
in C.
It follows immediately from (∗) that SL is stable under the formation ofretracts; similarly, SR is stable under the formation of retracts. To complete
the proof, it will suffice to show that f ⊥ g whenever f ∈ SL and g ∈ SR .
Fix a commutative diagram σ
/X
A
f

g



/Y
B
in C. In view of Lemma 5.2.8.22, it will suffice to show that the Kan complex
Fun(∆3 , C) ×Fun(Q,C) {σ} is contractible.
Let D denote the full subcategory of Fun(∆2 × ∆1 , C) spanned by thosediagrams
/Z
C
u0


C0

u00


C 00

v0


/ Z0

v 00


/ Z 00highertopoi

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379

for which u0 ∈ SL , v 00 ∈ SR , and the maps v 0 and u00 are equivalences. Let us
identify ∆3 with the full subcategory of ∆2 ×∆1 spanned by all those vertices
except for (2, 0) and (0, 1). Applying Proposition 4.3.2.15 twice, we deducethat the restriction functor Fun(∆2 × ∆1 , C) → Fun(∆3 , C) induces a trivial
Kan fibration from D to the full subcategory D0 ⊆ Fun(∆3 , C) spanned bythose diagrams
/ Z0
C
=
u0

v 00



/ Z 00C0such that u0 ∈ SL and v 00 ∈ SR . It will therefore suffice to show that thefiber D ×Fun(Q,C) {σ} is contractible.
By construction, the restriction functor D → Fun(Q, C) is equivalent tothe compositionq : D ⊆ Fun(∆2 × ∆1 , C) → Fun(∆{0,2} × ∆1 , C).
It will therefore suffice to show that q −1 {σ} is a contractible Kan complex.
Invoking assumption (2) and (∗), we deduce that q induces an equivalencefrom D to the full subcategory of Fun(∆{0,2} × ∆1 , C) spanned by thosediagrams
/Z
C
u

v



/ Z 00C 00
such that u ∈ SL and v ∈ SR . The desired result now follows from ourassumption that f ∈ SL and g ∈ SR .

5.3 ∞-CATEGORIES OF INDUCTIVE LIMITSLet C be a category. An Ind-object of C is a diagram f : I → C where I is asmall filtered category. We will informally denote the Ind-object f by[lim Xi ]−→where Xi = f (i). The collection of all Ind-objects of C forms a category inwhich the morphisms are given by the formula
HomInd(C) ([lim Xi ], [lim Yj ]) = lim lim HomC (Xi , Yj ).
−→
−→
←− −→
We note that C may be identified with a full subcategory of Ind(C) corresponding to diagrams indexed by the one-point category I = ∗. The idea isthat Ind(C) is obtained from C by formally adjoining colimits of filtered diagrams. More precisely, Ind(C) may be described by the following universal
property: for any category D which admits filtered colimits and any functor
F : C → D, there exists a functor Fe : Ind(C) → D whose restriction to C isisomorphic to F and which commutes with filtered colimits. Moreover, Fe isdetermined up to (unique) isomorphism.

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Example 5.3.0.1. Let C denote the category of finitely presented groups.
Then Ind(C) is equivalent to the category of groups. (More generally, one
could replace “group” by any type of mathematical structure described byalgebraic operations which are required to satisfy equational axioms.)
Our objective in this section is to generalize the definition of Ind(C) to thecase where C is an ∞-category. If we were to work in the setting of simplicial(or topological) categories, we could apply the definition given above directly.
However, this leads to a number of problems:(1) The construction of Ind-categories does not preserve equivalences between simplicial categories.
(2) The obvious generalization of the right hand side in the equation above
is given bylim lim MapC (Xi , Yj ).
←− −→While the relevant limits and colimits certainly exist in the category ofsimplicial sets, they are not necessarily the correct objects: one should
really replace the limit by a homotopy limit.
(3) In the higher-categorical setting, we should really allow the indexing
diagram I to be a higher category as well. While this does not resultin any additional generality (Corollary 5.3.1.18), the restriction to thediagrams indexed by ordinary categories is a technical inconvenience.
Although these difficulties are not insurmountable, it is far more convenient to proceed differently using the theory of ∞-categories. In §5.1, we
showed that if C is a ∞-category, then P(C) can be interpreted as an ∞category which is freely generated by C under colimits. We might therefore
hope to find Ind(C) inside P(C) as a full subcategory. The problem, then,is to characterize this subcategory and to prove that it has the appropriate
universal mapping property.
We will begin in §5.3.1 by introducing the definition of a filtered ∞category. Let C be a small ∞-category. In §5.3.5, we will define Ind(C) to be
the smallest full subcategory of P(C) which contains all representable presheaves on C and is stable under filtered colimits. There is also a more direct
characterization of which presheaves F : C → Sop belong to Ind(C): they are
precisely the right exact functors, which we will study in §5.3.2.
In §5.3.5, we will define the Ind-categories Ind(C) and study their properties. In particular, we will show that morphism spaces in Ind(C) are computedby the naive formula
HomInd(C) ([lim Xi ], [lim Yj ]) = lim lim HomC (Xi , Yj ).
−→
−→
←− −→
Unwinding the definitions, this amounts to two conditions:(1) The (Yoneda) embedding of j : C → Ind(C) is fully faithful (Proposition 5.1.3.1).

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381(2) For each object C ∈ C, the corepresentable functor
HomInd(C) (j(C), •)commutes with filtered colimits.
It is useful to translate condition (2) into a definition: an object D of an∞-category D is said to be compact if the functor D → S corepresented by
D commutes with filtered colimits. We will study this compactness conditionin §5.3.4.
One of our main results asserts that the ∞-category Ind(C) is obtained
from C by freely adjoining colimits of filtered diagrams (Proposition 5.3.5.10).
In §5.3.6, we will describe a similar construction in the case where the classof filtered diagrams has been replaced by any class of diagrams. We willrevisit this idea in §5.5.8, where we will study the ∞-category obtained fromC by freely adjoining colimits of sifted diagrams.
5.3.1 Filtered ∞-Categories
Recall that a partially ordered set A is filtered if every finite subset of A has
an upper bound in A. Diagrams indexed by directed partially ordered setsare extremely common in mathematics. For example, if A is the set
Z≥0 = {0, 1, . . .}of natural numbers, then a diagram indexed by A is a sequence
X0 → X1 → · · · .
The formation of direct limits for such sequences is one of the most basic
constructions in mathematics.
In classical category theory, it is convenient to consider not only diagrams
indexed by filtered partially ordered sets but also more general diagrams
indexed by filtered categories. A category C is said to be filtered if it satisfies
the following conditions:(1) For every finite collection {Xi } of objects of C, there exists an object
X ∈ C equipped with morphisms φi : Xi → X.
(2) Given any two morphisms f, g : X → Y in C, there exists a morphism
h : Y → Z such that h ◦ f = h ◦ g.
Condition (1) is analogous to the requirement that any finite part of C
admits an “upper bound,” while condition (2) guarantees that the upperbound is unique in some asymptotic sense.
If we wish to extend the above definition to the ∞-categorical setting, it
is natural to strengthen the second condition.
Definition 5.3.1.1. Let C be a topological category. We will say that C isfiltered if it satisfies the following conditions:highertopoi

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CHAPTER 5(10 ) For every finite set {Xi } of objects of C, there exists an object X ∈ C
and morphisms φi : Xi → X.
(20 ) For every pair X, Y ∈ C of objects of C, every nonnegative integern ≥ 0, and every continuous map S n → MapC (X, Y ), there exists amorphism Y → Z such that the induced map S n → MapC (X, Z) is
nullhomotopic.
Remark 5.3.1.2. It is easy to see that an ordinary category C is filteredin the usual sense if and only if it is filtered when regarded as a topologicalcategory with discrete mapping spaces. Conversely, if C is a filtered topological category, then its homotopy category hC is filtered (when viewed as an
ordinary category).
Remark 5.3.1.3. Condition (20 ) of Definition 5.3.1.1 is a reasonable analogue of condition (2) in the definition of a filtered category. In the special
case n = 0, condition (20 ) asserts that any pair of morphisms f, g : X → Y
become homotopic after composition with some map Y → Z.
Remark 5.3.1.4. Topological spheres S n need not play any distinguishedrole in the definition of a filtered topological category. Condition (20 ) is
equivalent to the following apparently stronger condition:(200 ) For every pair X, Y ∈ C of objects of C, every finite cell complex K,
and every continuous map K → MapC (X, Y ), there exists a morphism
Y → Z such that the induced map K → MapC (X, Z) is nullhomotopic.
Remark 5.3.1.5. The condition that a topological category C be filtereddepends only on the homotopy category hC (viewed as an H-enriched category). Consequently, if F : C → C0 is an equivalence of topological categories,
then C is filtered if and only if C0 is filtered.
Remark 5.3.1.6. Definition 5.3.1.1 has an obvious analogue for (fibrant)
simplicial categories: one simply replaces the topological n-sphere S n by thesimplicial n-sphere ∂ ∆n . It is easy to see that a topological category C isfiltered if and only if the simplicial category Sing C is filtered. Similarly,a (fibrant) simplicial category D is filtered if and only if the topologicalcategory | D | is filtered.
We now wish to study the analogue of Definition 5.3.1.1 in the settingof ∞-categories. It will be convenient to introduce a slightly more general
notion:Definition 5.3.1.7. Let κ be a regular cardinal and let C be a ∞-category.
We will say that C is κ-filtered if, for every κ-small simplicial set K andevery map f : K → C, there exists a map f : K . → C extending f . (Inother words, C is κ-filtered if it has the extension property with respect tothe inclusion K ⊆ K . for every κ-small simplicial set K.)
We will say that C is filtered if it is ω-filtered.

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383

Example 5.3.1.8. Let C be the nerve of a partially ordered set A. Then C
is κ-filtered if and only if every κ-small subset of A has an upper bound in
A.
Example 5.3.1.9. Let C is an ∞-category for which the inclusion C ,→ C.
admits a left inverse. Then C is κ-filtered for every regular cardinal κ. In
particular, the ∞-category Idem of Definition 4.4.5.2 is κ-filtered, for every
regular cardinal κ (see Lemma 4.4.5.9).
Remark 5.3.1.10. Let κ be a regular cardinal and suppose that C is an ∞category which admits small κ-filtered colimits. Combining Example 5.3.1.9
with Corollary 4.4.5.15, we deduce that C is idempotent complete.
Remark 5.3.1.11. One may rephrase Definition 5.3.1.7 as follows: an ∞category C is κ-filtered if and only if, for every diagram p : K → C, where
K is κ-small, the slice ∞-category Cp/ is nonempty. Let q : C → C0 be acategorical equivalence of ∞-categories. Proposition 1.2.9.3 asserts that theinduced map Cp/ → C0q◦p/ is a categorical equivalence. Consequently, Cp/ isnonempty if and only if C0q◦p/ is nonempty. It follows that C is κ-filtered if
and only if C0 is κ-filtered.
Remark 5.3.1.12. An ∞-category C is κ-filtered if and only if, for every
κ-small partially ordered set A, C has the right lifting property with respect
to the inclusion N(A) ⊆ N(A). ' N(A ∪ {∞}). The “only if” direction is
obvious. For the converse, we observe that for every κ-small diagram p : K →C, the ∞-category Cp/ is equivalent to Cq/ , where q denotes the compositionp0pN(A) → K → C. Here we have chosen p0 to be a cofinal map such that A
is a κ-small partially ordered set. (If κ is uncountable, the existence of p0
follows from Variant 4.2.3.15; otherwise, we use Variant 4.2.3.16.)
Remark 5.3.1.13. We will say that an arbitrary simplicial set S is κ-filteredif there exists a categorical equivalence j : S → C, where C is a κ-filtered∞-category. In view of Remark 5.3.1.11, this condition is independent of the
choice of j.
Our next major goal is to prove Proposition 5.3.1.15, which asserts thatan ∞-category C is filtered if and only if the associated topological category| C[C]| is filtered. First, we need a lemma.
Lemma 5.3.1.14. Let C be an ∞-category. Then C is filtered if and only if it
has the right extension property with respect to every inclusion ∂ ∆n ⊆ Λn+1
n+1 ,n ≥ 0.
Proof. The “only if” direction is clear: we simply take K = ∂ ∆n in Definition 5.3.1.7. For the converse, let us suppose that the assumption of Definition 5.3.1.7 is satisfied whenever K is the boundary of a simplex; we mustthen show that it remains satisfied for any K which has only finitely many
nondegenerate simplices.

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We work by induction on the dimension of K and the number of nondegenerate simplices of K. If K is empty, there is nothing to prove `(since it is
the boundary of a 0-simplex). Otherwise, we may write K = K 0 ∂ ∆n ∆n ,where n is the dimension of K.
Choose a map p : K → C; we wish to show that p may be extended
to a map pe : K ? {y} → C. We first consider the restriction p|K 0 ; by the
inductive hypothesis, it admits an extension q : K 0 ?{x} → C. The restriction
q| ∂ ∆n ? {x} and the map p|∆n assemble to give a map
ar : ∂ ∆n+1 ' (∂ ∆n ? {x})∆n → C .
∂ ∆n

By assumption, the map r admits an extension
re : ∂ ∆n+1 ? {y} → C .
Letas : (K 0 ? {x})(∂ ∆n+1 ? {y})∂ ∆n ?{x}denote the result of amalgamating r with pe. We note that the inclusionaa(K 0 ?{x})(∂ ∆n+1 ?{y}) ⊆ (K 0 ?{x}?{y})(∂ ∆n+1 ?{y})∂ ∆n ?{x}∂ ∆n ?{x}?{y}is a pushout of(K 0 ? {x})a(∂ ∆n ? {x} ? {y}) ⊆ K 0 ? {x} ? {y}∂ ∆n ?{x}and therefore a categorical equivalence by Lemma 2.1.2.3. It follows that s
admits an extensionase : (K 0 ? {x} ? {y})(∆n ? {y}) → C,
∂ ∆n ?{x}?{y}and we may now define pe = se|K ? {y}.
Proposition 5.3.1.15. Let C be a topological category. Then C is filtered if
and only if the ∞-category N(C) is filtered.
Proof. Suppose first that N(C) is filtered. We verify conditions (10 ) and (20 )of Definition 5.3.1.1:(10 ) Let {Xi }i∈I be a finite collection of objects of C corresponding to a map
p : I → N(C), where I is regarded as a discrete simplicial set. If N(C)is filtered, then p extends to a map pe : I ? {x} → N(C) corresponding
to an object X = p(x) equipped with maps Xi → X in C.
(20 ) Let X, Y ∈ C be objects, let n ≥ 0, and let S n → MapC (X, Y ) bea map. We note that this data may be identified with a topological
functor F : | C[K]| → C, where K is the simplicial set obtained from
∂ ∆n+2 by collapsing the initial face ∆n+1 to a point. If N(C) is filtered,then F extends to a functor Fe defined on | C[K ? {z}]|; this gives anobject Z = Fe(z) and a morphism Y → Z such that the induced map
S n → MapC (X, Z) is nullhomotopic.

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For the converse, let us suppose that C is filtered. We wish to show that
N(C) is filtered. By Lemma 5.3.1.14, it will suffice to prove that N(C) has theextension property with respect to the inclusion ∂ ∆n ⊆ Λn+1
n+1 for each n ≥ 0.
Equivalently, it suffices to show that C has the right extension property withrespect to the inclusion | C[∂ ∆n ]| ⊆ | C[Λn+1
n+1 ]|. If n = 0, this is simply theassertion that C is nonempty; if n = 1, this is the assertion that for any
pair of objects X, Y ∈ C, there exists an object Z equipped with morphisms
X → Z, Y → Z. Both of these conditions follow from part (1) of Definition
5.3.1.1; we may therefore assume that `n > 1.
Let A0 = | C[∂ ∆n ]|, A1 = | C[∂ ∆n Λnn Λnn ? {n + 1}]|, A2 = | C[Λn+1
n+1 ]|,and A3 = | C[∆n+1 ]|, so that we have inclusions of topological categories
A0 ⊆ A1 ⊆ A2 ⊆ A3 .
We will make use of the description of A3 given in Remark 1.1.5.2: its
objects are integers i satisfying 0 ≤ i ≤ n + 1, with MapA3 (i, j) given by
the cube of all functions p : {i, . . . , j} → [0, 1] satisfying p(i) = p(j) = 1 fori ≤ j and HomA3 (i, j) = ∅ for j < i. Composition is given by amalgamationof functions.
We note that A1 and A2 are subcategories of A3 having the same objects,
whose morphism spaces are may be described as follows:• MapA1 (i, j) = MapA2 (i, j) = MapA3 (i, j) unless i = 0 and j ∈ {n, n +1}.
• MapA1 (0, n) = MapA2 (0, n) is the boundary of the cube
MapA3 (0, n) ' [0, 1]n−1 .
• MapA1 (0, n + 1) consists of all functions p : [n + 1] → [0, 1] satisfyingp(0) = p(n + 1) = 1 and (∃i)[(1 ≤ i ≤ n − 1) ∧ p(i) ∈ {0, 1}].
• MapA2 (0, n + 1) is the union of MapA1 (0, n + 1) with the collection offunctions p : {0, . . . , n + 1} → [0, 1] satisfying p(0) = p(n) = p(n + 1) =1.
Finally, we note that A0 is the full subcategory of A1 (or A2 ) whose setof objects is {0, . . . , n}.
We wish to show that any topological functor F : A0 → C can be extended
to a functor Fe : A2 → C. Let X = F (0) and let Y = F (n). Then F induces
a map S n−1 ' MapA0 (0, n) → MapC (X, Y ). Since C is filtered, there exists
a map φ : Y → Z such that the induced map f : S n−1 → MapC (X, Z) is
nullhomotopic.
Now set Fe(n + 1) = Z; for p ∈ MapA1 (i, n + 1), we set Fe(p) = φ ◦ F (q),where q ∈ MapA1 (i, n) is such that q|{i, . . . , n−1} = p|{i, . . . , n−1}. Finally,
we note that the assumption that f is nullhomotopic allows us to extend Fe
from MapA1 (0, n + 1) to the whole of MapA2 (0, n + 1).

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Remark 5.3.1.16. Suppose that C is a κ-filtered ∞-category and let K bea simplicial set which is categorically equivalent to a κ-small simplicial set.
Then C has the extension property with respect to the inclusion K ⊆ K . .
This follows from Proposition A.2.3.1: to test whether or not a map K →S extends over K . , it suffices to check in the homotopy category of Set∆
(with respect to the Joyal model structure), where we may replace K by anequivalent κ-small simplicial set.
Remark 5.3.1.17. Let C be a ∞-category with a final object. Then C isκ-filtered for every regular cardinal κ.
A useful observation from classical category theory is that, if we are onlyinterested in using filtered categories to index colimit diagrams, then in fact
we do not need the notion of a filtered category at all: we can work insteadwith diagrams indexed by filtered partially ordered sets. We now prove an∞-categorical analogue of this statement.
Proposition 5.3.1.18. Suppose that C is a κ-filtered ∞-category. Then
there exists a κ-filtered partially ordered set A and a cofinal map N(A) → C.
Proof. The proof uses the ideas introduced in §4.2.3 and, in particular,
Proposition 4.2.3.8. Let X be a set of cardinality at least κ, and regard
X as a category with a unique isomorphism between any pair of objects.
We note that N(X) is a contractible Kan complex; consequently, the projection C × N(X) → C is cofinal. Hence, it suffices to produce a cofinal map
N(A) → C × N(X) with the desired properties.
Let {Kα }α∈A be the collection of all simplicial subsets of K = C × N(X)
which are κ-small and possess a final vertex. Regard A as S
a partially orderedby inclusion. We first claim that A is κ-filtered and that α∈A Kα = K. To
prove both of these assertions, it suffices to show that any κ-small simplicial
subset L ⊆ K is contained in a κ-small simplicial subset L0 which has a final
vertex.
Since C is κ-filtered, the composition
L → C × N(X) → C
extends to a map p : L. → C. Since X has cardinality at least κ, there exists
an element x ∈ X which is not in the image of L0 → N (X)0 = X. Liftp to a map pe : L. → K which extends the inclusion L ⊆ K × N(X) andcarries the cone point to the element x ∈ X = N (X)0 . It is easy to see that
pe is injective, so that we may regard L. as a simplicial subset of K × N(X).
Moreover, it is clearly κ-small and has a final vertex, as desired.
Now regard A as a category and let F : A → (Set∆ )/K be the functor
which carries each α ∈ A to the simplicial set Kα . For each α ∈ A, choose
a final vertex xα of Kα . Let KF be defined as in §4.2.3. We claim next that
there exists a retraction r : KF → K with the property that r(Xα ) = xα foreach I ∈ I.
The construction of r proceeds as in the proof of Proposition 4.2.3.4.
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0
r|KB
by induction on B. Moreover, we will select r so that it has the property
0that if B is nonempty with largest element β, then r(KB
) ⊆ Kβ .
0
If B is empty, then r|KB = r|K is the identity map. Otherwise, B hasa least element α and a largest element β. We are required to construct a
map Kα ? ∆B → Kβ or a map rB : ∆B → Kid |Kα / , where the values of thismap on ∂ ∆B have already been determined. If B is a singleton, we define
this map to carry the vertex ∆B to a final object of Kid |Kα / lying over xβ .
Otherwise, we are guaranteed that some extension exists by the fact thatrB | ∂ ∆B carries the final vertex of ∆B to a final object of Kid |Kα / .
Now let j : N(A) → K denote the restriction of the retraction of r to
N(A). Using Propositions 4.2.3.4 and 4.2.3.8, we deduce that j is a cofinal
map as desired.

A similar technique can be used to prove the following characterization of
κ-filtered ∞-categories:
Proposition 5.3.1.19. Let S be a simplicial set. The following conditions
are equivalent:(1) The simplicial set S is κ-filtered.
(2) There exists a diagram of simplicial sets {Yα }α∈I having colimit Y anda categorical equivalence S → Y , where each Yα is κ-filtered and theindexing category I is κ-filtered.
(3) There exists a categorical equivalence S → C, where C is a κ-filteredunion of simplicial subsets Cα ⊆ C such that each Cα is an ∞-category
with a final object.
Proof. Let T : Set∆ → Set∆ be the fibrant replacement functor given by
T (X) = N(| C[X]|).
There is a natural transformation jX : X → T (X) which is a categorical equivalence for every simplicial set X. Moreover, each T (X) is an ∞category. Furthermore, the functor T preserves inclusions and commuteswith filtered colimits.
It is clear that (3) implies (2). Suppose that (2) is satisfied. Replacing thediagram {Yα }α∈I by {T (Yα )}α∈I if necessary, we may suppose that each Yα
is an ∞-category. It follows that Y is an ∞-category. If p : K → Y is adiagram indexed by a κ-small simplicial set, then p factors through a map
pα : K → Yα for some α ∈ I, by virtue of the assumption that I is κ-filtered.
Since Yα is a κ-filtered ∞-category, we can find an extension K . → Yα of
pα , hence an extension K . → Y of p.
Now suppose that (1) is satisfied. Replacing S by T (S) if necessary, we
may suppose that S is an ∞-category. Choose a set X of cardinality at leastκ and let N(X) be defined as in the proof of Proposition 5.3.1.18. The proofof Proposition 5.3.1.18 shows that we may write S × N(X) as a κ-filteredunion of simplicial subsets {Yα }, where each Yα has a final vertex. We now

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take C = T (S × N(X)) and let Cα = T (Yα ): these choices satisfy (3), which
completes the proof.
By definition, an ∞-category C is κ-filtered if any map p : K → C whose
domain K is κ-small can be extended over the cone K . . We now consider
the possibility of constructing this extension uniformly in p. First, we needa few lemmas.
Lemma 5.3.1.20. Let C be a filtered ∞-category. Then C is weakly contractible.
Proof. Since C is filtered, it is nonempty. Fix an object C ∈ C. Let | C | denotethe geometric realization of C as a simplicial set. We identify C with a pointof the topological space | C |. By Whitehead’s theorem, to show that C isweakly contractible, it suffices to show that for every i ≥ 0, the homotopy
set πi (| C |, C) consists of a single point. If not, we can find a finite simplicial
subset K ⊆ C containing C such that the map f : πi (|K|, C) → πi (| C |, C)
has a nontrivial image. But C is filtered, so the inclusion K ⊆ C factors
through a map K . → C. It follows that f factors through πi (|K . |, C). But
this homotopy set is trivial since K . is weakly contractible.
Lemma 5.3.1.21. Let C be a κ-filtered ∞-category and let p : K → C be adiagram indexed by a κ-small simplicial set K. Then Cp/ is κ-filtered.
Proof. Let K 0 be a κ-small simplicial set and p0 : K 0 → Cp/ a κ-small
diagram. Then we may identify p0 with a map q : K ? K 0 → C, and we get
an isomorphism (Cp/ )p0 / ' Cq/ . Since K ? K 0 is κ-small, the ∞-category Cq/
is nonempty.
Proposition 5.3.1.22. Let C be an ∞-category and κ a regular cardinal.
Then C is κ-filtered if and only if, for each κ-small simplicial set K, thediagonal map d : C → Fun(K, C) is cofinal.
Proof. Suppose first that the diagonal map d : C → Fun(K, C) is cofinal forevery κ-small simplicial set K. Choose any map j : K → C; we wish to showthat j can be extended to K . . By Proposition A.2.3.1, it suffices to show thatj can be extended to the equivalent simplicial set K  ∆0 . In other words,
we must produce an object C ∈ C and a morphism j → d(C) in Fun(K, C).
It will suffice to prove that the ∞-category D = C ×Fun(K,C) Fun(K, C)j/
is nonempty. We now invoke Theorem 4.1.3.1 to deduce that D is weakly
contractible.
Now suppose that C is κ-filtered and that K is a κ-small simplicial set.
We wish to show that the diagonal map d : C → Fun(K, C) is cofinal. By
Theorem 4.1.3.1, it suffices to prove that for every object X ∈ Fun(K, C),
the ∞-category Fun(K, C)X/ ×Fun(K,C) C is weakly contractible. But if we
identify X with a map x : K → C, then we get a natural identification
Fun(K, C)X/ ×Fun(K,C) C ' Cx/ ,which is κ-filtered by Lemma 5.3.1.21 and therefore weakly contractible byLemma 5.3.1.20.

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5.3.2 Right Exactness
Let A and B be abelian categories. In classical homological algebra, a functor
F : A → B is said to be right exact if it is additive, and wheneverA0 → A → A00 → 0
is an exact sequence in A, the induced sequence
F (A0 ) → F (A) → F (A00 ) → 0
is exact in B.
The notion of right exactness generalizes in a natural way to functors
between categories which are not assumed to be abelian. Let F : A → B
be a functor between abelian categories as above. Then F is additive if andonly if F preserves finite coproducts. Furthermore, an additive functor F
is right exact if and only if it preserves coequalizer diagrams. Since every
finite colimit can be built out of finite coproducts and coequalizers, rightexactness is equivalent to the requirement that F preserve all finite colimits.
This condition makes sense whenever the category A admits finite colimits.
It is possible to generalize even further to the case of a functor between
arbitrary categories. To simplify the discussion, let us suppose that B =Setop . Then we may regard a functor F : A → B as a presheaf of sets on
the category A. Using this presheaf, we can define a new category AF whose
objects are pairs (A, η), where A ∈ A and η ∈ F (A), and morphisms from(A, η) to (A0 , η 0 ) are maps f : A → A0 such that f ∗ (η 0 ) = η, where f ∗ denotes
the induced map F (A0 ) → F (A). If A admits finite colimits, then the functor
F preserves finite colimits if and only if the category AF is filtered.
Our goal in this section is to adapt the notion of right exact functors to
the ∞-categorical context. We begin with the following:
Definition 5.3.2.1. Let F : A → B be a functor between ∞-categories andκ a regular cardinal. We will say that F is κ-right exact if, for any rightfibration B0 → B where B0 is κ-filtered, the ∞-category A0 = A ×B B0 is
also κ-filtered. We will say that F is right exact if it is ω-right exact.
Remark 5.3.2.2. We also have an dual theory of left exact functors.
Remark 5.3.2.3. If A admits finite colimits, then a functor F : A → B isright exact if and only if F preserves finite colimits (see Proposition 5.3.2.9
below).
We note the following basic stability properties of κ-right exact maps.
Proposition 5.3.2.4. Let κ be a regular cardinal.
(1) If F : A → B and G : B → C are κ-right exact functors between∞-categories, then G ◦ F : A → C is κ-right exact.
(2) Any equivalence of ∞-categories is κ-right exact.

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CHAPTER 5(3) Let F : A → B be a κ-right exact functor and let F 0 : A → B be
homotopic to F . Then F 0 is κ-right exact.
Proof. Property (1) is immediate from the definition. We will establish (2)and (3) as a consequence of the following more general assertion: if F : A →B and G : B → C are functors such that F is a categorical equivalence, then
G is κ-right exact if and only if G ◦ F is κ-right exact. To prove this, let
C0 → C be a right fibration. Proposition 3.3.1.3 implies that the induced map
A0 = A ×C C0 → B ×C C0 = B0
is a categorical equivalence. Thus A0 is κ-filtered if and only if B0 is κ-filtered.
We now deduce (2) by specializing to the case where G is the identity
map. To prove (3), we choose a contractible Kan complex K containing apair of vertices {x, y} and a map g : K → BA with g(x) = F , g(y) = F 0 .
Applying the above argument to the compositionG

A ' A ×{x} ⊆ A ×K → B,
we deduce that G is κ-right exact. Applying the converse to the diagramG

A ' A ×{y} ⊆ A ×K → B,
we deduce that F 0 is κ-right exact.
The next result shows that the κ-right exactness of a functor F : A → B
can be tested on a very small collection of right fibrations B0 → B.
Proposition 5.3.2.5. Let F : A → B be a functor between ∞-categoriesand κ a regular cardinal. The following are equivalent:(1) The functor F is κ-right exact.
(2) For every object B of B, the ∞-category A ×B B/B is κ-filtered.
Proof. We observe that for every object B ∈ B, the ∞-category B/B is rightfibered over B and is κ-filtered (since it has a final object). Consequently, (1)implies (2). Now suppose that (2) is satisfied. Let T : (Set∆ )/ B → (Set∆ )/ B
denote the composite functor
St(Set∆ )/ B →B (Set∆ )C[Bop

] Sing |•|→(Set∆ )C[B

op

] UnB

→ (Set∆ )/ B .

We will use the following properties of T :(i) There is a natural transformation jX : X → T (X), where jX is acontravariant equivalence in (Set∆ )/ B for every X ∈ (Set∆ )/ B .
(ii) For every X ∈ (Set∆ )/ B , the associated map T (X) → B is a rightfibration.
(iii) The functor T commutes with filtered colimits.
We will say that an object X ∈ (Set∆ )/ B is good if the ∞-category T (X) ×B
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391(A) If X → Y is a contravariant equivalence in (Set∆ )/ B , then X is good
if and only if Y is good. This follows from the fact that T (X) →T (Y ) is an equivalence of right fibrations, so that the induced map
T (X) ×B A → T (Y ) ×B A is an equivalence of right fibrations andconsequently a categorical equivalence of ∞-categories.
(B) If X → Y is a categorical equivalence in (Set∆ )/ B , then X is good if
and only if Y is good. This follows from (A) since every categoricalequivalence is a contravariant equivalence.
(C) The collection of good objects of (Set∆ )B is stable under κ-filteredcolimits. This follows from the fact that the functor X 7→ T (X) ×B A
commutes with κ-filtered colimits (in fact, with all filtered colimits)and Proposition 5.3.1.19.
(D) If X ∈ (Set∆ )/ B corresponds to a right fibration X → B, then X isgood if and only if X ×B A is κ-filtered.
(E) For every object B ∈ B, the overcategory B/B is a good object of(Set∆ )/ B . In view of (D), this is equivalent to assumption (2).
(F ) If X consists of a single vertex x, then X is good. To see this, let B ∈ B
denote the image of X. The natural map X → B/B can be identified
with the inclusion of a final vertex; this map is right anodyne andtherefore a contravariant equivalence. We now conclude by applying(A) and (E).
(G) If X ∈ (Set∆ )/ B is an ∞-category with a final object x, then X is good.
To prove this, we note that {x} is good by (F ) and the inclusion {x} ⊆X is right anodyne, hence a contravariant equivalence. We conclude by
applying (A).
(H) If X ∈ (Set∆ )/ B is κ-filtered, then X is good. To prove this, we apply
Proposition 5.3.1.19 to deduce the existence of a categorical equivalence i : X → C, where C is a κ-filtered union of ∞-categories withfinal objects. Replacing C by C ×K if necessary, where K is a contractible Kan complex, we may suppose that i is a cofibration. Since
B is an ∞-category, the lifting problem
/B
?

S
i


C

has a solution. Thus we may regard C as an object of (Set∆ )/ B . According to (B), it suffices to show that C is good. But C is a κ-filteredcolimit of good objects of (Set∆ )B (by (G)) and is therefore itself good(by (C)).

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Now let B0 → B be a right fibration, where B0 is κ-filtered. By (H), B0 is
a good object of (Set∆ )/ B . Applying (D), we deduce that A0 = B0 ×B A is
κ-filtered. This proves (1).
Our next goal is to prove Proposition 5.3.2.9, which gives a very concrete
characterization of right exactness under the assumption that there is asufficient supply of colimits. We first need a few preliminary results.
Lemma 5.3.2.6. Let B0 → B be a Cartesian fibration. Suppose that B has
an initial object B and that B0 is filtered. Then the fiber B0B = B0 ×B {B} isa contractible Kan complex.
Proof. Since B is an initial object of B, the inclusion {B}op ⊆ Bop is cofinal.
Proposition 4.1.2.15 implies that the inclusion (B0B )op ⊆ (B0 )op is also cofinal
and therefore a weak homotopy equivalence. It now suffices to prove that B0
is weakly contractible, which follows from Lemma 5.3.1.20.
Lemma 5.3.2.7. Let f : A → B be a right exact functor between ∞categories and let A ∈ A be an initial object. Then f (A) is an initial object
of B.
Proof. Let B be an object of B. Proposition 5.3.2.5 implies that A0 =B/B ×B A is filtered. We may identify MapB (f (A), B) with the fiber of theright fibration A0 → A over the object A. We now apply Lemma 5.3.2.6 todeduce that MapB (f (A), B) is contractible.
Lemma 5.3.2.8. Let κ be a regular cardinal, f : A → B a κ-right exact
functor between ∞-categories, and p : K → A a diagram indexed by a κsmall simplicial set K. The induced map Ap/ → Bf p/ is κ-right exact.
Proof. According to Proposition 5.3.2.5, it suffices to prove that for eachobject B ∈ Bf ◦p/ , the ∞-category A0 = Ap/ ×Bf p/ (Bf p/ )/B is κ-filtered. Let
B denote the image of B in B and let q : K 0 → A0 be a diagram indexed by a.
κ-small simplicial set K 0 ; we wish to show that q admits an extension to K 0 .
0We may regard p and q together as defining a diagram K ? K → A ×B B/B .
Since f is κ-filtered, we can extend this to a map (K ? K 0 ). → A ×B B/B ,.
which can be identified with an extension q : K 0 → A0 of q.
Proposition 5.3.2.9. Let f : A → B be a functor between ∞-categories
and let κ be a regular cardinal.
(1) If f is κ-right exact, then f preserves all κ-small colimits which exist
in A.
(2) Conversely, if A admits κ-small colimits and f preserves κ-small colimits, then f is κ-right exact.

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Proof. Suppose first that f is κ-right exact. Let K be a κ-small simplicial
set, and let p : K . → A be a colimit of p = p|K. We wish to show that f ◦ p
is a colimit diagram. Using Lemma 5.3.2.8, we may replace A by Ap/ andB by Bf p/ and thereby reduce to the case K = ∅. We are then reduced toproving that f preserves initial objects, which follows from Lemma 5.3.2.7.
Now suppose that A admits κ-small colimits and that f preserves κ-small
colimits. We wish to prove that f is κ-right exact. Let B be an object of B andset A0 = A ×B B/B . We wish to prove that A0 is κ-filtered. Let p0 : K → A0
be a diagram indexed by a κ-small simplicial set K; we wish to prove thatp0 extends to a map p0 : K . → A0 . Let p : K → A be the composition of
p0 with the projection A0 → A and let p : K . → A be a colimit of p. We
may identify f ◦ p and p0 with objects of Bf p/ . Since f preserves κ-small
colimits, f ◦ p is an initial object of Bf p/ , so that there exists a morphism
α : f ◦ p → p0 in Bf ◦p/ . The morphism α can be identified with the desired
extension p0 : K . → A0 .
Remark 5.3.2.10. The results of this section all dualize in an evident way:a functor G : A → B is said to be κ-left exact if the induced functor Gop :Aop → Bop is κ-right exact. In the case where A admits κ-small limits, thisis equivalent to the requirement that G preserve κ-small limits.
Remark 5.3.2.11. Let C be an ∞-category, let F : C → Sop be a functor,
e → C be the associated right fibration (the pullback of the universal
and let C
e is κ-filtered (since Q0right fibration Q0 → Sop ). If F is κ-right exact, then C
has a final object). If C admits κ-small colimits, then the converse holds: if
e is κ-filtered, then F preserves κ-small colimits by Proposition 5.3.5.3 andCis therefore κ-right exact by Proposition 5.3.2.5. The converse does not holde → C suchin general: it is possible to give an example of right fibration C
opethat C is filtered yet the classifying functor F : C → S is not right exact.
5.3.3 Filtered Colimits
Filtered categories tend not to be very interesting in themselves. Instead,they are primarily useful for indexing diagrams in other categories. This isbecause the colimits of filtered diagrams enjoy certain exactness properties
not shared by colimits in general. In this section, we will formulate and provethese exactness properties in the ∞-categorical setting. First, we need a few
definitions.
Definition 5.3.3.1. Let κ be a regular cardinal. We will say that an ∞category C is κ-closed if every diagram p : K → C indexed by a κ-small
simplicial set K admits a colimit p : K . → C.
In a κ-closed ∞-category, it is possible to construct κ-small colimits functorially. More precisely, suppose that C is an ∞-category and that K is a
simplicial set with the property that every diagram p : K → C has a colimit in C. Let D denote the full subcategory of Fun(K . , C) spanned by thehighertopoi

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colimit diagrams. Proposition 4.3.2.15 implies that the restriction functor
D → Fun(K, C) is a trivial fibration. It therefore admits a section s (whichis unique up to a contractible ambiguity). Let e : Fun(K . , C) → C be thefunctor given by evaluation at the cone point of K . . We will refer to thecompositionseFun(K, C) → D ⊆ Fun(K . , C) → C
as a colimit functor; it associates to each diagram p : K → C a colimit of p
in C. We will generally denote colimit functors by limK : Fun(K, C) → C.
−→
Lemma 5.3.3.2. Let F ∈ Fun(K, S) be a corepresentable functor (that is,
F lies in the essential image of the Yoneda embedding K op → Fun(K, S))
and let X ∈ S be a colimit of F . Then X is contractible.
Proof. Without loss of generality, we may suppose that K is an ∞-category.
e → K be a left fibration classified by F . Since F is corepresentable, Ke
Let K
has an initial object and is therefore weakly contractible. Corollary 3.3.4.6
e ' X in the homotopy category H,
implies that there is an isomorphism K
so that X is also contractible.
Proposition 5.3.3.3. Let κ be a regular cardinal and let I be an ∞-category.
The following conditions are equivalent:(1) The ∞-category I is κ-filtered.
(2) The colimit functor limI : Fun(I, S) → S preserves κ-small limits.
−→
Proof. Suppose that (1) is satisfied. According to Proposition 5.3.1.18, there
exists a κ-filtered partially ordered set A and a cofinal map i : N(A) → S.
Since i is cofinal, the colimit functor for I admits a factorizationi∗

Fun(I, S) → Fun(N(A), S)→ S .
Proposition 5.1.2.2 implies that i∗ preserves limits. We may therefore replace
I by N(A) and thereby reduce to the case where I is itself the nerve of aκ-filtered partially ordered set A.
We note that the functor limI : Fun(I, S) → S can be characterized as−→
the left adjoint to the diagonal functor δ : S → Fun(I, S). Let A denotethe category of all functors from A to Set∆ ; we regard A as a simplicial
model category with respect to the projective model structure described in§A.3.3. Let φ∗ : Set∆ → A denote the diagonal functor which associates
to each simplicial set K the constant functor A → Set∆ with value K,
and let φ! be a left adjoint of φ∗ , so that the pair (φ! , φ∗ ) gives a Quillen
adjunction between A and Set∆ . Proposition 4.2.4.4 implies that there isan equivalence of ∞-categories N(A◦ ) → Fun(I, S), and δ may be identified
with the right derived functor of φ∗ . Consequently, the functor limI may be−→
identified with the left derived functor of φ! . To prove that limI preserves−→
κ-small limits, it suffices to prove that limI preserves fiber products and−→κ-small products. According to Theorem 4.2.4.1, it suffices to prove that φ!

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preserves homotopy fiber products and κ-small homotopy products. For fiber
products, this reduces to the classical assertion that if we are given a familyof homotopy Cartesian squares
Wα

/ Xα


Yα


/ Zα

in the category of Kan complexes, indexed by a filtered partially ordered set
A, then the colimit square
W

/X


Y


/Z

is also homotopy Cartesian. The assertion regarding homotopy products ishandled similarly.
Now suppose that (2) is satisfied. Let K be a κ-small simplicial set andp : K → Iop a diagram; we wish to prove that Iop/p is nonempty. Suppose
otherwise. Let j : Iop → Fun(I, S) be the Yoneda embedding, let q = j ◦ p,
let q : K / → Fun(I, S) be a limit of q, and let X ∈ Fun(I, S) be the imageof the cone point of K / under q. Since j is fully faithful and Iop/p is empty,
we have MapSI (j(I), X) = ∅ for each I ∈ I. Using Lemma 5.1.5.2, we may
identify MapSI (j(I), X) with X(I) in the homotopy category H of spaces.
We therefore conclude that X is an initial object of Fun(I, S). Since thefunctor limI : Fun(I, S) → S is a left adjoint, it preserves initial objects.
−→
We conclude that limI X is an initial object of S. On the other hand, if−→
limI preserves κ-small limits, then limI ◦q exhibits limI X as the limit of−→−→−→the diagram limI ◦q : K → S. For each vertex k in K, Lemmas 5.1.5.2 and−→5.3.3.2 imply that limI q(k) is contractible and therefore a final object of S.
−→
It follows that limI X is also a final object of S. This is a contradiction since−→
the initial object of S is not final.
5.3.4 Compact Objects
Let C be a category which admits filtered colimits. An object C ∈ C is said
to be compact if the corepresentable functor
HomC (C, •)commutes with filtered colimits.
Example 5.3.4.1. Let C = Set be the category of sets. An object C ∈ C iscompact if and only if is finite.
Example 5.3.4.2. Let C be the category of groups. An object G of C iscompact if and only if it is finitely presented (as a group).

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Example 5.3.4.3. Let X be a topological space and let C be the category ofopen sets of X (with morphisms given by inclusions). Then an object U ∈ C
is compact if and only if U is compact when viewed as a topological space:that is, every open cover of U admits a finite subcover.
Remark 5.3.4.4. Because of Example 5.3.4.2, many authors call an object
C of a category C finitely presented if HomC (C, •) preserves filtered colimits.
Our terminology is motivated instead by Example 5.3.4.3.
Definition 5.3.4.5. Let C be an ∞-category which admits small κ-filteredcolimits. We will say a functor f : C → D is κ-continuous if it preserves
κ-filtered colimits.
Let C be an ∞-category containing an object C and let jC : C → b
S denotethe functor corepresented by C. If C admits κ-filtered colimits, then we will
say that C is κ-compact if jC is κ-continuous. We will say that C is compact
if it is ω-compact (and C admits filtered colimits).
Let κ be a regular cardinal and let C be an ∞-category which admits smalle → C is κ-compact if itκ-filtered colimits. We will say that a left fibration Cbis classified by a κ-continuous functor C → S.
Notation 5.3.4.6. Let C be an ∞-category and κ a regular cardinal. We
will generally let Cκ denote the full subcategory spanned by the κ-compact
objects of C.
Lemma 5.3.4.7. Let C be an ∞-category which admits small κ-filteredcolimits and let D ⊆ Fun(C, bS) be the full subcategory spanned by the κcontinuous functors f : C → b
S. Then D is stable under κ-small limits inCb
S .
Proof. Let K be a κ-small simplicial set, and let p : K → Fun(C, bS) bea diagram which we may identify with a map p0 : C → Fun(K, b
S). Using
Proposition 5.1.2.2, we may obtain a limit of the diagram p by composing
p0 with a limit functor
lim : Fun(K, b
S) → bS←−(that is, a right adjoint to the diagonal functor b
S → Fun(K, bS); see §5.3.3). It
therefore suffices to show that the functor lim is κ-continuous. This is simply←−
a reformulation of Proposition 5.3.3.3.
The basic properties of κ-compact left fibrations are summarized in thefollowing Lemma:
Lemma 5.3.4.8. Fix a regular cardinal κ.
(1) Let C be an ∞-category which admits small κ-filtered colimits and let
C ∈ C be an object. Then C is κ-compact if and only if the left fibration
CC/ → C is κ-compact.

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397

PRESENTABLE AND ACCESSIBLE ∞-CATEGORIES(2) Let f : C → D be a κ-continuous functor between ∞-categories whiche → D be a κ-compact left
admit small κ-filtered colimits and let D
e → C is also κfibration. Then the associated left fibration C ×D D
compact.
(3) Let C be an ∞-category which admits small κ-filtered colimits and let
A ⊆ (Set∆ )/ C denote the full subcategory spanned by the κ-compact left
fibrations over C. Then A is stable under κ-small homotopy limits (withrespect to the covariant model structure on (Set∆ )/ C ). In particular, A
is stable under the formation of homotopy pullbacks, κ-small products,and (if κ is uncountable) homotopy inverse limits of towers.
Proof. Assertions (1) and (2) are obvious. To prove (3), let us suppose thate is a κ-small homotopy limit of κ-compact left fibrations C
eα → C. Let J beC.
a small κ-filtered ∞-category and let p : J → C be a colimit diagram. We
wish to prove that the composition of p with the functor C → b
S classifying
e is a colimit diagram. Applying Proposition 5.3.1.18, we may reduce to theCcase where J is the nerve of a κ-filtered partially ordered set A. According
to Theorem 2.2.1.2, it will suffice to show that the collection of homotopy
colimit diagrams
A ∪ {∞} → Kan
is stable under κ-small homotopy limits in the category (Set∆ )A∪{∞} , which
follows easily from our assumption that A is κ-filtered.
Our next goal is to prove a very useful stability result for κ-compact
objects (Proposition 5.3.4.13). We first need to establish a few technicallemmas.
Lemma 5.3.4.9. Let κ be a regular cardinal, let C be an ∞-category which
admits small κ-filtered colimits, and let f : C → D be a morphism in C.
Suppose that C and D are κ-compact objects of C. Then f is a κ-compact
object of Fun(∆1 , C).
Proof. Let X = Fun(∆1 , C) ×Fun({1},C) Cf / , Y = Fun(∆1 , CC/ ) and Z =Fun(∆1 , C) ×Fun({1},C) CC/ , so that we have a (homotopy) pullback diagram
Fun(∆1 , C)f /

/X


Y


/Z

of left fibrations over Fun(∆1 , C). According to Lemma 5.3.4.8, it will suffice
to show that X, Y , and Z are κ-compact left fibrations. To show that X isa κ-compact left fibration, it suffices to show that Cf / → C is a κ-compact
left fibration, which follows since we have a trivial fibration Cf / → CD/ ,where D is κ-compact by assumption. Similarly, we have a trivial fibration
Y → Fun(∆1 , C) ×C(0) CC/ , so that the κ-compactness of C implies thathighertopoi

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Y is a κ-compact left fibration. Lemma 5.3.4.8 and the compactness of C
immediately imply that Z is a κ-compact left fibration, which completes theproof.
Lemma 5.3.4.10. Let κ be a regular cardinal and let {Cα } be a κ-smallfamily of ∞-categories having product C. Suppose that each Cα admits small
κ-filtered colimits. Then
(1) The ∞-category C admits κ-filtered colimits.
(2) If C ∈ C is an object whose image in each Cα is κ-compact, then C isκ-compact as an object of C.
Proof. The first assertion is obvious since colimits in a product can be computed pointwise. For the second, choose an object C ∈ C whose images{Cα ∈ Cα } are κ-compact.
The left fibration CC/ → C can be obtained as a κ-small product of theleft fibrations C ×Cα (Cα )Cα / → C. Lemma 5.3.4.8 implies that each factor is
κ-compact, so that the product is also κ-compact.
Lemma 5.3.4.11. Let S be a simplicial set and suppose we are given a
tower
f1

f0

· · · → X(1) → X(0) → S,
where each fi is a left fibration. Then the inverse limit X(∞) is a homotopy
inverse limit of the tower {X(i)} with respect to the covariant model structureon (Set∆ )/S .
Proof. Construct a ladder···/ X(1)f1/ X(0)

f0

/S

···


/ X 0 (1)f10

0
/ X 0 (0) f0

/S

where the vertical maps are covariant equivalences and the tower {X 0 (i)}is fibrant (in the sense that each of the maps fi0 is a covariant fibration).
We wish to show that the induced map on inverse limits X(∞) → X 0 (∞) isa covariant equivalence. Since both X(∞) and X 0 (∞) are left fibered overS, this can be tested by passing to the fibers over each vertex s of S. We
may therefore reduce to the case where S is a point, in which case the tower{X(i)} is already fibrant (since a left fibration over a Kan complex is a Kan
fibration; see Lemma 2.1.3.3).
Lemma 5.3.4.12. Let κ be an uncountable regular cardinal and let
f2

f1· · · → C2 → C1 → C0
be a tower of ∞-categories. Suppose that each Ci admits small κ-filteredcolimits and that each of the functors fi is a categorical fibration which preserves κ-filtered colimits. Let C denote the inverse limit of the tower. Thenhighertopoi

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399(1) The ∞-category C admits small κ-filtered colimits, and the projections
pn : C → Cn are κ-continuous.
(2) If C ∈ C has a κ-compact image in Ci for each i ≥ 0, then C is a
κ-compact object of C.
Proof. Let q : K . → C be a diagram indexed by an arbitrary simplicial set,
let q = q|K, and set q n = pn ◦ q, qn = pn ◦ q. Suppose that each q n is acolimit diagram in Cn . Then the map Cq/ → Cq/ is the inverse limit of a
tower of trivial fibrations Cnqn / → Cnqn / and therefore a trivial fibration.
To complete the proof of (1), it will suffice to show that if K is a κfiltered ∞-category, then any diagram q : K → C can be extended to a map
q : K . → C with the property described above. To construct q, it suffices
to construct a compatible family q n : K . → Cn . We begin by selecting
arbitrary colimit diagrams q 0n : K . → Cn which extend qn . We now explain
how to adjust these choices to make them compatible with one another using
induction on n. Set q 0 = q 00 . Suppose next that n > 0. Since fn preserves
κ-filtered colimits, we may identify q n−1 and fn ◦ q 0n with initial objects of0
Cn−1
qn−1 / . It follows that there exists an equivalence e : q n−1 → fn ◦ q n in
n−1
n
Cn−1
qn−1 / . The map fn induces a categorical fibration Cqn / → Cqn−1 / , so that en0lifts to an equivalence e : q n → q n in Cqn / . The existence of the equivalencee proves that q n is a colimit diagram in Cn , and we have q n−1 = fn ◦ q n by
construction. This proves (1).
Now suppose that C ∈ C is as in (2) and let C n = pn (C) ∈ Cn . The∞-category C/C is the inverse limit of a tower· · · → C1C 1 / ×C1 C → C0C 0 / ×C0 C .
Using Lemma 5.3.4.8, we deduce that each term in this tower is a κ-compact
left fibration over C. Proposition 2.1.2.1 implies that each map in the tower
is a left fibration, so that CC/ is a homotopy inverse limit of a tower of κcompact left fibrations by Lemma 5.3.4.11. We now apply Lemma 5.3.4.8
again to deduce that CC/ is a κ-compact left fibration, so that C ∈ C isκ-compact, as desired.
Proposition 5.3.4.13. Let κ be a regular cardinal, let C be an ∞-category
which admits small κ-filtered colimits, and let f : K → C be a diagram
indexed by a κ-small simplicial set K. Suppose that for each vertex x of K,
f (x) ∈ C is κ-compact. Then f is a κ-compact object of Fun(K, C).
Proof. Let us say that a simplicial set K is good if it satisfies the conclusionsof the lemma. We wish to prove that all κ-small simplicial sets are good.
The proof proceeds in several steps:(1) Suppose we are given a pushout square
/K
K0
i


L0


/ L,

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where i is a cofibration and the simplicial sets K 0 , K, and L0 are good.
Then the simplicial set L is also good. To prove this, we observe thatthe associated diagram of ∞-categories
Fun(L, C)/ Fun(L0 , C)
Fun(K, C)/ Fun(K 0 , C)is homotopy Cartesian and every arrow in the diagram preserves κfiltered colimits (by Proposition 5.1.2.2). Now apply Lemma 5.4.5.7.
(2) If K → K 0 is a categorical equivalence and K is good, then K 0 is
good: the forgetful functor Fun(K 0 , C) → Fun(K, C) is an equivalenceof ∞-categories and therefore detects κ-compact objects.
(3) Every simplex ∆n is good. To prove this, we observe that the inclusionaa∆{0,1}···∆{n−1,n} ⊆ ∆n
{1}

{n−1}

is a categorical equivalence. Applying (1) and (2), we can reduce tothe case n ≤ 1. If n = 0, there is nothing to prove, and if n = 1, weapply Lemma 5.3.4.9.
(4) If {Kα } is a κ-small collection of good simplicial sets having coproductK,
Q then K is also good. To prove this, we observe that Fun(K, C) 'α Fun(Kα , C) and apply Lemma 5.3.4.10.
(5) If K is a κ-small simplicial set of dimension at most n, then K isgood. The proof is by induction on n. Let K (n−1) ⊆ K denote the(n − 1)-skeleton of K, so that we have a pushout diagram`n/ K (n−1)σ∈Kn ∂ ∆`σ∈Kn ∆

n


/ K.

`
The inductive hypothesis implies that σ∈K`
∂ ∆n and K (n−1) arengood. Applying (3) and (4), we deduce that σ∈Kn ∆n is good. We
now apply (1) to deduce that K is good.
(6) Every κ-small simplicial set K is good. If κ = ω, then this follows immediately from (5) since every κ-small simplicial set is finite-dimensional.
If κ is uncountable, then we have an increasing filtration
K (0) ⊆ K (1) ⊆ · · ·which gives rise to a tower of ∞-categories· · · → Fun(K (1) , C) → Fun(K (0) , C)highertopoi

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having (homotopy) inverse limit Fun(K, C). Using Proposition 5.1.2.2,
we deduce that the hypotheses of Lemma 5.3.4.12 are satisfied, so that
K is good.

Corollary 5.3.4.14. Let κ be a regular cardinal and let C be an ∞-category
which admits small κ-filtered colimits. Suppose that p : K → C is a κ-small
diagram with the property that for every vertex x of K, p(x) is a κ-compact
object of C. Then the left fibration Cp/ → C is κ-compact.
Proof. It will suffice to show that the equivalent left fibration Cp/ → C isκ-compact. Let P be the object of Fun(K, C) corresponding to p. Then we
have an isomorphism of simplicial sets
Cp/ ' C ×Fun(K,C) Fun(K, C)P/ .
Proposition 5.3.4.13 asserts that P is a κ-compact object of Fun(K, C), sothat the left fibration
Fun(K, C)P/ → Fun(K, C)
is κ-compact. Proposition 5.1.2.2 guarantees that the diagonal map C →Fun(K, C) preserves κ-filtered colimits, so we can apply part (2) of Lemma
5.3.4.8 to deduce that Cp/ → C is κ-compact as well.
Corollary 5.3.4.15. Let C be an ∞-category which admits small κ-filteredcolimits and let Cκ denote the full subcategory of C spanned by the κ-compact
objects. Then Cκ is stable under the formation of all κ-small colimits which
exist in C.
Proof. Let K be a κ-small simplicial set and let p : K . → C be a colimit
diagram. Suppose that, for each vertex x of K, the object p(x) ∈ C is κcompact. We wish to show that C = p(∞) ∈ C is κ-compact, where ∞denotes the cone point of K . . Let p = p|K and consider the maps
Cp/ ← Cp/ → CC/ .
Both are trivial fibrations (the first because p is a colimit diagram and thesecond because the inclusion {∞} ⊆ K . is right anodyne). Corollary 5.3.4.14
asserts that the left fibration Cp/ → C is κ-compact. It follows that theequivalent left fibration CC/ is κ-compact, so that C is a κ-compact objectof C, as desired.
Remark 5.3.4.16. Let κ be a regular cardinal and let C be an ∞-category
which admits κ-filtered colimits. Then the full subcategory Cκ ⊆ C of κcompact objects is stable under retracts. To prove this, let C ∈ C be κcompact and let D be a retract of C. Let j : Cop → Fun(C, bS) denote theYoneda embedding, so that j(D) ∈ Fun(C, bS) is a retract of j(C). Since j(C)preserves κ-filtered colimits, Lemma 5.1.6.3 implies that j(D) preserves κfiltered colimits, so that D is κ-compact as desired.

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The following result gives a convenient description of the compact objectsof an ∞-category of presheaves:
Proposition 5.3.4.17. Let C be a small ∞-category, κ a regular cardinal,
and C ∈ P(C) an object. The following are equivalent:(1) There exists a diagram p : K → C indexed by a κ-small simplicial set,such that j ◦ p has a colimit D in P(C) and C is a retract of D.
(2) The object C is κ-compact.
Proof. Proposition 5.1.6.8 asserts that for every object A ∈ C, j(A) is
completely compact and, in particular, κ-compact. According to Corollary
5.3.4.15 and Remark 5.3.4.16, the collection of κ-compact objects of P(C) isstable under κ-small colimits and retracts. Consequently, (1) ⇒ (2).
Now suppose that (2) is satisfied. Let C/C = C ×P(C) P(C)/C . Lemma
5.1.5.3 implies that the composition
p : C./C → P(C)./C → P(C)is a colimit diagram. As in the proof of Corollary 4.2.3.11, we can write C
as the colimit of a κ-filtered diagram q : I → P(C), where each object q(I) is
the colimit of p| C0 , where C0 is a κ-small simplicial subset of C/C . Since C
is κ-compact, we may argue as in the proof of Proposition 5.1.6.8 to deducethat C is a retract of q(I) for some object I ∈ I. This proves (1).
We close with a result which we will need in §5.5. First, a bit of notation:
if C is a small ∞-category and κ a regular cardinal, we let Pκ (C) denote thefull subcategory consisting of κ-compact objects of P(C).
Proposition 5.3.4.18. Let C be a small idempotent complete ∞-categoryand κ a regular cardinal. The following conditions are equivalent:(1) The ∞-category C admits κ-small colimits.
(2) The Yoneda embedding j : C → Pκ (C) has a left adjoint.
Proof. Suppose that (1) is satisfied. For each object M ∈ P(C), let FM :P(C) → b
S denote the associated corepresentable functor. Let D ⊆ P(C) denote the full subcategory of P(C) spanned by those objects M such thatFM ◦ j : C → b
S is corepresentable. According to Proposition 5.1.2.2, composition with j induces a limit-preserving functor
Fun(P(C), b
S) → Fun(C, bS).
Applying Proposition 5.1.3.2 to Cop , we conclude that the collection of corepresentable functors on C is stable under retracts and κ-small limits. A second
application of Proposition 5.1.3.2 (this time to P(C)op ) now shows that D
is stable under retracts and κ-small colimits in P(C). Since j is fully faithful, D contains the essential image of j. It follows from Proposition 5.3.4.17

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that D contains Pκ (C). We now apply Proposition 5.2.4.2 to deduce thatj : C → Pκ (C) admits a left adjoint.
Conversely, suppose that (2) is satisfied. Let L denote a left adjoint to the
Yoneda embedding, let p : K → C be a κ-small diagram, and let q = j ◦ p.
Using Corollary 5.3.4.15, we deduce that q has a colimit q : K . → Pκ (C).
Since L is a left adjoint, L ◦ q is a colimit of L ◦ q. Since j is fully faithful,
the diagram p is equivalent to L ◦ q, so that p has a colimit as well.
5.3.5 Ind-Objects
Let S be a simplicial set. In §5.1.5, we proved that the ∞-category P(S) is
freely generated under small colimits by the image of the Yoneda embeddingj : S → P(S) (Theorem 5.1.5.6). Our goal in this section is to study theanalogous construction where we allow only filtered colimits.
Definition 5.3.5.1. Let C be a small ∞-category and let κ be a regularcardinal. We let Indκ (C) denote the full subcategory of P(C) spanned by
e → C, where thethose functors f : Cop → S which classify right fibrations C
e∞-category C is κ-filtered. In the case where κ = ω, we will simply write
Ind(C) for Indκ (C). We will refer to Ind(C) as the ∞-category of Ind-objects
of C.
Remark 5.3.5.2. Let C be a small ∞-category and κ a regular cardinal.
Then the Yoneda embedding j : C → P(C) factors through Indκ (C). This
follows immediately from Lemma 5.1.5.2, since j(C) classifies the right fibration C/C → C. The ∞-category C/C has a final object and is therefore
κ-filtered (Remark 5.3.1.17).
Proposition 5.3.5.3. Let C be a small ∞-category and let κ be a regular cardinal. The full subcategory Indκ (C) ⊆ P(C) is stable under κ-filteredcolimits.
Proof. Let P0∆ (C) denote the full subcategory of (Set∆ )/ C spanned by thee → C. According to Proposition 5.1.1.1, the ∞-category
right fibrations C
P(C) is equivalent to the simplicial nerve N(P0∆ (C)). Let Ind0κ (C) denote the
e → C, where C
e isfull subcategory of P0∆ (C) spanned by right fibrations C
0κ-filtered. It will suffice to prove that for any diagram p : I → N(Indκ (C))
indexed by a small κ-filtered ∞-category I, the colimit of p in N(P0∆ (C)) also
belongs to Ind0κ (C). Using Proposition 5.3.1.18, we may reduce to the casewhere I is the nerve of a κ-filtered partially ordered set A. Using Proposition
4.2.4.4, we may further reduce to the case where p is the simplicial nerveof a diagram taking values in the ordinary category Ind0κ (C). By virtue ofTheorem 4.2.4.1, it will suffice to prove that Ind0κ (C) ⊆ P0∆ (C) is stable underκ-filtered homotopy colimits. We may identify P0∆ (C) with the collection offibrant objects of (Set∆ )/ C with respect to the contravariant model structure.
Since the class of contravariant equivalences is stable under filtered colimits,any κ-filtered colimit in (Set∆ )/ C is also a homotopy colimit. Consequently, ithighertopoi

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will suffice to prove that Ind0κ (C) ⊆ P0∆ (C) is stable under κ-filtered colimits.
This follows immediately from the definition of a κ-filtered ∞-category.
Corollary 5.3.5.4. Let C be a small ∞-category, let κ be a regular cardinal,
and let F : Cop → S be an object of P(C). The following conditions areequivalent:(1) There exists a (small) κ-filtered ∞-category I and a diagram p : I → C
such that F is a colimit of the composition j ◦ p : I → P(C).
(2) The functor F belongs to Indκ (C).
If C admits κ-small colimits, then (1) and (2) are equivalent to(3) The functor F preserves κ-small limits.
Proof. Lemma 5.1.5.3 implies that F is a colimit of the diagramj

C/F → C → P(C),and Lemma 5.1.5.2 allows us to identify C/F = C ×P(C) P(C)/F with the rightfibration associated to F . Thus (2) ⇒ (1). The converse follows from Proposition 5.3.5.3 since every representable functor belongs to Indκ (C) (Remark
5.3.5.2).
Now suppose that C admits κ-small colimits. If (3) is satisfied, then F op :C → Sop is κ-right exact by Proposition 5.3.3.3. The right fibration associatedto F is the pullback of the universal right fibration by F op . Using Corollary
3.3.2.7, the universal right fibration over Sop is representable by the final
object of S. Since F is κ-right exact, the fiber product (Sop )/∗ ×Sop C isκ-filtered. Thus (3) ⇒ (2).
We now complete the proof by showing that (1) ⇒ (3). First supposethat F lies in the essential image of the Yoneda embedding j : C → P(C).
According to Lemma 5.1.5.2, j(C) is equivalent to the composition of the
opposite Yoneda embedding j 0 : Cop → Fun(C, S) with the evaluation functor
e : Fun(C, S) → S associated to the object C ∈ C. Propositions 5.1.3.2 and5.1.2.2 imply that j 0 and e preserve κ-small limits, so that j(C) preserves κsmall limits. To conclude the proof, it will suffice to show that the collectionof functors F : Cop → S which satisfy (3) is stable under κ-filtered colimits:
this follows easily from Proposition 5.3.3.3.
Proposition 5.3.5.5. Let C be a small ∞-category, let κ be a regular cardinal, and let j : C → Indκ (C) be the Yoneda embedding. For each object
C ∈ C, j(C) is a κ-compact object of Indκ (C).
Proof. The functor Indκ (C) → S corepresented by j(C) is equivalent to thecompositionIndκ (C) ⊆ P(C) → S,
where the first map is the canonical inclusion and the second is given byevaluation at C. The second map preserves all colimits (Proposition 5.1.2.2),
and the first preserves κ-filtered colimits since Indκ (C) is stable under κfiltered colimits in P(C) (Proposition 5.3.5.3).

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Remark 5.3.5.6. Let C be a small ∞-category and κ a regular cardinal.
Suppose that C is equivalent to an n-category, so that the Yoneda embeddingj : C → P(C) factors through P≤n−1 (C) = Fun(Cop , τ≤n−1 S), where τ≤n−1 S
denotes the full subcategory of S spanned by the (n − 1)-truncated spaces:that is, spaces whose homotopy groups vanish in dimensions n and above.
The class of (n − 1)-truncated spaces is stable under filtered colimits, sothat P≤n−1 (C) is stable under filtered colimits in P(C). Corollary 5.3.5.4
implies that Ind(C) ⊆ P≤n−1 (C). In particular, Ind(C) is itself equivalent toan n-category. In particular, if C is the nerve of an ordinary category I, then
Ind(C) is equivalent to the nerve of an ordinary category J, which is uniquelydetermined up to equivalence. Moreover, J admits filtered colimits, and there
is a fully faithful embedding I → J which generates J under filtered colimits
and whose essential image consists of compact objects of J. It follows that
J is equivalent to the category of Ind-objects of I in the sense of ordinary
category theory.
According to Corollary 5.3.5.4, we may characterize Indκ (C) as the smallest full subcategory of P(C) which contains the image of the Yoneda embedding j : C → P(C) and is stable under κ-filtered colimits. Our goal is toobtain a more precise characterization of Indκ (C): namely, we will show that
it is freely generated by C under κ-filtered colimits.
Lemma 5.3.5.7. Let D be an ∞-category (not necessarily small). Thereexists a fully faithful functor i : D → D0 with the following properties:(1) The ∞-category D0 admits small colimits.
(2) A small diagram K . → D is a colimit if and only if the composite map
K . → D0 is a colimit.
Proof. Let D0 = Fun(D, bS)op and let i be the opposite of the Yoneda embedding. Then (1) follows from Proposition 5.1.2.2 and (2) from Proposition
5.1.3.2.
We will need the following analogue of Lemma 5.1.5.5:
Lemma 5.3.5.8. Let C be a small ∞-category, κ a regular cardinal, j : C →Indκ (C) the Yoneda embedding, and C0 ⊆ Indκ (C) the essential image of j.
Let D be an ∞-category which admits small κ-filtered colimits. Then
(1) Every functor f0 : C0 → D admits a left Kan extension f : Indκ (C) →D.
(2) An arbitrary functor f : Indκ (C) → D is a left Kan extension of f | C0
if and only if f is κ-continuous.
Proof. Fix an arbitrary functor f0 : C0 → D. Without loss of generality, we
may assume that D is a full subcategory of a larger ∞-category D0 , satisfying
the conclusions of Lemma 5.3.5.7; in particular, D is stable under small κfiltered colimits in D0 . We may further assume that D coincides with its

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essential image in D0 . Lemma 5.1.5.5 guarantees the existence of a functor
F : P(C) → D0 which is a left Kan extension of f0 = F | C0 and such thatF preserves small colimits. Since Indκ (C) is generated by C0 under κ-filteredcolimits (Corollary 5.3.5.4), the restriction f = F | Indκ (C) factors through
D. It is then clear that f : Indκ (C) → D is a left Kan extension of f0 andthat f is κ-continuous. This proves (1) and the “only if” direction of (2)(since left Kan extensions of f0 are unique up to equivalence).
We now prove the “if” direction of (2). Let f : Indκ (C) → D be the functor
constructed above and let f 0 : Indκ (C) → D be an arbitrary κ-continuousfunctor such that f | C0 = f 0 | C0 . We wish to prove that f 0 is a left Kan
extension of f 0 | C0 . Since f is a left Kan extension of f | C0 , there exists anatural transformation α : f → f 0 which is an equivalence when restricted
to C0 . Let E ⊆ Indκ (C) be the full subcategory spanned by those objects
C for which the morphism αC : f (C) → f 0 (C) is an equivalence in D. By
hypothesis, C0 ⊆ E. Since both f and f 0 are κ-continuous, E is stable underκ-filtered colimits in Indκ (C). We now apply Corollary 5.3.5.4 to conclude
that E = Indκ (C). It follows that f 0 and f are equivalent, so that f 0 is a left
Kan extension of f 0 | C0 , as desired.
Remark 5.3.5.9. The proof of Lemma 5.3.5.8 is very robust and can beused to establish a number of analogous results. Roughly speaking, given
any class S of colimits, one can consider the smallest full subcategory C00 of
P(C) which contains the essential image C0 of the Yoneda embedding and isstable under colimits of type S. Given any functor f0 : C0 → D, where D
is an ∞-category which admits colimits of type S, one can show that there
exists a functor f : C00 → D which is a left Kan extension of f0 = f | C0 .
Moreover, f is characterized by the fact that it preserves colimits of type
S. Taking S to be the class of all small colimits, we recover Lemma 5.1.5.5.
Taking S to be the class of all small κ-filtered colimits, we recover Lemma
5.3.5.8. Other variations are possible as well: we will exploit this idea furtherin §5.3.6.
Proposition 5.3.5.10. Let C and D be ∞-categories and let κ be a regularcardinal. Suppose that C is small and that D admits small κ-filtered colimits.
Then composition with the Yoneda embedding induces an equivalence of ∞categories
Mapκ (Indκ (C), D) → Fun(C, D),
where the left hand side denotes the ∞-category of all κ-continuous functors
from Indκ (C) to D.
Proof. Combine Lemma 5.3.5.8 with Corollary 4.3.2.16.
In other words, if C is small and D admits κ-filtered colimits, then any
functor f : C → D determines an essentially unique extension F : Indκ (C) →D (such that f is equivalent to F ◦ j). We next give a criterion which willallow us to determine when F is an equivalence.

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Proposition 5.3.5.11. Let C be a small ∞-category, κ a regular cardinal,
and D an ∞-category which admits κ-filtered colimits. Let F : Indκ (C) → D
be a κ-continuous functor and f = F ◦ j its composition with the Yoneda
embedding j : C → Indκ (C). Then
(1) If f is fully faithful and its essential image consists of κ-compact objectsof D, then F is fully faithful.
(2) The functor F is an equivalence if and only if the following conditions
are satisfied:(i) The functor f is fully faithful.
(ii) The functor f factors through Dκ .
(iii) The objects {f (C)}C∈C generate D under κ-filtered colimits.
Proof. We first prove (1) using the argument of Proposition 5.1.6.10. Let C
and D be objects of Indκ (C). We wish to prove that the map
ηC,D : MapP(C) (C, D) → MapD (F (C), F (D))
is an isomorphism in the homotopy category H. Suppose first that C belongs
to the essential image of j. Let G : P(C) → S be a functor corepresented by
C and let G0 : D → S be a functor corepresented by F (C). Then we havea natural transformation of functors G → G0 ◦ F . Assumption (2) implies
that G0 preserves small κ-filtered colimits, so that G0 ◦ F preserves small κfiltered colimits. Proposition 5.3.5.5 implies that G preserves small κ-filteredcolimits. It follows that the collection of objects D ∈ Indκ (C) such that ηC,D
is an equivalence is stable under small κ-filtered colimits. If D belongs to theessential image of j, then the assumption that f is fully faithful implies that
ηC,D is a homotopy equivalence. Since the image of the Yoneda embedding
generates Indκ (C) under small κ-filtered colimits, we conclude that ηC,D isa homotopy equivalence for every object D ∈ Indκ (C).
We now drop the assumption that C lies in the essential image of j. Fix
D ∈ Indκ (C). Let H : Indκ (C)op → S be a functor represented by D and
let H 0 : Dop → S be a functor represented by F D. Then we have a natural transformation of functors H → H 0 ◦ F op which we wish to prove is anequivalence. By assumption, F op preserves small κ-filtered limits. Proposition 5.1.3.2 implies that H and H 0 preserve small limits. It follows that thecollection P of objects C ∈ P(S) such that ηC,D is an equivalence is stable under small κ-filtered colimits. The special case above established that
P contains the essential image of the Yoneda embedding. Since Indκ (C) is
generated under small κ-filtered colimits by the image of the Yoneda embedding, we deduce that ηC,D is an equivalence in general. This completes theproof of (1).
We now prove (2). Suppose first that F is an equivalence. Then (i) follows
from Proposition 5.1.3.1, (ii) from Proposition 5.3.5.5, and (iii) from Corollary 5.3.5.4. Conversely, suppose that (i), (ii), and (iii) are satisfied. Using(1), we deduce that F is fully faithful. The essential image of F contains thehighertopoi

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essential image of f and is stable under small κ-filtered colimits. Therefore
F is essentially surjective, so that F is an equivalence as desired.
According to Corollary 4.2.3.11, an ∞-category C admits small colimits if
and only if C admits κ-small colimits and κ-filtered colimits. Using Proposition 5.3.5.11, we can make a much more precise statement:
Proposition 5.3.5.12. Let C be a small ∞-category and κ a regular cardinal. The ∞-category Pκ (C) of κ-compact objects of P(C) is essentially small:that is, there exists a small ∞-category D and an equivalence i : D → Pκ (C).
Let F : Indκ (D) → P(C) be a κ-continuous functor such that the compositionof f with the Yoneda embedding
D → Indκ (D) → P(C)is equivalent to i (according to Proposition 5.3.5.10, F exists and is unique
up to equivalence). Then F is an equivalence of ∞-categories.
Proof. Since P(C) is locally small, to prove that Pκ (C) is small it will suffice
to show that the collection of isomorphism classes of objects in the homotopy
category h Pκ (C) is small. For this, we invoke Proposition 5.3.4.17: every κcompact object X of P(C) is a retract of some object Y , which is itself thecolimit of some compositionp

K → C → P(C),where K is κ-small. Since there is a bounded collection of possibilities for K
and p (up to isomorphism in Set∆ ) and a bounded collection of idempotent
maps Y → Y in h P(C), there is only a bounded number of possibilities forX.
To prove that F is an equivalence, it will suffice to show that F satisfies
conditions (i), (ii), and (iii) of Proposition 5.3.5.11. Conditions (i) and (ii)are obvious. For (iii), we must prove that every object of X ∈ P(C) can beobtained as a small κ-filtered colimit of κ-compact objects of C. Using Lemma
5.1.5.3, we can write X as a small colimit taking values in the essential
image of j : C → P(C). The proof of Corollary 4.2.3.11 shows that X can be
written as a κ-filtered colimit of a diagram with values in a full subcategory
E ⊆ P(C), where each object of E is itself a κ-small colimit of some diagram
taking values in the essential image of j. Using Corollary 5.3.4.15, we deducethat E ⊆ Pκ (C), so that X lies in the essential image of F , as desired.
Note that the construction C 7→ Indκ (C) is functorial in C. Given a functor
f : C → C0 , Proposition 5.3.5.10 implies that the composition of f with the
Yoneda embedding jC0 : C0 → Indκ C0 is equivalent to the composition
jC

F

C → Indκ C → Indκ C0 ,where F is a κ-continuous functor. The functor F is well-defined up to equivalence (in fact, up to contractible ambiguity). We will denote F by Indκ f(though this is perhaps a slight abuse of notation since F is uniquely determined only up to equivalence).

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Proposition 5.3.5.13. Let f : C → C0 be a functor between small ∞categories. The following are equivalent:(1) The functor f is κ-right exact.
(2) The map G : P(C0 ) → P(C) given by composition with f restricts to afunctor g : Indκ (C0 ) → Indκ (C).
(3) The functor Indκ f has a right adjoint.
Moreover, if these conditions are satisfied, then g is a right adjoint to Indκ f .
Proof. The equivalence (1) ⇔ (2) is just a reformulation of the definitionof κ-right exactness. Let P(f ) : P(C) → P(C0 ) be a functor which preserves
small colimits such that the diagram of ∞-categories
/ C0

f

C

P(C)

P(f )


/ P(C0 )is homotopy commutative. Then we may identify Indκ (f ) with the restriction P(f )| Indκ (C). Proposition 5.2.6.3 asserts that G is a right adjoint ofP(f ). Consequently, if (2) is satisfied, then g is a right adjoint to Indκ (f ).
We deduce in particular that (2) ⇒ (3). We will complete the proof byshowing that (3) implies (2). Suppose that Indκ (f ) admits a right adjoint
g 0 : Indκ (C0 ) → Indκ (C). Let X : (C0 )op → S be an object of Indκ (C0 ). Then
X op is equivalent to the composition
j 0

c

C
X
C0 −→
Indκ (C0 ) −→Sop ,where cX denotes the functor represented by X. Since g 0 is a left adjoint
to Indκ f , the functor cX ◦ Indκ (f ) is represented by g 0 X. Consequently, we
have a homotopy commutative diagram
C jC
f


C0

0

/ Indκ (C)


cg0 X

Indκ (f )/ Indκ (C0 )

cX

/ Sop

/ Sop ,

so that G(X)op = f ◦ X op ' cg0 X ◦ jC and therefore belongs to Indκ (C).
Proposition 5.3.5.14. Let C be a small ∞-category and κ a regular cardinal. The Yoneda embedding j : C → Indκ (C) preserves all κ-small colimits
which exist in C.
Proof. Let K be a κ-small simplicial set and p : K . → C a colimit diagram.
We wish to show that j ◦ p : K . → Indκ (C) is also a colimit diagram.
Let C ∈ Indκ (C) be an object and let F : Indκ (C)op → b
S be the functor

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represented by C. According to Proposition 5.1.3.2, it will suffice to showthat F ◦ (j ◦ p)op is a limit diagram in S. We observe that F ◦ j op is equivalent
to the object C ∈ Indκ (C) ⊆ Fun(Cop , S) and therefore κ-right exact. We now
conclude by invoking Proposition 5.3.2.9.
We conclude this section with a useful result concerning diagrams in ∞categories of Ind-objects:
Proposition 5.3.5.15. Let C be a small ∞-category, κ a regular cardinal,and j : C → Indκ (C) the Yoneda embedding. Let A be a finite partiallyordered set and let j 0 : Fun(N(A), C) → Fun(N(A), Indκ (C)) be the induced
map. Then j 0 induces an equivalenceIndκ (Fun(N(A), C)) → Fun(N(A), Indκ (C)).
In other words, every diagram N(A) → Indκ (C) can be obtained, in an
essentially unique way, as a κ-filtered colimit of diagrams N(A) → C.
Warning 5.3.5.16. The statement of Proposition 5.3.5.15 fails if we replace
N(A) by an arbitrary finite simplicial set. For example, we may identify thecategory of abelian groups with the category of Ind-objects of the categoryof finitely generated abelian groups. If n > 1, then the map q 7→ nq from thegroup of rational numbers Q to itself cannot be obtained as a filtered colimitof endomorphisms of finitely generated abelian groups.
Proof of Proposition 5.3.5.15. According to Proposition 5.3.5.11, it will suffice to prove the following:(i) The functor j 0 is fully faithful.
(ii) The essential image of j 0 is comprised of κ-compact objects of the∞-category Fun(N(A), Indκ (C)).
(iii) The essential image of j 0 generates Fun(N(A), Indκ (C)) under small
κ-filtered colimits.
Since the Yoneda embedding j : C → Indκ (C) satisfies the analogues of these
conditions, (i) is obvious and (ii) follows from Proposition 5.3.4.13. To prove(iii), we fix an object F ∈ Fun(N(A), Indκ (C)). Let C0 denote the essential
image of j and form a pullback diagram of simplicial sets
D

/ Fun(N(A), C0 )


Fun(N(A), Indκ (C))/F/ Fun(N(A), Indκ (C)).

Since D is essentially small, (iii) is a consequence of the following assertions:(a) The ∞-category D is κ-filtered.
(b) The canonical map D. → Fun(N(A), C) is a colimit diagram.

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To prove (a), we need to show that D has the right lifting property withrespect to the inclusion N(B) ⊆ N(B ∪ {∞}) for every κ-small partiallyordered set B (Remark 5.3.1.12). Regard B ∪ {∞, ∞0 } as a partially ordered
set with b < ∞ < ∞0 for each b ∈ B. Unwinding the definitions, we see that(a) is equivalent to the following assertion:(a0 ) Let F : N(A×(B∪{∞0 })) → Indκ (C) be such that F | N(A×{∞0 }) = F
00
and F | N(A × B) factors through C0 . Then there exists a map F :0
N(A×(B∪{∞, ∞0 })) → Indκ (C) which extends F , such that F | N(A×(B ∪ {∞})) factors through C0 .
0

To find F , we write A = {a1 , . . . , an }, where ai ≤ aj implies i ≤ j. We will
construct a compatible sequence of maps
F k : N((A × (B ∪ {∞0 })) ∪ ({a1 , . . . , ak } × {∞})) → C,0with F 0 = F and F n = F . For each a ∈ A, we let A≤a = {a0 ∈ A : a0 ≤a}, and we define A<a , A≥a , A>a similarly. Supposing that F k−1 has been
constructed, we observe that constructing F k amounts to constructing an
object of the ∞-category(C0/F | N(A≥a ) )F k−1 |M/ ,kwhere M = (A≤ak × B) ∪ (A<ak × {∞}). The inclusion {ak } ⊆ N(A≥ak ) isleft anodyne. It will therefore suffice to construct an object in the equivalent∞-category (C0/F (ak ) )F k−1 |M/ . Since M is κ-small, it suffices to show thatthe ∞-category C0/F (ak ) is κ-filtered. This is simply a reformulation of thefact that F (ak ) ∈ Indκ (C).
We now prove (b). It will suffice to show that for each a ∈ A, the composition
D. → Fun(N(A), Indκ (C)) → Indκ (C)is a colimit diagram, where the second map is given by evaluation at a. Let
D(a) = C0 ×Indκ (C) Indκ (C)/F (a) , so that D(a) is κ-filtered and the associatedmap D(a). → Indκ (C) is a colimit diagram. It will therefore suffice to showthat the canonical map D → D(a) is cofinal. According to Theorem 4.1.3.1,
it will suffice to show that for each object D ∈ D(a), the fiber product
E = D ×D(a) D(a)D/ is weakly contractible. In view of Lemma 5.3.1.20, it
will suffice to show that E is filtered. This can be established by a minor
variation of the argument given above.
5.3.6 Adjoining Colimits to ∞-Categories
Let C be a small ∞-category. According to Proposition 5.3.5.10, the ∞category Ind(C) enjoys the following properties, which characterize it up toequivalence:(1) There exists a functor j : C → Ind(C).

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CHAPTER 5(2) The ∞-category Ind(C) admits small filtered colimits.
(3) Let D be an ∞-category which admits small filtered colimits and let
Fun0 (Ind(C), D) be the full subcategory of Fun(Ind(C), D) spanned bythose functors which preserve filtered colimits. Then composition withj induces an equivalence Fun0 (Ind(C), D) → Fun(C, D).
We can summarize this characterization as follows: the ∞-category Ind(C) is
obtained from C by freely adjoining the colimits of all small filtered diagrams.
In this section, we will study a generalization of this construction whichallows us to freely adjoin to C the colimits of any collection of diagrams.
Notation 5.3.6.1. Let C and D be ∞-categories and let R be a collectionof diagrams {pα : Kα. → C}. We let FunR (C, D) denote the full subcategoryof Fun(C, D) spanned by those functors which carry each diagram in R to acolimit diagram in D.
Let K be a collection of simplicial sets. We will say that an ∞-category C
admits K-indexed colimits if it admits K-indexed colimits for each K ∈ K.
If f : C → D is a functor between ∞-categories which admit K-indexed
colimits, then we will say that f preserves K-indexed colimits if f preserves
K-indexed colimits for each K ∈ K. We let FunK (C, D) denote the full subcategory of Fun(C, D) spanned by those functors which preserve K-indexed
colimits.
Proposition 5.3.6.2. Let K be a collection of simplicial sets, C an ∞category, and R = {pα : Kα. → C} a collection of diagrams in C. Assume
that each Kα belongs to K. Then there exists a new ∞-category PK
R (C) and
(C)
with
the
followingproperties:
a map j : C → PK
R(1) The ∞-category PK
R (C) admits K-indexed colimits.
(2) For every ∞-category D which admits K-indexed colimits, composition
with j induces an equivalence of ∞-categories
FunK (PK
R (C), D) → FunR (C, D).
Moreover, if every member of R is already a colimit diagram in C, then we
have in addition:(3) The functor j is fully faithful.
Remark 5.3.6.3. In the situation of Proposition 5.3.6.2, assertion (2) (applied in the case D = PK
R (C)) guarantees that j carries each diagram in
R to a colimit diagram in PK
R (C). We can informally summarize conditions(1) and (2) as follows: the ∞-category PK
R (C) is freely generated by C under K-indexed colimits, subject only to the relation that each diagram in R
determines a colimit diagram in PK
R (C). It is clear that this property characterizes PK
R (C) (and the map j) up to equivalence.

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Example 5.3.6.4. Suppose that K is the collection of all small simplicial
sets, that the ∞-category C is small, and that the set of diagrams R is
empty. Then the Yoneda embedding j : C → P(C) satisfies the conclusions of
Proposition 5.3.6.2. This is precisely the assertion of Theorem 5.1.5.6 (savefor assertion (3), which follows from Proposition 5.1.3.1). This justifies thenotation of Proposition 5.3.6.2; in the general case we can think of PK
R (C)as a sort of generalized presheaf category C, and j as an analogue of the
Yoneda embedding.
Proof of Proposition 5.3.6.2: We will employ essentially the same argument
as in our proof of Proposition 5.3.5.10. First, we may enlarge the universe if
necessary to reduce to the case where every element of K is a small simplicial
set, the ∞-category C is small, and the collection of diagrams R is small.
Let j0 : C → P(C) denote the Yoneda embedding. For every diagram pα :K . → C, we let pα denote the restriction pα |K, Xα ∈ P(C) a colimit for
the induced diagram j ◦ pα : K → P(C), and Yα ∈ C the image of the conepoint under pα . The diagram j0 ◦ pα induces a map sα : Xα → j0 (Yα ) (welldefined up to homotopy); let S = {sα } be the set of all such morphisms. We
let S −1 P(C) ⊆ P(C) denote the ∞-category of S-local objects of P(C) andL : P(C) → S −1 P(C) a left adjoint to the inclusion. We define PK
R (C) to be
the smallest full subcategory of S −1 P(C) which contains the essential imageof the functor L ◦ j0 and is closed under K-indexed colimits and let j = L ◦ j0
be the induced map. We claim that the map j : C → PK
R (C) has the desired
properties.
Assertion (1) is obvious. We now prove (2). Let D be an ∞-category
which admits K-indexed colimits. In view of Lemma 5.3.5.7, we can assume
that there exists a fully faithful inclusion D ⊆ D0 , where D0 admits all
small colimits and D is stable under K-indexed colimits in D0 . We have acommutative diagram
FunK (PK
R (C), D)


φ

φ0
0
FunK (PK
R (C), D )/ FunR (C, D)/ FunR (C, D0 ).

We claim that this diagram is homotopy Cartesian. Unwinding the defini0
tions, this is equivalent to the assertion that a functor f ∈ FunK (PK
R (C), D )0factors through D if and only if f ◦ j : C → D factors through D. The“only if” direction is obvious. Conversely, if f ◦ j factors through D, thenf −1 D is a full subcategory of PK
R (C) which is stable under K-indexed limits(since f preserves K-indexed limits and D is stable under K-indexed limitsin D) and contains the essential image of j; by minimality, we conclude thatf −1 D = PK
R (C).
Our goal is to prove that the functor φ is an equivalence of ∞-categories.
In view of the preceding argument, it will suffice to show that φ0 is anequivalence of ∞-categories. In other words, we may replace D by D0 andthereby reduce to the case where D0 admits small colimits.

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Let E ⊆ P(C) denote the inverse image L−1 PK
R (C) and let S denote the collection of all morphisms α in E such that Lα is an equivalence. Composition
with L induces a fully faithful embedding Fun(PK
R (C), D) → Fun(E, D) whose
essential image consists of those functors E → D which carry every morphismin S to an equivalence in D. Furthermore, a functor f : PK
R (C) → D preserves
K-indexed colimits if and only if the composition f ◦ L : E → D preserves
K-indexed colimits. The functor φ factors as a composition
0
FunK (PK
R (C), D) → Fun (E, D) → FunR (C, D),ψwhere Fun0 (E, D) denotes the full subcategory of Fun(E, D) spanned by thosefunctors which carry every morphism in S to an equivalence and preserve
K-indexed colimits. It will therefore suffice to show that ψ is an equivalenceof ∞-categories.
In view of Proposition 4.3.2.15, we need only show that if F : E → D is afunctor such that F ◦j0 belongs to FunR (C, D), then F belongs to Fun0 (E, D)
if and only if F is a left Kan extension of F | C0 , where C0 ⊆ E denotes theessential image of the Yoneda embedding j0 : C → E. We first prove the“if” direction. Let F0 = F | C0 . Since D admits small colimits, the functor
F0 admits a left Kan extension F : P(C) → D; without loss of generality,
we may suppose that F = F | E. According to Lemma 5.1.5.5, the functor
F preserves small colimits. Since E is stable under K-indexed colimits in
P(C), it follows that F | E preserves K-indexed colimits. Furthermore, sinceF ◦ j0 belongs to FunR (C, D), the functor F carries each morphism in S toan equivalence in D. It follows that F factors (up to homotopy) throughthe localization functor L, so that F | E carries each morphism in S to an
equivalence in D.
For the converse, let us suppose that F ∈ Fun0 (E, D); we wish to showthat F is a left Kan extension of F | C0 . Let F 0 denote an arbitrary left Kan
extension of F | C0 , so that the identification F | C0 = F 0 | C0 induces a naturaltransformation α : F 0 → F . We wish to prove that α is an equivalence. Since
F 0 and F both carry each morphism in S to an equivalence, we may assume
without loss of generality that F = f ◦ L, F 0 = f 0 ◦ L, and α = β ◦ L,
where β : f 0 → f is a natural transformation of functors from PK
R (C) toD. Let X ⊆ PK
R (C) denote the full subcategory spanned by those objects
X such that βX : f 0 (X) → f (X) is an equivalence. Since both f 0 and fpreserve K-indexed colimits, we conclude that X is stable under K-indexed
colimits in PK
R (C). It is clear that X contains the essential image of theKfunctor j : C → PK
R (C). It follows by construction that X = PR (C), so thatβ is an equivalence, as desired. This completes the proof of (2).
It remains to prove (3). Suppose that every element of R is already acolimit diagram in C. We note that the functor j factors as a composition
L ◦ j0 , where the Yoneda embedding j0 : C → E is already known to be fully
faithful (Proposition 5.1.3.1). Since the functor L|S −1 P(C) is equivalent to
the identity, it will suffice to show that the essential image of j0 is containedin S −1 P(C). In other words, we must show that if sα : Xα → j0 Yα belongs

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIES

to S, and C ∈ C, then the induced map
MapP(C) (j0 Yα , j0 C) → MapP(C) (Xα , j0 C)
is a homotopy equivalence. Let p : Kα. → C be the corresponding diagram(so that p carries the cone point of Kα. to Yα ), let p = p|Kα , and let q :Kα. → P(C) be a colimit diagram extending q = q|Kα = j0 ◦ p. Consider the
diagram
g0

g1

g2

g3

P(C)j0 Yα / ← P(C)j0 p/ → P(C)j0 p/ ← P(C)q/ → P(C)Xα / .
The maps g0 and g3 are trivial Kan fibrations (since the inclusion of the conepoint into Kα. is cofinal), and the map g2 is a trivial Kan fibration since q isa colimit diagram. Moreover, for every object Z ∈ P(C), the above diagram
determines the map MapP(C) (j0 Yα , Z) → MapP(C) (Xα , Z). Consequently, toprove that this map is an equivalence, it suffices to show that g1 induces atrivial Kan fibration
P(C)j0 p/ ×P(C) {Z} → P(C)j0 p/ ×P(C) {Z}.
Assuming Z belongs to the essential image C0 of the Yoneda embedding j0 ,we may reduce to proving that the induced map C0j0 p/ → C0j0 p/ is a trivial
Kan fibration, which is equivalent to the assertion that j0 ◦ p is a colimit
diagram in C0 . This is clear since p is a colimit diagram by assumption andj0 induces an equivalence of ∞-categories from C to C0 .
Definition 5.3.6.5. Let K ⊆ K0 be collections of simplicial sets and let C
0
be an ∞-category which admits K-indexed limits. We let PK
K (C) denote the0.
∞-category PK
R (C), where R is the set of all colimit diagrams p : K → C
such that K ∈ K.
Example 5.3.6.6. Let K = ∅ and let K0 denote the class of all small simplicial sets. If C is a small ∞-category, then we have a canonical equivalence0PK
K (C) ' P(C) (Theorem 5.1.5.6).
Example 5.3.6.7. Let K = ∅ and let K0 denote the class of all small
κ-filtered simplicial sets for some regular cardinal κ. Then for any small ∞0
category C, we have a canonical equivalence PK
K (C) ' Indκ (C) (Proposition
5.3.5.10).
Example 5.3.6.8. Let K denote the collection of all κ-small simplicial sets
for some regular cardinal κ and let K0 be the class of all small simplicial
sets. Let C be a small ∞-category which admits κ-small colimits. Then we0
have a canonical equivalence PK
K (C) ' Indκ (C). This follows from Theorem
5.5.1.1 and Proposition 5.5.1.9.
Example 5.3.6.9. Let K = ∅ and let K0 = {Idem}, where Idem is the0
simplicial set defined in §4.4.5. Then, for any ∞-category C, PK
K (C) is an
idempotent completion of C.

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Corollary 5.3.6.10. Let K ⊆ K0 be classes of simplicial sets. Let CatK
d dedenote the ∞-category of (not necessarily small) ∞-categories, let Cat∞note the subcategory spanned by those ∞-categories which admit K-indexed
K0dcolimits and those functors which preserve K-indexed colimits, and let Cat∞
be defined likewise. Then the inclusion
K0

Kd ⊆ Cat
d
Cat
∞
∞
0

admits a left adjoint given by C 7→ PK
K (C).
Proof. Combine Proposition 5.3.6.2 with Proposition 5.2.2.12.
We conclude this section by noting the following transitivity property of
the construction C 7→ PK
R (C):
Proposition 5.3.6.11. Let K ⊆ K0 be collections of simplicial sets andlet C1 , . . . , Cn be a sequence of ∞-categories. For 1 ≤ i ≤ n, let Ri be acollection of diagrams {pα : Kα. → Ci }, where each Kα belongs to K, and let
R0i denote the collection of all colimit diagrams {q α : Kα. → PK
Ri (Ci )} suchthat Kα ∈ K. Then the canonical map
0

0

K
K
K
PK
R1 ···Rn (C1 × · · · × Cn ) → PR01 ···R0n (PR1 (C1 ) × · · · × PRn (Cn ))is an equivalence of ∞-categories. Here R1  · · ·  Rn denotes the collectionof all diagrams of the form
p

α
Kα. →
Ci ' {C1 } × · · · × {Ci−1 } × Ci × · · · × {Cn } ⊆ C1 × · · · × Cn ,where pα ∈ Ri and Cj is an object of Cj for j 6= i, and the collection
R01  · · ·  R0n is defined likewise.
Proof. Let D be an ∞-category which admits K0 -indexed colimits. It will
suffice to show that the functor0

KK
FunK0 (PKR01 ···R0n (PR1 (C1 ) × · · · × PRn (Cn )), D)0
FunK0 (PK(C
1 × · · · × Cn ), D)
R1 ···Rn
is an equivalence of ∞-categories. Unwinding the definitions, we are reduced
to proving that the functor
K
FunR01 ···R0n (PKR1 (C1 ) × · · · × PRn (Cn ), D)φFunR ···Rn (C1 × · · · × Cn , D)
is an equivalence of ∞-categories. The proof goes by induction on n. If n = 0,then both sides are equivalent to D and there is nothing to prove. If n > 0,highertopoi

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417

then set D0 = FunRn (Cn , D) and D00 = FunK (PK
R1 (Cn ), D). Proposition
5.3.6.2 implies that the canonical map D00 → D0 is an equivalence of ∞categories. We can identify φ with the functor
K00
FunR01 ···R0n−1 (PKR1 (C) × · · · × PRn−1 (C), D )FunR1 ···Rn−1 (C1 × · · · × Cn−1 , D0 ).
The desired result now follows from the inductive hypothesis.

5.4 ACCESSIBLE ∞-CATEGORIES
Many of the categories which commonly arise in mathematics can be realized
as categories of Ind-objects. For example, the category of sets is equivalent
to Ind(C), where C is the category of finite sets; the category of rings isequivalent to Ind(C), where C is the category of finitely presented rings. The
theory of accessible categories is an axiomatization of this situation. We refer
the reader to [1] for an exposition of the theory of accessible categories. In
this section, we will describe an ∞-categorical generalization of the theoryof accessible categories.
We will begin in §5.4.1 by introducing the notion of a locally small ∞category. A locally small ∞-category C need not be small but has small
morphism spaces MapC (X, Y ) for any fixed pair of objects X, Y ∈ C. Thisis analogous to the usual set-theoretic conventions taken in category theory:
one allows categories which have a proper class of objects but requires thatmorphisms between any pair of objects form a set.
In §5.4.2, we will introduce the definition of an accessible ∞-category.
An ∞-category C is accessible if it is locally small and has a good supplyof filtered colimits and compact objects. Equivalently, C is accessible if it
is equivalent to Indκ (C0 ) for some small ∞-category C0 and some regularcardinal κ (Proposition 5.4.2.2).
The theory of accessible ∞-categories will play an important technical role
throughout the remainder of this book. To understand the usefulness of the
hypothesis of accessibility, let us consider the following example. Supposethat C is an ordinary category, that F : Cop → Set is a functor, and that wewould like to prove that F is representable by an object C ∈ C. The functore = {(C, η) : C ∈ C, η ∈ F (C)}, which is fiberedF determines a category C
e is equivalent to C/C for someover C in sets. We would like to prove that C
C ∈ C. The object C can then be characterized as the colimit of the diagram
e → C. If C admits colimits, then we can attempt to construct C by
p : C
forming the colimit lim(p).
−→We now encounter a set-theoretic difficulty. Suppose that we try to ensure
the existence of lim(p) by assuming that C admits all small colimits. In this−→e iscase, it is not reasonable to expect C itself to be small. The category Chighertopoi

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roughly the same size as C (or larger), so our assumption will not allow us toe are small, then it
construct lim(p). On the other hand, if we assume C and C−→
is not reasonable to expect C to admit colimits of arbitrary small diagrams.
An accessibility hypothesis can be used to circumvent the difficulty described above. An accessible category C is generally not small but is “controlled” by a small subcategory C0 ⊆ C: it therefore enjoys the best featuresof both the “small” and “large” worlds. More precisely, the fiber producte ×C C0 is small enough that we might expect the colimit lim(p|C
e ×C C0 ) toC−→
exist on general grounds yet large enough to expect a natural isomorphism
e ×C C0 ).
lim(p) ' lim(p|C−→−→
We refer the reader to §5.5.2 for a detailed account of this argument, which
we will use to prove an ∞-categorical version of the adjoint functor theorem.
The discussion above can be summarized as follows: the theory of accessible ∞-categories is a tool which allows us to manipulate large ∞-categories
as if they were small without fear of encountering any set-theoretic paradoxes. This theory is quite useful because the condition of accessibility isvery robust: the class of accessible ∞-categories is stable under most of thebasic constructions of higher category theory. To illustrate this, we will prove
the following results:(1) A small ∞-category C is accessible if and only if C is idempotent complete (§5.4.3).
(2) If C is an accessible ∞-category and K is a small simplicial set, thenFun(K, C) is accessible (§5.4.4).
(3) If C is an accessible ∞-category and p : K → C is a small diagram,then Cp/ and C/p are accessible (§5.4.5 and §5.4.6).
(4) The collection of accessible ∞-categories is stable under homotopy fiber
products (§5.4.6).
We will apply these facts in §5.4.7 to deduce a miscellany of further stability results which will be needed throughout §5.5 and Chapter 6.
5.4.1 Locally Small ∞-Categories
In mathematical practice, it is very common to encounter categories C forwhich the collection of all objects is large (too big to form a set), but thecollection of morphisms HomC (X, Y ) is small for every X, Y ∈ C. The same
situation arises frequently in higher category theory. However, it is slightly
more difficult to describe because the formalism of ∞-categories blurs thedistinction between objects and morphisms. Nevertheless, there is an adequate notion of “local smallness” in the ∞-categorical setting, which we will
describe in this section.
Our first step is to give a characterization of the class of essentially small∞-categories. We will need the following lemma.

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Lemma 5.4.1.1. Let C be a simplicial category, n a positive integer, andf0 : ∂ ∆n → N(C) a map. Let X = f0 ({0}), Y = f0 ({n}), and g0 denote theinduced map∂(∆1 )n−1 → MapC (X, Y ).
Let f, f 0 : ∆n → N(C) be extensions of f0 , and let g, g 0 : (∆1 )n−1 →MapC (X, Y ) be the corresponding extensions of g0 . The following conditions
are equivalent:(1) The maps f and f 0 are homotopic relative to ∂ ∆n .
(2) The maps g and g 0 are homotopic relative to ∂(∆1 )n−1 .
Proof. It is not difficult to show that (1) is equivalent to the assertion thatf and f 0 are left homotopic in the model category (Set∆ )∂ ∆n / (with the
Joyal model structure) and that (2) is equivalent to the assertion that C[f ]and C[f 0 ] are left homotopic in the model category (Cat∆ )C[∂ ∆n ]/ . We now
invoke the Quillen equivalence of Theorem 2.2.5.1 to complete the proof.
Proposition 5.4.1.2. Let C be an ∞-category and κ an uncountable regularcardinal. The following conditions are equivalent:(1) The collection of equivalence classes of objects of C is κ-small, and forevery morphism f : C → D in C and every n ≥ 0, the homotopy set
πi (HomR
C (C, D), f ) is κ-small.
(2) If C0 ⊆ C is a minimal model for C, then C0 is κ-small.
(3) There exists a κ-small ∞-category C0 and an equivalence C0 → C of∞-categories.
(4) There exists a κ-small simplicial set K and a categorical equivalence
K → C.
(5) The ∞-category C is κ-compact when regarded as an object of Cat∞ .
Proof. We begin by proving that (1) ⇒ (2). Without loss of generality, we
may suppose that C = N(D), where D is a topological category. Let C0 ⊆ C
be a minimal model for C. We will prove by induction on n ≥ 0 that the set
HomSet∆ (∆n , C0 ) is κ-small. If n = 0, this reduces to the assertion that C has
fewer than κ equivalence classes of objects. Suppose therefore that n > 0.
By the inductive hypothesis, the set HomSet∆ (∂ ∆n , C0 ) is κ-small. Since κ
is regular, it will suffice to prove that for each map f0 : ∂ ∆n → C0 , the set
S = {f ∈ HomSet∆ (∆n , C0 ) : f | ∂ ∆n = f0 } is κ-small. Let C = f0 ({0}), let
D = f0 ({n}), and let g0 : ∂(∆1 )n−1 → MapD (C, D) be the corresponding
map. Assumption (1) ensures that there are fewer than κ extensions g :(∆1 )n−1 → MapD (C, D) modulo homotopy relative to ∂(∆1 )n−1 . Invoking
Lemma 5.4.1.1, we deduce that there are fewer than κ maps f : ∆n → C
modulo homotopy relative to ∂ ∆n . Since C0 is minimal, no two distinct

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elements of S are homotopic in C relative to ∂ ∆n ; therefore S is κ-small, as
desired.
It is clear that (2) ⇒ (3) ⇒ (4). We next show that (4) ⇒ (3). Let K → C
be a categorical equivalence, where K is κ-small. We construct a sequenceof inner anodyne inclusions
K = K(0) ⊆ K(1) ⊆ · · · .
Supposing that K(n) has been defined, we form a pushout diagram` n / ` ∆n
Λi
 

K(n)


/ K(n + 1),

where the coproduct is taken over all 0 < i < n and all maps Λni → K(n).
It follows by induction on n that each S
K(n) is κ-small. Since κ is regular and uncountable, the limit K(∞) = n K(n) is κ-small. The inclusion
K ⊆ K(∞) is inner anodyne; therefore the map K → C factors through anequivalence K(∞) → C of ∞-categories; thus (3) is satisfied.
We next show that (3) ⇒ (5). Suppose that (3) is satisfied. Without
loss of generality, we may replace C by C0 and thereby suppose that C isitself κ-small. Let F : Cat∞ → S denote the functor corepresented by C.
According to Lemma 5.1.5.2, we may identify F with the simplicial nerveof the functor f : Cat∆∞ → Kan, which carries an ∞-category D to thelargest Kan complex contained in DC . Let I be a κ-filtered ∞-category andp : I → Cat∞ a diagram. We wish to prove that p has a colimit p : I. → Cat∞such that F ◦ p is a colimit diagram in S. According to Proposition 5.3.1.18,
we may suppose that I is the nerve of a κ-filtered partially ordered set A.
Using Proposition 4.2.4.4, we may further reduce to the case where p is the+simplicial nerve of a diagram P : A → Cat∆∞ ⊆ Set∆ taking values in theordinary category of marked simplicial sets. Let P be a colimit of P . Since
the class of weak equivalences in Set+
∆ is stable under filtered colimits, P isa homotopy colimit. Theorem 4.2.4.1 implies that p = N(P ) is a colimit ofp. It therefore suffices to show that F ◦ p = N(f ◦ P ) is a colimit diagram.
Using Theorem 4.2.4.1, it suffices to show that f ◦ P is a homotopy colimit
diagram in Set∆ . Since the class of weak homotopy equivalences in Set∆ isstable under filtered colimits, it will suffice to prove that f ◦ P is a colimit
diagram in the ordinary category Set∆ . It now suffices to observe that f
preserves κ-filtered colimits because C is κ-small.
We now complete the proof by showing that (5) ⇒ (1). Let A denotethe collection of all κ-small simplicial subsets Kα ⊆ C and let A0 ⊆ A be
the subcollection consisting of indices α such that Kα is an ∞-category.
ItS
is clear that A is a κ-filtered partially ordered set and that C = α∈A Kα .
Using the fact that κ > S
ω, it is easy to see that A0 is cofinal in A, so that A0 is
also κ-filtered and C = α∈A0 Kα . We may therefore regard C as the colimitof a diagram P : A0 → Set+
∆ in the ordinary category of fibrant objects ofhighertopoi

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0
Set+
∆ . Since A is filtered, we may also regard C as a homotopy colimit of P .
The above argument shows that CC = f C can be identified with a homotopy
colimit of the diagram f ◦ P : A0 → Set∆ . In particular, the vertex idC ∈ CC
must be homotopic to the image of some map KαC → CC for some α ∈ A0 .
It follows that C is a retract of Kα in the homotopy category hCat∞ . Since
Kα is κ-small, we easily deduce that Kα satisfies condition (1). Therefore C,
being a retract of Kα , satisfies condition (1) as well.

Definition 5.4.1.3. An ∞-category C is essentially κ-small if it satisfies theequivalent conditions of Proposition 5.4.1.2. We will say that C is essentiallysmall if it is essentially κ-small for some (small) regular cardinal κ.
The following criterion for essential smallness is occasionally useful:
Proposition 5.4.1.4. Let p : C → D be a Cartesian fibration of ∞categories and κ an uncountable regular cardinal. Suppose that D is essentially κ-small and that, for each object D ∈ D, the fiber CD = C ×D {D} isessentially κ-small. Then C is essentially κ-small.
Proof. We will apply criterion (1) of Proposition 5.4.1.2. Choose a κ-small
set of representatives {Dα } for the equivalence classes of objects of D. For
each α, choose a κ-small set of representatives {Cα,β } for the equivalenceclasses of objects of CDα . The collection of all objects Cα,β is κ-small (since κ
is regular) and contains representatives for all equivalence classes of objects
of C.
Now suppose that C and C 0 are objects of C having images D, D0 ∈ D.
Since D is essentially κ-small, the set π0 MapD (D, D0 ) is κ-small. Let f :e →D → D0 be a morphism and choose a p-Cartesian morphism fe : C0
D covering f . According to Proposition 2.4.4.2, we have a homotopy fiber
sequencee → MapC (C, C 0 ) → MapD (D, D0 )MapCD (C, C)
in the homotopy category H. In particular, we see that MapC (C, C 0 ) contains
fewer than κ connected components lying over f ∈ π0 MapD (D, D0 ) andtherefore fewer than κ components in total (since κ is regular). Moreover,the long exact sequence of homotopy groups shows that for every f : C → C 0
lifting f , the homotopy sets πi (HomrC (C, C 0 ), f ) are κ-small as desired.
By restricting our attention to Kan complexes, we obtain an analogue of
Proposition 5.4.1.2 for spaces:
Corollary 5.4.1.5. Let X be a Kan complex and κ an uncountable regularcardinal. The following conditions are equivalent:(1) For each vertex x ∈ X and each n ≥ 0, the homotopy set πn (X, x) isκ-small.
(2) If X 0 ⊆ X is a minimal model for X, then X 0 is κ-small.

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CHAPTER 5(3) There exists a κ-small Kan complex X 0 and a homotopy equivalence
X 0 → X.
(4) There exists a κ-small simplicial set K and a weak homotopy equivalence K → X.
(5) The Kan complex X is κ-compact when regarded as an object of S.
(6) The Kan complex X is essentially small (when regarded as an ∞category).
Proof. The equivalences (1) ⇔ (2) ⇔ (3) ⇔ (6) follow from Proposition
5.4.1.2. The implication (3) ⇒ (4) is obvious. We next prove that (4) ⇒ (5).
Let p : K → S be the constant diagram taking the value ∗, let p : K . → S bea colimit of p and let X 0 ∈ S be the image under p of the cone point of K . . It
follows from Proposition 5.1.6.8 that ∗ is a κ-compact object of S. Corollarye → K . denote
5.3.4.15 implies that X 0 is a κ-compact object of S. Let Ke denote the fiber lying
the left fibration associated to p, and let X 00 ⊆ Kover the cone point of K . . The inclusion of the cone point in K . is righteanodyne. It follows from Proposition 4.1.2.15 that the inclusion X 00 ⊆ K
is right anodyne. Since p is a colimit diagram, Proposition 3.3.4.5 impliese ⊆K
e is a weak homotopy equivalence. We
that the inclusion K ' K ×K . K
therefore have a chain of weak homotopy equivalencese ← X 00 ← X 0 ,X←K⊆K
so that X and X 0 are equivalent objects of S. Since X 0 is κ-compact, it
follows that X is κ-compact.
To complete the proof, we will show that (5) ⇒ (1). We employ the argument used in the proof of Proposition 5.4.1.2. Let F : S → S be thefunctor corepresented by X. Using Lemma 5.1.5.2, we can identify F withthe simplicial nerve of the functor f : Kan → Kan given by
Y 7→ Y X .
Let A denote the collection of κ-small simplicial subsets Xα ⊆ X S
which areKan complexes. Since κ is uncountable, A is κ-filtered and X = α∈A Kα .
We may regard X as the colimit of a diagram P : A → Set∆ . Since A isfiltered, X is also a homotopy colimit of this diagram. Since F preserves
κ-filtered colimits, f preserves κ-filtered homotopy colimits; therefore X X isa homotopy colimit of the diagram f ◦ P . In particular, the vertex idX ∈ X X
must be homotopic to the image of the map XαX → X X for some α ∈ A. It
follows that X is a retract of Xα in the homotopy category H. Since Xα isκ-small, we can readily verify that Xα satisfies (1). Because X is a retractof Xα , X satisfies (1) as well.
Remark 5.4.1.6. When κ = ω, the situation is quite a bit more complicated. Suppose that X is a Kan complex representing a compact object of S.
Then there exists a simplicial set Y with only finitely many nondegenerate
simplices and a map i : Y → X which realizes X as a retract of Y in thehighertopoi

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423

homotopy category H of spaces. However, one cannot generally assume that
Y is a Kan complex or that i is a weak homotopy equivalence. The latter canbe achieved if X is connected and simply connected, or more generally if acertain K-theoretic invariant of X (the Wall finiteness obstruction) vanishes:
we refer the reader to [82] for a discussion.
For many applications, it is important to be able to slightly relax thecondition that an ∞-category be essentiall small.
Proposition 5.4.1.7. Let C be an ∞-category. The following conditions areequivalent:(1) For every pair of objects X, Y ∈ C, the space MapC (X, Y ) is essentiallysmall.
(2) For every small collection S of objects of C, the full subcategory of C
spanned by the elements of S is essentially small.
Proof. This follows immediately from criterion (1) in Propositions 5.4.1.2
and 5.4.1.5.
We will say that an ∞-category C is locally small if it satisfies the equivalent conditions of Proposition 5.4.1.7.
Example 5.4.1.8. Let C and D be ∞-categories. Suppose that C is locallysmall and that D is essentially small. Then CD is essentially small. To provethis, we may assume without loss of generality that C and D are minimal.
Let {Cα } denote the collection of all full subcategories of C spanned by smallcollections of objects. Since D is small, every finite collection of functors
D → C factors through some small Cα ⊆ C. It follows that Fun(D, C) is theunion of small full subcategories Fun(D, Cα ) and is therefore locally small. In
particular, for every small ∞-category D, the ∞-category P(D) of presheaves
is locally small.
5.4.2 Accessibility
In this section, we will begin our study of the class of accessible ∞-categories.
Definition 5.4.2.1. Let κ be a regular cardinal. An ∞-category C is κaccessible if there exists a small ∞-category C0 and an equivalenceIndκ (C0 ) → C .
We will say that C is accessible if it is κ-accessible for some regular cardinalκ.
The following result gives a few alternative characterizations of the classof accessible ∞-categories.
Proposition 5.4.2.2. Let C be an ∞-category and κ a regular cardinal. The
following conditions are equivalent:highertopoi

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CHAPTER 5(1) The ∞-category C is κ-accessible.
(2) The ∞-category C is locally small and admits κ-filtered colimits, thefull subcategory Cκ ⊆ C of κ-compact objects is essentially small, andCκ generates C under small, κ-filtered colimits.
(3) The ∞-category C admits small κ-filtered colimits and contains an essentially small full subcategory C00 ⊆ C which consists of κ-compact
objects and generates C under small κ-filtered colimits.
The main obstacle to proving Proposition 5.4.2.2 is in verifying that if C0
is small, then Indκ (C0 ) has only a bounded number of κ-compact objects
up to equivalence. It is tempting to guess that any such object must be
equivalent to an object of C0 . The following example shows that this is notnecessarily the case.
Example 5.4.2.3. Let R be a ring and let C0 denote the (ordinary) category of finitely generated free R-modules. Then C = Ind(C0 ) is equivalent
to the category of flat R-modules (by Lazard’s theorem; see, for example,
the appendix of [47]). The compact objects of C are precisely the finitely
generated projective R-modules, which need not be free.
Nevertheless, the naive guess is not far off, by virtue of the following result:Lemma 5.4.2.4. Let C be a small ∞-category, κ a regular cardinal, andC0 ⊆ Indκ (C) the full subcategory of Indκ (C) spanned by the κ-compact objects. Then the Yoneda embedding j : C → C0 exhibits C0 as an idempotent
completion of C. In particular, C0 is essentially small.
Proof. Remark 5.3.1.10 implies that Indκ (C) is idempotent complete. Since
C0 is stable under retracts in Indκ (C), C0 is also idempotent complete. Proposition 5.1.3.1 implies that j is fully faithful. It therefore suffices to prove thatevery object C 0 ∈ C0 is a retract of j(C) for some C ∈ C.
Let C/C 0 = C ×Indκ (C) Indκ (C)/C 0 . Lemma 5.1.5.3 implies that the diagram
p : C./C 0 → Indκ (C)./C 0 → Indκ (C)is a colimit of p = p| C/C 0 . Let F : Indκ (C) → S be the functor corepresented by C 0 ; we note that the left fibration associated to F is equivalent toIndκ (C)C 0 / . Since F preserves κ-filtered colimits, Proposition 3.3.4.5 implies
that the inclusion
C/C 0 ×Indκ (C) Indκ (C)C 0 / ⊆ C./C 0 ×Indκ (C) Indκ (C)C 0 /is a weak homotopy equivalence. The simplicial set on the right has a canonical vertex, corresponding to the identity map idC 0 . It follows that there
exists a vertex on the left hand side belonging to the same path component.
Such a vertex classifies a diagram
j(C)
=
C0

f

!
/ C 0,

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where f is homotopic to the identity, which proves that C 0 is a retract ofj(C) in Indκ (C).
Proof of Proposition 5.4.2.2. Suppose that (1) is satisfied. Without loss ofgenerality, we may suppose that C = Indκ C0 , where C0 is small. Since C is afull subcategory of P(C0 ), it is locally small (see Example 5.4.1.8). Proposition 5.3.5.3 implies that C admits small κ-filtered colimits. Corollary 5.3.5.4
shows that C is generated under κ-filtered colimits by the essential imageof the Yoneda embedding j : C0 → C, which consists of κ-compact objects
by Proposition 5.3.5.5. Lemma 5.4.2.4 implies that the full subcategory ofIndκ (C0 ) consisting of compact objects is essentially small. We conclude that(1) ⇒ (2).
It is clear that (2) ⇒ (3). Suppose that (3) is satisfied. Choose a small∞-category C0 and an equivalence i : C0 → C00 . Using Proposition 5.3.5.10,
we may suppose that i factors as a compositionjf

C0 → Indκ (C0 ) → C,
where f preserves small κ-filtered colimits. Proposition 5.3.5.11 implies thatf is a categorical equivalence. This shows that (3) ⇒ (1) and completes theproof.
Definition 5.4.2.5. If C is an accessible ∞-category, then a functor F :C → C0 is accessible if it is κ-continuous for some regular cardinal κ (andtherefore for all regular cardinals τ ≥ κ).
Remark 5.4.2.6. Generally, we will only speak of the accessibility of afunctor F : C → C0 in the case where both C and C0 are accessible. However,
it is occasionally convenient to use the terminology of Definition 5.4.2.5 inthe case where C is accessible and C0 is not (or C0 is not yet known to beaccessible).
Example 5.4.2.7. The ∞-category S of spaces is accessible. More generally,
for any small ∞-category C, the ∞-category P(C) is accessible: this follows
immediately from Proposition 5.3.5.12.
If C is a κ-accessible ∞-category and τ > κ, then C is not necessarilyτ -accessible. Nevertheless, this is true for many values of τ . In what follows,
we write κ  τ if, for every τ0 < τ and every κ0 < κ, we have τ0κ0 < τ (seeDefinition A.2.6.3).
Lemma 5.4.2.8. If κ0  κ, then any κ0 -filtered partially ordered set I may
be written as a union of κ-filtered subsets which are κ0 -small. Moreover, thefamily of all such subsets is κ0 -filtered.
Proof. It will suffice to show that every κ0 -small subset S ⊆ I can be included
in a larger κ0 -small subset S 0 ⊆ I, where S 0 is κ-filtered.
We define a transfinite sequence of subsets SαS⊆ I by induction. Let
S0 = S. When λ is a limit ordinal, we let Sλ = α<λ Sα . Finally, we lethighertopoi

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Sα+1 denote a set obtained from Sα by adjoining an upper bound for every
κ-small subset of Sα (which exists because I is κ0 -filtered). It follows fromthe assumption κ0  κ that if Sα is κ0 -small, then so is Sα+1 . Since κ0 isregular, we deduce easily by induction that |Sα | < κ0 for all α < κ0 . It iseasy to check that the set S 0 = Sκ has the desired properties.
Proposition 5.4.2.9. Let C be a κ-accessible ∞-category. Then C is κ0 accessible for any κ0  κ.
Proof. Let Cκ ⊆ C denote the full subcategory consisting of κ-compact objects and let C0 ⊆ C denote the full subcategory spanned by the colimits of all
κ0 -small κ-filtered diagrams in Cκ . Since C is locally small and the collectionof all equivalence classes of such diagrams is bounded, we conclude that C0
is essentially small. Corollary 5.3.4.15 implies that C0 consists of κ0 -compact
objects of C. According to Proposition 5.4.2.2, it will suffice to prove that
C0 generates C under small κ0 -filtered colimits. Let X be an object of C andlet p : I → Cκ be a small κ-filtered diagram with colimit X. Using Proposition 5.3.1.18, we may reduce to the case where I is the nerve of a κ-filteredpartially ordered set A. Lemma 5.4.2.8 implies that A can be written as a
κ0 -filtered union of κ0 -small κ-filtered subsets {Aβ ⊆ A}β∈B . Using Propositions 4.2.3.4 and 4.2.3.8, we deduce that X can also be obtained as thecolimit of a diagram indexed by N(B) which takes values in C0 .
Remark 5.4.2.10. If C is a κ-accessible ∞-category and κ0 > κ, then C
is generally not κ0 -accessible. There are counterexamples even in ordinary
category theory: see [1].
Remark 5.4.2.11. Let C be an accessible ∞-category and κ a regular cardinal. Then the full subcategory Cκ ⊆ C consisting of κ-compact objects isessentially small. To prove this, we are free to enlarge κ. Invoking Proposition 5.4.2.9, we can reduce to the case where C is κ-accessible, in which casethe desired result is a consequence of Proposition 5.4.2.2.
Notation 5.4.2.12. If C and D are accessible ∞-categories, we will write
FunA (C, D) to denote the full subcategory of Fun(C, D) spanned by accessible functors from C to D.
Remark 5.4.2.13. Accessible ∞-categories are usually not small. However,they are determined by a “small” amount of data: namely, they always havethe form Indκ (C), where C is a small ∞-category. Similarly, an accessiblefunctor F : C → D between accessible categories is determined by a “small”
amount of data in the sense that there always exists a regular cardinal κ suchthat F is κ-continuous and maps Cκ into Dκ . The restriction F | Cκ then determines F up to equivalence (Proposition 5.3.5.10). To prove the existenceof κ, we first choose a regular cardinal τ such that F is τ -continuous. Enlarging τ if necessary, we may suppose that C and D are τ -accessible. The
collection of equivalence classes of τ -compact objects of C is small; consequently, by Remark 5.4.2.11, there exists a (small) regular cardinal τ 0 suchhighertopoi

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0that F carries Cτ into Dτ . We may now choose κ to be any regular cardinalsuch that κ  τ 0 .
d∞Definition 5.4.2.14. Let κ be a regular cardinal. We let Accκ ⊆ Cat
denote the subcategory defined as follows:(1) The objects of Accκ are the κ-accessible ∞-categories.
(2) A functor F : C → D between accessible ∞-categories belongs to Acc
if and only if F is κ-continuous and preserves κ-compact objects.
S
Let Acc = κ Accκ . We will refer to Acc as the ∞-category of accessible∞-categories.
d∞
Proposition 5.4.2.15. Let κ be a regular cardinal and let θ : Accκ → Cat
be the simplicial nerve of the functor which associates to each C ∈ Accκ thefull subcategory of C spanned by the κ-compact objects. Then
(1) The functor θ is fully faithful.
d∞ belongs to the essential image of θ if and(2) An ∞-category C ∈ Cat
only if C is essentially small and idempotent complete.
Proof. Assertion (1) follows immediately from Proposition 5.3.5.10. If C ∈d∞ belongs to the essential image of θ, then C is essentially small andCatidempotent complete (because C is stable under retracts in an idempotent
complete ∞-category). Conversely, suppose that C is essentially small andidempotent complete and choose a minimal model C0 ⊆ C. Then Indκ (C0 ) is
κ-accessible. Moreover, the collection of κ-compact objects of Indκ (C0 ) is an
idempotent completion of C0 (Lemma 5.4.2.4) and therefore equivalent to C
(since C0 is already idempotent complete).
Let Cat∨∞ denote the full subcategory of Cat∞ spanned by the idempotent
complete ∞-categories.
Proposition 5.4.2.16. The inclusion Cat∨∞ ⊆ Cat∞ has a left adjoint.
Proof. Combine Propositions 5.1.4.2, 5.1.4.9, and 5.2.7.8.
We will refer to a left adjoint to the inclusion Cat∨∞ ⊆ Cat∞ as the idempotent completion functor. Proposition 5.4.2.15 implies that we have fully faithd∞ ←- Cat∨ with the same essential image. Conful embeddings Accκ → Cat∞
sequently, there is a (canonical) equivalence of ∞-categories e : Cat∨
∞ ' Accκwhich is well-defined up to homotopy. We let Indκ : Cat∞ → Accκ denotethe composition of e with the idempotent completion functor. In summary:
Proposition 5.4.2.17. There is a functor Indκ : Cat∞ → Accκ which
exhibits Accκ as a localization of the ∞-category Cat∞ .

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Remark 5.4.2.18. There is a slight danger of confusion with our terminology. The functor Indκ : Cat∞ → Accκ is well-defined only up to a contractible space of choices. Consequently, if C is an ∞-category which admits
finite colimits, then the image of C under Indκ is well-defined only up toequivalence. Definition 5.3.5.1 produces a canonical representative for thisimage.
5.4.3 Accessibility and Idempotent Completeness
Let C be an accessible ∞-category. Then there exists a regular cardinal κsuch that C admits κ-filtered colimits. It follows from Remark 5.3.1.10 that
C is idempotent complete. Our goal in this section is to prove a converse tothis result: if C is small and idempotent complete, then C is accessible.
Let C be a small ∞-category and suppose we want to prove that C isaccessible. The main problem is to show that C admits κ-filtered colimits
provided that κ is sufficiently large. The idea is that if κ is much larger
than the size of C, then any κ-filtered diagram J → C is necessarily very
“redundant” (Proposition 5.4.3.4). Before making this precise, we will needa few preliminary results.
Lemma 5.4.3.1. Let κ < τ be uncountable regular cardinals, let A a τ filtered partially ordered set, and let F : A → Kan a diagram of Kan complexes indexed by A. Suppose that for each α ∈ A, the Kan complex F (α) isessentially κ-small. For every τ -small subset A0 ⊆ A, there exists a filteredτ -small subset A00 ⊆ A containing A0 , with the property that the map
limα∈A0 F (α) → limα∈A F (α)−→−→
0is a homotopy equivalence.
Proof. Let X = limα∈A F (α). Since F is a filtered diagram, X is also a Kan−→complex. Let K be a simplicial set with only finitely many nondegenerate
simplices. Our first claim is that the set [K, X] of homotopy classes of maps
from K into X is κ-small. Suppose we are given a collection {gβ : K → X}of pairwise nonhomotopic maps, having cardinality κ. Since A is τ -filtered,
we may suppose that there is a fixed index α ∈ A such that each gβ factors
as a composition
0gβ

K → F (α) → X.
The maps gβ0 are also pairwise nonhomotopic, which contracts our assumption that F (α) is weakly homotopy equivalent to a κ-small simplicial set.
We now define an increasing sequenceα0 ≤ α1 ≤ · · ·of elements of A. Let α0 be any upper bound for A0 . Assuming that αi has
already been selected, choose a representative for every homotopy class ofhighertopoi

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429

diagrams/ F (αi )∂ ∆ _n



hγ
/ X.
∆n
The argument above proves that we can take the set of all such representatives to be κ-small, so that there exists αi+1 ≥ αi such that each hγ factors
as a composition
h0γ∆n → F (αi+1 ) → X
and the associated diagram/ F (αi )∂ ∆ _n

∆n

h0γ


/ F (αi+1 )is commutative. We now set A00 = A0 ∪ {α0 , α1 , . . .}; it is easy to check that
this set has the desired properties.
Lemma 5.4.3.2. Let κ < τ be uncountable regular cardinals, let A be a
τ -filtered partially ordered set, and let {Fβ }β∈B be a collection of diagrams
A → Set∆ indexed by a τ -small set B. Suppose that for each α ∈ A and eachβ ∈ B, the Kan complex Fβ (α) is essentially κ-small. Then there exists a
filtered τ -small subset A0 ⊆ A such that for each β ∈ B, the map
limA0 Fβ (α) → limA Fβ (α)−→−→
is a homotopy equivalence of Kan complexes.
Proof. Without loss of generality, we may suppose that B = {β : β < β0 }is a set of ordinals. We will define a sequence of filtered τ -small subsets
A(n) ⊆ A by induction on n. For n = 0, choose an element α ∈ A and set
A(0) = {α}. Suppose next that A(n) has been defined. We define a sequenceof enlargements{A(n)β }β≤β0 by induction on β. Let A(n)0 = A(n), letSA(n)λ = β<λ A(n)β when λ is a nonzero limit ordinal, and let A(n)β+1 bea τ -small filtered subset of A such that the maplimA(n)Fβ (α) → limA Fβ (α)
−→
−→
β+1
is a weak homotopy equivalence (such a subset existsSby virtue of Lemma
5.4.3.1). We now take A(n + 1) = A(n)β0 and A0 = n A(n); it is easy tocheck that A0 ⊆ A has the desired properties.
Lemma 5.4.3.3. Let κ < τ be uncountable regular cardinals. Let C be a
τ -small ∞-category with the property that each of the spaces MapC (C, D)
is essentially κ-small and let j : C → P(C) denote the Yoneda embedding.
Let p : K → C be a diagram indexed by a τ -filtered ∞-category K and let
p : K. → P(C) be a colimit of j ◦ p. Then there exists a map i : K → K suchthat K is τ -small and the composition p ◦ i. : K . → K. → P(C) is a colimit
diagram.

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Proof. In view of Proposition 5.3.1.18, we may suppose that K is the nerveof a τ -filtered partially ordered set A. According to Proposition 5.1.2.2, p
induces a colimit diagrameCS,
pC : K. → P(C) →where eC denotes the evaluation functor associated to an object C ∈ C.
We will identify K. with the nerve of the partially ordered set A ∪ {∞}.
Proposition 4.2.4.4 implies that we may replace pC with the simplicial nerveof a functor FC : A ∪ {∞} → Kan. Our hypothesis on C implies that FC |Atakes values in κ-small simplicial sets. Applying Theorem 4.2.4.1, we see thatthe map limA FC (α) → FC (∞) is a homotopy equivalence. We now apply−→
Lemma 5.4.3.1 to deduce the existence of a filtered τ -small subset A0 ⊆ Asuch that each of the maps
limA0 FC (α) → FC (∞)−→
is a homotopy equivalence. Let K = N(A0 ) and let i : K → K denotethe inclusion. Using Theorem 4.2.4.1 again, we deduce that the compositioneC ◦p◦i. : K . → S is a colimit diagram for each C ∈ C. Applying Proposition
5.1.2.2, we deduce that p ◦ i. is a colimit diagram, as desired.
Proposition 5.4.3.4. Let κ < τ be uncountable regular cardinals. Let C be
an ∞-category which is τ -small, such that the morphism spaces MapC (C, D)
are essentially κ-small. Let j : C → P(C) denote the Yoneda embedding,let p : K → C be a diagram indexed by a τ -filtered ∞-category K, and let
X ∈ P(C) be a colimit of j ◦ p : K → P(C). Then there exists an object C ∈ Csuch that X is a retract of j(C).
Proof. Let i : K → K be a map satisfying the conclusions of Lemma 5.4.3.3.
Since K is τ -small and K is τ -filtered, there exists an extension i : K . → K
e ∈ Cp◦i/of i. Let C be the image of the cone point of K . under p ◦ i and let C.
be the corresponding lift. Let p : K → P(C) be a colimit of j ◦ p carryinge ∈ P(C)q/ be
the cone point of K. to X. Let q = j ◦ p ◦ i : K → P(C), let Xethe corresponding lift of X, and let Y ∈ P(C)q/ be a colimit of q. Since Ye is
an initial object of P(C)q/ , there is a commutative triangle
e
j(C)
>
e
/X
Ye
in the ∞-category P(C)q/ . Moreover, Lemma 5.4.3.3 asserts that the horie is a retract of j(C)e in the homotopy
zontal map is an equivalence. Thus X
category of P(C)q/ , so that X is a retract of j(C) in P(C).
Corollary 5.4.3.5. Let κ < τ be uncountable regular cardinals and let C
be a τ -small ∞-category whose morphism spaces MapC (C, D) are essentiallyκ-small. Then the Yoneda embedding j : C → Indτ (C) exhibits Indτ (C) as an
idempotent completion of C.

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Proof. Since Indτ (C) admits τ -filtered colimits, it is idempotent complete byRemark 5.3.1.10. Proposition 5.4.3.4 implies that every object of Indτ (C) isa retract of j(C) for some object C ∈ C.
Corollary 5.4.3.6. A small ∞-category C is accessible if and only if it is
idempotent complete. Moreover, if these conditions are satisfied and D is anany accessible ∞-category, then every functor f : C → D is accessible.
Proof. The “only if” direction follows from Remark 5.3.1.10, and the “if”
direction follows from Corollary 5.4.3.5. Now suppose that C is small andaccessible, and let D be a κ-accessible ∞-category and f : C → D any
functor; we wish to prove that f is accessible. By Proposition 5.3.5.10, we
may suppose that f = F ◦j, where j : C → Indκ (C) is the Yoneda embedding
and F : Indκ (C) → D is a κ-continuous functor and therefore accessible.
Enlarging κ if necessary, we may suppose that j is an equivalence of ∞categories, so that f is accessible as well.
5.4.4 Accessibility of Functor ∞-Categories
Let C be an accessible ∞-category and let K be a small simplicial set. Our
goal in this section is to prove that Fun(K, C) is accessible (Proposition
5.4.4.3). In §5.4.7, we will prove a much more general stability result of thiskind (Corollary 5.4.7.17), but the proof of that result ultimately rests on theideas presented here.
Our proof proceeds roughly as follows. If C is accessible, then C has many
τ -compact objects provided that τ is sufficiently large. Using Proposition
5.3.4.13, we deduce the existence of many τ -compact objects in Fun(K, C).
Our main problem is to show that these objects generate Fun(K, C) under τ filtered colimits. To prove this, we will use a rather technical cofinality result(Lemma 5.4.4.2 below). We begin with the following preliminary observation:
Lemma 5.4.4.1. Let τ be a regular cardinal and let q : Y → X be a coCartesian fibration with the property that for every vertex x of X, the fiber
Yx = Y ×X {x} is τ -filtered. Then q has the right lifting property with respect
to K ⊆ K . for every τ -small simplicial set K.
Proof. Using Proposition A.2.3.1, we can reduce to the problem of showingthat q has the right lifting property with respect to the inclusion K ⊆ K ∆0 .
In other words, we must show that given any edge e : C → D in X K , where
e of Y K lifting C, there exists an edge
D is a constant map, and any vertex Ceeeee : C → D lifting ee, where D is a constant map from K to Y . We first choosee→D
e 0 lifting e (since the map q K : Y K → X K is aan arbitrary edge ee0 : C
coCartesian fibration, we can even choose ee0 to be q K -coCartesian, though
we will not need this). Suppose that D takes the constant value x : ∆0 → X.
e0 → De in YxK , whereSince the fiber Yx is τ -filtered, there exists an edge ee00 : D
e is a constant map from K to Yx . We now invoke the fact that q K is anDhighertopoi

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inner fibration to supply the dotted arrow in the diagram(ee0 ,•,e
e00 )

Λ21
_

/ YK
7

σ


∆2

s1 e


/ XK.

We now define ee = σ|∆{0,2} .
Lemma 5.4.4.2. Let κ and τ be regular cardinals such that κ < τ andκ  τ . Let q : Y → X be a map of simplicial sets with the following
properties:(i) The simplicial set X is τ -small.
(ii) The map q is a coCartesian fibration.
(iii) For every vertex x ∈ X, the fiber Yx = Y ×X {x} is τ -filtered andadmits τ -small κ-filtered colimits.
(iv) For every edge e : x → y in X, the associated functor Yx → Yy preserves τ -small κ-filtered colimits.
Then(1) The ∞-category C = Map/X (X, Y ) of sections of q is τ -filtered.
(2) For each vertex x of X, the evaluation map ex : C → Yx is cofinal.
Proof. Using Proposition 5.4.1.2 (and the assumption that τ is uncountable),
we can choose a categorical equivalence X → X 0 , where X 0 is a τ -small ∞category. According to Corollary 3.3.1.2, Y is equivalent to the pullback of
a coCartesian fibration Y 0 → X 0 . We may therefore replace X by X 0 andthereby reduce to the case where X is an ∞-category.
Let K be a τ -small simplicial set equipped with a map f : K → Y .
We define a new object T (K) ∈ (Set∆ )/X so that the following universal
property is satisfied: for every simplicial set A, giving a map A → K 0 is
equivalent to giving the following data:• A map of simplicial sets A → ∆1 .
• A map ξ : A ×∆1 {0} → K.
• A map χ : A → X such that χ|(A ×∆1 {0}) = q ◦ f ◦ ξ.
We will prove the existence of a dotted arrow F as indicated in the diagram
f

K
F


T (K)

/Y
<
q


/ X.

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIESSpecializing to the case K = Z × X where Z is a τ -small simplicial set, we
will deduce that any diagram Z → C extends to a map Z .`→ C, which proves(1). Similarly, by specializing to the case K = (Z × X) Z×{x} (Z / × {x}),we will deduce that for every object y ∈ Y with q(y) = x, the ∞-category
C ×Yx (Yx )y/ is τ -filtered and therefore weakly contractible. Applying Theorem 4.1.3.1, we deduce (2).
To prove the existence of F , there is no harm in enlarging K (so longas it remains τ -small). We may therefore assume that K is an ∞-category.
We construct a transfinite sequence of objects fα : K(α) → Y of (Set∆ )/Y
for α < κ, where each K(α) is τ -small, as follows. Let f0 = f , and for alimit ordinal λ we let fλ denote the amalgam of the maps fα for α < λ.
Assume that α < κ and that fα : K(α) → Y has been defined. Let S be a
set of representatives for isomorphism classes of pairs s = (x, φ) where x is avertex of X and φ : As → K(α) ×X X/x is map of simplicial sets, where As
is κ-small. Applying Lemma 5.4.4.1 to the fibrations X/x ×X Y → X/x , wededuce that for every pair (x, φ) ∈ S, there exists a map φ : A.s → X/x ×X Y
compatible with fα and φ. Let K(α)0 denote the pushout
a a
(
A/s ,
K(α)
`

s∈S Ass∈S

so that the maps φ induce a map fα0 : K(α)0 → Y . Our assumptions on κand τ guarantee that the set S is τ -small so that K(α)0 is τ -small. We may
therefore factor fα0 as a composition
fα+1

K(α)0 → K(α + 1) → Y
where K(α + 1) is a τ -small ∞-category. This completes the construction.
Let K 0 = limα<κ K(α) and let f 0 : K 0 → Y be the amalgam of the maps−→
fα . We will prove that the lifting problem indicated in the diagram
f0

K0
F


T (K 0 )

/Y
<

0

q


/ X.

admits a solution. In fact, we claim that we can choose F 0 to be a q-left Kan
extension of f 0 . According to Lemma 4.3.2.13, it will suffice to show that forevery object x ∈ X, the induced map fx0 : K 0 ×X X/x → Y can be extended.
to a q-colimit diagram lifting the projection (K 0 ×X X/x ). → X/x→ X.
Using assumptions (ii), (iii), (iv), and Corollary 4.3.1.11, we are reduced toproving that K 0 ×X X/x is τ -small and κ-filtered. The first claim is obvious.
For the second, suppose we are given a map φ : A → K 0 ×X X/x . If A isκ-small, then φ factors through φ0 : A → K(α) ×X X/x for some α < κ. By
construction, the map φ0 extends to a map φ0 : A. → K(α + 1) ×X X/x , sothat φ extends to φ : A. → K 0 ×X X/x .

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Proposition 5.4.4.3. Let C be an accessible ∞-category and let K be asmall simplicial set. Then Fun(K, C) is accessible.
Proof. Without loss of generality, we may suppose that K is an ∞-category.
Choose a regular cardinal κ such that C admits small κ-filtered colimits
and choose a second regular cardinal τ with τ  κ, τ > κ, K is τ small, and C is τ -accessible. We will prove that Fun(K, C) is τ -accessible.
Let C0 = Fun(K, Cτ ) ⊆ Fun(K, C). It is clear that C0 is essentially small.
Proposition 5.1.2.2 implies that Fun(K, C) admits small τ -filtered colimits,and Proposition 5.3.4.13 asserts that C0 consists of τ -compact objects ofFun(K, C). According to Proposition 5.4.2.2, it will suffice to prove that C0
generates Fun(K, C) under small τ -filtered colimits.
Without loss of generality, we may suppose that C = Indτ D0 , where0
D is a small ∞-category. Let D ⊆ C denote the essential image of the
Yoneda embedding. Let F : K → C be an arbitrary object of CK and let
Fun(K, D)/F = Fun(K, D) ×Fun(K,C) Fun(K, C)/F . Consider the composite
diagramp : Fun(K, D)/F  ∆0 → Fun(K, C)/F  ∆0 → Fun(K, C).
The ∞-category Fun(K, D)/F is equivalent toFun(K, D0 ) ×Fun(K,C) Fun(K, C)/Fand therefore essentially small. To complete the proof, it will suffice to showthat Fun(K, D)/F is τ -filtered and that p is a colimit diagram.
We may identify F with a map fK : K → C ×K in (Set∆ )/K . According
to Proposition 4.2.2.4, we obtain a coCartesian fibration q : (C ×K)/fK →K, and the q-coCartesian morphisms are precisely those which project toequivalences in C. Let X denote the full subcategory of (C ×K)/fK consistingof those objects whose projection to C belongs to D. It follows that q 0 = q|X :X → K is a coCartesian fibration. We may identify the fiber of q 0 over a
vertex x ∈ K with D/F (x) = D ×C C/F (x) . It follows that the fibers of q 0 areτ -filtered ∞-categories; Lemma 5.4.4.2 now guarantees that Fun(K, D)/F 'Map/K (K, X) is τ -filtered.
According to Proposition 5.1.2.2, to prove that p is a colimit diagram,it will suffice to prove that for every vertex x of K, the composition of pwith the evaluation map ex : Fun(K, C) → C is a colimit diagram. The
composition ex ◦ p admits a factorization
Fun(K, D)/F  ∆0 → D/F (x) ∆0 → C
where D/F (x) = D ×C C/F (x) and the second map is a colimit diagram in
C by Lemma 5.1.5.3. It will therefore suffice to prove that the map gx :Fun(K, D)/F → D/F (x) is cofinal, which follows from Lemma 5.4.4.2.
5.4.5 Accessibility of Undercategories
Let C be an accessible ∞-category and let p : K → C be a small diagram.
Our goal in this section is to prove that the ∞-category Cp/ is accessible(Corollary 5.4.5.16).

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Remark 5.4.5.1. The analogous result for the ∞-category C/p will be
proven in §5.4.6 using Propositions 5.4.4.3 and 5.4.6.6. It is possible to use
the same argument to give a second proof of Corollary 5.4.5.16; however, we
will need Corollary 5.4.5.16 in our proof of Proposition 5.4.6.6.
We begin by studying the behavior of colimits with respect to (homotopy)
fiber products of ∞-categories.
Lemma 5.4.5.2. Let
X0

q0

p0


Y0

/X
p

q


/Y

be a diagram of ∞-categories which is homotopy Cartesian (with respect to
the Joyal model structure). Suppose that X and Y0 have initial objects andthat p and q preserve initial objects. An object X 0 ∈ X0 is initial if and only if
p0 (X 0 ) is an initial object of Y0 and q 0 (X 0 ) is an initial object of X. Moreover,there exists an initial object of X0 .
Proof. Without loss of generality, we may suppose that p and q are categorical fibrations and that X0 = X ×Y Y0 . Suppose first that X 0 is an object of
X0 with the property that X = q 0 (X 0 ) and Y 0 = p0 (X 0 ) are initial objects of
X and Y0 . Then Y = p(X) = q(Y 0 ) is an initial object of Y. Let Z be anotherobject of X0 . We have a pullback diagram of Kan complexes0HomR
X0 (X , Z)0
/ HomR
X (X, q (Z))


0 0
HomR
(Y
, p (Z))
0Y/ HomR(Y,(q◦ p0 )(Z)).
Y

Since the maps p and q are inner fibrations, Lemma 2.4.4.1 implies that thisdiagram is homotopy Cartesian (with respect to the usual model structure on
Set∆ ). Since X, Y 0 , and Y are initial objects, each one of the Kan complexes
R
R
0
0
0 0
HomR
X (X, q (Z)), HomY0 (Y , p (Z)), and HomY (Y, (q ◦p )(Z)) is contractible.
R0
It follows that HomX0 (X , Z) is contractible as well, so that X 0 is an initial
object of X0 .
We now prove that there exists an object X 0 ∈ X0 such that p0 (X 0 ) andq 0 (X 0 ) are initial. The above argument shows that X 0 is an initial object of
X0 . Since all initial objects of X0 are equivalent, this will prove that for any
initial object X 00 ∈ X0 , the objects p0 (X 00 ) and q 0 (X 00 ) are initial.
We begin by selecting arbitrary initial objects X ∈ X and Y ∈ Y0 . Thenp(X) and q(Y ) are both initial objects of Y, so there is an equivalence e :p(X) → q(Y ). Since q is a categorical fibration, there exists an equivalencee : Y 0 → Y in Y such that q(e) = e. It follows that Y 0 is an initial objectof Y0 with q(Y 0 ) = p(X), so that the pair (X, Y 0 ) can be identified with anobject of X0 which has the desired properties.

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Lemma 5.4.5.3. Let p : X → Y be a categorical fibration of ∞-categories,
and let f : K → X be a diagram. Then the induced map p0 : Xf / → Ypf / isa categorical fibration.
Proof. It suffices to show that p0 has the right lifting property with respect
to every inclusion A ⊆ B which is a categorical equivalence. Unwinding
the definitions, it suffices to show that p has the right lifting property withrespect to i : K ? A ⊆ K ? B. This is immediate since p is a categoricalfibration and i is a categorical equivalence.
Lemma 5.4.5.4. Let
X0

q0

p0


Y0

/X
p

q


/Y

be a diagram of ∞-categories which is homotopy Cartesian (with respect to
the Joyal model structure) and let f : K → X0 be a diagram in X0 . Then the
induced diagram
X0f // Xq0 f /

Y0p0 f // Yqp0 f /is also homotopy Cartesian.
Proof. Without loss of generality, we may suppose that p and q are categorical fibrations and that X0 = X ×Y Y0 . Then X0f / ' Xq0 f / ×Yqp0 f / Y0p0 f / , so theresult follows immediately from Lemma 5.4.5.3.
Lemma 5.4.5.5. Let
X0

q0

p0


Y0

/X
p

q


/Y

be a diagram of ∞-categories which is homotopy Cartesian (with respect to
the Joyal model structure) and let K be a simplicial set. Suppose that X and
Y0 admit colimits for all diagrams indexed by K and that p and q preserve
colimits of diagrams indexed by K. Then
(1) A diagram f : K . → X0 is a colimit of f = f |K if and only if p0 ◦f andq 0 ◦ f are colimit diagrams. In particular, p0 and q 0 preserve colimits
indexed by K.
(2) Every diagram f : K → X0 has a colimit in X0 .

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437

PRESENTABLE AND ACCESSIBLE ∞-CATEGORIESProof. Replacing X0 by X0f / , X by Xq0 f / , Y0 by Y0p0 f / , and Y by Yqp0 f / , we
may apply Lemma 5.4.5.4 to reduce to the case K = ∅. Now apply Lemma
5.4.5.2.
Lemma 5.4.5.6. Let C be a small filtered category and let C. be the category
obtained by adjoining a (new) final object to C. Suppose we are given ahomotopy pullback diagram
/F
F0

G0

p

q


/G

in the diagram category SetC
∆ (which we endow with the projective model
structure). Suppose further that the diagrams F, G, G0 : C. → Set∆ are homotopy colimits. Then F 0 is also a homotopy colimit diagram.
.

Proof. Without loss of generality, we may suppose that G is fibrant, that p
and q are fibrations, and that F 0 = F ×G G0 . Let ∗ denote the cone pointof C. and let F (∞), G(∞), F 0 (∞), and G0 (∞) denote the colimits of the
diagrams F | C, G| C, F 0 | C, and G0 | C. Since fibrations in Set∆ are stable
under filtered colimits, the pullback diagram/ F (∞)F 0 (∞)G0 (∞)/ G(∞)

exhibits F 0 (∞) as a homotopy fiber product of F (∞) and G0 (∞) over G(∞)
in Set∆ . Since weak homotopy equivalences are stable under filtered colimits,
the natural maps G(∞) → G(∗), F 0 (∞) → F 0 (∗), and G0 (∞) → G0 (∗) are
weak homotopy equivalences. Consequently, the diagram
F 0 (∞)f$
F 0 (∗)/ F (∗)

G0 (∗)

/ G(∗)

exhibits both F 0 (∞) and F 0 (∗) as homotopy fiber products of F (∗) and G0 (∗)
over G(∗). It follows that f is a weak homotopy equivalence, so that F is a
homotopy colimit diagram, as desired.
Lemma 5.4.5.7. Let
X0

q0

p0


Y0

/X
p

q


/Y

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be a diagram of ∞-categories which is homotopy Cartesian (with respect to
the Joyal model structure) and let κ be a regular cardinal. Suppose that X and
Y0 admit small κ-filtered colimits and that p and q preserve small κ-filteredcolimits. Then
(1) The ∞-category X0 admits small κ-filtered colimits.
(2) If X 0 is an object of X0 such that Y 0 = p0 (X 0 ) and X = q 0 (X 0 ), andY = p(X) = q(Y 0 ) are κ-compact, then X 0 is a κ-compact object of X0 .
Proof. Claim (1) follows immediately from Lemma 5.4.5.5. To prove (2), consider a colimit diagram f : I. → X0 . We wish to prove that the composition
S corepresented by X 0 is also a colimit diagram.
of f with the functor X0 → b
Using Proposition 5.3.1.18, we may assume without loss of generality that I
is the nerve of a κ-filtered partially ordered set A. We may further supposethat p and q are categorical fibrations and that X0 = X ×Y Y0 . Let I.X 0 / denote the fiber product I. ×X0 XX 0 / and define I.X/ , I.Y 0 / , and I.Y / similarly.
We have a pullback diagram
I.X 0 // I.X/


I.Y 0 /


/ I.Y /of left fibrations over I. . Proposition 2.1.2.1 implies that every arrow in thisdiagram is a left fibration, so that Corollary 3.3.1.6 implies that I.X 0 / is a
homotopy fiber product of I.X/ and I.Y 0 / over I.Y / in the covariant model category (Set∆ )/ I. . Let G : (Set∆ )A∪{∞} → (Set∆ )I. denote the unstraightening
functor of §2.1.4. Since G is the right Quillen functor of a Quillen equivalence, the above diagram is weakly equivalent to the image under G of ahomotopy pullback diagram
/ FX
FX 0

FY 0


/ FY

of (weakly) fibrant objects of (Set∆ )A∪{∞} . Moreover, the simplicial nerveof each FZ can be identified with the composition of f with the functor
corepresented by Z. According to Theorem 4.2.4.1, it will suffice to showthat FX 0 is a homotopy colimit diagram. We now observe that FX , FY 0 , andFY are homotopy colimit diagrams (since X, Y 0 , and Y are assumed to beκ-compact) and conclude by applying Lemma 5.4.5.6.
In some of the arguments below, it will be important to be able to replacefcolimits of a diagram J → C by colimits of some composition I → J → C.
According to Proposition 4.1.1.8, this maneuver is justified provided that f
is cofinal. Unfortunately, the class of cofinal morphisms is not sufficiently

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robust for our purposes. We will therefore introduce a property somewhat
stronger than cofinality which has better stability properties.
Definition 5.4.5.8. Let f : I → J denote a functor between filtered ∞categories. We will say that f is weakly cofinal if, for every object J ∈ J,
there exists an object I ∈ I and a morphism J → f (I) in J. We will say thatf is κ-cofinal if, for every diagram p : K → I where K is κ-small and weakly
contractible, the induced functor Ip/ → Jf p/ is weakly cofinal.
Example 5.4.5.9. Let I be a τ -filtered ∞-category and let p : K → I
be a τ -small diagram. Then the projection Ip/ → I is τ -cofinal. To provethis, consider a τ -small diagram K 0 → Ip/ , where K 0 is weakly contractible,
corresponding to a map q : K ? K 0 → I. According to Lemma 4.2.3.6, theinclusion K 0 ⊆ K ? K 0 is right anodyne, so that the map Iq/ → Iq|K 0 / is atrivial fibration (and therefore weakly cofinal).
Lemma 5.4.5.10. Let A, B, and C be simplicial sets and suppose that B
is weakly contractible. Then the inclusiona(A ? B) (B ? C) ⊆ A ? B ? C
B

is a categorical equivalence.
`
Proof. Let F (A, B, C) = (A ? B) B (B ? C) and let G(A, B, C) = A ? B ? C.
We first observe that both F and G preserve filtered colimits and homotopy
pushout squares separately in each argument. Using standard arguments(see, for example, the proof of Proposition 2.2.2.7), we can reduce to thecase where A and C are simplices.
Let us say that a simplicial set B is good if the inclusion F (A, B, C) ⊆G(A, B, C) is a categorical equivalence. We now make the following observations:(1) Every simplex is good. Unwinding the definitions, this is equivalent tothe assertion that for 0 ≤ m ≤ n ≤ p, the diagram/ ∆{0,...,n}∆{m,...n}
_
_


∆{m,...,p}/ ∆{0,...,p}is a homotopy pushout square (with respect to the Joyal model structure). For 0 ≤ i ≤ j ≤ p, setaaXij = ∆{i,i+1}···∆{j−1,j} ⊆ ∆{i,...,j}
{i}

{j−1}

(by convention, we agree that Xij = {i} if i = j). Since each of the
inclusions Xij ⊆ ∆{i,...,j} is inner anodyne, it will suffice to show thathighertopoi

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the diagram
Xmn

/ X0n


Xmp


/ X0p

is a homotopy pushout square, which is clear.
(2) Given a pushout diagram of simplicial setsB _/ B 0


B 00


/ B 000_

in which the vertical arrows are cofibrations, if B, B 0 , and B 00 are good,then B 000 is good. This follows from the compatibility of the functors
F and G with homotopy pushouts in B.
(3) Every horn Λni is good. This follows by induction on n using (1) and(2).
(4) The collection of good simplicial sets is stable under filtered colimits;
this follows from the compatibility of F and G with filtered colimits
and the stability of categorical equivalences under filtered colimits.
(5) Every retract of a good simplicial set is good (since the collection of
categorical equivalences is stable under the formation of retracts).
(6) If i : B → B 0 is an anodyne map of simplicial sets and B is good, thenB 0 is good. This follows by combining observations (1) through (5).
(7) If B is weakly contractible, then B is good. To see this, choose a vertex
b of B. The simplicial set {b} ' ∆0 is good (by (1) ), and the inclusion{b} ⊆ B is anodyne. Now apply (6).

Lemma 5.4.5.11. Let κ and τ be regular cardinals, let f : I → J be aκ-cofinal functor between τ -filtered ∞-categories, and let p : K → J be aκ-small diagram. Then
(1) The ∞-category Ip/ = I ×J Jp/ is τ -filtered.
(2) The induced functor Ip/ → Jp/ is κ-cofinal.
Proof. We first prove (1). Let qe : K 0 → Ip/ be a τ -small diagram classifying acompatible pair of maps q : K 0 → I and q 0 : K ? K 0 → J. Since I is τ -filtered,highertopoi

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we can find an extension q : (K 0 ). → I of q. To find a compatible extensionof qe, it suffices to solve the lifting problem`/8 J(K ? K 0 )  K 0 (K 0 ).
_i
(K ? K 0 ). ,which is possible since i is a categorical equivalence (Lemma 5.4.5.10) and J
is an ∞-category.
To prove (2), we consider a map qe : K 0 → Ip/ as above, where K is now
κ-small and weakly contractible. We have a pullback diagram(Ip/ )q// Iq/

Jq 0 /

/ Jq0 |K 0 / .

Lemma 4.2.3.6 implies that the inclusion K 0 ⊆ K ? K 0 is right anodyne,
so that the lower horizontal map is a trivial fibration. It follows that theupper horizontal map is also a trivial fibration. Since f is κ-cofinal, the rightvertical map is weakly cofinal, so that the left vertical map is weakly cofinal
as well.
Lemma 5.4.5.12. Let κ be a regular cardinal and let f : I → J be anκ-cofinal map of filtered ∞-categories. Then f is cofinal.
Proof. According to Theorem 4.1.3.1, to prove that f is cofinal it suffices toshow that for every object J ∈ J, the fiber product IJ/ = I ×J JJ/ is weakly
contractible. Lemma 5.4.5.11 asserts that IJ/ is κ-filtered; now apply Lemma
5.3.1.20.
Lemma 5.4.5.13. Let κ be a regular cardinal, let C be an ∞-category which
admits κ-filtered colimits, let p : K . → Cτ be a κ-small diagram in the ∞category of κ-compact objects of C, and let p = p|K. Then p is a κ-compact
object of Cp/ .
Proof. Let p0 denote the compositionp

K  ∆0 → K . → Cκ ;it will suffice to prove that p0 is a τ -compact object of Cp/ . Consider the
pullback diagramCp// Fun(K × ∆1 , C)∗
/ Fun(K × {0}, C).

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Corollary 2.4.7.12 implies that the f is a Cartesian fibration, so we can apply Proposition 3.3.1.3 to deduce that the diagram is homotopy Cartesian(with respect to the Joyal model structure). Using Proposition 5.1.2.2, wededuce that f preserves κ-filtered colimits and that any functor ∗ → D preserves filtered colimits (since filtered ∞-categories are weakly contractible;
see §4.4.4). Consequently, Lemma 5.4.5.7 implies that p0 is a κ-compact object of Cp/ provided that its images in ∗ and Fun(K × ∆1 , C) are κ-compact.
The former condition is obvious, and the latter follows from Proposition
5.3.4.13.
Lemma 5.4.5.14. Let C be an ∞-category which admits small τ -filteredcolimits and let p : K → C be a small diagram. Then Cp/ admits small
τ -filtered colimits.
Proof. Without loss of generality, we may suppose that K is an ∞-category.
Let I be a τ -filtered ∞-category and let q0 : I → Cp/ be a diagram corresponding to a map q : K ? I → C. We observe that K ? I is small andτ -filtered, so that q admits a colimit q : (K ? I). → C. The map q can also
be identified with a colimit of q0 .
Proposition 5.4.5.15. Let τ  κ be regular cardinals, let C be a τ -accessible∞-category, and let p : K → Cτ be a κ-small diagram. Then Cp/ is τ accessible, and an object of Cp/ is τ -compact if and only if its image in C is
τ -compact.
Proof. Let D = Cp/ ×C Cτ be the full subcategory of Cp/ spanned by thoseobjects whose images in C are τ -compact. Since Cp/ is idempotent complete
and the collection of τ -compact objects of C is stable under the formation ofretracts, we conclude that D is idempotent complete. We also note that D isessentially small; replacing C by a minimal model if necessary, we may suppose that D is actually small. Proposition 5.3.5.10 and Lemma 5.4.5.14 imply
that there is an (essentially unique) τ -continuous functor F : Indτ (D) → Cp/
Fsuch that the composition D → Indτ (D) → Cp/ is equivalent to the inclusion of D in Cp/ . To complete the proof, it will suffice to show that F is anequivalence of ∞-categories. According to Proposition 5.3.5.11, it will suffice
to show that D consists of τ -compact objects of Cp/ and generates Cp/ underτ -filtered colimits. The first assertion follows from Lemma 5.4.5.13.
To complete the proof, choose an object p : K . → C of Cp/ and let C ∈ C
denote the image under p of the cone point of K . . Then we may identify
p with a diagram pe : K → Cτ/C . Since C is τ -accessible, the ∞-category
E = Cτ/C is τ -filtered. It follows that Epe/ is τ -filtered and essentially small;to complete the proof, it will suffice to show that the associated map
E.pe/ → Cp/
is a colimit diagram. Equivalently, we must show that the compositition
θ.

θ

0
1
K ? E.pe/ →E. →Chighertopoi

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is a colimit diagram. Since θ1 is a colimit diagram, it suffices to prove that
θ0 is cofinal. For this, we consider the composition
θ

i

0
E.
q : Epe/ → K ? Epe/ →The ∞-category E is τ -filtered, so that Epe/ is also τ -filtered and therefore
weakly contractible (Lemma 5.3.1.20). It follows that i is right anodyne(Lemma 4.2.3.6) and therefore cofinal. Applying Proposition 4.1.1.3, we conclude that θ0 is cofinal if and only if q is cofinal. We now observe that thatq is τ -cofinal (Example 5.4.5.9) and therefore cofinal (Lemma 5.4.5.12).
Corollary 5.4.5.16. Let C be an accessible ∞-category and let p : K → C
be a diagram indexed by a small simplicial set K. Then Cp/ is accessible.
Proof. Choose appropriate cardinals τ  κ and apply Proposition 5.4.5.15.

5.4.6 Accessibility of Fiber Products
Our goal in this section is to prove that the class of accessible ∞-categories isstable under (homotopy) fiber products (Proposition 5.4.6.6). The strategy
of proof should now be familiar from §5.4.4 and §5.4.5. Suppose we are givena homotopy Cartesian diagram
X0

q0

p0


Y0

/X
p

q


/Y

of ∞-categories, where X, Y0 , and Y are accessible ∞-categories, and the functors p and q are likewise accessible. If κ is a sufficiently large regular cardinal,
then we can use Lemma 5.4.5.7 to produce a good supply of κ-compact objects of X0 . Our problem is then to prove that these objects generate X0 underκ-filtered colimits. This requires some rather delicate cofinality arguments.
Lemma 5.4.6.1. Let τ  κ be regular cardinals and let f : C → D bea τ -continuous functor between τ -accessible ∞-categories which carries τ compact objects of C to τ -compact objects of D. Let C be an object of C,
let Cτ/C denote the full subcategory of C/C spanned by those objects C 0 → C,
where C 0 is τ -compact, and let Dτ/f (C) the full subcategory of D/f (C) spannedby those objects D → f (C), where D ∈ D is τ -compact. Then f induces aκ-cofinal functor f 0 : Cτ/C → Dτ/f (C) .
Proof. Let pe : K → Cτ/C be a diagram indexed by a τ -small weakly contractible simplicial set K and let p : K → C be the underlying map. We
need to show that the induced functor (Cτ/C )pe/ → (Dτ/f (C) )f 0 pe/ is weakly
cofinal. Using Proposition 5.4.5.15, we may replace C by Cp/ and D by Df p/
and thereby reduce to the problem of showing that f is weakly cofinal. Let
φ : D → f (C) be an object of Dτ/f (C) and let FD : D → S be the functor

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corepresented by D. Since D is τ -compact, the functor FD is τ -continuous,
so that FD ◦ f is τ -continuous. Consequently, the space FD (f (C)) can beobtained as a colimit of the τ -filtered diagramF

DS.
p : Cτ/C → Dτ/f (C) → D →In particular, the path component of FD (f (C)) containing φ lies in theimage of p(η) for some η : C 0 → C as above. It follows that there exists
a commutative diagram
φ

D

f (η)

/ f (C);!
f (C 0 )in D, which can be identified with a morphism in Dτ/f (C) having the desired
properties.
Lemma 5.4.6.2. Let A = A0 ∪ {∞} be a linearly ordered set containing alargest element ∞ and let B ⊆ A0 be a cofinal subset (in other words, forevery α ∈ A0 , there exists β ∈ B such that α ≤ β). The inclusionaφ : N(A0 )N(B ∪ {∞}) ⊆ N(A)
N(B)

is a categorical equivalence.
Proof. For each β ∈ B, let φβ denote the inclusion ofaN({α ∈ A0 : α ≤ β})N({α ∈ B : α ≤ β} ∪ {∞})
N({α∈B:α≤β})0into N({α ∈ A : α ≤ β} ∪ {∞}). Since B is cofinal in A0 , φ is a filteredcolimit of the inclusions φβ . Replacing A0 by {α ∈ A0 : α ≤ β} and B by{α ∈ B : α ≤ β}, we may reduce to the case where A0 has a largest element(which we will continue to denote by β).
We have a categorical equivalenceaN(B)N({β, ∞}) ⊆ N(B ∪ {∞}).
{β}Consequently, to prove that φ is a categorical equivalence, it will suffice toshow that the compositionaaN(A0 )
N({β, ∞}) ⊆ N(A0 )N(B ∪ {∞}) ⊆ N(A)
{β}

N(B)

is a categorical equivalence, which is clear.
Lemma 5.4.6.3. Let τ > κ be regular cardinals and letpp0

X → Y ← X0
be functors between ∞-categories. Assume that the following conditions are
satisfied:highertopoi

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445

PRESENTABLE AND ACCESSIBLE ∞-CATEGORIES(1) The ∞-categories X, X0 , and Y are κ-filtered, and admit τ -small κfiltered colimits.
(2) The functors p and p0 preserve τ -small κ-filtered colimits.
(3) The functors p and p0 are κ-cofinal.
Then there exist objects X ∈ X, X 0 ∈ X0 such that p(X) and p0 (X 0 ) areequivalent in Y.
Proof. For every ordinal α, we let [α] = {β : β ≤ α} and (α) = {β : β < α}.
Let us say that an ordinal α is even if it is of the form λ + n, where λ is alimit ordinal and n is an even integer; otherwise, we will say that α is odd.
Let A denote the set of all even ordinals smaller than κ and A0 the set of all
odd ordinals smaller than κ. We regard A and A0 as subsets of the linearlyordered set A ∪ A0 = (κ). We will construct a commutative diagram
N(A)/ N(κ) o

q

Q


X

p


/Yo

N(A0 )


p0

q0

X0 .

Supposing that this is possible, we choose colimits X ∈ X, X 0 ∈ X0 , andY ∈ Y for q, q 0 , and Q, respectively. Since the inclusion N(A) ⊆ N(κ) is
cofinal and p preserves κ-filtered colimits, we conclude that p(X) and Y areequivalent. Similarly, p0 (X 0 ) and Y are equivalent, so that p(X) and p0 (X 0 )are equivalent, as desired.
The construction of q, q 0 , and Q is given by induction. Let α < κ andsuppose that q| N({β ∈ A : β < α}), q 0 | N({β ∈ A0 : β < α}) and Q| N(α)
have already been constructed. We will show how to extend the definitionsof q, q 0 , and Q to include the ordinal α. We will suppose that α is even; thecase where α is odd is similar (but easier).
Suppose first that α is a limit ordinal. In this case, define q| N({β ∈ A :β ≤ α}) to be an arbitrary extension of q| N({β ∈ A : β < α}): such an
extension exists by virtue of our assumption that X is κ-filtered. In order todefine Q| N(α), it suffices to verify that Y has the extension property withrespect to the inclusionaN(α)
N({β ∈ A : β ≤ α}) ⊆ N[α].
N({β∈A:β<α})Since Y is an ∞-category, this follows immediately from Lemma 5.4.6.2.
We now treat the case where α = α0 + 1 is a successor ordinal. Let q<α =
q|{β ∈ A : β < α}, and regard Q| N({α0 } ∪ {β ∈ A : β < α}) as an
object of Yf q<α / . We now observe that N({β ∈ A : β < α}) is κ-small andweakly contractible. Since p is κ-cofinal, we can construct q|{β ∈ A : β ≤ α}extending q<α and a compatible map Q| N({α0 } ∪ {β ∈ A : β ≤ α}). To

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complete the construction of Q, it suffices to show that Y has the extension
property with respect to the inclusionaN(α)
N({β ∈ A : β ≤ α} ∪ {α0 }) ⊆ N[α].
N({β∈A:β<α}∪{α0 })Once again, this follows from Lemma 5.4.6.2.
Lemma 5.4.6.4. Let κ and τ be regular cardinals, let f : I → J be aκ-cofinal functor between τ -filtered ∞-categories, and let p : K → I be adiagram indexed by a τ -small simplicial set K. Then the induced functor
Ip/ → Jf p/
is κ-cofinal.
Proof. Let K 0 be a simplicial set which is κ-small and weakly contractible
and let q : K ? K 0 → I be a diagram. We have a commutative diagram/ Jf q/Iq/
Iq|K 0 // If q|K 0 / .

Lemma 4.2.3.6 implies that K 0 ⊆ K ? K 0 is a right anodyne inclusion, sothat the vertical maps are trivial fibrations. Since f is κ-cofinal, the lower
horizontal map is weakly cofinal; it follows that the upper horizontal map isweakly cofinal as well.
Lemma 5.4.6.5. Let τ > κ be regular cardinals and let
I0

q0

p0


J0

/I
p

q


/J

be a diagram of ∞-categories which is homotopy Cartesian (with respect
to the Joyal model structure). Suppose that I, J, and J0 are τ -filtered ∞categories which admit τ -small κ-filtered colimits. Suppose further that p
and q are κ-cofinal functors which preserve τ -small κ-filtered colimits. Then
I0 is τ -filtered, and the functors p0 and q 0 are κ-cofinal.
Proof. Without loss of generality, we may suppose that p and q are categorical fibrations and that I0 = I ×J J0 . To prove that I0 is τ -filtered, wemust show that I0f / is nonempty for every diagram f : K → I0 indexed by a
τ -small simplicial set K. We have a (homotopy) pullback diagramI0f // Iq0 f /g

J0p0 f /h/ Jpq0 f / .

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Lemma 5.3.1.21 implies that the ∞-categories Iq0 f / , J0p0 f / , and Jpq0 f / areτ -filtered, and Lemma 5.4.6.4 implies that g and h are κ-cofinal. We may
therefore apply Lemma 5.4.6.3 to deduce that I0f / is nonempty, as desired.
We now prove that q 0 is κ-cofinal; the analogous assertion for p0 is provenby the same argument. We must show that for every diagram f : K → I0 ,where K is κ-small and weakly contractible, the induced map I0f / → Iq0 f / isweakly cofinal. Replacing I0 by I0f / as above, we may reduce to the problemof showing that q 0 itself is weakly cofinal. Let I be an object of I, let J =p(I) ∈ J and consider the (homotopy) pullback diagram
I0I/

/ II/





u
v

J0J/

/ JJ/ .

We wish to show that I0I/ is nonempty. This follows from Lemma 5.4.6.3
because u and v are τ -cofinal (Lemmas 5.4.6.4 and 5.4.5.11, respectively).
Proposition 5.4.6.6. Let
X0

q0

p0


Y0

/X
p

q


/Y

be a diagram of ∞-categories which is homotopy Cartesian (with respect to
the Joyal model structure). Suppose further that X, Y, and Y0 are accessible
and that both p and q are accessible functors. Then X0 is accessible. Moreover,
for any accessible ∞-category C and any functor f : C → X, f is accessible
if and only if the compositions p0 ◦ f and q 0 ◦ f are accessible. In particular(taking f = idX ), the functors p0 and q 0 are accessible.
Proof. Choose a regular cardinal κ such that X, Y0 , and Y are κ-accessible.
Enlarging κ if necessary, we may suppose that p and q are κ-continuous.
It follows from Lemma 5.4.5.5 that X0 admits small κ-filtered colimits andthat for any κ0 > κ, a functor f : C → X is κ0 -continuous if and only if
p0 ◦ f and q 0 ◦ f are κ0 -continuous. This proves the second claim; it now
suffices to show that X0 is accessible. For this, we will use characterization(3) of Proposition 5.4.2.2. Without loss of generality, we may suppose thatp and q are categorical fibrations and that X0 = X ×Y Y0 . It then follows
easily that X0 is locally small. It will therefore suffice to show that there
exists a regular cardinal τ such that X0 is generated by a small collection ofτ -compact objects under small τ -filtered colimits.
Since the ∞-categories of κ-compact objects of X and Y0 are essentiallyκsmall, there exists τ > κ such that p| Xκ ⊆ Yτ and q|Y0 ⊆ Yτ . Enlarging τ
if necessary, we may suppose that τ  κ. The proof of Proposition 5.4.2.9
shows that every τ -compact object of X can be written as a τ -small κ-filteredhighertopoi

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colimit of objects belonging to Xκ . Since p is κ-continuous, it follows thatττp(Xτ ) ⊆ Yτ , and similarly q(Y0 ) ⊆ Yτ . Let X00 = Xτ ×Yτ Y0 . Then X00 isan essentially small full subcategory of X0 . Lemma 5.4.5.7 implies that X00
consists of τ -compact objects of X0 . To complete the proof, it will suffice toshow that X00 generates X0 under small τ -filtered colimits.
Let X 0 = (X, Y 0 ) be an object of X0 and set Y = pX = qY 0 . We have a(homotopy) pullback diagram
f0

X00/X 0

/ Xτ/Xg0g/ Yτ/Yfτ

Y0 /Y 0of essentially small ∞-categories. Lemma 5.4.6.1 asserts that f and g areκ-cofinal. We apply Lemma 5.4.6.5 to conclude that X00/X 0 is τ -filtered andthat f 0 and g 0 are κ-cofinal. Now consider the diagram/ (Xτ/X ).

(X00/X 0 ).
h

#

X0

q0

p0



q

τ

(Y0 /Y ).


/X
p


/ Y0


/ Y.

Lemma 5.4.5.12 allows us to conclude that f 0 and g 0 are cofinal, so that p0 ◦ h
and q 0 ◦ h are colimit diagrams. Lemma 5.4.5.5 implies that h is a colimit
diagram as well, so that X 0 is the colimit of an essentially small τ -filtereddiagram taking values in X00 .
Corollary 5.4.6.7. Let C be an accessible ∞-category and let p : K → C bea diagram indexed by a small simplicial set K. Then the ∞-category C/p isaccessible.
Proof. Since the map C/p → C/p is a categorical equivalence, it will suffice
to prove that C/p is accessible. We have a pullback diagram
C/p/ Fun(K × ∆1 , C)∗
/ Fun(K × {1}, C)pqof ∞-categories. Since p is a coCartesian fibration, Proposition 3.3.1.3 implies that this diagram is homotopy Cartesian. According to Proposition
1
5.4.4.3, the ∞-categories CK×∆ and CK×{1} are accessible. Using Proposition 5.1.2.2, we conclude that for every regular cardinal κ such that C admits

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κ-filtered colimits, p is κ-continuous; in particular, p is accessible. Corollary
5.4.3.6 implies that ∗ is accessible and that q is an accessible functor. Applying Proposition 5.4.6.6, we deduce that C/p is accessible.
5.4.7 Applications
In §5.4.4 through §5.4.6, we established some of the basic stability properties
enjoyed by the class of accessible ∞-categories. In this section, we will reap
some of the rewards for our hard work.
Lemma 5.4.7.1. Let {C`α }α∈A be a family of ∞-categories indexed by asmall set A and let C = α∈A Cα be their coproduct. Then C is an accessible
if and only if each Cα is accessible.
Proof. This is immediate from the definitions.
Lemma 5.4.7.2. Let {CαQ}α∈A be a family of ∞-categories indexed by asmall set A and let C = α∈A Cα be their product. If each Cα is accessible, then C is accessible. Moreover, if D is an accessible ∞-category, then afunctor D → C is accessible if and only if each of the compositions
D → C → Cα
is accessible.
`
Proof. Let D = α∈A Cα . By Lemma 5.4.7.1, D is accessible. Let N(A)denote the constant simplicial set with value A. Proposition 5.4.4.3 implies
that Fun(N(A), D) is accessible. We now observe that Fun(N(A), D) can be
written as a disjoint union of C with another ∞-category; applying Lemma
5.4.7.1 again, we deduce that C is accessible. The second claim follows immediately from the definitions.
Proposition 5.4.7.3. The ∞-category Acc of accessible ∞-categories add∞ preserves small limits.
mits small limits, and the inclusion i : Acc ⊆ Cat
Proof. By Proposition 4.4.2.6, it suffices to prove that Acc admits pullbacks
and small products and that i preserves pullbacks and (small) products. Let[
Acc∆ be the (simplicial) subcategory of Set∆ defined as follows:(1) The objects of Acc∆ are the accessible ∞-categories.
(2) If C and D are accessible ∞-categories, then MapAcc∆ (C, D) is thesubcategory of Fun(C, D) whose objects are accessible functors andwhose morphisms are equivalences of functors.
The ∞-category Acc is isomorphic to the simplicial nerve N(Acc∆ ). In viewof Theorem 4.2.4.1, it will suffice to prove that the simplicial category Acc∆
admits homotopy fiber products and (small) homotopy products and that the+ ◦[inclusion Acc ⊆ (Set
) preserves homotopy fiber products and homotopy∆∆

products. The case of homotopy fiber products follows from Proposition
5.4.6.6, and the case of (small) homotopy products follows from Lemma
5.4.7.2.

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If C is an accessible ∞-category, then C is the union of full subcategories{Cτ ⊆ C}, where τ ranges over all (small) regular cardinals. It seems reasonable to expect that if τ is sufficiently large, then the properties of C are
mirrored by properties of Cτ . The following result provides an illustration ofthis philosophy:
Proposition 5.4.7.4. Let C be a κ-accessible ∞-category and let τ  κ bean uncountable regular cardinal such that Cκ is essentially τ -small. Then thefull subcategory Cτ ⊆ C is stable under all κ-small limits which exist in C.
Before giving the proof, we will need to establish a few lemmas.
Lemma 5.4.7.5. Let τ  κ be regular cardinals and assume that τ is
uncountable. Let C be a τ -small ∞-category and let D be an object of Indκ (C).
The following are equivalent:(1) The object D is τ -compact in Indκ (C).
(2) For every C ∈ C, the space MapIndκ (C) (j(C), D) is essentially τ -small,where j : C → Indκ (C) denotes the Yoneda embedding.
Proof. Suppose first that (1) is satisfied. Using Lemma 5.1.5.3, we can write
D as the colimit of the κ-filtered diagram
C/D = C ×Indκ (C) Indκ (C)/D → Indκ (C).
Since τ  κ, we also write D as a small τ -filtered colimit of objects {Dα },where each Dα is the colimit of a τ -small κ-filtered diagram
e → C → Indκ (C).
C
Since D is τ -compact, we conclude that D is a retract of Dα . Let F :Indκ (C) → S denote the functor corepresented by j(C). According to Proposition 5.3.5.5, F is κ-continuous. It follows that F (D) is a retract of F (Dα ),which is itself a τ -small colimit of spaces equivalent to
MapIndκ (C) (j(C), j(C 0 )) ' MapC (C, C 0 ),which is essentially τ -small by assumption and therefore a τ -compact objectof S. It follows that D is also a τ -compact object of S.
Now assume (2). Once again, we observe that D can be obtained as thecolimit of a diagram C/D → Indκ (C). By assumption, C is τ -small and thefibers of the right fibration C/D → C are essentially τ -small. Proposition
5.4.1.4 implies that C/D is essentially τ -small, so that D is a τ -small colimitof κ-compact objects of Indκ (C) and therefore τ -compact.
Lemma 5.4.7.6. Let τ  κ be regular cardinals such that τ is uncountable
and let Sτ be the full subcategory of S consisting of essentially τ -small spaces.
Then Sτ is stable under κ-small limits in S.
Proof. In view of Proposition 4.4.2.6, it suffices to prove that Sτ is stable
under pullbacks and κ-small products. Using Theorem 4.2.4.1, it will sufficehighertopoi

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451

to show that the full subcategory of Kan spanned by essentially τ -small
spaces is stable under κ-small products and homotopy fiber products. This
follows immediately from characterization (1) given in Proposition 5.4.1.5.

Proof of Proposition 5.4.7.4. Let K be a κ-small simplicial set and let p :K → Cτ be a diagram which admits a limit X ∈ C. We wish to show that
X is τ -compact. According to Lemma 5.4.7.5, it suffices to prove that the
space F (X) is essentially τ -small, where F : C → S denotes the functor
corepresented by a κ-compact object C ∈ C. Since F preserves limits, wenote that F (X) is a limit of F ◦ p. Lemma 5.4.7.5 implies that the diagram
F ◦p takes values in Sτ ⊆ S. We now conclude by applying Lemma 5.4.7.6.
We note the following useful criterion for establishing that a functor isaccessible:
Proposition 5.4.7.7. Let G : C → C0 be a functor between accessible ∞categories. If G admits a right or a left adjoint, then G is accessible.
Proof. If G is a left adjoint, then G commutes with all colimits which exist
in C. Therefore G is κ-continuous for any cardinal κ having the property that
C is κ-accessible. Let us therefore assume that G is a right adjoint; choose aleft adjoint F for G.
Choose a regular cardinal κ such that C0 is κ-accessible. We may suppose
without loss of generality that C0 = Indκ D, where D is a small ∞-category.
Consider the composite functorjF

D → Indκ (D) → C .
Since D is small, there exists a regular cardinal τ  κ such that C is τ accessible and the essential image of F ◦ j consists of τ -compact objects of
C. We will show that G is τ -continuous.
Since Indκ (D) ⊆ P(D) is stable under small τ -filtered colimits, it will
suffice to prove that the compositionG

G0 : C → Indκ (D) → P(D)is τ -continuous. For each object D ∈ D, let G0D : C → b
S denote the composition of G0 with the functor given by evaluation at D. According to Proposition 5.1.2.2, it will suffice to show that each G0D is τ -continuous. Lemma
5.1.5.2 implies that G0D is equivalent to the composition of G with the functor C0 → b
S corepresented by j(D). Since F is left adjoint to G, we may
identify this with the functor corepresented by F (j(D)). Since F (j(D)) isτ -compact by construction, this functor is τ -continuous.
Definition 5.4.7.8. Let C be an accessible category. A full subcategory
D ⊆ C is an accessible subcategory of C if D is accessible, and the inclusionof D into C is an accessible functor.

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Example 5.4.7.9. Let C be an accessible ∞-category and K a simplicial
set. Suppose that every diagram K → C has a limit in C. Let D ⊆ Fun(K / , C)
be the full subcategory spanned by the limit diagrams. Then D is equivalent
to Fun(K, C) and is therefore accessible (Proposition 5.4.4.3). The inclusion D ⊆ Fun(K / , C) is a right adjoint and therefore accessible (Proposition
5.4.7.7). Thus D is an accessible subcategory of Fun(K / , C). Similarly, if every diagram K → C has a colimit, then the full subcategory D0 ⊆ Fun(K . , C)spanned by the colimit diagrams is an accessible subcategory of Fun(K . , C).
Proposition 5.4.7.10. Let C be an accessible category and letT{Dα ⊆ C}α∈A
be a (small) collection of accessible subcategories of C. Then α∈A Dα is anaccessible subcategory of C.
Proof. We have a homotopy Cartesian diagramT

α∈A Dαα∈A Dα

Q

i0

i

/C


f

/ CA .

Q
Lemma 5.4.7.2 implies that α∈A Dα and CA are accessible, and it is easy
to see that f and
T i are accessible functors. Applying Proposition 5.4.6.6, weconclude that α∈A Dα is accessible and that i0 is an accessible functor, as
desired.
We conclude this chapter by establishing a generalization of Proposition
5.4.4.3.
d∞ ofProposition 5.4.7.11. Let C be a subcategory of the ∞-category Cat(not necessarily small) ∞-categories satisfying the following conditions:d∞(a) The ∞-category C admits small limits, and the inclusion C ⊆ Cat
preserves small limits.
(b) If X belongs to C, then Fun(∆1 , X) belongs to C.
(c) If X and Y belong to C, then a functor X → Fun(∆1 , Y ) is a morphismof C if and only if, for every vertex v of ∆1 , the composite functor
X → Fun(∆1 , Y ) → Fun({v}, Y ) ' Y is a morphism of C.
Let p : X → S be a map of simplicial sets, where S is small. Assume that(i) The map p is a categorical fibration and a locally coCartesian fibration.
(ii) For each vertex s in S, the fiber Xs belongs to C.
(iii) For each edge s → s0 in S, the associated functor Xs → Xs0 is amorphism in C.

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Let E be a set of edges of S and let Y be the full subcategory of MapS (S, X)
spanned by those sections f : S → X of p which satisfy the following condition:(∗) For every edge e : ∆1 → S belonging to E, f carries e to a pe coCartesian edge of ∆1 ×S X, where pe : ∆1 ×S X → ∆1 is the
projection.
Then Y belongs to C. Moreover, if Z ∈ C, then a functor Z → Y belongs to
C if and only if, for every vertex s in S, the composite map Z → Y → Xs
belongs to C.
Remark 5.4.7.12. Hypotheses (i) through (iii) of Proposition 5.4.7.11 aresatisfied, in particular, if p : X → S is a coCartesian fibration classified byd∞ .
a functor S → C ⊆ Cat
Remark 5.4.7.13. Hypotheses (a), (b), and (c) of Proposition 5.4.7.11 ared∞ :satisfied for the following subcategories C ⊆ Cat• Fix a class of simplicial sets {Kα }α∈A . Then we can take C to be thed∞ whose objects are ∞-categories which admit Kα subcategory of Cat
indexed (co)limits for each α ∈ A, and whose morphisms are functors
which preserve Kα -indexed (co)limits for each α ∈ A.
• We can take the objects of C to be accessible ∞-categories and the morphisms in C to be accessible functors (in view of Propositions 5.4.4.3
and 5.4.7.3).
We will meet some other examples in §5.5.
Remark 5.4.7.14. In the situation of Proposition 5.4.7.11, we can replace“coCartesian” by “Cartesian” everywhere to obtain a dual result. This follows by applying Proposition 5.4.7.11 to the map X op → S op after replacingd∞ .
C by its preimage under the “opposition” involution of hCat
The proof of Proposition 5.4.7.11 makes use of the following observation:
Lemma 5.4.7.15. Let p : M → ∆1 be a coCartesian fibration classifying a functor F : C → D, where C = p−1 {0} and D = p−1 {1}. Let
X = Map∆1 (∆1 , M) be the ∞-category of sections of p. Then X can be
identified with a homotopy limit of the diagram
F

C → Fun({0}, D) ← Fun(∆1 , D).
Proof. We first replace the diagram in question by a fibrant one. Let C0
denote the ∞-category of coCartesian sections of p. Then the evaluation
map e : C0 → C is a trivial fibration of simplicial sets. Moreover, since F isassociated to the correspondence M, the map e admits a section s such thatthe compositions

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coincides with F . It follows that we have a weak equivalence of diagrams
CF/ Fun({0}, D) o

Fun(∆1 , D)

F0

/ Fun({0}, D) o

Fun(∆1 , D),

s


C0

where F 0 is given by evaluation at {1} and is therefore a categorical fibration.
Let X0 denote the pullback of the lower diagram, which we can identify withthe full subcategory of Map∆1 (Λ21 , M) spanned by those functors which carry
the first edge of Λ21 to a coCartesian edge of M.
Regard ∆2 as an object of (Set∆ )/∆1 via the unique retraction r : ∆2 →1
∆ onto the simplicial subset ∆{0,1} ⊆ ∆{0,1,2} . Let X00 denote the fullsubcategory of Map∆1 (∆2 , M) spanned by those maps ∆2 → M which carry
the initial edge of ∆2 to a p-coCartesian edge of M.
Let T denote the marked simplicial set whose underlying simplicial set
is ∆2 , whose sole nondegenerate marked edge is ∆1 ⊆ ∆2 , and let T 0 =T ×(∆2 )] (Λ21 )] . Since the opposites of the inclusions T 0 ⊆ T , (∆{0,2} )[ ⊆ Tare marked anodyne, we conclude that the evaluation maps
X ← X00 → X0
are trivial fibrations of simplicial sets. It follows that X and X0 are (canonically) homotopy equivalent, as desired.
Remark 5.4.7.16. In the situation of Lemma 5.4.7.15, the full subcategoryof X spanned by the coCartesian sections of p is equivalent (via evaluation
at {0}) to C.
Proof of Proposition 5.4.7.11. Let us first suppose that E = ∅. Let skn S
denote the n-skeleton of S. We observe that MapS (S, X) coincides with the(homotopy) inverse limit
lim{MapS (skn S, X)}.
←−
In view of assumption (a), it will suffice to prove the result after replacing S
by skn S. In other words, we may reduce to the case where S is n-dimensional.
We now work by induction on n and observe that there is a homotopy
pushout diagram of simplicial sets
/ Sn × ∆n
S n × ∂ ∆n



/ S.
skn−1 S 
We therefore obtain a homotopy pullback diagram of ∞-categories
MapS (S, X)/ MapS (skn−1 S, X)

MapS (Sn × ∆n , X)
/ MapS (Sn × ∂ ∆n , X).

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Invoking assumption (a) again, we are reduced to proving the same result
after replacing S by skn−1 S, Sn × ∂ ∆n , and Sn × ∆n . The first two cases
follow from the inductive hypothesis; we may therefore assume that S is adisjoint union of copies of ∆n . Applying (a) once more, we can reduce to thecase S = ∆n .
If n = 0, there is nothing to prove. If n > 1, then we have a trivial fibration
MapS (S, X) → MapS (Λn1 , X).
Since the horn Λn1 is of dimension less than n, we may conclude by applying
the inductive hypothesis. We are therefore reduced to the case S = ∆1 .
According to Lemma 5.4.7.15, the ∞-category Map∆1 (∆1 , X) can be identified with a homotopy limit of the diagramF

1∆
X{0} → X{1} ← X{1}.

In view of (a), it will suffice to prove that all of the ∞-categories and functors
in the above diagram belong to C. This follows immediately from (b) and(c).
We now consider the general case where E is not required to be empty.
For each edge e ∈ E, let Y (e) denote the full subcategory of MapS (S, X)
spanned by those sections f : S → X which satisfy the condition (∗) for theedge e. We wish to prove:T(1) The intersection e∈E Y (e) belongs to C.
T(2) If Z ∈ C, then a functor Z → e∈E Y (e) is a morphism of C if andonly if the induced map Z → MapS (S, X) is a morphism of C.
In view Tof (a), it will suffice to prove the corresponding results when the intersection e∈E Y (e) is replaced by a single subcategory Y (e) ⊆ MapS (S, X).
Let e : s → s0 be an edge belonging to E. Lemma 5.4.7.15 implies theexistence of a homotopy pullback diagram We now observe that there is a
homotopy pullback diagramY (e)/ MapS (S, X)


Fun0 (∆1 , Xs0 )/ Fun(∆1 , Xs0 ),where Fun0 (∆1 , Xs0 ) ' Xs0 is the full subcategory of Fun(∆1 , Xs0 ) spannedby the equivalences. In view of (a), it suffices to prove the following analoguesof (1) and (2):(10 ) For each s0 ∈ S, the ∞-categories Fun0 (∆1 , Xs0 ) and Fun(∆1 , Xs0 )belong to C.
(20 ) Given an object Z ∈ C, a functor Z → Fun0 (∆1 , Xs0 ) is a morphism in
C if and only if the induced map Z → Fun(∆1 , Xs0 ) is a morphism ofC.

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These assertions follow immediately from (b) and (c), respectively.
Corollary 5.4.7.17. Let p : X → S be a map of simplicial sets which is a
coCartesian fibration (or a Cartesian fibration). Assume that the following
conditions are satisfied:(1) The simplicial set S is small.
(2) For each vertex s of S, the ∞-category Xs = X ×S {s} is accessible.
(3) For each edge e : s → s0 of S, the associated functor Xs → Xs0 (orXs0 → Xs ) is accessible.
Then MapS (S, X) is an accessible ∞-category. Moreover, if C is accessible,
then a functor
C → MapS (S, X)
is accessible if and only if, for every vertex s of S, the induced map C → Xs
is accessible.

5.5 PRESENTABLE ∞-CATEGORIESOur final object of study in this chapter is the theory of presentable ∞categories.
Definition 5.5.0.1. An ∞-category C is presentable if C is accessible andadmits small colimits.
We will begin in §5.5.1 by giving a number of equivalent reformulations ofDefinition 5.5.0.1. The main result, Theorem 5.5.1.1, is due to Carlos Simpson: an ∞-category C is presentable if and only if it arises as an (accessible)
localization of an ∞-category of presheaves.
Let C be an ∞-category and let F : C → Sop be a functor. If F isrepresentable by an object of C, then F preserves colimits (Proposition
5.1.3.2). In §5.5.2, we will prove that the converse holds when C is presentable. This representability criterion has a number of consequences: it implies
that C admits (small) limits (Corollary 5.5.2.4) and leads to an ∞-categoricalanalogue of the adjoint functor theorem (Corollary 5.5.2.9).
In §5.5.3, we will see that the collection of all presentable ∞-categories
can be organized into an ∞-category PrL . Moreover, we will explain how tocompute limits and colimits in PrL . In the course of doing so, we will provethat the class of presentable ∞-categories is stable under most of the basic
constructions of higher category theory.
In view of Theorem 5.5.1.1, the theory of localizations plays a central rolein the study of presentable ∞-categories. In §5.5.4, we will show that thecollection of all (accessible) localizations of a presentable ∞-category C canbe parametrized in a very simple way. Moreover, there is a good supply ofhighertopoi

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localizations of C: given any (small) collection of morphisms S of C, one canconstruct a corresponding localization functor
L

C → S −1 C ⊆ C,
where S −1 C is the full subcategory of C spanned by the S-local objects. These
ideas are due to Bousfield, who works in the setting of model categories; wewill give an exposition here in the language of ∞-categories. In §5.5.5, we will
employ the same techniques to produce examples of factorization systems on
the ∞-category C.
Let C be an ∞-category and let C ∈ C be an object. We will say that C ∈ C
is discrete if, for every D ∈ C, the nonzero homotopy groups of the mapping
space MapC (D, C) vanish. If we let τ≤0 C denote the full subcategory of C
spanned by the discrete objects, then τ≤0 C is (equivalent to) an ordinary
category. If C is the ∞-category of spaces, then we can identify the discrete
objects of C with the ordinary category of sets. Moreover, the inclusion
τ≤0 S ⊆ S has a left adjoint given by
X 7→ π0 X.
In §5.5.6, we will show that the preceding remark generalizes to an arbitrary presentable ∞-category C: the discrete objects of C constitute an (accessible) localization of C. We will also consider a more general conditionof k-truncatedness (which specializes to the condition of discreteness whenk = 0). The truncation functors which we construct will play an importantrole throughout Chapter 6.
In §5.5.7, we will study the theory of compactly generated ∞-categories:∞-categories which are generated (under colimits) by their compact objects.
This class of ∞-categories includes some of the most important examples,such as S and Cat∞ . In fact, the ∞-category S satisfies an even stronger condition: it is generated by compact projective objects (see Definition 5.5.8.18).
The presence of enough compact projective objects in an ∞-category allows us to construct projective resolutions, which gives rise to the theoryof nonabelian homological algebra (or “homotopical algebra”). We will review the rudiments of this theory in §5.5.8. Finally, in §5.5.9 we will present
the same ideas in a more classical form following Quillen’s manuscript [64].
The comparison of these two perspectives is based on a rectification result(Proposition 5.5.9.2) which is of some independent interest.
Remark 5.5.0.2. We refer the reader to [1] for a study of presentability inthe setting of ordinary category theory. Note that [1] uses the term “locallypresentable category” for what we have chosen to call a presentable category.
5.5.1 Presentability
Our main goal in this section is to establish the following characterizationof presentable ∞-categories:
Theorem 5.5.1.1 (Simpson [71]). Let C be an ∞-category. The following
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CHAPTER 5(1) The ∞-category C is presentable.
(2) The ∞-category C is accessible, and for every regular cardinal κ thefull subcategory Cκ admits κ-small colimits.
(3) There exists a regular cardinal κ such C is κ-accessible and Cκ admits
κ-small colimits.
(4) There exists a regular cardinal κ, a small ∞-category D which admits
κ-small colimits, and an equivalence Indκ D → C.
(5) There exists a small ∞-category D such that C is an accessible localization of P(D).
(6) The ∞-category C is locally small and admits small colimits, and there
exists a regular cardinal κ and a (small) set S of κ-compact objects of
C such that every object of C is a colimit of a small diagram taking
values in the full subcategory of C spanned by S.
Before giving the proof, we need a few preliminary remarks. We firstobserve that condition (5) is potentially ambiguous: it is unclear whether
the accessibility hypothesis is on C or on the associated localization functor
L : P(D) → P(D). The distinction turns out to be irrelevant by virtue of thefollowing:
Proposition 5.5.1.2. Let C be an accessible ∞-category and let L : C → C
be a functor satisfying the equivalent conditions of Proposition 5.2.7.4. The
following conditions are equivalent:(1) The essential image L C of L is accessible.
(2) There exists a localization f : C → D, where D is accessible, and an
equivalence L ' g ◦ f .
(3) The functor L is accessible (when regarded as a functor from C toitself).
Proof. Suppose (1) is satisfied. Then we may take D = L C, f = L, and g to
be the inclusion L C ⊆ C; this proves (2). If (2) is satisfied, then Proposition
5.4.7.7 shows that f and g are accessible functors, so their composite g◦f ' L
is also accessible; this proves (3). Now suppose that (3) is satisfied. Choosea regular cardinal κ such that C is κ-accessible and L is κ-continuous. The
full subcategory Cκ consisting of κ-compact objects of C is essentially small,so there exists a regular cardinal τ  κ such that LC is τ -compact for every
C ∈ Cκ . Let C0 denote the full subcategory of C spanned by the colimitsof all τ -small κ-filtered diagrams in Cκ and let L C0 denote the essential
image of C0 under L. We note that L C0 is essentially small. Since L is κcontinuous, L C is stable under small κ-filtered colimits in C. It follows thatany τ -compact object of C which belongs to L C is also τ -compact whenhighertopoi

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viewed as an object of L C, so that L C0 consists of τ -compact objects of L C.
According to Proposition 5.4.2.2, to complete the proof that L C is accessible,
it will suffice to show that L C0 generates L C under small τ -filtered colimits.
Let X be an object of C. Then X can be written as a small κ-filteredcolimit of objects of Cκ . The proof of Proposition 5.4.2.9 shows that we
can also write X as the colimit of a small τ -filtered diagram in C0 . Since L
preserves colimits, it follows that LX can be obtained as the colimit of asmall τ -filtered diagram in L C0 .
The proof of Theorem 5.5.1.1 will require a few easy lemmas:
Lemma 5.5.1.3. Let f : C → D be a functor between small ∞-categories
which exhibits D as an idempotent completion of C and let κ be a regular cardinal. Then Indκ (f ) : Indκ (C) → Indκ (D) is an equivalence of ∞-categories.
Proof. We first apply Proposition 5.3.5.11 to conclude that Indκ (f ) is fully
faithful. To prove that Indκ (f ) is an equivalence, we must show that it generates Indκ (D) under κ-filtered colimits. Since Indκ (D) is generated underκ-filtered colimits by the essential image of the Yoneda embedding jD : D →Indκ (D), it will suffice to show that the essential image of jD is contained in
the essential image of Indκ (f ). Let D be an object of D. Then D is a retractof f (C) for some object C ∈ C. Then jD (D) is a retract of (Indκ (f ) ◦ jC )(C).
Since Indκ (C) is idempotent complete (Remark 5.3.1.10), we conclude thatjD (D) belongs to the essential image of Indκ (f ).
Lemma 5.5.1.4. Let F : C → D be a functor between ∞-categories which
admit small κ-filtered colimits and let G be a right adjoint to F . Supposethat G is κ-continuous. Then F carries κ-compact objects of C to κ-compact
objects of D.
Proof. Let C be a κ-compact object of C, eC : C → b
S the functor corepresented by C, and eF (C) : D → b
S the functor corepresented by F (C). Since
G is a right adjoint to F , we have an equivalence eF (C) = eC ◦ G. Since eC
and G are both κ-continuous, eF C is κ-continuous. It follows that F (C) isκ-compact, as desired.
Proof of Theorem 5.5.1.1. Corollary 5.3.4.15 asserts that the full subcategory Cκ is stable under all κ-small colimits which exist in C. This provesthat (1) implies (2). The implications (2) ⇒ (3) ⇒ (4) are obvious. We next
prove that (4) implies (5). According to Lemma 5.5.1.3, we may suppose
without loss of generality that D is idempotent complete. Let Pκ (D) denotethe full subcategory of P(D) spanned by the κ-compact objects, let D0 be aminimal model for Pκ (D), and let g denote the compositionj

D → Pκ (D) → D0 ,where the second map is a homotopy inverse to the inclusion D0 ⊆ Pκ (D).
Proposition 5.1.3.1 implies that g is fully faithful, and Proposition 5.3.4.18
implies that g admits a left adjoint f . It follows that F = Indκ (f ) andhighertopoi

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G = Indκ (g) are adjoint functors, and Proposition 5.3.5.11 implies that
G is fully faithful. Moreover, Proposition 5.3.5.12 implies that Indκ D0 is
equivalent to P(D), so that C is equivalent to an accessible localization ofP(D0 ).
We now prove that (5) implies (6). Let D be a small ∞-category and
L : P(D) → C an accessible localization. Remark 5.2.7.5 implies that C
admits small colimits and that C is generated under colimits by the essential
image of the compositionjL

T : D → P(D) → C .
To complete the proof of (6), it will suffice to show that there exists a regularcardinal κ such that the essential image of T consists of κ-compact objects.
Let G denote a left adjoint to L. By assumption, G is an accessible functor
so that there exists a regular cardinal κ such that G is κ-continuous. For
each object D ∈ D, the Yoneda image j(D) is a completely compact objectof P(D) and, in particular, κ-compact. Lemma 5.5.1.4 implies that T (D) isa κ-compact object of C.
We now complete the proof by showing that (6) ⇒ (1). Assume that
there exists a regular cardinal κ and a set S of κ-compact objects of C suchthat every object of C is a colimit of objects in S. Let C0 ⊆ C be the fullsubcategory of C spanned by S and let C00 ⊆ C be the full subcategory of C
spanned by the colimits of all κ-small diagrams with values in C00 . Since C0
is essentially small, there is only a bounded number of such diagrams up toequivalence, so that C00 is essentially small. Moreover, since every object of
C is a colimit of a small diagram with values in C0 , the proof of Corollary
4.2.3.11 shows that every object of C can also be obtained as the colimit of asmall κ-filtered diagram with values in C00 . Corollary 5.3.4.15 implies that C00
consists of κ-compact objects of C (a slightly more refined argument showsthat, if κ > ω, then C00 consists of precisely the κ-compact objects of C). We
may therefore apply Proposition 5.4.2.2 to deduce that C is accessible.
Remark 5.5.1.5. The characterization of presentable ∞-categories as localizations of presheaf ∞-categories was established by Simpson in [71] (using
a somewhat different language). The theory of presentable ∞-categories isessentially equivalent to the theory of combinatorial model categories (see§A.3.7 and Proposition A.3.7.6). Since most of the ∞-categories we will meetare presentable, our study could also be phrased in the language of model
categories. However, we will try to avoid this language since for many purposes the restriction to presentable ∞-categories seems unnatural and isoften technically inconvenient.
Remark 5.5.1.6. Let C be a presentable ∞-category and let D be an accessible localization of C. Then D is presentable: this follows immediatelyfrom characterization (5) of Proposition 5.5.1.1.
Remark 5.5.1.7. Let C be a presentable ∞-category. Since C admits arbitrary colimits, it is “tensored over spaces,” as we explained in §4.4.4. In

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particular, the homotopy category of C is naturally tensored over the homotopy category H: for each object C of C and every simplicial set S, there
exists an object C ⊗ S of C, well-defined up to equivalence, equipped withisomorphisms
MapC (C ⊗ S, C 0 ) ' MapC (C, C 0 )S
in the homotopy category H.
Example 5.5.1.8. The ∞-category S of spaces is presentable. This follows
from characterization (1) of Theorem 5.5.1.1 since S is accessible (Example
5.4.2.7) and admits (small) colimits by Corollary 4.2.4.8.
According to Theorem 5.5.1.1, if C is κ-accessible, then C admits small
colimits if and only if the full subcategory Cκ ⊆ C admits κ-small colimits.
Roughly speaking, this is because arbitrary colimits in C can be rewritten
in terms of κ-filtered colimits and κ-small colimits of κ-compact objects.
Our next result is another variation on this idea; it may also be regarded
as an analogue of Theorem 5.5.1.1 (which describes functors rather than∞-categories):
Proposition 5.5.1.9. Let f : C → D be a functor between presentable∞-categories. Suppose that C is κ-accessible. The following conditions areequivalent:(1) The functor f preserves small colimits.
(2) The functor f is κ-continuous, and the restriction f | Cκ preserves κsmall colimits.
Proof. Without loss of generality, we may suppose C = Indκ (C0 ), where C0 is
a small idempotent complete ∞-category which admits κ-small colimits. The
proof of Theorem 5.5.1.1 shows that the inclusion Indκ (C0 ) ⊆ P(C0 ) admits
a left adjoint L. Let α : idP(C0 ) → L be a unit for the adjunction and let f 0 :C0 → D denote the composition of f with the Yoneda embedding j : Indκ (C0 ).
According to Theorem 5.1.5.6, there exists a colimit-preserving functor F :P(C0 ) → D and an equivalence of f 0 with F ◦ j. Proposition 5.3.5.10 implies
that f and F | Indκ (C) are equivalent; we may therefore assume without lossof generality that f = F | Indκ (C). Let F 0 = f ◦ L, so that α induces anatural transformation β : F → F 0 of functors from P(C0 ) to D. We willshow that β is an equivalence. Consequently, we deduce that the functor F 0
is colimit-preserving. It then follows that f is colimit-preserving. To see this,we consider an arbitrary diagram p : K → Indκ (C0 ) and choose a colimit
p : K . → P(C0 ). Then q = L ◦ p is a colimit diagram in Indκ (C0 ), andf ◦ q = F 0 ◦ p is a colimit diagram in D. Since q = q|K is equivalent (via α)
to the original diagram p, we conclude that f preserves the colimit of p inIndκ (C0 ) as well.
It remains to prove that β is an equivalence of functors. Let E ⊆ P(C0 )denote the full subcategory spanned by those objects X ∈ P(C0 ) for whichhighertopoi

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CHAPTER 5β(X) : F (X) → F 0 (X) is an equivalence in D. We wish to prove thatE = P(C0 ). Since F and F 0 are both κ-continuous functors, E is stable underκ-filtered colimits in P(C0 ). It will therefore suffice to prove that E contains
Pκ (C0 ).
It is clear that E contains Indκ (C0 ); in particular, E contains the essential
image E0 of the Yoneda embedding j : C0 → P(C0 ). According to Proposition
5.3.4.17, every object of Pκ (C0 ) is a retract of the colimit of a κ-small diagram
p : K → E0 . Since C0 is idempotent complete, we may identify E0 with thefull subcategory of Indκ (C0 ) consisting of κ-compact objects. In particular,
E0 is stable under κ-small colimits and retracts in Indκ (C0 ). It follows that
L restricts to a functor L0 : Pκ (C) → E0 which preserves κ-small colimits.
To complete the proof that Pκ (C0 ) ⊆ E, it will suffice to prove that0
F | Pκ (C) preserves κ-small colimits. To see this, we write F | Pκ (C0 ) as acomposition
L0

F | E0Pκ (C0 ) → E0 → C,
where L0 preserves κ-small colimits (as noted above) and F | E0 = f | Cκ preserves κ-small colimits by assumption.
5.5.2 Representable Functors and the Adjoint Functor Theorem
An object F of the ∞-category P(C) of presheaves on C is representable if
it lies in the essential image of the Yoneda embedding j : C → P(C). If
F : Cop → S is representable, then F preserves limits: this follows from thefact that F is equivalent to the composite mapjCop → P(Cop ) → S,
where j denotes the Yoneda embedding for Cop (which is limit-preserving by
Proposition 5.1.3.2) and the right map is given by evaluation at C (which islimit-preserving by Proposition 5.1.2.2). If C is presentable, then the converse
holds.
Lemma 5.5.2.1. Let S be a small simplicial set, let f : S → S be an objectof P(S op ), and let F : P(S op ) → b
S be the functor corepresented by f . Thenthe compositionjF
S → P(S op ) → b
Sis equivalent to f .
Proof. According to Corollary 4.2.4.7, we can choose a (small) fibrant simplicial category C and a categorical equivalence φ : S → N(Cop ) such that f
is equivalent to the composition of φop with the nerve of a simplicial functor
f 0 : C → Kan. Without loss of generality, we may suppose that f 0 ∈ SetC
∆ is
projectively cofibrant. Using Proposition 4.2.4.4, we have an equivalence of∞-categories◦ψ : N((SetC∆ ) ) → P(S).

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We observe that the composition F ◦ ψ can be identified with the simplicial◦
0
nerve of the functor G : (SetC∆ ) → Kan corepresented by f . The Yoneda
embedding factors through ψ via the adjoint of the composition◦
j 0 : C[S] → Cop → (SetC∆) .

It follows that F ◦ j can be identified with the adjoint of the composition
j0◦
C[S] → (SetC∆ ) → Kan .
GThis composition is equal to the functor f 0 , so its simplicial nerve coincides
with the original functor f .
Proposition 5.5.2.2. Let C be a presentable ∞-category and let F : Cop → S
be a functor. The following are equivalent:(1) The functor F is representable by an object C ∈ C.
(2) The functor F preserves small limits.
Proof. The implication (1) ⇒ (2) was proven above (for an arbitrary ∞category C). For the converse, we first treat the case where C = P(D) for somesmall ∞-category D. Let f : Dop → S denote the composition of F with the(opposite) Yoneda embedding j op : Dop → P(D)op and let F 0 : P(D)op → b
Sdenote the functor represented by f ∈ P(D). We will prove that F and
F 0 are equivalent. We observe that F and F 0 both preserve small limits;
consequently, according to Theorem 5.1.5.6, it will suffice to show that thecompositions f = F ◦ j op and f 0 = F 0 ◦ j op are equivalent. This follows
immediately from Lemma 5.5.2.1.
We now consider the case where C is an arbitrary presentable ∞-category.
According to Theorem 5.5.1.1, we may suppose that C is an accessible localization of a presentable ∞-category C0 which has the form P(D), so thatthe assertion for C0 has already been established. Let L : C0 → C denote thelocalization functor. The functor F ◦ Lop : (C0 )op → S preserves small limits
and is therefore representable by an object C ∈ C0 . Let S denote the set of
all morphisms φ in C0 such that L(φ) is an equivalence in C. Without lossof generality, we may identify C with the full subcategory of C0 consisting of
S-local objects. By construction, C ∈ C0 is S-local and therefore belongs to
C. It follows that C represents the functor (F ◦ Lop )| C, which is equivalent
to F .
The representability criterion provided by Proposition 5.5.2.2 has many
consequences, as we now explain.
Lemma 5.5.2.3. Let X and Y be simplicial sets, let q : C → D be a categorical fibration of ∞-categories, and let p : X . × Y . → C be a diagram.
Suppose that
(1) For every vertex x of X . , the associated map px : Y . → C is a q-colimit
diagram.

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CHAPTER 5(2) For every vertex y of Y , the associated map py : X . → C is a q-colimit
diagram.
Let ∞ denote the cone point of Y . . Then the restriction p∞ : X . → C is aq-colimit diagram.
Proof. Without loss of generality, we can suppose that X and Y are ∞categories. Since the inclusion X × {∞} ⊆ X × Y . is cofinal, it will suffice
to show that the restriction p|(X × Y . ). is a q-colimit diagram. According
to Proposition 4.3.2.9, p|(X × Y . ) is a q-left Kan extension of p|(X × Y ).
By transitivity, it suffices to show that p|(X × Y ). is a q-colimit diagram.
For this, it will suffice to prove the stronger assertion that p|(X . × Y ). isa q-left Kan extension of p|(X × Y ). Since Proposition 4.3.2.9 also implies
that p|(X . × Y ) is a q-left Kan extension of p|(X × Y ), we may again apply
transitivity and reduce to the problem of showing that p|(X . × Y ). is aq-colimit diagram. Let ∞0 denote the cone point of X . . Since the inclusion{∞0 } × Y ⊆ X . × Y is cofinal, we are reduced to proving that p∞0 : Y . → C
is a q-colimit diagram, which follows from (1).
Corollary 5.5.2.4. A presentable ∞-category C admits all (small) limits.
b
Proof. Let P(C)= Fun(Cop , bS), where b
S denotes the ∞-category of spacesbwhich are not necessarily small, and let j : C → P(C)be the Yoneda embedding. Since j is fully faithful, it will suffice to show that the essentialbimage of j admits small limits. The ∞-category P(C)admits all small limits(in fact, even limits which are not necessarily small); it therefore suffices toshow that the essential image of j is stable under small limits. This follows
immediately from Proposition 5.5.2.2 and Lemma 5.5.2.3.
Remark 5.5.2.5. Let A be a (small) partially ordered set. The ∞-category
N(A) is presentable if and only if every subset of A has a least upper bound.
Corollary 5.5.2.4 can then be regarded as a generalization of the followingclassical observation: if every subset of A has a least upper bound, then every
subset of A has a greatest lower bound (namely, a least upper bound for thecollection of all lower bounds).
Remark 5.5.2.6. Now that we know that every presentable ∞-category
C has arbitrary limits, we can apply an argument dual to that of Remark
5.5.1.7 to show that C is cotensored over S. In other words, for any C ∈ C
and every simplicial set X, there exists an object C X ∈ C (well-defined up
to equivalence) together with a collection of natural isomorphisms
MapC (C 0 , C X ) ' MapC (C 0 , C)X
in the homotopy category H.
We can now formulate a dual version of Proposition 5.5.2.2, which requiresa slightly stronger hypothesis.

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Proposition 5.5.2.7. Let C be a presentable ∞-category and let F : C → S
be a functor. Then F is corepresentable by an object of C if and only if F isaccessible and preserves small limits.
Proof. The “only if” direction is clear since every object of C is κ-compactfor κ  0. We will prove the converse. Without loss of generality, we may
suppose that C is minimal (this assumption is a technical convenience which
will guarantee that various constructions below stay in the world of small ∞e → C denote the left fibration represented by F . Choose acategories). Let Ceκregular cardinal κ such that C is κ-accessible and F is κ-continuous and let Cκκe ×C C , where C ⊆ C denotes the full subcategory
denote the fiber product Ceκ is small (since C
spanned by the κ-compact objects of C. The ∞-category Ceκ → Cis assumed minimal). Corollary 5.5.2.4 implies that the diagram p : Cκe )/ → C. Since the functor F preserves small limits,
admits a limit p : (Ceκ )/ → Ce which extends
Corollary 3.3.3.3 implies that there exists a map q : (C
κ
e
e
e
e
the inclusion q : C ⊆ C and covers p. Let X0 ∈ C denote the image of thee0 determines a connected
cone point under q and X0 its image in C. Then X
component of the space F (X0 ). Since C is κ-accessible, we can write X0 as a
κ-filtered colimit of κ-compact objects {Xα } of C. Since F is κ-continuous,
there exists a κ-compact object X ∈ C such that the induced map F (X) →e0 .
F (X0 ) has nontrivial image in the connected component classified by X
e lying over X and a morphisme ∈CIt follows that there exists an object X
e Since C
e/q → Ce is a right fibration, we can pull q back to
e →X
e0 in C.
f :Xeκ )/ → Ce which extends q and carries the cone point toobtain a map q 0 : (C0eκ . We have a homotopy commutativee
X. It follows that q factors through C
diagram
e
?C

e
{X}

i


e/ ,
/C

where i denotes the inclusion of the cone point. The map i is left anodyneeκ is aand therefore a covariant equivalence in (Set∆ )/ C . It follows that C
retract of {X} in the homotopy category of the covariant model category(Set∆ )/ Cκ . Proposition 5.1.1.1 implies that F | Cκ is a retract of the Yoneda
image j(X) in P(Cκ ). Since the ∞-category Cκ is idempotent complete andthe Yoneda embedding j : Cκ → P(Cκ ) is fully faithful, we deduce that
F | Cκ is equivalent to j(X 0 ), where X 0 ∈ Cκ is a retract of X. Let F 0 : C → S
denote the functor corepresented by X 0 . We note that F | Cκ and F 0 | Cκ areequivalent and that both F and F 0 are κ-continuous. Since C is equivalent
to Indκ (Cκ ), Proposition 5.3.5.10 guarantees that F and F 0 are equivalent,so that F is representable by X 0 .
Remark 5.5.2.8. It is not difficult to adapt our proof of Proposition 5.5.2.7
to obtain an alternative proof of Proposition 5.5.2.2.

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From Propositions 5.5.2.2 and 5.5.2.7, we can deduce a version of theadjoint functor theorem:
Corollary 5.5.2.9 (Adjoint Functor Theorem). Let F : C → D be a functor
between presentable ∞-categories.
(1) The functor F has a right adjoint if and only if it preserves small
colimits.
(2) The functor F has a left adjoint if and only if it is accessible andpreserves small limits.
Proof. The “only if” directions follow from Propositions 5.2.3.5 and 5.4.7.7.
We now prove the converse direction of (2); the proof of (1) is similar but
easier. Suppose that F is accessible and preserves small limits. Let F 0 : D →S be a corepresentable functor. Then F 0 is accessible and preserves small
limits (Proposition 5.5.2.7). It follows that the composition F 0 ◦ F : C → S
is accessible and preserves small limits. Invoking Proposition 5.5.2.7 again,
we deduce that F 0 ◦ F is representable. We now apply Proposition 5.2.4.2 todeduce that F has a left adjoint.
Remark 5.5.2.10. The proof of (1) in Corollary 5.5.2.9 does not requirethat D be presentable but only that D be (essentially) locally small.
5.5.3 Limits and Colimits of Presentable ∞-Categories
In this section, we will introduce and study an ∞-category whose objects arepresentable ∞-categories. In fact, we will introduce two such ∞-categories
which are (canonically) antiequivalent to one another. The basic observationis the following: given a pair of presentable ∞-categories C and D, the proper
notion of “morphism” between them is a pair of adjoint functors
Co

F
G.

/D

Of course, either one of F and G determines the other up to canonicalequivalence. We may therefore think of either one as encoding the data of amorphism.
d∞ denote the ∞-category of (not necessarily
Definition 5.5.3.1. Let Cat
d∞ as follows:
small) ∞-categories. We define subcategories PrR , PrL ⊆ Cat(1) The objects of both PrR and PrL are the presentable ∞-categories.
(2) A functor F : C → D between presentable ∞-categories is a morphismin PrL if and only if F preserves small colimits.
(3) A functor G : C → D between presentable ∞-categories is a morphismin PrR if and only if G is accessible and preserves small limits.

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As indicated above, the ∞-categories PrR and PrL are antiequivalent to
one another. To prove this, it is convenient to introduce the following definition:
Definition 5.5.3.2. A map of simplicial sets p : X → S is a presentablefibration if it is both a Cartesian fibration and a coCartesian fibration andif each fiber Xs = X ×S {s} is a presentable ∞-category.
The following result is simply a reformulation of Corollary 5.5.2.9:
Proposition 5.5.3.3. (1) Let p : X → S be a Cartesian fibration of simd∞ . Then p is a presentableplicial sets classified by a map χ : S op → Catd∞ .
fibration if and only if χ factors through PrR ⊆ Cat(2) Let p : X → S be a coCartesian fibration of simplicial sets classified
d∞ . Then p is a presentable fibration if and only
by a map χ : S → Catd∞ .
if χ factors through PrL ⊆ Cat
Corollary 5.5.3.4. For every simplicial set S, there is a canonical bijection[S, PrL ] ' [S op , PrR ],where [S, C] denotes the collection of equivalence classes of objects of the ∞category Fun(S, C). In particular, there is a canonical isomorphism PrL '(PrR )op in the homotopy category of ∞-categories.
Proof. According to Proposition 5.5.3.3, both [S, PrL ] and [S op , PrR ] can be
identified with the collection of equivalence classes of presentable fibrations
X → S.
We now commence our study of the ∞-category PrL (or equivalently, theantiequivalent ∞-category PrR ). The next few results express the idea thatd∞ is stable under a variety of categorical constructions.
PrL ⊆ Cat
Proposition 5.5.3.5. Let
Q {Cα }α∈A be a family of ∞-categories indexed by asmall set A and let C = α∈A Cα be their product. If each Cα is presentable,
then C is presentable.
Proof. It follows from Lemma 5.4.7.2 that C is accessible. Let p : K → C bea diagram indexed by a small simplicial set K corresponding to a family of
diagrams {pα : K → Cα }α∈A . Since each Cα is presentable, each pα has acolimit pα : K . → Cα . These colimits determine a map p : K . → C which isa colimit of p.
Proposition 5.5.3.6. Let C be an presentable ∞-category and let K be asmall simplicial set. Then Fun(K, C) is presentable.
Proof. According to Proposition 5.4.4.3, Fun(K, C) is accessible. It follows
from Proposition 5.1.2.2 that if C admits small colimits, then Fun(K, C)
admits small colimits.

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Remark 5.5.3.7. Let S be a (small) simplicial set. It follows from Example 5.4.2.7 and Corollary 5.1.2.4 that P(S) is a presentable ∞-category.
Moreover, Theorem 5.1.5.6 has a natural interpretation in the language ofpresentable ∞-categories: informally speaking, it asserts that the construction
S 7→ P(S)is left adjoint to the inclusion functor from presentable ∞-categories to all∞-categories.
The following is a variant on Proposition 5.5.3.6:
Proposition 5.5.3.8. Let C and D be presentable ∞-categories. The ∞category FunL (C, D) is presentable.
Proof. Since D admits small colimits, the ∞-category Fun(C, D) admits
small colimits (Proposition 5.1.2.2). Using Lemma 5.5.2.3, we conclude that
FunL (C, D) ⊆ Fun(C, D)
is stable under small colimits. To complete the proof, it will suffice to showthat FunL (C, D) is accessible.
Choose a regular cardinal κ such that C is κ-accessible and let Cκ be thefull subcategory of C spanned by the κ-compact objects. Propositions 5.5.1.9
and 5.3.5.10 imply that the restriction functor
FunL (C, D) → Fun(Cκ , D)is fully faithful, and its essential image is the full subcategory E ⊆ Fun(Cκ , D)spanned by those functors which preserve κ-small colimits.
Since Cκ is essentially small, the ∞-category Fun(Cκ , D) is accessible(Proposition 5.4.4.3). To complete the proof, we will show that E is an accessible subcategory of Fun(Cκ , D). For each κ-small diagram p : K → Cκ ,
κlet E(p) denote the full subcategory of Fun(CT, D) spanned by those functors
which preserve the colimit of p. Then E = p E(p), where the intersection
is taken over a set of representatives for all equivalence classes of κ-small
diagrams in Cκ . According to Proposition 5.4.7.10, it will suffice to showthat each E(p) is an accessible subcategory of Fun(Cκ , D). We now observethat there is a (homotopy) pullback diagram of ∞-categoriesE(p)_/ E0 (p)_
Fun(Cκ , D)
/ Fun(K . , D),

where E0 denotes the full subcategory of Fun(K . , D) spanned by the colimit
diagrams. According to Proposition 5.4.4.3, it will suffice to prove that E0 (p)is an accessible subcategory of Fun(K . , D), which follows from Example
5.4.7.9.

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Remark 5.5.3.9. In the situation of Proposition 5.5.3.8, the presentable∞-category FunL (C, D) can be regarded as an internal mapping object in
PrL . For every presentable ∞-category C0 , a colimit-preserving functor C0 →FunL (C, D) can be identified with a bifunctor C × C0 → D, which is colimitpreserving separately in each variable. There exists a universal recipient forsuch a bifunctor: a presentable category which we may denote by C ⊗ C0 .
The operation ⊗ endows PrL with the structure of a symmetric monoidal∞-category. Proposition 5.5.3.8 can be interpreted as asserting that thismonoidal structure is closed.
Proposition 5.5.3.10. Let C be an ∞-category and let p : K → C be adiagram in C indexed by a (small) simplicial set K. If C is presentable, then
the ∞-category C/p is also presentable.
Proof. According to Corollary 5.4.6.7, C/p is accessible. The existence ofsmall colimits in C/p follows from Proposition 1.2.13.8.
Proposition 5.5.3.11. Let C be an ∞-category and let p : K → C be adiagram in C indexed by a small simplicial set K. If C is presentable, then
the ∞-category Cp/ is also presentable.
Proof. It follows from Corollary 5.4.5.16 that Cp/ is accessible. It therefore
suffices to prove that every diagram q : K 0 → Cp/ has a colimit in C. We
now observe that (Cp/ )q/ ' Cq0 / , where q 0 : K ? K 0 → C is the map classified
by q. Since C admits small colimits, Cq0 / has an initial object.
Proposition 5.5.3.12. Let
X0

q0

p0


Y0

/X
p

q


/Y

be a diagram of ∞-categories which is homotopy Cartesian (with respect to
the Joyal model structure). Suppose further that X, Y, and Y0 are presentable
and that the functors p and q preserve small colimits. Then X0 is presentable.
Moreover, for any presentable ∞-category C and any functor f : C → X, f
preserves small colimits if and only i the compositions p0 ◦f and q 0 ◦f preserve
small colimits. In particular (taking f = idX ), we deduce that the functors
p0 are q 0 preserve small colimits.
Proof. Proposition 5.4.6.6 implies that X0 is accessible. It therefore suffices
to prove that any diagram f : K → X0 indexed by a small simplicial set
K has a colimit in X0 . Without loss of generality, we may suppose that p
and q are categorical fibrations and that X0 = X ×Y Y0 . Let X be an initial
object of Xq0 ◦f / and let Y 0 be an initial object of Y0p0 f / . Since p and q preserve
colimits, the images p(X) and q(Y 0 ) are initial objects in Ypq0 f / and therefore
equivalent to one another. Choose an equivalence η : p(X) → q(Y 0 ). Since q

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is a categorical fibration, η lifts to an equivalence η : Y → Y 0 in Y0p0 f / suchthat q(η) = η. Replacing Y 0 by Y , we may suppose that p(X) = q(Y ), so thatthe pair (X, Y ) may be considered as an object of X0f / = Y0p0 f / ×Ypqf / Xqf / .
According to Lemma 5.4.5.2, it is an initial object of X0f / , so that f has
a colimit in X0 . This completes the proof that X0 is accessible. The second
assertion follows immediately from Lemma 5.4.5.5.
Proposition 5.5.3.13. The ∞-category PrL admits all small limits, andd∞ preserves all small limits.
the inclusion functor PrL ⊆ Cat
Proof. The proof of Proposition 4.4.2.6 shows that it will suffice to consider
the case of pullbacks and small products. The desired result now follows bycombining Propositions 5.5.3.12 and 5.5.3.5.
Corollary 5.5.3.14. Let p : X → S be a presentable fibration of simplicial
sets, where S is small. Then the ∞-category C of coCartesian sections of p
is presentable.
Proof. According to Proposition 5.5.3.3, p is classified by a functor χ : S →PrL . Using Proposition 5.5.3.13, we deduce that the limit of the composite
diagramd∞
S → PrL → Cat
is presentable. Corollary 3.3.3.2 allows us to identify this limit with the ∞category C.
Our goal in the remainder of this section is to prove the analogue of
Proposition 5.5.3.13 for the ∞-category PrR (which will show that PrL isequipped with all small colimits as well as all small limits). The main step isto prove that for every small diagram S → PrR , the limit of the composite
functord∞
S → PrR → Cat
is presentable. As in the proof of Corollary 5.5.3.14, this is equivalent to theassertion that for any presentable fibration p : X → S, the ∞-category C of
Cartesian sections of p is presentable. To prove this, we will show that the ∞category MapS (S, X) is presentable and that C is an accessible localizationof MapS (S, X).
Lemma 5.5.3.15. Let p : M → ∆1 be a Cartesian fibration, let C denote the∞-category of sections of p, and let e : X → Y and e0 : X 0 → Y 0 be objects ofC. If e0 is p-Cartesian, then the evaluation map MapC (e, e0 ) → MapM (Y, Y 0 )is a homotopy equivalence.
Proof. There is a homotopy pullback diagram of simplicial sets whose image
in the homotopy category H is isomorphic to/ MapM (Y, Y 0 )MapC (e, e0 )MapM (X, X 0 )/ MapM (X, Y 0 ).

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If e0 is p-Cartesian, then the lower horizonal map is a homotopy equivalence,
so the upper horizonal map is a homotopy equivalence as well.
Lemma 5.5.3.16. Let p : M → ∆1 be a Cartesian fibration. Let C denotethe ∞-category of sections of p and C0 ⊆ C the full subcategory spanned byCartesian sections of p. Then C0 is a reflective subcategory of C.
Proof. Let e : X → Y be an arbitrary section of p and choose a Cartesiansection e0 : X 0 → Y with the same target. Since e0 is Cartesian, there exists
a diagram
X

/Y


X0


/Y

idY

in M which we may regard as a morphism φ from e ∈ C to e0 ∈ C0 . Inview of Proposition 5.2.7.8, it will suffice to show that φ exhibits e0 as a
C0 -localization of C. In other words, we must show that for any Cartesiansection e00 : X 00 → Y 00 , composition with φ induces a homotopy equivalence
MapC (e0 , e00 ) → MapC (e, e00 ). This follows immediately from Lemma 5.5.3.15.
Proposition 5.5.3.17. Let p : X → S be a presentable fibration, where S
is a small simplicial set. Then
(1) The ∞-category C = MapS (S, X) of sections of p is presentable.
(2) The full subcategory C0 ⊆ C spanned by Cartesian sections of p is anaccessible localization of C.
Proof. The accessibility of C follows from Corollary 5.4.7.17. Since p is a
Cartesian fibration and the fibers of p admit small colimits, C admits small
colimits by Proposition 5.1.2.2. This proves (1).
For each edge e of S, let C(e) denote the full subcategory of C spanned bythoseTmaps S → X which carry e to a p-Cartesian edge of X. By definition,
C0 = C(e). According to Lemma 5.5.4.18, it will suffice to show that eachC(e) is an accessible localization of C. Consider the map
θe : C → MapS (∆1 , X).
Proposition 5.1.2.2 implies that θe preserves all limits and colimits. Moreover, C(e) = θe−1 Map0S (∆1 , X), where Map0S (∆1 , X) denotes the full subcategory of MapS (∆1 , X) spanned by p-Cartesian edges. According to Lemma
5.5.4.17, it will suffice to show that Map0S (∆1 , X) ⊆ MapS (∆1 , X) is anaccessible localization of MapS (∆1 , X). In other words, we may supposeS = ∆1 . It then follows that evaluation at {1} induces a trivial fibration
C0 → X ×S {1}, so that C0 is presentable. It therefore suffices to show that
C0 is a reflective subcategory of C, which follows from Lemma 5.5.3.16.

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Theorem 5.5.3.18. The ∞-category PrR admits small limits, and the ind∞ preserves small limits.
clusion functor PrR ⊆ Cat
d∞ be
Proof. Let χ : S op → PrR be a small diagram and let χ : (S . )op → CatRda limit of χ in Cat∞ . We will show that χ factors through Pr ⊆ Cat∞ andthat χ is a limit when regarded as a diagram in PrR .
We first show that χ carries each vertex to a presentable ∞-category. Thisis clear with the exception of the cone point of (S . )op . Let p : X → S bea presentable fibration classified by χ. According to Corollary 3.3.3.2, we
may identify the image of the cone point under χ with the ∞-category C of
Cartesian sections of p. Proposition 5.5.3.17 implies that this ∞-category ispresentable.
We next show that χ carries each edge of (S . )op to an accessible limitpreserving functor. This is clear for edges which are degenerate or belong to
S op . The remaining edges are in bijection with the vertices of s and connect
those vertices to the cone point. The corresponding functors can be identified
with the composition
C ⊆ MapS (S, X) → Xs ,where the second functor is given by evaluation at s. Proposition 5.5.3.17
implies that the inclusion i : C ⊆ MapS (S, X) is accessible and preserves
small limits, and Proposition 5.1.2.2 implies that the evaluation map
MapS (S, X) → Xspreserves all limits and colimits. This completes the proof that χ factors
through PrR .
We now show that χ is a limit diagram in PrR . Since PrR is a subcategoryd∞ and χ is already a limit diagram in Cat
d∞ , it will suffice to verify
of Catthe following assertion:• If D is a presentable ∞-category and F : D → C has the property thateach of the composite functors
Fi

D → C ⊆ MapS (S, X) → Xs
is accessible and limit-preserving, then F is accessible and preserves
limits.
Applying Proposition 5.5.3.17, we see that F is accessible and preserves
limits if and only if i ◦ F is accessible and preserves limits. We now conclude
by applying Proposition 5.1.2.2.
5.5.4 Local Objects
According to Theorem 5.5.1.1, every presentable ∞-category arises as an(accessible) localization of some presheaf ∞-category P(X). Consequently,
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the study of presentable ∞-categories. In this section, we will classify theaccessible localizations of an arbitrary presentable ∞-category C. The basic observation is that a localization functor L : C → C is determined, up
to equivalence, by the collection S of all morphisms f such that Lf is anequivalence. Moreover, a collection of morphisms S arises from an accessible
localization functor if and only if S is strongly saturated (Definition 5.5.4.5)
and of small generation (Remark 5.5.4.7). Given any small collection of morphisms S in C, there is a smallest strongly saturated collection containing
S: this permits us to define a localization S −1 C ⊆ C. The ideas presented inthis section go back (at least) to Bousfield; we refer the reader to [12] for a
discussion in a more classical setting.
Definition 5.5.4.1. Let C be an ∞-category and S a collection of morphisms of C. We say that an object Z of C is S-local if, for every morphisms : X → Y belonging to S, composition with s induces an isomorphism
MapC (Y, Z) → MapC (X, Z) in the homotopy category H of spaces.
A morphism f : X → Y of C is an S-equivalence if, for every S-local
object Z, composition with f induces a homotopy equivalence MapC (Y, Z) →MapC (X, Z).
The following result provides a dictionary for relating localization functors
to classes of morphisms:
Proposition 5.5.4.2. Let C be an ∞-category and let L : C → C be a
localization functor. Let S denote the collection of all morphisms f in C
such that Lf is an equivalence. Then
(1) An object C of C is S-local if and only if it belongs to L C.
(2) Every S-equivalence in C belongs to S.
(3) Suppose that C is accessible. The following conditions are equivalent:(i) The ∞-category L C is accessible.
(ii) The functor L : C → C is accessible.
(iii) There exists a (small) subset S0 ⊆ S such that every S0 -local
object is S-local.
Proof of (1) and (2). By assumption, L is left adjoint to the inclusion L C ⊆C; let α : idC → L be a unit map for the adjunction. We begin by proving(1). Suppose that X ∈ L C. Let f : Y → Z belong to S. Then we have acommutative diagram
MapC (LZ, X)/ MapC (LY, X)


MapC (Z, X)/ MapC (Y, X),highertopoi

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in the homotopy category H, where the vertical maps are given by composition with α and are homotopy equivalences by assumption. Since Lf is
an equivalence, the top horizontal map is also a homotopy equivalence. It
follows that the bottom horizontal map is a homotopy equivalence, so that
X is S-local. Conversely, suppose that X is S-local. According to Proposition 5.2.7.4, the map α(X) : X → LX belongs to S, so that composition
with α(X) induces a homotopy equivalence MapC (LX, X) → MapC (X, X).
In particular, there exists a map LX → X whose composition with α(X) is
homotopic to idX . Thus X is a retract of LX. Since α(LX) is an equivalence,
we conclude that α(X) is an equivalence, so that X ' LX and therefore X
belongs to the essential image of L, as desired. This proves (1).
Suppose that f : X → Y is an S-equivalence. We have a commutative
diagram
X

f

α(X)


LX

Lf

/Y
α(Y )


/ LY

where the vertical maps are S-equivalences (by Proposition 5.2.7.4), so that
Lf is also an S-equivalence. Therefore LX and LY corepresent the samefunctor on the homotopy category hL C. Yoneda’s lemma implies that Lf is
an equivalence, so that f ∈ S. This proves (2).
The proof of (3) is more difficult and will require a few preliminaries.
Lemma 5.5.4.3. Let τ  κ be regular cardinals and suppose that τ is
uncountable. Let A be a κ-filtered partially ordered set, A0 ⊆ A a τ -small
subset, and{fγ : Xγ → Yγ }γ∈Ca τ -small collection of natural transformations of diagrams in KanA . Supposethat for each α ∈ A, γ ∈ C, the Kan complexes Xγ (α) and Yγ (α) areessentially τ -small. Suppose further that, for each γ ∈ C, the map of Kancomplexes limA fγ is a homotopy equivalence. Then there exists a τ -small−→
κ-filtered subset A00 ⊆ A such that A0 ⊆ A00 , and limA00 fγ |A00 is a homotopy−→
equivalence for each γ ∈ C.
Proof. Replacing each fγ by an equivalent transformation if necessary, we
may suppose that for each γ ∈ C, α ∈ A, the map fγ (α) is a Kan fibration.
Let α ∈ A, let γ ∈ C, and let σ(α, γ) be a diagram∂ ∆ _n/ Xγ (α)

∆n/ Yγ (α).

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We will say that α0 ≥ α trivializes σ(α, γ) if the lifting problem depicted in
the induced diagram∂ ∆ _n/ Xγ (α0 )
;


∆n


/ Yγ 0 (α)

admits a solution. Observe that, if B ⊆ A is filtered, then limB fγ |B is a Kan−→
fibration, which is trivial if and only if for every diagram σ(α, γ) as above,
where α ∈ B, there exists α0 ∈ B such that α0 ≥ α and α0 trivializes σ(α, γ).
In particular, since limA fγ is a homotopy equivalence, every such diagram−→
σ(α, γ) is trivialized by some α0 ≥ α.
We now define a sequence of τ -smallSsubsets A(λ) ⊆ A indexed by ordinals
λ ≤ κ. Let A(0) = A0 and let A(λ) = λ0 <λ A(λ0 ) when λ is a limit ordinal.
Supposing that λ < κ and that A(λ) has been defined, we choose a set ofrepresentatives Σ = {σ(α, γ)} for all homotopy classes of diagrams as above,
where α ∈ A(λ) and γ ∈ C. Since the Kan complexes Xγ (α), Yγ (α) areessentially τ -small, we may choose the set Σ to be τ -small. Each σ ∈ Σis trivialized by some ασ0 ∈ A. Let B = {ασ0 }σ∈Σ ; then B is τ -small. Now
choose a τ -small κ-filtered subset A(λ + 1) ⊆ A containing A(λ) ∪ B (theexistence of A(λ + 1) is guaranteed by Lemma 5.4.2.8).
We now define A00 to be A(κ); it is easy to see that A00 has the desired
properties.
Lemma 5.5.4.4. Let τ  κ be regular cardinals and suppose that τ is
uncountable. Let A be a κ-filtered partially ordered set and for every subset
B ⊆ A, letlimB : Fun(N(B), S) → S−→
denote a left adjoint to the diagonal functor. Let A0 ⊆ A be a τ -small subset
and {fγ : Xγ → Yγ }γ∈C a τ -small collection of morphisms in the ∞-category
Fun(N(A), Sτ ) of diagrams N(A) → Sτ . Suppose that limA (fγ ) is an equiva−→
lence for each γ ∈ C. Then there exists a τ -small κ-filtered subset A00 ⊆ A
which contains A0 such that each of the morphisms limA00 (fγ | N(A00 )) is an−→equivalence in S.
Proof. Using Proposition 4.2.4.4, we may assume without loss of generality that each fγ is the simplicial nerve of a natural transformation offunctors from A to Kan. According to Theorem 4.2.4.1, we can identify
limB (Xγ | N(B)) and limB (Yγ | N(B)) with homotopy colimits in Kan. If B is−→−→filtered, then these homotopy colimits reduce to ordinary colimits (since theclass of weak homotopy equivalences in Kan is stable under filtered colimits),
and we may apply Lemma 5.5.4.3.
Proof of part (3) of Proposition 5.5.4.2. If L C is accessible, then Proposition 5.4.7.7 implies that both the inclusion L C → C and the functor L : C →highertopoi

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L C are accessible functors, so that their composition is accessible. Thus(i) ⇒ (ii). Suppose next that (ii) is satisfied. Let α : idC → L denote aunit for the adjunction between L and the inclusion L C ⊆ C and let κ be a
regular cardinal such that C is κ-accessible and L is κ-continuous. Without
loss of generality, we may suppose that C is minimal, so that Cκ is a small∞-category. Let S0 = {α(X) : X ∈ Cκ } and let Y ∈ C be S0 -local. We wish
to prove that Y is S-local. Let FY : C → Sop denote the functor represented
by Y . Then α induces a natural transformation FY → FY ◦ L. The functors
FY and FY ◦ L are both κ-continuous. By assumption, α induces an equivalence of functors FY | Cκ → (FY ◦ L)| Cκ when both sides are restricted toκ-compact objects. Proposition 5.3.5.10 now implies that FY and FY ◦ L areequivalent, so that Y is S-local. This proves (iii).
We complete the proof by showing that (iii) implies (i). Let κ be a regularcardinal such that C is κ-accessible and S0 is a set of morphisms between
κ-compact objects of C. We claim that L C is stable under κ-filtered colimits
in C. To prove this, let p : K . → C be a colimit diagram, where K is smalland κ-filtered and p = p|K factors through L C ⊆ C. Let s : X → Y be amorphism which belongs to S0 and let s0 : FX → FY be the corresponding
map of corepresentable functors C → b
S. Since X and Y are κ-compact byassumption, both pX : FX ◦ p and pY : FY ◦ p are colimit diagrams in b
S. Themap s0 induces a transformation pX → pY , which is an equivalence whenrestricted to K and is therefore an equivalence in general. It follows that
MapC (Y, p(∞)) ' MapC (X, p(∞)), where ∞ denotes the cone point of K . .
Thus p(∞) is S0 -local as desired.
Now choose an uncountable regular cardinal τ  κ such that Cκ is essentially τ -small. According to Proposition 5.4.2.2, to complete the proof that
C is accessible it will suffice to show that L C is generated by τ -compact
objects under τ -filtered colimits. Let X be an object of L C. Lemma 5.1.5.3
implies that X can be written as the colimit of a small diagram p : I → Cκ ,where I is κ-filtered. Using Proposition 5.3.1.18, we may suppose that I is
the nerve of a κ-filtered partially ordered set A. Let B denote the collectionof all κ-filtered τ -small subsets Aβ ⊆ A for which the colimit of p| N(Aβ )is S0 -local. Lemma 5.5.4.4 asserts that every τ -small subset of A is contained in Aβ for some β ∈ B. It follows that BSis τ -filtered when regarded
as partially ordered by inclusion and that A = β∈B Aβ . Using Proposition
4.2.3.8 and Corollary 4.2.3.10, we can obtain X as the colimit of a diagramq : N(B) → C, where each q(β) is a colimit Xβ of p| N(Aβ ). The objects{Xβ }β∈B are S0 -local and τ -compact by construction.
According to Proposition 5.5.4.2, every localization L of an ∞-category C
is determined by the class S of morphisms f such that Lf is an equivalence.
This raises the question: which classes of morphisms S arise in this way? To
answer this question, we will begin by isolating some of the most obvious
properties enjoyed by S.
Definition 5.5.4.5. Let C be a ∞-category which admits small colimits
and let S be a collection of morphisms of C. We will say that S is strongly

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saturated if it satisfies the following conditions:(1) Given a pushout diagram
/D

f

C

C0


/ D0

f0

in C, if f belongs to S, then so does f 0 .
(2) The full subcategory of Fun(∆1 , C) spanned by S is stable under small
colimits.
(3) Suppose we are given a 2-simplex of C corresponding to a diagram
/Y

f

X
g

h

Z.

~

If any two of f , g, and h belong to S, then so does the third.
Remark 5.5.4.6. Let C be an ∞-category which admits small colimits andlet S be a strongly saturated class of morphisms of C. Let ∅ be an initial
object of C. Condition (2) of Definition 5.5.4.5 implies that id∅ : ∅ → ∅belongs to S, since it is an initial object of Fun(∆1 , C). Any equivalence in C
is a pushout of id∅ , so condition (1) implies that S contains all equivalencesin C. It also follows from condition (1) that if f : C → D belongs to S andf 0 : C → D is homotopic to f , then f 0 belongs to S (since f 0 is a pushoutof f ). Note also that condition (2) implies that S is stable under retracts
because any retract of a morphism f can be written as a colimit of copies off (Proposition 4.4.5.12).
Remark 5.5.4.7. Let C be an ∞-category which admits colimits. Given
any collection {Sα T}α∈A of strongly saturated classes of morphisms of C, theintersection S = α∈A Sα is also strongly saturated. It follows that any
collection S0 of morphisms in C is contained in a smallest strongly saturated
class of morphisms S. In this case we will also write S = S0 ; we refer to it
as the strongly saturated class of morphisms generated by S0 . We will saythat S is of small generation if S = S0 , where S0 ⊆ S is small.
Remark 5.5.4.8. Let C be an ∞-category which admits small colimits. Let
S be a strongly saturated class of morphisms of C. If f : X → Y lies in S
and K is a simplicial set, then the induced map X ⊗ K → Y ⊗ K (which
is well-defined up to equivalence) lies in S. This follows from the closure of
S under colimits. We will use this observation in the proof of Proposition
5.5.4.15 in the case where K = ∂ ∆n is a (simplicial) sphere.

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Example 5.5.4.9. Let C be an ∞-category which admits small colimits andlet S denote the class of all equivalences in C. Then S is strongly saturated;
it is clearly the smallest strongly saturated class of morphisms of C.
Remark 5.5.4.10. Let F : C0 → C be a functor between ∞-categories.
Suppose that C and C0 admit small colimits and that F preserves small
colimits. Let S be a strongly saturated class of morphisms in C. Then F −1 S isa strongly saturated class of morphisms of C. In particular, if we let S denotethe collection of all morphisms f of C0 such that F (f ) is an equivalence, then
S is strongly saturated.
Lemma 5.5.4.11. Let C be an ∞-category which admits small colimits,
let S0 be a class of morphisms in C, and let S denote the collection of all
S0 -equivalences. Then S is strongly saturated.
Proof. For each object X ∈ C, let FX : C → Sop denote the functor represented by X and let S(X) denote the collection of all morphisms f such that
FX (f ) is an equivalence. Since FX preserves small colimits, Remark 5.5.4.10
implies thatT S(X) is strongly saturated. We now observe that S is the intersection S(X), where X ranges over the class of all S0 -local objects ofC.
Lemma 5.5.4.12. Let C be an ∞-category which admits small colimits, let
S be a strongly saturated collection of morphisms of C, and let C ∈ C be an
object. Let D ⊆ CC/ be the full subcategory of CC/ spanned by those objects
C → C 0 which belong to S. Then D is stable under small colimits in CC/ .
Proof. The proofs of Corollaries 4.2.3.11 and 4.4.2.4 show that it will suffice
to prove that D is stable under filtered colimits and pushouts, and contains
the initial objects of CC/ . The last condition is equivalent to the requirementthat S contain all equivalences, which follows from Remark 5.5.4.6. Now
suppose that p : K . → CC/ is a colimit of p = p|K, where K is either
filtered or equivalent to Λ20 , and that p(K) ⊆ D. We can identify p with a
map P : K . × ∆1 → C such that P |K . × {0} is the constant map taking thevalue C ∈ C. Since K is weakly contractible, P |K . ×{0} is a colimit diagramin C. The map P |K . × {1} is the image of a colimit diagram under theleft fibration CC/ → C; since K is weakly contractible, Proposition 4.4.2.9
implies that P |K . × {1} is a colimit diagram. We now apply Proposition1
5.1.2.2 to deduce that P : K . → C∆ is a colimit diagram. Since S is stable1
under colimits in C∆ , we conclude that P carries the cone point of K . to amorphism belonging to S, as we wished to show.
Lemma 5.5.4.13. Let C be an ∞-category which admits small filtered colimits, let κ be an uncountable regular cardinal, let A0 and A1 be κ-filteredpartially ordered sets, and let p0 : N(A0 ) → Cκ and p1 : A1 → Cκ be two
diagrams which have the same colimit. Let A000 ⊆ A0 , A001 ⊆ A1 be κ-small
subsets. Then there exist κ-small filtered subsets A00 ⊆ A0 , A01 ⊆ A1 suchhighertopoi

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that A000 ⊆ A00 , A001 ⊆ A01 and the diagrams p0 | N(A00 ), p1 | N(A01 ) have thesame colimit in C.
Proof. Let p0 , p1 be colimits of p0 and p1 , respectively, which carry the conepoints to the same object C ∈ C. Let B = A000 ∪ A001 ∪ Z≥0 ∪ {∞}, which weregard as a partially ordered set so thataN(B) ' ((N(A000 )N(A001 )) ? N(Z≥0 )). .
We will construct sequences of elements{a0 ≤ a2 ≤ . . .} ⊆ {a ∈ A0 : (∀a00 ∈ A000 )[a00 ≤ a]}{a1 ≤ a3 ≤ . . .} ⊆ {a ∈ A1 : (∀a00 ∈ A001 )[a00 ≤ a]}
and a diagram q : N(B) → C such thatq|(N(A000 ) ∪ N{0, 2, 4, . . .}). = p0 | N(A000 ∪ {a0 , a2 , . . .}).
q|(N(A001 ) ∪ N{1, 3, 5, . . .}). = p1 | N(A001 ∪ {a1 , a3 , . . .}). .
Supposing that this has been done, we take A00 = A000 ∪ {a0 , a2 , . . .}, A01 =A001 ∪ {a1 , a3 , . . .} and observe that the colimits of p0 | N(A00 ) and p1 | N(A01 )are both equivalent to the colimit of q| N(Z≥0 ).
The construction is by recursion. Let us suppose that the sequencea0 , a1 , . . . , an−1`
N(A001 )) ? (N{0, . . . , n − 1}). have alreadybeen constructed (when n = 0, we observe that q 0 is uniquely determinedby p0 and p1 ). For simplicity we will treat only the case where n is even; thecase where n is odd can`be handled by a similar argument.
Let qn = q n |(N(A000 ) N(A001 )) ? N{0, . . . , n − 1} and qn0 = q n | N(A000 ) ?
N{0, 2, . . . , n−2}. According to Corollary 5.3.4.14, the left fibrations Cqn / →C and Cqn0 / → C are κ-compact. Set
and the map q n = q|((N(A000 )A0 (n) = {a ∈ A0 : (∀a00 ∈ A000 ∪ {a0 , . . . , an−2 })[a00 ≤ a]}
X = Cqn / ×C N(A0 (n)).

X 0 = Cqn0 / ×C N(A0 (n)).

so that X and X 0 are left fibrations classified by colimit diagrams
N(A0 (n)). → S .
Form a pullback diagramY/X


N(A0 (n)).

/ X 0,

where the left vertical map is a left fibration (by Proposition 2.1.2.1) and thebottom horizontal map is determined by p| N(A000 ∪{0, . . . , n−2})?N(A0 (n)). .
It follows that the diagram is a homotopy pullback, so that Y → N(A0 (n)).

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is also a left fibration classified by a colimit diagram N(A0 (n)). → S. The
map q n determines a vertex v of Y lying over the cone point of N(A0 (n)). .
According to Proposition 3.3.4.5, the inclusion Y ×N(A0 (n)). N(A0 (n)) ⊆ Y
is a weak homotopy equivalence of simplicial sets. It follows that there exists
an edge e : v 0 → v of Y which joins v to some vertex v 0 lying over an elementa ∈ A0 (n). We now set an = a and observe that the edge e corresponds tothe desired extension q n+1 of q n .
Lemma 5.5.4.14. Let C be a presentable ∞-category, let S be a strongly
saturated collection of morphisms in C, and let D ⊆ Fun(∆1 , C) be the fullsubcategory spanned by S. The following conditions are equivalent:(1) The ∞-category D is accessible.
(2) The ∞-category D is presentable.
(3) The collection S is of small generation (as a strongly saturated classof morphisms).
Proof. We observe that D is stable under small colimits in Fun(∆1 , C) andtherefore admits small colimits; thus (1) ⇒ (2). To see that (2) implies (3),we choose a small collection S0 of morphisms in C which generates D undercolimits; it is then obvious that S0 generates S as a strongly saturated classof morphisms.
Now suppose that (3) is satisfied. Choose a small collection of morphisms{fβ : Xβ → Yβ } which generates S and an uncountable regular cardinal κsuch that C is κ-accessible and each of the objects Xβ , Yβ is κ-compact. We
will prove that D is κ-accessible.
It is clear that D is locally small and admits κ-filtered colimits. Let D0 ⊆ D
be the collection of all morphisms f : X → Y such that f belongs to S, where
both X and Y are κ-compact. Lemma 5.3.4.9 implies that each f ∈ D0 is a
κ-compact object of Fun(∆1 , C) and, in particular, a κ-compact object of D.
Assume for simplicity that C is a minimal ∞-category, so that D0 is small.
According to Proposition 5.3.5.10, the inclusion D0 ⊆ D is equivalent to j ◦F ,where j : D0 → Indκ D0 is the Yoneda embedding and F : Indκ D0 → D isκ-continuous. Proposition 5.3.5.11 implies that F is fully faithful; let D00
denote its essential image. To complete the proof, it will suffice to show that
D00 = D. Let S 00 ⊆ S denote the collection of objects of D00 (which we may
identify with morphisms in C). By construction, S 00 contains the collectionof morphisms {fβ } which generates S. Consequently, to prove that S 00 = S,
it will suffice to show that S 00 is strongly saturated.
It follows from Proposition 5.5.1.9 that D00 ⊆ Fun(∆1 , C) is stable undersmall colimits. We next verify that S 00 is stable under pushouts. Let K = Λ20
and let p : K . → C be a colimit of p = p|K,
X

X0

f

f0

/Y

/ Y0

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such that f belongs to S 00 . The proof of Proposition 5.4.4.3 shows that we
can write p as a colimit of a diagram q : N(A) → Fun(Λ20 , Cκ ), where A is a κfiltered partially ordered set. For α ∈ A, we let pα denote the corresponding
diagram, which we may depict asfα

Xα0 ← Xα → Yα .
For each A0 ⊆ A, we let pA0 denote a colimit of q| N(A0 ), which we will
denote by
f

0

A
0
XA
0 ← XA0 → YA0 .
Let B denote the collection of κ-small filtered subsets A0 ⊆ A such that thefA0 belongs to S 00 . Since f ∈ S 00 , we conclude that f can be obtained as0
the colimit of a κ-filtered diagram N(A0 ) → DS
. Applying Lemma 5.5.4.13,
we deduce that B is κ-filtered and that A = A0 ∈B A0 . Using Proposition
4.2.3.4 and Corollary 4.2.3.10, we deduce that p is the colimit of a diagram
q 0 : N(B) → Fun(Λ20 , C), where q 0 (A0 ) = pA0 . Replacing A by B, we may
suppose that each fα belongs to S 0 .
Let lim : Fun(Λ20 , C) → Fun(∆1 ×∆1 , C) be a colimit functor (that is, a left−→
adjoint to the restriction functor). Lemma 5.5.2.3 implies that we may identify p with a colimit of the diagram lim ◦q. Consequently, the morphism f 0−→
can be written as a colimit of morphisms fα0 which fit into pushout diagrams

/ Yα

fα

Xα

Xα0


/ Yα0 .

0fα

Since fα ∈ S 00 ⊆ S, we conclude that fα0 ∈ S. Since Xα0 and Yα0 are κcompact, we deduce that fα0 ∈ S 00 . Since D00 is stable under colimits, we
deduce that f 0 ∈ S 00 , as desired.
We now complete the proof by showing that S 00 has the two-out-of-threeproperty, using the same style of argument. Let σ : ∆2 → C be a simplex
corresponding to a diagram
/Y

f

X
g

 h
Z
in C. We will show that if f, g ∈ S 00 , then h ∈ S 00 : the argument in theother two cases is the same. The proof of Proposition 5.4.4.3 shows that we
can write σ as the colimit of a diagram q : N(A) → Fun(∆2 , Cκ ), where
A is a κ-filtered partially ordered set. For each α ∈ A, we will denote thecorresponding diagram by
/ Yα

fα

Xα
gα

!

Zα .

}

hα

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Arguing as above, we may assume (possibly after changing A and q) thateach fα belongs to S 00 . Repeating the same argument, we may suppose thatgα belongs to S 00 . Since S has the two-out-of-three property, we conclude
that each hα belongs to S. Since Xα and Zα are κ-compact, we then havehα ∈ S 00 . The stability of D00 under colimits now implies that h ∈ S 00 , asdesired.
Proposition 5.5.4.15. Let C be a presentable ∞-category and let S be a(small) collection of morphisms of C. Let S denote the strongly saturated
class of morphisms generated by S. Let C0 ⊆ C denote the full subcategory of
C consisting of S-local objects. Then
(1) For each object C ∈ C, there exists a morphism s : C → C 0 such thatC 0 is S-local and s belongs to S.
(2) The ∞-category C0 is presentable.
(3) The inclusion C0 ⊆ C has a left adjoint L.
(4) For every morphism f of C, the following are equivalent:(i) The morphism f is an S-equivalence.
(ii) The morphism f belongs to S.
(iii) The induced morphism Lf is an equivalence.
Proof. Assertion (1) is a consequence of Lemma 5.5.5.14, which we will provein §5.5.5. The equivalence (1) ⇔ (3) follows immediately from Proposition
5.2.7.8. We now prove (4). Lemma 5.5.4.11 implies that the collection ofS-equivalences is a strongly saturated class of morphisms containing S; it
therefore contains S, so that (ii) ⇒ (i). Now suppose that f : X → Y issuch that Lf is an equivalence and consider the diagram
X

/Y


LX

" 
/ LY.

Our proof of (1) shows that the vertical morphisms belong to S, and thelower horizontal arrow belongs to S by Remark 5.5.4.6. Two applications of
the two-out-of-three property now show that f ∈ S, so that (iii) ⇒ (ii). If f
is an S-equivalence, then we may again use the above diagram and the twoout-of-three property to conclude that Lf is an equivalence. It follows that
LX and LY corepresent the same functor on the homotopy category h C0 ,so that Yoneda’s lemma implies that Lf is an equivalence. Thus (i) ⇒ (iii),
and the proof of (4) is complete.
It remains to prove (2). Remark 5.2.7.5 implies that L C admits small colimits, so it will suffice to prove that L C is accessible. According to Proposition 5.5.4.2, this follows from the implication (iii) ⇒ (i) of assertion (4).

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483

Proposition 5.5.4.15 gives a clear picture of the collection of all accessible
localizations of a presentable ∞-category C. For any (small) set of morphisms
S in C, the full subcategory S −1 C ⊆ C consisting of S-local objects is alocalization of C, and every localization arises in this way. Moreover, thesubcategories S −1 C and T −1 C coincide if and only if S and T generate thesame strongly saturated class of morphisms. We will also write S −1 C for
the class of S-local objects of C in the case where S is not small; however,this is generally only a well-behaved object in the case where there is a(small) subset S0 ⊆ S which generates the same strongly saturated class ofmorphisms.
Proposition 5.5.4.16. Let f : C → D be a colimit-preserving functor between presentable ∞-categories and let S be a strongly saturated class ofmorphisms of D which is of small generation. Then f −1 S is of small generation (as a strongly saturated class of morphisms of C).
Proof. Replacing D by S −1 D if necessary, we may suppose that S consistsof precisely the equivalences in D. Let ED ⊆ Fun(∆1 , D) denote the fullsubcategory spanned by those morphisms which are equivalences in D and
let EC ⊆ Fun(∆1 , C) denote the full subcategory spanned by those morphisms which belong to f −1 S. We have a homotopy Cartesian diagram of∞-categories
/ ED
EC

Fun(∆1 , C)
/ Fun(∆1 , D).

The ∞-category ED is equivalent to D and therefore presentable. The ∞categories Fun(∆1 , C) and Fun(∆1 , D) are presentable by Proposition 5.5.3.6.
It follows from Proposition 5.5.3.12 that EC is presentable. In particular,
there is a small collection of objects of EC which generates EC under colimits,
as desired.
Let C be a presentable ∞-category. We will say that a full subcategory
C0 ⊆ C is strongly reflective if it is the essential image of an accessible
localization functor. Equivalently, C0 is strongly reflective if it is presentable,
it is stable under equivalence in C, and the inclusion C0 ⊆ C admits a left
adjoint. According to Proposition 5.5.4.15, C0 is strongly reflective if andonly if there exists a (small) set S of morphisms of C such that C0 is the fullsubcategory of C spanned by the S-local objects. For later use, we record a
few easy stability properties enjoyed by the collection of strongly reflective
subcategories of C:
Lemma 5.5.4.17. Let f : C → D be a colimit-preserving functor between
presentable ∞-categories and let C0 ⊆ C be a strongly reflective subcategory.
Let f ∗ denote a right adjoint of f and let D0 ⊆ D be the full subcategory
spanned by those objects D ∈ D such that f ∗ D ∈ C0 . Then D0 is a strongly
reflective subcategory of D.

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Proof. Let S be a (small) set of morphisms of C such that C0 is the full subcategory of C spanned by the S-local objects. Then D0 is the full subcategoryof D spanned by the f (S)-local objects.
Lemma 5.5.4.18. Let C be a presentable ∞-category and let {Cα }α∈A be
a family of full subcategories of C indexed
by a (small) set A. Suppose thatT
each Cα is strongly reflective. Then α∈A Cα is strongly reflective.
Proof. For each α ∈ A, choose a (small) set S(α) of morphisms of C suchthat
S(α)-local objects. ThenT Cα is the full subcategory of C spanned by the S
C
is
the
full
subcategoryofC
spanned
by
the
α
α∈A
α∈A S(α)-local objects.
Lemma 5.5.4.19. Let C be a presentable ∞-category and K a small simplicial set. Let D denote the full subcategory of Fun(K / , C) spanned by thosediagrams p : K / → C which are limits of p = p|K. Then D is a strongly
reflective subcategory of Fun(K / , C).
Proof. The restriction functor D → Fun(K, C) is an equivalence of ∞categories. This proves that D is accessible. Let s : Fun(K, C) → D be a
homotopy inverse to the restriction map. Then the compositions

Fun(K / , C) → Fun(K, C) → D
is left adjoint to the inclusion.
We conclude this section by giving a universal property which characterizes
the localization S −1 C.
Proposition 5.5.4.20. Let C be a presentable ∞-category and D an arbitrary ∞-category. Let S be a (small) set of morphisms of C, and L : C →S −1 C ⊆ C an associated (accessible) localization functor. Composition with
L induces a functor
η : FunL (S −1 C, D) → FunL (C, D).
The functor η is fully faithful, and the essential image of η consists of thosefunctors f : C → D such that f (s) is an equivalence in D for each s ∈ S.
Proof. Let α : idC → L be a unit for the adjunction between L and theinclusion S −1 C ⊆ C. We first observe that every functor f0 : S −1 C → D
admits a right Kan extension f : C → D. To prove this, we may first replacef0 by the equivalent diagram g0 = f0 ◦ (L|S −1 C) and define g = f0 ◦ L. To
prove that g is a right Kan extension of g0 , it suffices to show that for eachobject X ∈ C, the diagramLf0

p : (S −1 C)/X/ → C → S −1 C → D
exhibits f0 (LX) as a limit of p = p|(S −1 C)/X ). For this, we note that an Slocalization map α(X) : X → LX is an initial object of (S −1 C)X/ (Remark
5.2.7.7) and that f0 (Lα(X)) is an equivalence by Proposition 5.2.7.4.

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485

Let X denote the full subcategory of DC spanned by those functors f :C → D which are right Kan extensions of f |S −1 C. According to Proposition
4.3.2.15, the restriction map X → Fun(S −1 C, D) is a trivial fibration. Let
η : Fun(S −1 C, D) → Fun(C, D) be given by composition with L. The above
argument shows that η factors through X. Moreover, the composition of ηwith the restriction map is homotopic to the identity on Fun(S −1 C, D). It
follows that η is an equivalence of ∞-categories.
We have a commutative diagram
FunL (S −1 C, D)
Fun(S −1 C, D)ηη/ FunL (C, D)/ Fun(C, D),

where the vertical maps are inclusions of full subcategories and the lower horizontal map is fully faithful. It follows that η is fully faithful. To complete
the proof, we must show that a functor f : C → D belongs to the essential
image of η if and only if f (s) is an equivalence for each s ∈ S. The “only if”
direction is clear since the functor L carries each element of S to an equivalence in C. Conversely, suppose that f carries each s ∈ S to an equivalence.
The natural transformation α gives a map of functors α(f ) : f → f ◦ L; wewish to show that α(f ) is an equivalence. Equivalently, we wish to show thatfor each object X ∈ C, f carries the map α(X) : X → LX to an equivalencein D. Let S 0 denote the class of all morphisms φ in C such that f (φ) is anequivalence in D. By assumption, S ⊆ S 0 . Lemma 5.5.4.11 implies that S 0 isstrongly saturated, so that Proposition 5.5.4.15 asserts that α(X) ∈ S 0 , asdesired.
5.5.5 Factorization Systems on Presentable ∞-Categories
Let C be a presentable ∞-category. In §5.5.4, we saw that it is easy to
produce localizations of C: for any small collection of morphisms S in C, thefull subcategory S −1 C of S-local objects of C is a presentable localizationof C which depends only on the strongly saturated class of morphisms S
generated by S. Our goal in this section is to prove a similar result forfactorization systems on C. The first step is to introduce the analogue of thenotion of “strongly saturated”:
Definition 5.5.5.1. Let S be a collection of morphisms in a presentable∞-category C. We will say that S is saturated if the following conditions are
satisfied:(1) The collection S is closed under small colimits in Fun(∆1 , C).
(2) The collection S contains all equivalences and is stable under composition.

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CHAPTER 5(3) The collection S is closed under the formation of pushouts. That is,given a pushout diagram
/ X0

X

Y

f0

f


/ Y0

in C, if f belongs to S, then f 0 also belongs to S.
Remark 5.5.5.2. Let C be a presentable ∞-category. Then any intersectionof saturated collections of morphisms in C is again saturated. It follows thatfor any class of morphisms S of C, there exists a smallest saturated collectionof morphisms S containing S. We will refer to S as the saturated collectionof morphisms generated by S. We will say that a saturated collection ofmorphisms S is of small generation if it is generated by some (small) subset
S ⊆ S.
Remark 5.5.5.3. If S is a saturated collection of morphisms of C, then S
is closed under retracts.
Remark 5.5.5.4. Let C be (the nerve of) a presentable category and let S
be a saturated class of morphisms in C. Then S is also weakly saturated in
the sense of Definition A.1.2.2.
Example 5.5.5.5. Let C be a presentable ∞-category. Then every strongly
saturated class of morphisms in C is also saturated.
Example 5.5.5.6. Let C be a presentable ∞-category and let S be anycollection of morphisms of C. Then ⊥ S is saturated; this follows immediatelyfrom Proposition 5.2.8.6. In particular, if (SL , SR ) is a factorization system
on C, then SL is saturated.
The main result of this section is the following converse to Example 5.5.5.6:
Proposition 5.5.5.7. Let C be a presentable ∞-category and let S be asaturated collection of morphisms in C which is of small generation. Then
(S, S ⊥ ) is a factorization system on C.
Corollary 5.5.5.8. Let C be a presentable ∞-category, let S be a saturated
collection of morphisms of C, and suppose that S is of small generation. Let
>Y
f

X

g
h



/Z

be a commutative diagram in C. If f and h belong to S, then g belongs to S.
Proof. Combine Propositions 5.5.5.7, 5.2.8.11, and 5.2.8.6.

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In the situation of Proposition 5.5.5.7, we will refer to the elements of S ⊥as S-local morphisms of C. Note that an object X ∈ C is S-local if and only
if a morphism X → 1C is S-local, where 1C denotes a final object of C.
The proof of Proposition 5.5.5.7 will be given at the end of this section
after we have established a series of technical lemmas.
Lemma 5.5.5.9. Let C be a presentable ∞-category and S a saturated collection of morphisms of C. The following conditions are equivalent:(1) The collection S is of small generation.
(2) The full subcategory D ⊆ Fun(∆1 , C) spanned by the elements of S ispresentable.
Proof. If D is presentable, then D is generated under small colimits by asmall set of objects; these objects clearly generate S as a saturated collectionof morphisms. This proves that (2) ⇒ (1). To prove the reverse implication,choose a small collection of morphisms {fβ : Xβ → Yβ } which generates S
as a semisaturated class of morphisms and an uncountable regular cardinal
κ such that C is κ-accessible and each of the objects Xβ , Yβ is κ-compact.
Let D0 ⊆ D be the collection of all morphisms f : X → Y such that fbelongs to S, where both X and Y are κ-compact. Lemma 5.3.4.9 implies
that each f ∈ D0 is a κ-compact object of Fun(∆1 , C) and, in particular, a κcompact object of D. Assume for simplicity that C is a minimal ∞-category,so that D0 is small. According to Proposition 5.3.5.10, the inclusion D0 ⊆ D
is equivalent to j ◦ F , where j : D0 → Indκ D0 is the Yoneda embedding
and F : Indκ D0 → D is κ-continuous. Proposition 5.3.5.11 implies that F
is fully faithful; let D00 denote its essential image. To complete the proof, it
will suffice to show that D00 = D.
Let S 0 ⊆ S denote the collection of objects of D00 (which we may identify
with morphisms in C). By construction, S 0 contains the collection of morphisms {fβ } which generates S. Consequently, to prove that S 0 = S, it will
suffice to show that S 0 is saturated.
It follows from Proposition 5.5.1.9 that D00 ⊆ Fun(∆1 , C) is stable undersmall colimits. We next verify that S 0 is stable under pushouts. Let K = Λ20
and let p : K . → C be a colimit of p = p|K,
X

X0

f

f0

/Y

/ Y 0,

such that f belongs to S 0 . Using Proposition 5.3.5.15, we can write p as thecolimit of a diagram q : N(A) → Fun(Λ20 , Cκ ), where A is a κ-filtered partiallyordered set. For α ∈ A, we let pα denote the corresponding diagram, which
we may depict asfα

Xα0 ← Xα → Yα .

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For each A0 ⊆ A, we let pA0 denote a colimit of q| N(A0 ), which we will
denote by
f

0

A
0
XA
0 ← XA0 → YA0 .

Let B denote the collection of κ-small filtered subsets A0 ⊆ A such that thefA0 belongs to S 0 . Since f ∈ S 0 , we conclude that f can be obtained as the0colimit of a κ-filtered diagram in D
Lemma 5.5.4.13, we deduce
S. Applyingthat B is κ-filtered and that A = A0 ∈B A0 . Using Proposition 4.2.3.4 andCorollary 4.2.3.10, we deduce that p is the colimit of a diagram q 0 : N(B) →Fun(Λ20 , C), where q 0 (A0 ) = pA0 . Replacing A by B, we may suppose thateach fα belongs to S 0 .
Let lim : Fun(Λ20 , C) → Fun(∆1 ×∆1 , C) be a colimit functor (that is, a left−→
adjoint to the restriction functor). Lemma 5.5.2.3 implies that we may identify p with a colimit of the diagram lim ◦q. Consequently, the morphism f 0−→
can be written as a colimit of morphisms fα0 which fit into pushout diagrams
/ Yα

fα

Xα

Xα0


/ Yα0 .

0fα

Since fα ∈ S 0 ⊆ S, we conclude that fα0 ∈ S. Since Xα0 and Yα0 are κ-compact,
we deduce that fα0 ∈ S 0 . Since D00 is stable under colimits, we deduce thatf 0 ∈ S 0 , as desired.
We now complete the proof by showing that S 0 is stable under composition.
Let σ : ∆2 → C be a simplex corresponding to a diagram
/Y

f

X
g

Z



h

in C. We will show that if f, g ∈ S , then h ∈ S 0 . Using Proposition 5.3.5.15,
we can write σ as the colimit of a diagram q : N(A) → Fun(∆2 , Cκ ), where
A is a κ-filtered partially ordered set. For each α ∈ A, we will denote thecorresponding diagram by
0

/ Yα

fα

Xα
gα

!

Zα .

}

hα

Arguing as above, we may assume (possibly after changing A and q) thateach fα belongs to S 0 . Repeating the same argument, we may suppose thatgα belongs to S 0 . Since S is stable under composition, we conclude that eachhα belongs to S. Since each Xα and each Zα is κ-compact, we have hα ∈ S 0 .
The stability of D00 under colimits now implies that h ∈ S 0 , as desired.

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Lemma 5.5.5.10. Let C be a presentable ∞-category and let S be a saturated
collection of morphisms in C. For every object X ∈ C, let SX denote thecollection of all morphisms of C/X whose image in C belongs to S. Theneach SX is strongly saturated in C/X . Moreover, if S is of small generation,then each SX is also of small generation.
Proof. The first assertion follows immediately from the definitions together
with Proposition 1.2.13.8. To prove the second, let D be the full subcategoryof Fun(∆1 , C) spanned by the elements of S, and D0 the full subcategoryof Fun(∆1 , C/X ) spanned by the elements of SX . We have a (homotopy)
pullback diagram of ∞-categories/ Fun(∆1 , C/X )
D0
ψ


D

φ


/ Fun(∆1 , C).

The functors φ and ψ preserve small colimits, and the ∞-categories D,
Fun(∆1 , C/X ), and Fun(∆1 , C) are all presentable (the first in view of Lemma
5.5.5.9). Using Proposition 5.5.3.12, we deduce that D0 is presentable andtherefore generated under small colimits by a (small) set of elements of SX .
This proves that SX is of small generation, as desired.
Lemma 5.5.5.11. Let C be a presentable ∞-category, S a saturated collection of morphisms of C, and X an object of C. Then the full subcategory
D ⊆ CX/ spanned by the elements which belong to S is closed under small
colimits.
Proof. In view of Corollaries 4.2.3.11 and 4.4.2.4, it will suffice to show that
D is closed under small filtered colimits and pushouts, and contains the initial
objects of CX/ . The last condition follows from the fact that S contains allequivalences. Now suppose that p : K . → CC/ is a colimit of p = p|K, where
K is either filtered or equivalent to Λ20 , and that p(K) ⊆ D. We can identify
p with a map P : K . × ∆1 → C such that P |K . × {0} is the constant
map taking the value C ∈ C. Since K is weakly contractible, P |K . × {0}is a colimit diagram in C. The map P |K . × {1} is the image of a colimit
diagram under the left fibration CC/ → C; since K is weakly contractible,
Proposition 4.4.2.9 implies that P |K . × {1} is a colimit diagram. We now
1apply Proposition 5.1.2.2 to deduce that P : K . → C∆ is a colimit diagram.
1
Since S is stable under colimits in C∆ , we conclude that P carries the conepoint of K . to a morphism belonging to S, as we wished to show.
Lemma 5.5.5.12. Let C be an ∞-category and let f : C → D, g : C → E
be morphisms in C. Then there is a natural identification of MapCC/ (f, g)
with the homotopy fiber of the map
MapC (D, E) → MapC (C, E)
induced by composition with f , where the fiber is taken over the point corresponding to g.

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Proof. We have a commutative diagram of simplicial sets
/ Cf / ×C {E}/ Cf /
Cf / ×C {g}
C/

φ0

φ


{g}


/ CC/ ×C {E}φ00

/ CC/ ,

where both squares are pullbacks. Proposition 2.1.2.1 asserts that φ00 is aleft fibration, so that φ0 and φ are left fibrations as well. Since CC/ ×C {E} =HomLC (C, E) is a Kan complex, the map φ0 is actually a Kan fibration(Lemma 2.1.3.3), so that the square on the left is a homotopy pullback
and identifies
Cf / ×CC/ {g} ' MapCC/ (f, g)
with the homotopy fiber of φ0 over g; we conclude by observing that φ0 is amodel for the map MapC (D, E) → MapC (C, E) given by composition withf.
Lemma 5.5.5.13. Let
X

f

/ X0

g



f0
/ Y0
Y
be a pushout diagram in an ∞-category C. Then there exists an isomorphism
MapCX/ (f, g) ' MapCY / (f 0 , idY )in the homotopy category H.
Proof. According to Corollary 4.2.4.7, we can assume without loss of generality that C is the nerve of a fibrant simplicial category D and that the
diagram in question is the nerve of a commutative diagram
X

f

/ X0

g



f0
/ Y0
Y
in D. Theorem 4.2.4.1 implies that this diagram is homotopy coCartesian inD, so that we have a homotopy pullback diagram
MapD (Y 0 , Y )MapD (X 0 , Y )φφ0/ MapD (Y, Y )/ MapD (X, Y )of Kan complexes. Consequently, we obtain an isomorphism in H between
the homotopy fiber of φ over idY and the homotopy fiber of φ0 over g.
According to Lemma 5.5.5.12, these homotopy fibers may be identified with
MapCY / (f 0 , idY ) and MapCX/ (f, g), respectively.

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Lemma 5.5.5.14. Let C be a presentable ∞-category and let S be a saturated
collection of morphisms of C which is of small generation. Then, for every
object X ∈ C, there exists a morphism f : X → Y in C such that f ∈ S and
Y is S-local.
Proof. Let D be the full subcategory of Fun(∆1 , C) spanned by the elementsof S and form a fiber diagram
/D
DX

{X}


/ Fun({0}, C).

Since S is stable under pushouts, the right vertical map is a coCartesianfibration, so that the above diagram is homotopy Cartesian by Proposition
3.3.1.3. Lemma 5.5.5.9 asserts that D is accessible, so that DX is accessible by Proposition 5.4.6.6. Using Lemma 5.5.5.11, we conclude that DX ispresentable, so that DX has a final object f : X → Y . To complete theproof, it will suffice to show that Y is S-local.
Let t : A → B be an arbitrary morphism in C which belongs to S. We
wish to show that composition with t induces a homotopy equivalence φ :MapC (B, Y ) → MapC (A, Y ). Let g : A → Y be an arbitrary morphism; using
Lemma 5.5.5.12, we may identify MapCA/ (t, g) with the homotopy fiber of
φ over the base point g of MapC (A, Y ). We wish to show that this space iscontractible. Form a pushout diagram
A

t

/B

g



t0 /
Y
Z
in the ∞-category C. Lemma 5.5.5.13 implies the existence of a homotopy
equivalence MapCA/ (t, g) ' MapCY / (t0 , idY ). It will therefore suffice to provethat MapCY / (t0 , idY ) is contractible. Since t0 is a pushout of t, it belongs to
S. Let σ be a 2-simplex of C classifying a diagram
?Y
t0

s

X

s0


/ Z,

so that s0 is a composition of the morphisms s and t0 in C and therefore also
belongs to S. Applying Lemma 5.5.5.12 again, we may identify
MapCY / (t0 , idC 0 ) ' MapCs/ (σ, s1 (s))
with the homotopy fiber of the map MapCY / (s0 , s) → MapCY / (s, s) given bycomposition with σ. By construction, DX is a full subcategory of CX/ whichcontains s and s0 , and s is a final object of DX . In view of the equivalence of
CX/ with CX/ , we conclude that the spaces MapCX/ (s0 , s) and MapCX/ (s, s)
are contractible, so that φ is a homotopy equivalence, as desired.

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Proof of Proposition 5.5.5.7. Let h : X → Z be a morphism in C; we wish
to show that h admits a factorization
?Y
f

X

g
h


/ Z,

where f ∈ S and g ∈ S ⊥ . Using Remark 5.2.8.3, we deduce that a morphism
g : Y → Z belongs to S ⊥ if and only if it is an SZ -local object of C/Z , where
SZ is defined as in Lemma 5.5.5.10. The existence of h then follows from
Lemma 5.5.5.14.
5.5.6 Truncated Objects
Let X be a topological space. The first step in the homotopy-theoretic analysis of the space X is to divide X into path components. The situation canbe described as follows: we associate to X a set π0 X, which we may view as
a discrete topological space. There is a map f : X → π0 X which collapses
each component of X to a point. If X is a sufficiently nice space (for example, a CW complex), then the path components of X are open, so f iscontinuous. Moreover, f is universal among continuous maps from X into a
discrete topological space.
The next step in the analysis of X is to consider its fundamental group
π1 X, which (provided that X is sufficiently nice) may be studied by means
e of X. However, it is important to realize that neitherof a universal cover X
e is invariantly associated to X: both require a choice of base
π1 X nor X
point. The situation can be described more canonically as follows: to X
we can associate a fundamental groupoid π(X) and a map φ from X toe of X can be identified
the classifying space Bπ(X). The universal cover X(up to homotopy equivalence) with the homotopy fibers of the map φ. The
classifying space Bπ(X) can be regarded as a “quotient” of X obtained bykilling all of the higher homotopy groups of X. Like π0 X, it can be describedby a universal mapping property.
To continue the analysis, we first recall that a space Y is said to be ktruncated if the homotopy groups of Y vanish in dimensions larger than k(see Definition 2.3.4.15). Every (sufficiently nice) topological space X admits
an essentially unique Postnikov tower
X → · · · → τ≤n X → · · · → τ≤−1 X,
where τ≤i X is i-truncated, and is universal (in a suitable homotopy-theoretic
sense) among i-truncated spaces which admit a map from X. For example, we
can take τ≤0 X = π0 X, considered as a discrete space, and τ≤1 X = Bπ(X).
Moreover, we can recover the space X (up to weak homotopy equivalence)by taking the homotopy limit of the tower.
The objective of this section is to construct an analogous theory in thecase where X is not a space but instead an object of some (abstract) ∞category C. We begin by observing that the condition that a space X behighertopoi

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k-truncated can be reformulated in more categorical terms: a Kan complex
X is k-truncated if and only if, for every simplicial set S, the mapping space
MapSet∆ (S, X) is k-truncated. This motivates the following:
Definition 5.5.6.1. Let C be an ∞-category and k ≥ −1 an integer. We
will say that an object C of C is k-truncated if, for every object D ∈ C,
the space MapC (D, C) is k-truncated. By convention, we will say that C is(−2)-truncated if it is a final object of C. We will say that an object of C
is discrete if it is 0-truncated. We will generally let τ≤k C denote the fullsubcategory of C spanned by the k-truncated objects.
Notation 5.5.6.2. Let C be an ∞-category. Using Propositions 2.3.4.18
and 2.3.4.5, we conclude that the full subcategory τ≤0 C is equivalent to thenerve of its homotopy category. We will denote this homotopy category byDisc(C) and refer to it as the category of discrete objects of C.
Lemma 5.5.6.3. Let C be an object of an ∞-category C and let k ≥ −2.
The following conditions are equivalent:(1) The object C is k-truncated.
(2) For every n ≥ k + 3 and every diagram/=Cf∂ ∆ _n∆nfor which f carries the final vertex of ∆n to C, there exists a dottedarrow rendering the diagram commutative.
Proof. Suppose first that (2) is satisfied. Then for every object D ∈ D,
the Kan complex HomR
C (D, C) has the extension property with respect to∂ ∆n−1 ⊆ ∆n−1 for all n ≥ k + 3, and is therefore k-truncated. For theconverse, suppose that (1) is satisfied and choose a categorical equivalence
g : C → N D, where D is a topological category. According to Proposition
A.2.3.1, it will suffice to show that for every n ≥ k + 3 and every diagramn| C[∂ ∆
 _ ]|F/D;| C[∆n ]|having the property that F carries the final object of | C[∆n ]| to g(C),there exists a dotted arrow as indicated, rendering the diagram commutative. Let D ∈ D denote the image of the initial object of ∂ ∆n under F .
Then, constructing the desired extension is equivalent to extending a map∂[0, 1]n−1 → MapD (D, g(C)) to a map defined on all of [0, 1]n−1 , which ispossibly by virtue of assumption (1).

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Remark 5.5.6.4. A Kan complex X is k-truncated if and only if it is ktruncated when regarded as an object in the ∞-category S of spaces.
Proposition 5.5.6.5. Let C be an ∞-category and k ≥ −2 an integer. The
full subcategory τ≤k C ⊆ C of k-truncated objects is stable under all limits
which exist in C.
Proof. Let j : C → P(C) be the Yoneda embedding. By definition, τ≤k C is
the preimage of Fun(Cop , τ≤k S) under j. Since j preserves all limits which
exist in C, it will suffice to prove that Fun(Cop , τ≤k S) ⊆ Fun(Cop , S) is stable
under limits. Using Proposition 5.1.2.2, it suffices to prove that the inclusioni : τ≤k S ⊆ S is stable under limits. In other words, we must show thatτ≤k S admits small limits and that i preserves small limits. According to
Propositions 4.4.2.6 and 4.4.2.7, it will suffice to show that τ≤k S ⊆ S isstable under the formation of pullbacks and small products. According to
Theorem 4.2.4.1, this is equivalent to the assertion that the full subcategoryof Kan spanned by the k-truncated Kan complexes is stable under homotopy
products and the formation of homotopy pullback squares. Both assertions
can be verified easily by computing homotopy groups.
Remark 5.5.6.6. Let p : C → D be a coCartesian fibration of ∞-categories.
Let C and C 0 be objects of C, let f : p(C 0 ) → p(C) be a morphism in D,
and let f : C 0 → C 00 be a p-coCartesian morphism lifting f . According to
Proposition 2.4.4.2, we may identify MapCp(C) (C 00 , C) with the homotopy
fiber of MapC (C 0 , C) → MapD (p(C 0 ), p(C)) over the base point determinedby f . By examining the associated long exact sequences of homotopy groups(as f varies), we conclude that if C is a k-truncated object of the fiber Cp(C)and p(C) is a k-truncated object of D, then C is a k-truncated object of C.
This can be considered as a generalization of Lemma 2.4.4.7 (which treats
the case k = −2).
Remark 5.5.6.7. Let p : M → ∆1 be a coCartesian fibration of simplicial
sets, which we regard as a correspondence from the ∞-category C = p−1 {0}to D = p−1 {1}. Suppose that D is a k-truncated object of D. Remark
5.5.6.6 implies that D is a k-truncated object of M. Let C, C 0 ∈ C and
let f : C → D be a p-Cartesian morphism of M. Then composition with f
induces a homotopy equivalence MapC (C 0 , C) → MapM (C 0 , D); we conclude
that C is a k-truncated object of M.
Definition 5.5.6.8. We will say that a map f : X → Y of Kan complexes
is k-truncated if the homotopy fibers of f (taken over any base point of Y )are k-truncated. We will say that a morphism f : C → D in an arbitrary∞-category C is k-truncated if composition with f induces a k-truncatedmap MapC (E, C) → MapC (E, D) for every object E ∈ C.
Remark 5.5.6.9. There is an apparent potential for ambiguity in Definition 5.5.6.8 in the case where C is an ∞-category whose objects are Kancomplexes. However, there is no cause for concern: a map f : X → Y of Kan

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complexes is k-truncated if and only if it is k-truncated as a morphism in
the ∞-category S.
Remark 5.5.6.10. Let f : C → D and g : E → D be morphisms in an∞-category C, and let φ : MapC (E, C) → MapC (E, D) be the map (in thehomotopy category H) given by composition with f . Lemma 5.5.5.12 implies
that the homotopy fiber of φ over g is homotopy equivalent to MapC/D (g, f ).
Consequently, we deduce that f : C → D is k-truncated in the sense of
Definition 5.5.6.8 if and only if it is k-truncated when viewed as an objectof the ∞-category D/D .
Lemma 5.5.6.11. Let p : C → D be a right fibration of ∞-categories andlet f : X → Y be a morphism in C. Then f is n-truncated if and only ifp(f ) : p(X) → p(Y ) is n-truncated.
Proof. The map C/Y → D/p(Y ) is a trivial fibration and therefore an equivalence of ∞-categories.
Remark 5.5.6.12. A morphism f : C → D in an ∞-category C is ktruncated if and only if it is k-truncated when regarded as an object of
the ∞-category C/D (since the natural map C/D → C/D is an equivalenceof ∞-categories). We may identify C/D with p−1 {D}, where p denotes the1evaluation map C∆ → C{1} . Corollary 2.4.7.12 implies that p is a coCartesian fibration. Consequently, Remark 5.5.6.7 translates into the following
assertion: if
C0

C

f0

f

/ D0

/D

is a pullback diagram in C and f is k-truncated, then f 0 is k-truncated.
Example 5.5.6.13. A morphism f : C → D in an ∞-category C is (−2)truncated if and only if it is an equivalence.
We will say that a morphism f : C → D is a monomorphism if it is (−1)truncated; this is equivalent to the assertion that the functor C/f → C/D isfully faithful.
Lemma 5.5.6.14. Let C be an ∞-category and f : X → Y a morphism in
C. Suppose that Y is n-truncated. Then X is n-truncated if and only if f isn-truncated.
Proof. Unwinding the definitions, we reduce immediately to the following
statement in classical homotopy theory: given a map f : X → Y of Kancomplexes, where Y is n-truncated, X is n-truncated if and only if the homotopy fibers of f are n-truncated. This can be established easily using the longexact sequence of homotopy groups associated to f .

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The following lemma gives a recursive characterization of the class of ntruncated morphisms:
Lemma 5.5.6.15. Let C be an ∞-category which admits finite limits andlet k ≥ −1 be an integer. A morphism f : C → C 0 is k-truncated if and only
if the diagonal δ : C → C ×C 0 C (which is well-defined up to homotopy) is(k − 1)-truncated.
Proof. For each object D ∈ C, let FD : C → S denote the functor corepresented by D. Then each FD preserves finite limits, and a morphism f in
C is k-truncated if and only if each FD (f ) is a k-truncated morphism in S.
We may therefore reduce to the case where C = S. Without loss of generality, we may suppose that f : C → C 0 is a Kan fibration. Then Theorem
4.2.4.1 allows us to identify the fiber product C ×C 0 C in S with the same
fiber product formed in the ordinary category Kan. We now reduce to thefollowing assertion in classical homotopy theory (applied to the fibers of f ):if X is a Kan complex, then X is k-truncated if and only if the homotopy
fibers of the diagonal map X → X × X are (k − 1)-truncated. This can beproven readily by examining homotopy groups.
We immediately deduce the following:
Proposition 5.5.6.16. Let F : C → C0 be a left exact functor between ∞categories which admit finite limits. Then F carries k-truncated objects into
k-truncated objects and k-truncated morphisms into k-truncated morphisms.
Proof. An object C is k-truncated if and only if the morphism C → 1 tothe final object is k-truncated. Since F preserves final objects, it suffices
to prove the assertion concerning morphisms. Since F commutes with fiber
products, Lemma 5.5.6.15 allows us to use induction on k, thereby reducing
to the case where k = −2. But the (−2)-truncated morphisms are precisely
the equivalences, and these are preserved by any functor.
We now specialize to the case of a presentable ∞-category C. In this setting,
we can construct an analogue of the Postnikov tower.
Lemma 5.5.6.17. Let X be a Kan complex and let k ≥ −2. The following
conditions are equivalent:(1) The Kan complex X is k-truncated.
(2) The diagonal map δ : X → X ∂ ∆
k+2

k+2

is a homotopy equivalence.

Proof. If k = −2, then X ∂ ∆is a point and the assertion is obvious.
Assuming k > −2, we can choose a vertex v of ∂ ∆k+2 , which gives rise to
k+2
an evaluation map e : X ∂ ∆→ X. Since e ◦ δ = idX , (2) is equivalent tothe assertion that e is a homotopy equivalence. We observe that e is a Kan
k+2
×X {x} denote the fiber of e overfibration. For each x, let Yx = X ∂ ∆highertopoi

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the vertex x. Then Yx has a canonical base point given by the constant map
δ(x). Moreover, we have a natural isomorphism
πi (Yx , δ(x)) ' πi+k+1 (X, x).
Condition (1) is equivalent to the assertion that πi+k+1 (X, x) vanishes for
all i ≥ 0 and all x ∈ X. In view of the above isomorphism, this is equivalent
to the assertion that each Yx is contractible, which is true if and only if the
Kan fibration e is trivial.
Proposition 5.5.6.18. Let C be a presentable ∞-category, let k ≥ −2, andlet τ≤k C denote the full subcategory of C spanned by the k-truncated objects.
Then the inclusion τ≤k C ⊆ C has an accessible left adjoint, which we will
denote by τ≤k .
Proof. Let f : ∂ ∆k+2 → Fun(C, C) denote the constant diagram taking thevalue idC . Let f : (∂ ∆k+2 ). → Fun(C, C) be a colimit of f and let F : C → C
be the image of the cone point under f . Informally, F is given by the formula
C 7→ C ⊗ S k+1 ,where S k+1 denotes the (k + 1)-sphere and we regard C as tensored overspaces (see Remark 5.5.1.7).
0
Let f : (∂ ∆k+2 ). → CC be the constant diagram taking the value idC . It
follows that there exists an essentially unique map f → f 0 in (CC )f / , which
induces a natural transformation of functors α : F → idC . Let S = {α(C) :C ∈ C}. Since F is a colimit of functors which preserve small colimits, F itself preserves small colimits (Lemma 5.5.2.3). Applying Proposition 5.1.2.2,
we conclude that α : C → Fun(∆1 , C) also preserves small colimits. Consequently, there exists a small subset S0 ⊆ S which generates S under colimits
in Fun(∆1 , C). According to Proposition 5.5.4.15, the collection of S-local
objects of C is an accessible localization of C. It therefore suffices to provethat an object X ∈ C is S-local if and only if X is k-truncated.
According to Proposition 5.1.2.2, for each C ∈ C we may identify F (C)with the colimit of the constant diagram ∂ ∆k+2 → C taking the value C.
Corollary 4.4.4.9 implies that we have a homotopy equivalence
MapC (F (C), X) ' MapC (C, X)∂ ∆k+2.

The map α(C) induces a map
α(C)X : MapC (C, X) → MapC (C, X)∂ ∆k+2which can be identified with the inclusion of MapC (C, X) as the space ofconstant maps from ∂ ∆k+2 into MapC (C, X). According to Lemma 5.5.6.17,
the map α(C)X is an equivalence if and only if MapC (C, X) is k-truncated.
Thus X is k-truncated if and only if X is S-local.
Remark 5.5.6.19. The notation of Proposition 5.5.6.18 is self-consistentin the following sense: the existence of the localization functor τ≤k implies
that the collection of k-truncated objects of C may be identified with theessential image of τ≤k .

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Remark 5.5.6.20. If the ∞-category C is potentially unclear in context,
CC
then we will write τ≤kfor the truncation functor in C. Note also that τ≤kiswell-defined up to equivalence (in fact, up to a contractible ambiguity).
Remark 5.5.6.21. It follows from Proposition 5.5.6.18 that if C is a presentable ∞-category, then the full subcategory τ≤k C of k-truncated objects is
also presentable. In particular, the ordinary category Disc(C) of discrete objects of C is a presentable category in the sense of Definition A.1.1.2.
Recall that, if C and D are ∞-categories, then FunL (C, D) denotes the fullsubcategory of Fun(C, D) spanned by those functors which are left adjoints.
The following result gives a characterization of τ≤n C by a universal mapping
property:
Corollary 5.5.6.22. Let C and D be presentable ∞-categories. Suppose that
D is equivalent to an (n + 1)-category. Then composition with τ≤n induces
an equivalence s : FunL (τ≤n C, D) → FunL (C, D).
Proof. According to Proposition 5.5.4.20, the functor s is fully faithful. A
functor f : C → D belongs to the essential image of s if and only if f has
a right adjoint g which factors through τ≤n C. Since g preserves limits, it
automatically carries D = τ≤n D into τ≤n C (Proposition 5.5.6.16).
In classical homotopy theory, every space X can be recovered (up to weakhomotopy equivalence) as the homotopy inverse limit of its Postnikov tower{τ≤n X}n≥0 . The analogous statement is not true in an arbitrary presentable∞-category but often holds in specific examples. We now make a general
study of this phenomenon.
Definition 5.5.6.23. Let Z∞≥0 denote the union Z≥0 ∪ {∞} regarded asa linearly ordered set with largest element ∞. Let C be a presentable ∞op
category. Recall that a tower in C is a functor N(Z∞
→ C, which we view
≥0 )as a diagram
X∞ → · · · → X2 → X1 → X0 .
A Postnikov tower is a tower with the property that for each n ≥ 0, the map
X∞ → Xn exhibits Xn as an n-truncation of X∞ . We define a pretower tobe a functor from N(Z≥0 )op → C. A Postnikov pretower is a pretower· · · → X2 → X1 → X0
which exhibits each Xn as an n-truncation of Xn+1 . We let Post+ (C) denoteopthe full subcategory of Fun(N(Z∞≥0 ) , C) spanned by the Postnikov towers, and Post(C) the full subcategory of Fun(N(Z≥0 )op , C) spanned by the
Postnikov pretowers. We have an evident forgetful functor φ : Post+ (C) →Post(C). We will say that Postnikov towers in C are convergent if φ is anequivalence of ∞-categories.
Remark 5.5.6.24. Let C be a presentable ∞-category and let E denote theopfull subcategory of C × N(Z∞
spanned by those pairs (C, n) where C ∈ C≥0 )highertopoi

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is n-truncated (by convention, we agree that this condition is always satisfied
opwhere C = ∞). Then we have a coCartesian fibration p : E → N(Z∞≥0 ) ,
which classifies a tower of ∞-categories
τ≤1

τ≤0

C → · · · → τ≤2 C → τ≤1 C → τ≤0 C .
We can identify Postnikov towers with coCartesian sections of p (and Postnikov pretowers with coCartesian sections of the induced fibrationop
E ×N(Z∞→ N(Z≥0 )op ).
)op N(Z≥0 )≥0According to Proposition 3.3.3.1, Postnikov towers in C converge if and only
if the tower above exhibits C as the homotopy limit of the sequence of ∞categories· · · → τ≤2 C → τ≤1 C → τ≤0 C .
Remark 5.5.6.25. Let C be a presentable ∞-category and assume that
Postnikov towers in C are convergent. Then every Postnikov tower in C is alimit diagram. Indeed, given objects X, Y ∈ C, we have natural homotopy
equivalences
MapC (X, Y ) ' holim MapC (τ≤n X, τ≤n Y ) ' holim MapC (X, τ≤n Y ),so that Y is the limit of the pretower {τ≤n Y }.
Proposition 5.5.6.26. Let C be a presentable ∞-category. Then Postnikov
optowers in C are convergent if and only if, for every tower X : N(Z∞
→ C,
≥0 )
the following conditions are equivalent:(1) The diagram X is a Postnikov tower.
(2) The diagram X is a limit in C, and the restriction X| N(Z≥0 )op is a
Postnikov pretower.
op
Proof. Let Post0 (C) be the full subcategory of Fun(N(Z∞≥0 ) , C) spannedby those towers which satisfy condition (2). Using Proposition 4.3.2.15, wededuce that the restriction functor Post0 (C) → Post(C) is a trivial Kan fibration. If conditions (1) and (2) are equivalent, then Post0 (C) = Post+ (C), sothat Postnikov towers in C are convergent. Conversely, suppose that Postnikov towers in C are convergent. Using Remark 5.5.6.25, we deduce that
Post+ (C) ⊆ Post0 (C), so we have a commutative diagram/ Post0 (C)Post+ (C)%yPost(C).

Since both of the vertical arrows are trivial Kan fibrations, we conclude thatthe inclusion Post+ (C) ⊆ Post0 (C) is an equivalence, so that Post+ (C) =Post0 (C). This proves that (1) ⇔ (2).

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Remark 5.5.6.27. Let C be a presentable ∞-category. We will say thatopa tower X : N(Z∞
→ X is highly connected if, for every n ≥ 0, there≥0 )exists an integer k such that the induced map τ≤n X(∞) → τ≤n X(k 0 ) is anequivalence for k 0 ≥ k. We will say that a pretower Y : N(Z≥0 ) → X is highly
connected if, for every n ≥ 0, there exists an integer k such that the map
τ≤n Y (k 00 ) → τ≤n Y (k 0 ) is an equivalence for k 00 ≥ k 0 ≥ k. It is clear thatevery Postnikov (pre)tower is highly connected. Conversely, if X is a highly
connected tower and its underlying pretower is highly connected, then X isa Postnikov tower. Indeed, for each n ≥ 0 we can choose k ≥ n such thatthe map τ≤n X(∞) → τ≤n X(k) is an equivalence. Since X is a Postnikov
pretower, this induces an equivalence τ≤n X(∞) ' X(n). Consequently, toestablish the implication (2) ⇒ (1) in the criterion of Proposition 5.5.6.26,
it suffices to verify the following:(∗) Let X : N(Z∞≥0 ) → C be a tower in C. Assume that X is a limit diagram
and that the underlying pretower is highly connected. Then X is highly
connected.
In §7.2.1, we will apply this criterion to prove that Postnikov towers areconvergent in a large class of ∞-topoi.
We conclude this section with a useful compatibility property between
truncation functors in different ∞-categories:
Proposition 5.5.6.28. Let C and D be presentable ∞-categories and let
F : C → D be a left exact colimit-preserving functor. Then there is anCD
equivalence of functors F ◦ τ≤k
' τ≤k
◦ F.
Proof. Since F is left exact, it restricts to a functor from τ≤k C to τ≤k D by
Proposition 5.5.6.16. We therefore have a diagram
C


F

C
τ≤k

τ≤k C

F

/D


D
τ≤k

/ τ≤k D

which we wish to prove is commutative up to homotopy. Let G denote aright adjoint to F ; then G is left exact and so induces a functor τ≤k D →τ≤k C. Using Proposition 5.2.2.6, we can reduce to proving that the associateddiagram of right adjoints
CO o

G

DO

τ≤k C oGτ≤k D

commutes up to homotopy, which is obvious (since the diagram strictly commutes).

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5.5.7 Compactly Generated ∞-Categories
Definition 5.5.7.1. Let κ be a regular cardinal. We will say that an ∞category C is κ-compactly generated if it is presentable and κ-accessible.
When κ = ω, we will simply say that C is compactly generated.
The proof of Theorem 5.5.1.1 shows that an ∞-category C is κ-compactly
generated if and only if there exists a small ∞-category D which admits
κ-small colimits and an equivalence C ' Indκ (D). In fact, we can choose
D to be (a minimal model of) the ∞-category of κ-compact objects of C.
We would like to assert that this construction establishes an equivalence
between two sorts of ∞-categories. In order to make this precise, we need to
introduce the appropriate notion of functor between κ-compactly generated∞-categories.
Proposition 5.5.7.2. Let κ be a regular cardinal and let C oFG/ D be apair of adjoint functors, where C and D admit small κ-filtered colimits.
(1) If G is κ-continuous, then F carries κ-compact objects of C to κcompact objects of D.
(2) Conversely, if C is κ-accessible and F preserves κ-compactness, then
G is κ-continuous.
Proof. Suppose first that G is κ-continuous and let C ∈ C be a κ-compact
object. Let e : C → b
S be a functor corepresented by C. Then e ◦ G : D → b
S
is corepresented by F (C). Since e and G are κ-continuous, so is e ◦ G; thisproves (1).
Conversely, suppose that F preserves κ-compact objects and that C is κaccessible. Without loss of generality, we may suppose that there is a small∞-category C0 such that C = Indκ (C0 ) ⊆ P(C0 ). We wish to prove that G is
κ-continuous. Since Indκ (C0 ) is stable under κ-filtered colimits in P(C0 ), it
will suffice to prove that the composite map
Gθ : D → C ⊆ P(C0 )is κ-continuous. In view of Proposition 5.1.2.2, it will suffice to prove thatfor every object C ∈ C0 , the composition of θ with evaluation at C is aκ-continuous functor. We conclude by observing that this functor is corepresentable by the image under F of j(C) ∈ C (here j : C0 → Indκ (C0 ) denotes
the Yoneda embedding).
Corollary 5.5.7.3. Let C be a κ-compactly generated ∞-category and let
L : C → C be a localization functor. The following conditions are equivalent:(1) The functor L is κ-continuous.
(2) The full subcategory L C ⊆ C is stable under κ-filtered colimits.
Suppose that these conditions are satisfied. Thenhighertopoi

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CHAPTER 5(3) The functor L carries κ-compact objects of C to κ-compact objects of
L C.
(4) The ∞-category L C is κ-compactly generated.
(5) An object D ∈ L C is κ-compact (in L C) if and only if there exists a
compact object C ∈ C such that D is a retract of LC.
Proof. Suppose that (1) is satisfied. Let p : K → L C be a κ-filtered diagram.
Then the natural transformation p → Lp is an equivalence. Using (1), we
conclude that the induced map lim(p) → L lim(p) is an equivalence, so that−→−→lim(p) ∈ L C. This proves (2).
−→Conversely, if (2) is satisfied, then the inclusion L C ⊆ C is κ-continuous,
so that L : C → C is a composition of κ-continuous functors
L

C → L C → C,
which proves (1).
Assume that (1) and (2) are satisfied. Then L is accessible, so that L C is apresentable ∞-category. Assertion (3) follows from Proposition 5.5.7.2. Let
D ∈ L C. Since C is κ-compactly generated, D can be written as the colimitof a κ-filtered diagram p : K → C taking values in the κ-compact objects of
C. Then D ' LD can be written as the colimit of L ◦ p, which takes values
among the κ-compact objects of L C. This proves (4). If D is a κ-compact
object of D, then we deduce that the identity map idD : D → D factors
through (L ◦ p)(k) for some vertex k ∈ K, which proves (5).
Corollary 5.5.7.4. Let C be a κ-compactly generated ∞-category and letn ≥ −2. Then
(1) The full subcategory τ≤n C is stable under κ-filtered colimits in C.
(2) The truncation functor τ≤n : C → C is κ-continuous.
(3) The truncation functor τ≤n carries compact objects of C to compact
objects of C≤n .
(4) The full subcategory τ≤n C is κ-compactly generated.
(5) An object C ∈ τ≤n C is compact (in τ≤n C) if and only if there exists a
compact object C 0 ∈ C such that C is a retract of τ≤n C 0 .
Proof. Corollary 5.5.7.3 shows that condition (1) implies (2), (3), (4), and(5). Consequently, it will suffice to prove that (1) is satisfied.
Let C be an object of C. We will show that C is n-truncated if and only
if the space MapC (D, C) is n-truncated for every κ-compact object D ∈ C.
The “only if” direction is obvious. For the converse, let FC : Cop → S be thefunctor represented by C and let C0 ⊆ C be the full subcategory of C spannedby those objects D such that FC (D) is n-truncated. Since FC preserves
limits, C0 is stable under colimits in C. If C0 contains every κ-compact objectof C, then C0 = C (since C is κ-compactly generated).

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503

Now suppose that D is a κ-compact object of C, let GD : C → S be
the functor corepresented by D, and let C(D) ⊆ C be the full subcategoryof C spannedby those objects C for which GD (C) is n-truncated. ThenTτ≤n C = D C(D). To complete the proof, it will suffice to show that each
C(D) is stable under κ-filtered colimits. Since GD is κ-continuous, it suffices
to observe that τ≤n S is stable under κ-filtered colimits in S.
Definition 5.5.7.5. If κ is a regular cardinal, we let PrRκ denote the subcatd∞ whose objects are κ-compactly generated ∞-categories andegory of Cat
whose morphisms are κ-continuous limit-preserving functors.
Proposition 5.5.7.6. The ∞-category PrRκ admits small limits, and theRdinclusion Prκ ⊆ Cat∞ preserves small limits.
Proof. In view of Theorem 5.5.3.18, the only nontrivial point is to verify that
if p : K → PrR
κ is a diagram of κ-compactly generated ∞-categories {Cα },d∞ is κ-compactly generated. In other words,
then the limit C = lim(p) in Cat←−
we must show that C is generated under colimits by its κ-compact objects.
For each vertex α of K, let
Cα o

Fα
Gα

/C

denote the corresponding adjunction. Lemma 6.3.3.7 implies that the identity functor idC can be obtained as the colimit of a diagram q : K →Fun(C, C), where q(α) ' Fα ◦ Gα . In particular, C is generated (under small
colimits) by the essential images of the functors Fα . Since each Cα is generated under colimits by κ-compact objects, and the functors Fα preserve
colimits and κ-compact objects (Proposition 5.5.7.2), we conclude that C isgenerated under colimits by its κ-compact objects, as desired.
Notation 5.5.7.7. Let κ be a regular cardinal. We let PrLκ denote the fulld∞ whose objects are κ-compactly generated ∞-categories
subcategory of Cat
and whose morphisms are functors which preserve small colimits and κcompact objects. In view of Proposition 5.5.7.2, the equivalence PrL 'op(PrR )op of Corollary 5.5.3.4 restricts to an equivalence PrLκ ' (PrRκ) .
Rex(κ)dd∞ whose objects are (not necLet Cat
denote the subcategory of Cat∞
essarily small) ∞-categories which admit κ-small colimits and whose morphisms are functors which preserve κ-small colimits, and let CatRex(κ)=∞
Rex(κ)dCat∩ Cat∞ .
∞Proposition 5.5.7.8. Let κ be a regular cardinal and let
Rex(κ)dθ : PrLκ → Cat∞be the nerve of the simplicial functor which associates to a κ-compactly generated ∞-category C the full subcategory Cκ ⊆ C spanned by the κ-compact
objects of C. Thenhighertopoi

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CHAPTER 5(1) The functor θ is fully faithful.
d∞(2) The essential image of θ consists precisely of those objects of Cat
which are essentially small and idempotent complete.
Proof. Combine Propositions 5.4.2.15 and 5.5.1.9.
Remark 5.5.7.9. If κ > ω, then Remark 5.3.1.10 shows that the hypothesisof idempotent completeness in (2) is superfluous.
The proof of Proposition 5.4.2.17 yields the following analogue:
Proposition 5.5.7.10. Let κ be a regular cardinal. The functor Indκ :Rex(κ)
Cat∞ → Accκ exhibits PrL. If κ > ω, thenκ as a localization of Cat∞
Rex(κ)
Indκ induces an equivalence of ∞-categories Cat∞→ PrLκ.
Proof. The only additional ingredient needed is the following observation:
if C is an ∞-category which admits κ-small colimits, then the idempotent
completion C0 of C also admits κ-small colimits. To prove this, we observethat C0 can be identified with the collection of κ-compact objects of Indκ (C)(Lemma 5.4.2.4). Since C admits all small colimits (Theorem 5.5.1.1), we
conclude that C0 admits κ-small colimits.
We conclude this section with a remark about the structure of the ∞.
category CatRex(κ)∞
Proposition 5.5.7.11. Let κ be a regular cardinal. Then the ∞-category
Rex(κ)⊆
admits small κ-filtered colimits and the inclusion Cat∞
CatRex(κ)∞
Cat∞ preserves small κ-filtered colimits.
be a
Proof. Let I be a small κ-filtered ∞-category and let p : I → CatRex(κ)∞
diagram. Let C be a colimit of the induced diagram I → Cat∞ . To complete
the proof we must prove the following:(i) The ∞-category C admits κ-small colimits.
(ii) For each I ∈ I, the associated functor p(I) → C preserves κ-small
colimits.
(iii) Let f : C → D be an arbitrary functor. If each of the compositionsp(I) → C → D preserves κ-small colimits, then f preserves κ-small
colimits.
Since I is κ-filtered, any κ-small diagram in C factors through one of themaps p(I) → C (Proposition 5.4.1.2). Thus (ii) ⇒ (i) and (ii) ⇒ (iii).
To prove (ii), we first use Proposition 5.3.1.18 to reduce to the case where
I ' N(A), where A is a κ-filtered partially ordered set. Using Proposition
4.2.4.4, we can reduce to the case where p is the nerve of a functor fromq : A → Set∆ . In view of Theorem 4.2.4.1, we can identify C with a homotopy
colimit of q. Since the collection of categorical equivalences is stable underhighertopoi

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505

filtered colimits, we can assume that C is actually the filtered colimit of afamily of ∞-categories {Cα }α∈A .
Let K be a κ-small simplicial set and let g α : K . → Cα be a colimit
diagram. We wish to show that the induced map g : K . → C is a colimit
diagram. Let g = g|K; we need to show that the map θ : Cg/ → Cg/ is atrivial Kan fibration. We observe that θ is a filtered colimit of maps θβ :(Cβ )gβ / → (Cβ )gβ / , where β ranges over the set {β ∈ A : β ≥ α}. Using thefact that each of the associated maps Cα → Cβ preserves κ-small colimits,
we conclude that each θβ is a trivial fibration, so that θ is a trivial fibration,
as desired.
5.5.8 Nonabelian Derived Categories
According to Corollary 4.2.3.11, we can analyze arbitrary colimits in an ∞category C in terms of finite colimits and filtered colimits. In particular, suppose that C admits finite colimits and that we construct a new ∞-category
Ind(C) by formally adjoining filtered colimits to C. Then Ind(C) admits all
small colimits (Theorem 5.5.1.1), and the Yoneda embedding C → Ind(C)preserves finite colimits (Proposition 5.3.5.14). Moreover, we can identify
Ind(C) with the ∞-category of functors Cop → S which carry finite colimits
in C to finite limits in S. In this section, we will introduce a variation on
the same theme. Instead of assuming C admits all finite colimits, we willassume only that C admits finite coproducts. We will construct a coproductpreserving embedding of C into a larger ∞-category PΣ (C) which admits all
small colimits. Moreover, we can characterize PΣ (C) as the ∞-category obtained from C by formally adjoining colimits of sifted diagrams (Proposition
5.5.8.15).
Our first goal in this section is to introduce the notion of a sifted simplicial
set. We begin with a bit of motivation. Let C denote the (ordinary) categoryof groups. Then C admits arbitrary colimits. However, colimits of diagramsin C can be very difficult to analyze even if the diagram itself is quite simple.
For example, the coproduct of a pair of groups G and H is the amalgamatedproduct G ? H. The group G ? H is typically very complicated even when
G and H are not. For example, the amalgamated product Z/2Z ? Z/3Z isisomorphic to the arithmetic
group PSL2 (Z). In general, G?H is much larger`
than the coproduct G H of the underlying sets of G and H. In other words,
the forgetful functor U : C → Set does not preserve coproducts. However, U
does preserve some colimits: for example, the colimit of a sequence of groups
G0 → G1 → · · ·can be obtained by taking the colimit of the underlying sets and equipping
the result with an appropriate group structure.
The forgetful functor U from groups to sets preserves another important
type of colimit: namely, the formation of quotients by equivalence relations.
If G is a group, then a subgroup R ⊆ G × G is an equivalence relation on G
if and only if there exists a normal subgroup H ⊆ G such that R = {(g, g 0 ) :highertopoi

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g −1 g 0 ∈ H}. In this case, the set of R-equivalence classes in G is in bijection
with the quotient G/H, which inherits a group structure from G. In otherwords, the quotient of G by the equivalence relation R can be computed
either in the category of groups or in the category of sets; the result is thesame.
Each of the examples given above admits a generalization: the colimit of asequence is a special case of a filtered colimit, and the quotient by an equivalence relation is a special case of a reflexive coequalizer. The forgetful functor
C → Set preserves filtered colimits and reflexive coequalizers; moreover, thesame is true if we replace the category of groups by any other category of setswith some sort of finitary algebraic structure (for example, abelian groups
or commutative rings). The following definition, which is taken from [67], isan attempt to axiomatize the essence of the situation:
Definition 5.5.8.1 ([67]). A simplicial set K is sifted if it satisfies thefollowing conditions:(1) The simplicial set K is nonempty.
(2) The diagonal map K → K × K is cofinal.
Warning 5.5.8.2. In [67], Rosicki uses the term “homotopy sifted” to describe the analogue of Definition 5.5.8.1 for simplicial categories and reserves
the term “sifted” for analogous notion in the setting of ordinary categories.
There is some danger of confusion with our terminology: if C is an ordinary
category and N(C) is sifted (in the sense of Definition 5.5.8.1), then C issifted in the sense of [67]. However, the converse is false in general.
Example 5.5.8.3. Every filtered ∞-category is sifted (this follows from
Proposition 5.3.1.22).
Lemma 5.5.8.4. The simplicial set N(∆)op is sifted.
Proof. Since N(∆)op is clearly nonempty, it will suffice to show that the
diagonal map N(∆)op → N(∆)op ×N(∆)op is cofinal. According to Theorem
4.1.3.1, this is equivalent to the assertion that for every object ([m], [n]) ∈∆ × ∆, the category
C = ∆/[m] ×∆ ∆/[n]has weakly contractible nerve. Let C0 be the full subcategory of C spanned bythose objects which correspond to monomorphisms of partially ordered sets
J → [m] × [n]. The inclusion of C0 into C has a left adjoint, so the inclusion
N(C0 ) ⊆ N(C) is a weak homotopy equivalence. It will therefore suffice toshow that N(C0 ) is weakly contractible. We now observe that N(C0 ) canbe identified with the barycentric subdivision of ∆m × ∆n and is therefore
weakly homotopy equivalent to ∆m × ∆n and so weakly contractible.
Remark 5.5.8.5. The formation of the geometric realizations of simplicial
objects should be thought of as the ∞-categorical analogue of the formationof reflexive coequalizers.

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIESOur next pair of results captures some of the essential features of thetheory of sifted simplicial sets:
Proposition 5.5.8.6. Let K be a sifted simplicial set, let C, D, and E be∞-categories which admit K-indexed colimits, and let f : C × D → E be amap which preserves K-indexed colimits separately in each variable. Then fpreserves K-indexed colimits.
Proof. Let p : K → C and q : K → D be diagrams indexed by a small
simplicial set K and let δ : K → K × K be the diagonal map. Using thefact that f preserves K-indexed colimits separately in each variable andLemma 5.5.2.3, we conclude that lim(f ◦ (p × q)) is a colimit for the diagram−→f ◦ (p × q) ◦ δ. Consequently, f preserves K-indexed colimits provided thatthe diagonal δ is cofinal. We conclude by invoking the assumption that K is
sifted.
Proposition 5.5.8.7. Let K be a sifted simplicial set. Then K is weakly
contractible.
Proof. Choose a vertex x in K. According to Whitehead’s theorem, it will
suffice to show that for each n ≥ 0, the homotopy set πn (|K|, x) consists ofa single element. Let δ : K → K × K be the diagonal map. Since δ is cofinal,
Proposition 4.1.1.3 implies that the induced map
πn (|K|, x) → πn (|K × K|, δ(x)) ' πn (|K|, x) × πn (|K|, x)is bijective. Since πn (|K|, x) is nonempty, we conclude that it is a singleton.
We now return to the problem introduced in the beginning of this section.
Definition 5.5.8.8. Let C be a small ∞-category which admits finite coproducts. We let PΣ (C) denote the full subcategory of P(C) spanned by thosefunctors Cop → S which preserve finite products.
Remark 5.5.8.9. The ∞-categories of the form PΣ (C) have been studied
in [67], where they are called homotopy varieties. Many of the results provenbelow can also be found in [67].
Proposition 5.5.8.10. Let C be a small ∞-category which admits finite
coproducts. Then
(1) The ∞-category PΣ (C) is an accessible localization of P(C).
(2) The Yoneda embedding j : C → P(C) factors through PΣ (C). Moreover,j carries finite coproducts in C to finite coproducts in PΣ (C).
(3) Let D be a presentable ∞-category and let
P(C) oFG/D

be a pair of adjoint functors. Then G factors through PΣ (C) if and only
if f = F ◦ j : C → D preserves finite coproducts.

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CHAPTER 5(4) The full subcategory PΣ (C) ⊆ P(C) is stable under sifted colimits.
(5) Let L : P(C) → PΣ (C) be a left adjoint to the inclusion. Then L preserves sifted colimits (when regarded as a functor from P(C) to itself).
(6) The ∞-category PΣ (C) is compactly generated.
Before giving the proof, we need a preliminary result concerning the interactions between products and sifted colimits.
Lemma 5.5.8.11. Let K be a sifted simplicial set. Let X be an ∞-category
which admits finite products and K-indexed colimits and suppose that theformation of products in X preserves K-indexed colimits separately in eachvariable. Then the colimit functor lim : Fun(K, X) → X preserves finite−→
products.
Remark 5.5.8.12. The hypotheses of Lemma 5.5.8.11 are satisfied when X
is the ∞-category S of spaces: see Lemma 6.1.3.14. More generally, Lemma
5.5.8.11 applies whenever the ∞-category X is an ∞-topos (see Definition
6.1.0.2).
Proof. Since the simplicial set K is weakly contractible (Proposition 5.5.8.7),
Corollary 4.4.4.9 implies that the functor lim preserves final objects. To−→
complete the proof, it will suffice to show that the functor lim preserves−→
pairwise products. Let X and Y be objects of Fun(K, X). We wish to provethat the canonical map
lim(X × Y ) → lim(X) × lim(Y )−→−→−→is an equivalence. In other words, we must show that the formation of products commutes with K-indexed colimits, which follows immediately by applying Proposition 5.5.8.6 to the Cartesian product functor X × X → X.
Proof of Proposition 5.5.8.10. Assertion (1) is an immediate consequence ofLemmas 5.5.4.17, 5.5.4.18, and 5.5.4.19. To prove (2), it will suffice to showthat for every representable functor e : PΣ (C)op → S, the compositionj ope

Cop → PΣ (C)op → S
preserves finite products (Proposition 5.1.3.2). This is obvious since the composition can be identified with the object of PΣ (C) ⊆ Fun(Cop , S) representing e.
We next prove (3). We note that f preserves finite coproducts if and only
if, for every object D ∈ D, the composition
f op

e

D
Cop → Dop →S

preserves finite products, where eD denotes the functor represented by D.
This composition can be identified with G(D), so that f preserves finite
coproducts if and only if G factors through PΣ (C).

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Assertion (4) is an immediate consequence of Lemma 5.5.8.11 and Remark
5.5.8.12, and (5) follows formally from (4). To prove (6), we first observe that
P(C) is compactly generated (Proposition 5.3.5.12). Let E ⊆ P(C) be the fullsubcategory spanned by the compact objects and let L : P(C) → PΣ (C)be a localization functor. Since E generates PΣ (C) under filtered colimits,
L(D) generates PΣ (C) under filtered colimits. Consequently, it will suffice
to show that for each E ∈ E, the object LE ∈ PΣ (C) is compact. Let
f : PΣ (C) → S be the functor corepresented by LE and let f 0 : P(C) → S
be the functor corepresented by E. Then the map E → LE induces anequivalence f → f 0 | PΣ (C). Since f 0 is continuous and PΣ (C) is stable underfiltered colimits in P(C), we conclude that f is continuous, so that LE is acompact object of PΣ (C), as desired.
Our next goal is to prove a converse to part (4) of Proposition 5.5.8.10.
Namely, we will show that PΣ (C) is generated by the essential image of
the Yoneda embedding under sifted colimits. In fact, we will need to use
only special types of sifted colimits: namely, filtered colimits and geometricrealizations (Lemma 5.5.8.14). The proof is based on the following technicalresult:Lemma 5.5.8.13. Let C be a small ∞-category and let X be an object ofP(C). Then there exists a simplicial object Y• : N(∆)op → P(C) with thefollowing properties:(1) The colimit of Y• is equivalent to X.
(2) For each n ≥ 0, the object Yn ∈ P(C) is equivalent to a small coproductof objects lying in the image of the Yoneda embedding j : C → P(C).
We will defer the proof until the end of this section.
Lemma 5.5.8.14. Let C be a small ∞-category which admits finite coproducts and let X ∈ P(C). The following conditions are equivalent:(1) The object X belongs to PΣ (C).
(2) There exists a simplicial object U• : N(∆)op → Ind(C) whose colimitin P(C) is X.
Proof. The full subcategory PΣ (C) contains the essential image of the Yoneda
embedding and is stable under filtered colimits and geometric realizations(Proposition 5.5.8.10); thus (2) ⇒ (1). We will prove that (1) ⇒ (2).
We first choose a simplicial object Y• of P(C) which satisfies the conclusionsof Lemma 5.5.8.13. Let L be a left adjoint to the inclusion PΣ (C) ⊆ P(C).
Since X is a colimit of Y• , LX ' X is a colimit of LY• (part (5) of Proposition
5.5.8.10). It will therefore suffice to prove that each LYn belongs` to Ind(C).
By hypothesis, each Yn can be written as a small coproduct α∈A j(Cα ),where j : C → P(C) denotes the Yoneda embedding. Using the results of§4.2.3, we see that Yn can also be obtained as a filtered colimit of coproducts

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α∈A0 j(Cα ), where A0 ranges over the finite subsets of A. Since L preserves
filtered colimits (Proposition 5.5.8.10), it will suffice to show that each of theobjectsaj(Cα ))
L(
α∈A0belongs to Ind(C). We` now invoke part (2) of Proposition 5.5.8.10 to identify
this object with j( α∈A0 Cα ).
Proposition 5.5.8.15. Let C be a small ∞-category which admits finite
coproducts and let D be an ∞-category which admits filtered colimits and geometric realizations. Let FunΣ (PΣ (C), D) denote the full subcategory spannedby those functors PΣ (C) → D which preserve filtered colimits and geometricrealizations. Then
(1) Composition with the Yoneda embedding j : C → PΣ (C) induces anequivalence of categories
θ : FunΣ (PΣ (C), D) → Fun(C, D).
(2) Any functor g ∈ FunΣ (PΣ (C), D) preserves sifted colimits.
(3) Assume that D admits finite coproducts. A functor g ∈ FunΣ (PΣ (C), D)
preserves small colimits if and only if g ◦ j preserves finite coproducts.
Proof. Lemma 5.5.8.14 and Proposition 5.5.8.10 imply that PΣ (C) is thesmallest full subcategory of P(C) which is closed under filtered colimits, isclosed under geometric realizations, and contains the essential image of the
Yoneda embedding. Consequently, assertion (1) follows from Remark 5.3.5.9
and Proposition 4.3.2.15.
We now prove (2). Let g ∈ FunΣ (PΣ (C), D); we wish to show that g
preserves sifted colimits. It will suffice to show that for every representablefunctor e : D → Sop , the composition e ◦ g preserves sifted colimits. In otherwords, we may replace D by Sop and thereby reduce to the case where D
itself admits sifted colimits. Let Fun0Σ (PΣ (C), D) denote the full subcategoryof FunΣ (PΣ (C), D) spanned by those functors which preserve sifted colimits.
Since PΣ (C) is also the smallest full subcategory of P(C) which contains theessential image of the Yoneda embedding and is stable under sifted colimits,
Remark 5.3.5.9 implies that θ induces an equivalence
Fun0Σ (PΣ (C), D) → Fun(C, D).
Combining this observation with (1), we deduce that the inclusion
Fun0Σ (PΣ (C), D) ⊆ FunΣ (PΣ (C), D)
is an equivalence of ∞-categories and therefore an equality.
The “only if” direction of (3) is immediate since the Yoneda embeddingj : C → PΣ (C) preserves finite coproducts (Proposition 5.5.8.10). To provethe converse, we first apply Lemma 5.3.5.7 to reduce to the case where D isa full subcategory of an ∞-category D0 with the following properties:

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511(i) The ∞-category D0 admits small colimits.
(ii) A small diagram K . → D is a colimit if and only if the induced diagram
K . → D0 is a colimit.
Let C0 denote the essential image of the Yoneda embedding j : C → P(C).
Using Lemma 5.1.5.5, we conclude that there exists a functor G : P(C) → D0
which is a left Kan extension of G| C0 = g| C0 and that G preserves small
colimits. Let G0 = G| PΣ (C). Then G0 is a left Kan extension of g| C0 , sothere is a canonical natural transformation G0 → g. Let C00 denote thefull subcategory of PΣ (C) spanned by those objects C for which the map
G0 (C) → g(C) is an equivalence. Then C00 contains C0 and is stable under filtered colimits and geometric realizations and therefore contains all of PΣ (C).
We may therefore replace g by G0 and thereby assume that G| PΣ (C) = g.
Since G ◦ j = g ◦ j preserves finite coproducts, the right adjoint to G factors through PΣ (C) (Proposition 5.5.8.10), so that G is equivalent to thecomposition
L

G0

P(C) → PΣ (C) → D0
for some colimit-preserving functor G0 : PΣ (C) → D0 . Restricting to thesubcategory PΣ (C) ⊆ P(C), we deduce that G0 is equivalent to g, so that g
preserves small colimits, as desired.
Remark 5.5.8.16. Let C be a small ∞-category which admits finite coproducts. It follows from Proposition 5.5.8.15 that we can identify PΣ (C)0with PK
K (C) in each of the following three cases (for an explanation of thisnotation, we refer the reader to §5.3.6):(1) The collection K is empty, and the collection K0 consists of all small
filtered simplicial sets together with N(∆)op .
(2) The collection K is empty, and the collection K0 consists of all small
sifted simplicial sets.
(3) The collection K consists of all finite discrete simplicial sets, and thecollection K0 consists of all small simplicial sets.
Corollary 5.5.8.17. Let f : C → D be a functor between ∞-categories.
Assume that C admits small colimits. Then f preserves sifted colimits if andonly if f preserves filtered colimits and geometric realizations.
Proof. The “only if” direction is clear. For the converse, suppose that f
preserves filtered colimits and geometric realizations. Let I be a small sifted∞-category and p : I. → C a colimit diagram; we wish to prove that f ◦ p
is also a colimit diagram. Let p = p| I. Let J ⊆ P(I) denote a small fullsubcategory which contains the essential image of the Yoneda embeddingj : I → P(I) and is closed under finite coproducts. It follows from Remark
5.3.5.9 that the functor p is homotopic to a composition q◦j, where q : J → C

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is a functor which preserves finite coproducts. Proposition 5.5.8.15 implies
that q is homotopic to a composition
j0

q0

J → PΣ (J) → C,
where j 0 denotes the Yoneda embedding and q 0 preserves small colimits. The
composition f ◦ q 0 preserves filtered colimits and geometric realization andtherefore preserves sifted colimits (Proposition 5.5.8.15).
Let p0 : I. → PΣ (J) be a colimit of the diagram j 0 ◦ j. Since q 0 preserves
colimits, the composition q 0 ◦ p0 is a colimit of q 0 ◦ j 0 ◦ j ' p and is therefore
equivalent to p. Consequently, it will suffice to show that f ◦ q 0 ◦ p0 is acolimit diagram. Since I is sifted, we need only verify that f ◦ q 0 preserves
sifted colimits. By Proposition 5.5.8.15, it will suffice to show that f ◦ q 0
preserves filtered colimits and geometric realizations. Since q 0 preserves all
colimits, this follows from our assumption that f preserves filtered colimits
and geometric realizations.
In the situation of Proposition 5.5.8.15, every functor f : C → D extends(up to homotopy) to a functor F : PΣ (C) → D, which preserves sifted
colimits. We will sometimes refer to F as the left derived functor of f . In
§5.5.9 we will explain the connection of this notion of derived functor withthe more classical definition provided by Quillen’s theory of homotopicalalgebra.
Our next goal is to characterize those ∞-categories which have the formPΣ (C).
Definition 5.5.8.18. Let C be an ∞-category which admits geometric realizations of simplicial objects. We will say that an object P ∈ C is projective if
the functor C → S corepresented by P commutes with geometric realizations.
Remark 5.5.8.19. Let C be an ∞-category which admits geometric realizations of simplicial objects. Then the collection of projective objects of C isstable under all finite coproducts which exist in C. This follows immediatelyfrom Lemma 5.5.8.11 and Remark 5.5.8.12.
Remark 5.5.8.20. Let C be an ∞-category which admits small colimits andlet X be an object of C. Then X is compact and projective if and only if X
corepresents a functor C → Set∆ which preserves sifted colimits. The “onlyif” direction is obvious, and the converse follows from Corollary 5.5.8.17.
Example 5.5.8.21. Let A be an abelian category. Then an object P ∈A is projective in the sense of classical homological algebra (that is, thefunctor HomA (P, •) is exact) if and only if P corepresents a functor A → Set
which commutes with geometric realizations of simplicial objects. This is
not equivalent to the condition of Definition 5.5.8.18 since the fully faithfulembedding Set → S does not preserve geometric realizations. However, it isequivalent to the requirement that P be a projective object (in the senseof Definition 5.5.8.18) in the ∞-category underlying the homotopy theory

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513

of simplicial objects of A (equivalently, the theory of nonpositively gradedchain complexes with values in A; we will discuss this example in greater
detail in [51]).
Proposition 5.5.8.22. Let C be a small ∞-category which admits finite
coproducts, D an ∞-category which admits filtered colimits and geometricrealizations, and F : PΣ (C) → D a left derived functor of f = F ◦ j : C → D,
where j : C → PΣ (C) denotes the Yoneda embedding. Consider the following
conditions:(i) The functor f is fully faithful.
(ii) The essential image of f consists of compact projective objects of D.
(iii) The ∞-category D is generated by the essential image of f under filtered colimits and geometric realizations.
If (i) and (ii) are satisfied, then F is fully faithful. Moreover, F is an equivalence if and only if (i), (ii), and (iii) are satisfied.
Proof. If F is an equivalence of ∞-categories, then (i) follows from Proposition 5.1.3.1 and (iii) from Lemma 5.5.8.14. To prove (ii), it suffices to showthat for each C ∈ C, the functor e : PΣ (C) → S corepresented by C preserves filtered colimits and geometric realizations. We can identify e withthe compositione0

e00

PΣ (C) ⊆ P(C) → S,
where e00 denotes evaluation at C. It now suffices to observe that e0 and e00
preserve filtered colimits and geometric realizations (Lemma 5.5.8.14 andProposition 5.1.2.2).
For the converse, let us suppose that (i) and (ii) are satisfied. We willshow that F is fully faithful. First fix an object C ∈ C and let P0Σ (C) be thefull subcategory of PΣ (C) spanned by those objects M for which the map
MapPΣ (C) (j(C), M ) → MapD (f (C), F (M ))is an equivalence. Condition (i) implies that P0Σ (C) contains the essential image of j, and condition (ii) implies that P0Σ (C) is stable under filtered colimits
and geometric realizations. Lemma 5.5.8.14 now implies that P0Σ (C) = PΣ (C).
We now define P00Σ (C) to be the full subcategory of PΣ (C) spanned by thoseobjects M such that for all N ∈ PΣ (C), the map
MapPΣ (C) (M, N ) → MapD (F (M ), F (N ))is a homotopy equivalence. The above argument shows that P00Σ (C) contains
the essential image of j. Since F preserves filtered colimits and geometric realizations, P00Σ (C) is stable under filtered colimits and geometric realizations.
Applying Lemma 5.5.8.14, we conclude that P00Σ (C) = PΣ (C); this proves that
F is fully faithful.
If F is fully faithful, then the essential image of F contains f (C) and isstable under filtered colimits and geometric realizations. If (iii) is satisfied,
it follows that F is an equivalence of ∞-categories.

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Definition 5.5.8.23. Let C be an ∞-category which admits small colimits
and let S be a collection of objects of C. We will say that S is a set of compact
projective generators for C if the following conditions are satisfied:(1) Each element of S is a compact projective object of C.
(2) The full subcategory of C spanned by the elements of S is essentiallysmall.
(3) The set S generates C under small colimits.
We will say that C is projectively generated if there exists a set S of compact
projective generators for C.
Example 5.5.8.24. The ∞-category S of spaces is projectively generated.
The compact projective objects of S are precisely those spaces which arehomotopy equivalent to finite sets (endowed with the discrete topology).
Proposition 5.5.8.25. Let C be an ∞-category which admits small colimits
and let S be a set of compact projective generators for C. Then
(1) Let C0 ⊆ C be the full subcategory spanned by finite coproducts of theobjects S, let D ⊆ C0 be a minimal model for C0 , and let F : PΣ (D) →C be a left derived functor of the inclusion. Then F is an equivalenceof ∞-categories. In particular, C is a compactly generated presentable∞-category.
(2) Let C ∈ C be an object. The following conditions are equivalent:(i) The object C is compact and projective.
(ii) The functor e : C → b
S corepresented by C preserves sifted colimits.
(iii) There exists an object C 0 ∈ C0 such that C is a retract of C 0 .
Proof. Remark 5.5.8.19 implies that C0 consists of compact projective objects of C. Assertion (1) now follows immediately from Proposition 5.5.8.22.
We now prove (2). The implications (iii) ⇒ (i) and (ii) ⇒ (i) are obvious. To complete the proof, we will show that (i) ⇒ (iii). Using (1), weare free to assume C = PΣ (D). Let C ∈ C be a compact projective object.
Using Lemma 5.5.8.14, we conclude that there exists a simplicial object X•
of Ind(D) and an equivalence C ' |X• |. Since C is projective, we deducethat MapC (C, C) is equivalent to the geometric realization of the simplicial
space MapC (C, X• ). In particular, idC ∈ MapC (C, C) is homotopic to theimage of some map f : C → X0 . Using our assumption that C is compact,
we conclude that f factors as a composition
f0

C → j(D) → X0 ,where j : D → Ind(D) denotes the Yoneda embedding. It follows that C isa retract of j(D) in C, as desired.

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Remark 5.5.8.26. Let C be a small ∞-category which admits finite coproducts. Since the truncation functor τ≤n : S → S preserves finite products, it
induces a map τ : PΣ (C) → PΣ (C), which is easily seen to be a localization
functor. The essential image of τ consists of those functors F ∈ PΣ (C) whichtake n-truncated values. We claim that these are precisely the n-truncatedobjects of PΣ (C). Consequently, we can identify τ with the n-truncation
functor on PΣ (C).
One direction is clear: if F ∈ PΣ (C) is n-truncated, then for each C ∈ C thespace MapPΣ (C) (j(C), F ) ' F (C) must be n-truncated. Conversely, supposethat F : Cop → S takes n-truncated values. We wish to prove that the
space MapPΣ (C) (F 0 , F ) is n-truncated for each F 0 ∈ PΣ (C). The collectionof all objects F 0 which satisfy this condition is stable under small colimits inPΣ (C) and contains the essential image of the Yoneda embedding. It therefore
contains the entirety of PΣ (C), as desired.
We conclude this section by giving the proof of Lemma 5.5.8.13. Our
argument uses some concepts and results from Chapter 6 and may be omitted
at first reading.
Proof of Lemma 5.5.8.13. For n ≥ 0, let ∆≤n denote the full subcategory of∆ spanned by the objects {[k]}k≤n . We will construct a compatible sequenceof functors fn : N(∆≤n )op → P(C)/X with the following properties:(A) For n ≥ 0, let Ln denote a colimit of the composite diagram
fn−1

N(∆≤n−1 )op ×N(∆)op N(∆[n]/ )op → N(∆≤n−1 )op → P(C)/X → P(C)(the nth latching object). Then there exists an object Zn ∈ P(C), which
is a small coproduct of objects in the essential image of the Yoneda
embedding C → P(C), and a map Zn → fn ([n]), which together` withthe canonical map Ln → fn ([n]) determines an equivalence Ln Zn 'fn ([n]).
(B) For n ≥ 0, let M n denote the limit of the diagram
fn−1

N(∆≤n−1 )op ×N(∆)op N(∆/[n] )op → N(∆≤n−1 )op → P(C)/X(the nth matching object) and let Mn denote its image in P(C). Thenthe canonical map fn ([n]) → Mn is an effective epimorphism in P(C)(see §6.2.3).
The construction of the functors fn proceeds by induction on n, the casen < 0 being trivial. For n ≥ 0, we invoke Remark A.2.9.16: to extend fn−1
to a functor fn satisfying (A) and (B), it suffices to produce an object` Zn anda morphism ψ : Zn → Mn in P(C), such that the coproduct Ln Zn → Mn
is an effective epimorphism. This is satisfied in particular if ψ itself is aneffective epimorphism.
The maps fn together determine a simplicial object Y • of P(C)/X , which
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lim Y• → X. Assumption (B) guarantees that θ is a hypercovering of X (see−→§6.5.3), so that the map θ is ∞-connective (Lemma 6.5.3.11). The ∞-topos
P(C) has enough points (given by evaluation at objects of C) and is therefore
hypercomplete (Remark 6.5.4.7). It follows that θ is an equivalence. We now
complete` the proof by observing that for n ≥ 0, we have an equivalence
Yn ' [n]→[k] Zk where the coproduct is taken over all surjective maps oflinearly ordered sets [n] → [k], so that Yn is itself a small coproduct of objects
lying in the essential image of the Yoneda embedding j : C → P(C).
5.5.9 Quillen’s Model for PΣ (C)Let C be a small category which admits finite products. Then N(C)op is
an ∞-category which admits finite coproducts. In §5.5.8, we studied the∞-category PΣ (N(C)op ), which we can view as the full subcategory of thepresheaf ∞-category Fun(N(C), S) spanned by those functors which preserve
finite products. According to Proposition 4.2.4.4, Fun(N(C), S) can be identified with the ∞-category underlying the simplicial model category of diagrams SetC
∆ (which we will endow with the projective model structure described in §A.3.2). It follows that every functor f : N(C) → S is equivalent
to the (simplicial) nerve of a functor F : C → Kan. Moreover, f belongs
to PΣ (N(C)op ) if and only if the functor F is weakly product-preserving inthe sense that for any finite collection of objects {Ci ∈ C}1≤i≤n , the naturalmap
F (C1 × · · · × Cn ) → F (C1 ) × · · · × F (Cn )is a homotopy equivalence of Kan complexes. Our goal in this section isto prove a refinement of Proposition 4.2.4.4: if f preserves finite products,
then it is possible to arrange that F preserves finite products (up to isomorphism rather than up to homotopy equivalence). This result is mostnaturally phrased as an equivalence between model categories (Proposition
5.5.9.2) and is due to Bergner (see [9]). We begin by recalling the following
result of Quillen (for a proof, we refer the reader to [64]):
Proposition 5.5.9.1 (Quillen). Let C be a category which admits finite
products and let A denote the category of functors F : C → Set∆ which
preserve finite products. Then A admits a simplicial model structure which
may be described as follows:(W ) A natural transformation α : F → F 0 of functors is a weak equivalence in A if and only if α(C) : F (C) → F 0 (C) is a weak homotopy
equivalence of simplicial sets for every C ∈ C.
(F ) A natural transformation α : F → F 0 of functors is a fibration in A if
and only if α(C) : F (C) → F 0 (C) is a Kan fibration of simplicial setsfor every C ∈ C.
Suppose that C and A are as in the statement of Proposition 5.5.9.1. Then
we may regard A as a full subcategory of the category SetC
∆ of all functors

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from C to Set∆ , which we regard as endowed with the projective model
structure (so that fibrations and weak equivalences are given pointwise).
The inclusion G : A ⊆ SetC
∆ preserves fibrations and trivial fibrations andtherefore determines a Quillen adjunction
SetC
∆o

F

/A.

G

(A more explicit description of the adjoint functor F will be given below.)
Our goal in this section is to prove the following result:
Proposition 5.5.9.2 (Bergner). Let C be a small category which admits
finite products and let
SetC
∆o

F

/A

G

be as above. Then the right derived functor
RG : hA → hSetC∆
is fully faithful, and an object f ∈ hSetC∆ belongs to the essential image of RG
if and only if f preserves finite products up to weak homotopy equivalence.
Corollary 5.5.9.3. Let C be a small category which admits finite products
and let A be as in Proposition 5.5.9.2. Then the natural map N(A◦ ) →PΣ (N(C)op ) is an equivalence of ∞-categories.
The proof of Proposition 5.5.9.2 is somewhat technical and will occupy
the rest of this section. We begin by introducing some preliminaries.
Notation 5.5.9.4. Let C be a small category. We define a pair of categories
Env(C) ⊆ Env+ (C) as follows:(i) An object of Env+ (C) is a pair C = (J, {Cj }j∈J ), where J is a finite
set and each Cj is an object of C. The object C belongs to Env(C) if
and only if J is nonempty.
(ii) Given objects C = (J, {Cj }j∈J ) and C 0 = (J 0 , {Cj0 0 }j 0 ∈J 0 ) of Env+ (C),a morphism C → C 0 consists of the following data:(a) A map f : J 0 → J of finite sets.
(b) For each j 0 ∈ J 0 , a morphism Cf (j 0 ) → Cj0 0 in the category C.
Such a morphism belongs to Env(C) if and only if both J and J 0 are
nonempty and f is surjective.
There is a fully faithful embedding functor θ : C → Env(C) given by
C 7→ (∗, {C}). We can view Env+ (C) as the category obtained from C by
freely adjoining finite products. In particular, if C admits finite products,
then θ admits aQ(product-preserving) left inverse φ+
C given by the formula(J, {Cj }j∈J ) 7→ j∈J Cj . We let φC denote the restricton φ+
C | Env(C).

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CHAPTER 5+
Given a functor F ∈ SetC
∆ , we let E (F) ∈ Set∆
sition+Env (C)

denote the compo-

φ+Set

Env+ (F)Env+ (Set∆ ) →∆ Set∆
Y
(J, {Cj }j∈J ) 7→f (Cj ).

Env+ (C)→

Env(C)We let E(F) denote the restriction E + (F)| Env(C) ∈ Set∆.
C
If the category C admits finite products, then we let L, L+ : SetC
∆ → Set∆denote the compositions
SetC∆ → Set∆E

Env(C) (φC )!

→ SetC∆E+

Env+ (C) (φC )!

→ SetC∆,SetC∆ → Set∆+where (φC )! and (φ+C )! indicate left Kan extension functors. There is a
canonical isomorphism θ∗ ◦ E ' id which induces a natural transformation
α : id → L. Let β : L → L+ indicate the natural transformation induced by
the inclusion Env(C) ⊆ Env+ (C).
Remark 5.5.9.5. Let C be a small category. The functor E + : SetC
∆ →Env+ (C)Set∆

is fully faithful and has a left adjoint given by θ∗ .

We begin by constructing the left adjoint which appears in the statementof Proposition 5.5.9.2.
Lemma 5.5.9.6. Let C be a simplicial category which admits finite products
and let F ∈ SetC
∆ . Then(1) The object L+ (F) ∈ SetC
∆ is product-preserving.
(2) If F0 ∈ SetC
∆ is product-preserving, then composition with β ◦ α induces
an isomorphism of simplicial sets
MapSetC∆ (L+ (F), F0 ) → MapSetC∆ (F, F0 ).
Proof. Suppose we are given a finite collection of objects {C1 , . . . , Cn } in C
and letu : L+ (F)(C1 × · · · × Cn ) → L+ (F)(C1 ) × · · · × L+ (F)(Cn )be the product of the projection maps. We wish to show that u is an isomorphism of simplicial sets. We will give an explicit construction of an inverse tou. For C ∈ C, we let Env+ (C)/C denote the fiber product Env+ (C) ×D C/C .
For 1 ≤ i ≤ n, let Gi denote the restriction of E + (F) to Env+ (C)/Ci and let
YG:
Env+ (D)/Ci → Set∆
be the product of the functors Gi . We observe that L+ (F)(Ci ) ' lim(Gi ), so−→Q
that the product L+ (F)(Ci ) ' lim(G). We now observe that the formation−→of products in E+ (C) gives an identification of G with the composition
Y
E + (F)
Env+ (C)/Ci → Env+ (D)/C1 ×···×Cn → Set∆ .

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We thereby obtain a morphismv : lim(G) → lim(E + (F)| Env+ (D)/C1 ×···×Cn ' L+ (F)(C1 × · · · × Cn )).
−→−→It is not difficult to check that v is an inverse to u.
We observe that (2) is equivalent to the assertion that composition with
θ∗ induces an isomorphism
Map

Env+ (C)Set∆

0∗(E + (F), (φ+(F, F0 ).
C ) (F )) → MapSetC∆

0∗
Because G is product-preserving, there is a natural isomorphism (φ+C ) (F ) '0+
E (F ). The desired result now follows from Remark 5.5.9.5.
C
It follows that the functor L+ : SetC
∆ → Set∆ factors through A andcan be identified with a left adjoint to the inclusion A ⊆ SetC
∆ . In order toprove Proposition 5.5.9.2, we need to be able to compute the functor L+ .
We will do this in two steps: first, we show that (under mild hypotheses) thenatural transformation L → L+ is a weak equivalence. Second, we will seethat the colimit defining L is actually a homotopy colimit and therefore has
good properties. More precisely, we have the following pair of lemmas whose
proofs will be given at the end of this section.

Lemma 5.5.9.7. Let C be a small category which admits finite products andlet F ∈ SetC
∆ be a functor which carries the final object of C to a contractible
Kan complex K. Then the canonical map β : L(F) → L+ (F) is a weakequivalence in SetC∆.
Lemma 5.5.9.8. Let C be a small simplicial category. If F is a projectively cofibrant object of SetC
∆ , then E(F) is a projectively cofibrant object of
Env(C)Set∆.
We are now almost ready to give the proof of Proposition 5.5.9.2. The
essential step is contained in the following result:Lemma 5.5.9.9. Let C be a simplicial category that admits finite products
and let
SetC
∆o

F

/A

G

be as in the statement of Proposition 5.5.9.2. Then
(1) The functors F and G are Quillen adjoints.
(2) If F ∈ SetC
∆ is projectively cofibrant and weakly product-preserving,then the unit map F → (G ◦ F )(F) is a weak equivalence.
Proof. Assertion (1) is obvious since G preserves fibrations and trivial cofibrations. It follows that F preserves weak equivalences between projectively
cofibrant objects. Let K ∈ Set∆ denote the image under F of the final object of D. In proving (2), we are free to replace F by any weakly equivalentdiagram which is also projectively cofibrant. Choosing a fibrant replacement

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for F, we may suppose that K is a Kan complex. Since F is weakly productpreserving, K is contractible.
In view of Lemma 5.5.9.6, we can identify the composition G ◦ F with L+
and the unit map with the compositionαβ

F → L(F) → L+ (F).
Lemma 5.5.9.7 implies that β is a weak equivalence. Consequently, it will
suffice to show that α is a weak equivalence.
We recall the construction of α. Let θ : C → Env(C) be as in Notation
5.5.9.4, so that there is a canonical isomorphism F ' θ∗ E(F). This isomorphism induces a natural transformation α : θ! F → E(F). The functor α is
obtained from α by applying the functor (φC )! and identifying ((φC )! ◦ θ! )(F)with F. We observe that (φC )! preserves weak equivalences between projectively cofibrant objects. Since θ! preserves projective cofibrations, θ! F is
projectively cofibrant. Lemma 5.5.9.8 asserts that E(F) is projectively cofibrant. Consequently, it will suffice to prove that α is a weak equivalence in
Env(C)Set∆. Unwinding the definitions, this reduces to the condition that F be
weakly compatible with (nonempty) products.
Proof of Proposition 5.5.9.2. Lemma 5.5.9.9 shows that (F, G) is a Quillen
adjunction. To complete the proof, we must show:(i) The counit transformation LF ◦RG → id is an isomorphism of functors
from the homotopy category hA to itself.
(ii) The essential image of RG : hA → hSetC∆ consists precisely of thosefunctors which are weakly product-preserving.
We observe that G preserves weak equivalences, so we can identify RG
with G. Since G also detects weak equivalences, (i) will follow if we canshow that the induced transformation θ : G ◦ LF ◦ G → G is an isomorphismof functors from the homotopy category hA to itself. This transformation
has a right inverse given by composing the unit transformation id → G ◦ LF
with G. Consequently, (i) follows immediately from Lemma 5.5.9.9.
The image of G consists precisely of the product-preserving diagrams C →Set∆ ; it follows immediately that every diagram in the essential image of
G is weakly product-preserving. Lemma 5.5.9.9 implies the converse: every
weakly product-preserving functor belongs to the essential image of G. This
proves (ii).
Remark 5.5.9.10. Proposition 5.5.9.2 can be generalized to the situation
where C is a simplicial category which admits finite products. We leave thenecessary modifications to the reader.
It remains to prove Lemmas 5.5.9.8 and 5.5.9.7.
Proof of Lemma 5.5.9.8. For every object C ∈ C and every simplicial set
CK
K, we let FK
C ∈ Set∆ denote the functor given by the formula F C (D) =highertopoi

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521

MapC (C, D) × K. A cofibration K → K 0 induces a projective cofibration
K0
FK
C → F C . We will refer to a projective cofibration of this form as a generating projective cofibration.
The small object argument implies that if F ∈ SetC
∆ , then there is a transfinite sequence
F0 ⊆ F1 ⊆ · · · ⊆ Fαwith the following properties:(a) The functor F0 : D → Set∆ is constant, with value ∅.
S(b) If λ ≤ α is a limit ordinal, then Fλ = β<λ Fβ .
(c) For each β < α, the inclusion Fβ ⊆ Fβ+1 is a pushout of a generating
projective cofibration.
(d) The functor F is a retract of Fα .
The functor G 7→ E(G) preserves initial objects, filtered colimits, and retracts. Consequently, to show that E(F) is projectively cofibrant, it will
suffice to prove the following assertion:(∗) Suppose we are given a cofibration K → K 0 of simplicial sets and apushout diagram
FK
C

/G


0
FK
C


/ G0

0
in SetC∆ . If E(G) is projectively cofibrant, then E(G ) is projectively
cofibrant.

To prove this, we will need to analyze the structure of E(G0 ). Given an
object C 0 = (J, {Cj0 }j∈J ) of Env(C), we haveYaE(G0 )(C 0 ) =(G(Cj0 )(K 0 × MapC (C, Cj0 ))).
j∈J

K×MapC (C,Cj0 )Let σ : ∆n → E(G0 )(C 0 ) be a simplex and let Jσ ⊆ J be the collection of all
indices j for which the corresponding simplex σ(j) : ∆n → G0 (Cj0 ) does notfactor through G(Cj0 ). In this case, we can identify σ(j) with an n-simplex of
K 0 which does not belong to K. We will say that σ is of index ≤ k if the set{σ(j) : j ∈ Jσ } has cardinality ≤ k. Note that σ can be of index smaller thanthe cardinality of Jσ since it is possible for σ(j) = σ(j 0 ) ∈ MapSet∆ (∆n , K 0 )even if j 6= j 0 .
Let E(G0 )(k) (C 0 ) be the full simplicial subset of E(G0 )(C 0 ) spanned bythose simplices which are of index ≤ k. It is easy to see that that E(G0 )(k) (C 0 )highertopoi

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CHAPTER 5
Env(C)

depends functorially on C 0 , so we can view E(G0 )(k) as an object of Set∆
We observe that.

E(G) ' E(G0 )(0) ⊆ E(G0 )(1) ⊆ · · ·and that the union of this sequence is E(G0 ). Consequently, it will suffice
to prove that each of the inclusions E(G0 )(k−1) ⊆ E(G0 )(k) is a projective
cofibration.
kFirst, we need a bit of notation. Let us say that a simplex of K 0 is new if it0
consists of k distinct simplices of K , none of which belong to K. We will saykthat a simplex of K 0 is old if it is not new. The collection of old simplices of(k)
0k
K determines a simplicial subset which we will denote by K 0 . We define
a functor ψ : Env(C) → Env(C) by the formulaψ(J, {Cj0 }j∈J ) = (J ∪ {1, . . . , k}, {Cj0 }j∈J ∪ {C}{1...k} ).
Env(C)

Env(C)

Let ψ ∗ : Set∆
→ Set∆
be given by composition with ψ and let
Env(C)
Env(C)
ψ! : Set∆→ Set∆
be a left adjoint to ψ ∗ (a functor of left Kan∗
extension). Since ψ preserves projective fibrations and weak equivalences,
ψ! preserves projective cofibrations.
Env(C)
Recall that Set∆
is tensored over the category of simplicial sets: given
Env(C)
Env(C)
and a simplicial set A, we let M ⊗A ∈ Set∆
an object M ∈ Set∆
be defined by the formula (M ⊗A)(D0 ) = M(D0 ) × A. If M is projectively
cofibrant, then the operation M 7→ M ⊗A preserves cofibrations in A.
k
There is an obvious map E(G)⊗K 0 → ψ ∗ E(G0 )(k) which restricts to a map(k)E(G) ⊗ K 0 → ψ ∗ E(G0 )(k−1) . Passing to adjoints, we obtain a commutative
diagram)/ E(G0 )(k−1)

kψ! (E(G) ⊗ K 0 )/ E(G0 )(k) .

ψ! (E(G) ⊗ K 0(k)

An easy computation shows that this diagram is coCartesian. Since E(G) is
projectively cofibrant, the above remarks imply that the left vertical map is
a projective cofibration. It follows that the right vertical map is a projective
cofibration as well, which completes the proof.
The proof of Lemma 5.5.9.7 is somewhat more difficult and will require
some preliminaries.
Notation 5.5.9.11. Let M : Set → Set be the covariant functor whichassociates to each set S the collection of M(S) of nonempty finite subsets ofS. If K is a simplicial set, we let M(K) denote the composition of K with M,
so that an m-simplex of M(K) is a finite nonempty collection of m-simplices
of K.
Lemma 5.5.9.12. Let K be a finite simplicial set and let X ⊆ M(K . ) × ∆n
be a simplicial subset with the following properties:

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523(i) The projection X → ∆n is surjective.
(ii) If τ = (τ 0 , τ 00 ) : ∆m → M(K . )×∆n belongs to X and τ 0 ⊆ τ 0 as subsets
of HomSet∆ (∆m , K . ), then (τ 0 , τ 00 ) : ∆m → M(K . ) × ∆n belongs to
X.
Then X is weakly contractible.
Proof. Let X 0 ⊆ X be the simplicial subset spanned by those simplicesτ = (τ 0 , τ 00 ) : ∆m → M(K . ) × ∆n which factor through X and for whichτ 0 ⊆ HomSet∆ (∆m , K . ) includes the constant simplex at the cone point of
K . . Our first step is to show that X 0 is a deformation retract of X. Moreprecisely, we will construct a map
h : M(K . ) × ∆n × ∆1 → M(K . ) × ∆nwith the following properties:(a) The map h carries X × ∆1 into X and X 0 × ∆1 into X 0 .
(b) The restriction h| M(K . ) × ∆n × {0} is the identity map.
(c) The restriction h|X × {1} factors through X 0 .
The map h will be the product of a map h0 : M(K . ) × ∆1 → M(K . ) andthe identity map on ∆n . To define h0 , we consider an arbitrary simplex
τ : ∆m → M(K . ) × ∆1 corresponding to a subset S ⊆ HomSet∆ (∆m , K . )and a decomposition [m] = {0, . . . , i} ∪ {i + 1, . . . , m}. The subset h0 (τ ) ⊆HomSet∆ (∆m , K . ) is defined as follows: an arbitrary simplex σ : ∆m → K .
belongs to h0 (τ ) if there exists σ 0 ∈ S, i < j ≤ n such that σ 0 |∆{0,...,j−1} =σ|∆{0,...,j−1} , and σ|∆{j,...,m} is constant at the cone point of K . . It is easy
to check that h0 has the desired properties.
It remains to prove that X 0 is weakly contractible. At this point, it isconvenient to work in the setting of semisimplicial sets: that is, we will
ignore the degeneracy operations. Let X 00 be the semisimplicial subset ofM(K . ) × ∆n spanned by those maps τ = (τ 0 , τ 00 ) : ∆m → M(K . ) × ∆n forwhich τ 0 = HomSet∆ (∆m , K . ) (we observe that X 00 is not stable under thedegeneracy operators on M(K . ) × ∆n ). Assumptions (i) and (ii) guaranteethat X 00 ⊆ X 0 . Moreover, the projection X → ∆n induces an isomorphismof semisimplicial sets X 00 → ∆n . Consequently, it will suffice to prove that
X 00 is a deformation retract of X 0 .
The proof now proceeds by a variation on our earlier construction. Namely,
we will define a map of semisimplicial setsg : M(K . ) × ∆n × ∆1 → M(K . ) × ∆nwith the following properties:(a) The map g carries X 0 × ∆1 into X 0 and X 00 × ∆1 into X 00 .
(b) The restriction g|X 0 × {1} is the identity map.

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CHAPTER 5(c) The restriction g| M(K . ) × ∆n × {0} factors through X 0 .
As before, g is the product of a map g 0 : M(K . ) × ∆1 → M(K . ) with
the identity map on ∆n . To define g 0 , we consider an arbitrary simplex
τ : ∆m → M(K . ) × ∆1 , corresponding to a subset S ⊆ HomSet∆ (∆m , K . )and a decomposition [m] = {0, . . . , i} ∪ {i + 1, . . . , m}. We let g 0 (τ ) ⊆HomSet∆ (∆m , K . ) = S ∪ S 0 , where S 0 is the collection of all simplices
σ : ∆m → K . such that σ|∆{i+1,...,m} is the constant map at the conepoine of K . . It is readily checked that g 0 has the desired properties.
Lemma 5.5.9.13. Let K be a contractible Kan complex and let
X ⊆ M(K) × ∆n
be a simplicial subset with the following properties:(i) The projection X → ∆n is surjective.
(ii) If τ = (τ 0 , τ 00 ) : ∆m → M(K) × ∆n belongs to X and τ 0 ⊆ τ 0 as subsets
of HomSet∆ (∆m , K), then (τ 0 , τ 00 ) : ∆m → M(K) × ∆n belongs to X.
Then X is weakly contractible.
Proof. It will suffice to show that for every finite simplicial subset X 0 ⊆ X,
the inclusion of X 0 into X is weakly nullhomotopic. Enlarging X 0 if necessary,
we may assume that X 0 = (M(K 0 ) × ∆n ) ∩ X, where K 0 is a finite simplicial
subset of K. By further enlargement, we may suppose that the map X 0 → ∆n
is surjective. Since K is a contractible Kan complex, the inclusion K 0 ⊆ K.
.
extends to a map i : K 0 → K. Let X ⊆ M(K 0 ) × ∆n denote the inverse0
image of X. Then the inclusion X ⊆ X factors through X, and Lemma
5.5.9.12 implies that X is weakly contractible.
Proof of Lemma 5.5.9.7. Fix an object C ∈ C. The simplicial set L(F)(C)
can be described as follows:(∗) For every n ≥ 1, every map f : C1 × · · · × Cn → C in C, and every
collection of simplices {σi : ∆k → F(Ci )}, there is a simplex f ({σi }) :∆k → L(F)(C).
The simplices f ({σi }) satisfy relations which are determined by morphisms
in the simplicial category Env(C).
To every k-simplex τ : ∆k → L(F)(D) we can associate a nonempty finite
subset Sτ ⊆ HomSet∆ (∆k , K). If τ = f ({σi }), we assign the set of imagesof the simplices σi under the canonical maps F(Ci ) → F(1) = K. It is easy
to see that Sτ is independent of the representation f ({σi }) chosen for τ anddepends functorially on τ . Consequently, we obtain a map of simplicial sets
L(F)(C) → M(K). Moreover, this map has the following properties:(i) The product map β 0 : L(F)(C) → M(K) × L+ (F)(C) is a monomorphism of simplicial sets.

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PRESENTABLE AND ACCESSIBLE ∞-CATEGORIES(ii) If a k-simplex τ = (τ 0 , τ 00 ) : ∆k → M(K) → L+ (F)(C) belongs to
the image of β and τ 0 ⊆ τ 0 as finite subsets of HomSet∆ (∆k , K), then(τ 0 , τ 00 ) : ∆k → M(K) × L+ (F)(C) belongs to the image of β 0 .
We wish to show that β : L(F)(C) → L+ (F)(C) is a weak homotopy
equivalence. It will suffice to show that for every simplex ∆k → L+ (F)(C),
the fiber product L(F)(C)×L+ (F)(C) ∆k is weakly contractible. In view of (i),we can identify this fiber product with a simplicial subset X ⊆ ∆k × M(K).
The surjectivity of β and condition (ii) imply that X satisfies the hypothesesof Lemma 5.5.9.13, so that X is weakly contractible, as desired.
The model category A appearing in Proposition 5.5.9.1 is very well suited
to certain calculations, such as the formation of homotopy colimits of simplicial objects. The following result provides a precise formulation of thisidea:
Proposition 5.5.9.14. Let C be a category which admits finite products,C
the full subcategories spanned by the productand A ⊆ SetC
∆ , A ⊆ Set
preserving functors. Let F : ∆op → A be a simplicial object of A which we
can identify with a bisimplicial object F : ∆op × ∆op → A. Composition
with the diagonalF∆op → ∆op × ∆op → A
gives a simplicial object of A which we can identify with an object | F | ∈ A.
Then the homotopy colimit of F is canonically isomorphic to | F | in thehomotopy category hA.
The proof requires the following lemma:
Lemma 5.5.9.15. Let C be a category which admits finite products and let
A ⊆ SetC∆ be the full subcategory spanned by the product-preserving functors.
For every object C ∈ C, the evaluation map A → Set∆ preserves homotopy
colimits of simplicial objects.
Proof. In view of Corollary 5.5.9.3 and Theorem 4.2.4.1, it will suffice to showthat the evaluation functor PΣ (N(C)op ) → Set∆ preserves N(∆)op -indexed
colimits. This follows from Proposition 5.5.8.10 since N(∆)op is sifted (seeLemma 5.5.8.4).
Proof of Proposition 5.5.9.14. Since A is a combinatorial simplicial model
category, Corollary A.2.9.30 implies the existence of a canonical map
γ : hocolim F → | F |in the homotopy category hA; we wish to prove that γ is an isomorphism.
To prove this, it will suffice to show that the induced map
γC : (hocolim F)(C) → | F |(C)highertopoi

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is an isomorphism in the homotopy category of simplicial sets for each object
C ∈ C. This map fits into a commutative diagram
hocolim(F(C))
hocolim(F)(C)0
γC/ | F(C)|/ | F |(C).

The left vertical map is an isomorphism in the homotopy category of simplicial sets by Lemma 5.5.9.15, the right vertical map is evidently an iso0morphism, and the map γC
is an isomorphism in the homotopy category by
Example A.2.9.31; it follows that γC is also an isomorphism, as desired.

highertopoi

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Chapter Six∞-TopoiIn this chapter, we come to the main subject of this book: the theory of ∞topoi. Roughly speaking, an ∞-topos is an ∞-category which “looks like” the∞-category of spaces, just as an ordinary topos is a category which “looks
like” the category of sets. As in classical topos theory, there are various waysof making this precise. We will begin in §6.1 by reviewing several possible
definitions and proving that they are equivalent to one another.
The main result of §6.1 is Theorem 6.1.0.6, which asserts that an ∞category X is an ∞-topos if and only if X arises as an (accessible) left exact
localization of an ∞-category of presheaves. In §6.2, we consider the problem
of constructing left exact localizations. In classical topos theory, there is abijective correspondence between left exact localizations of a presheaf category P(C) and Grothendieck topologies on C. In the ∞-category categoricalcontext, one can again use Grothendieck topologies to construct examples ofleft exact localizations. Unfortunately, not every ∞-topos arises in this way.
Nevertheless, the construction of an ∞-category of sheaves Shv(C) from aGrothendieck topology on C is an extremely useful construction, which willplay an important role throughout Chapter 7.
In order to understand higher topos theory, we will need to consider ∞topoi not only individually but in relation to one another. In §6.3, we will
introduce the notion of a geometric morphism of ∞-topoi. The collection of
all ∞-topoi and geometric morphisms between them can be organized into
an ∞-category RTop. We will study the problem of constructing colimitsand (certain) limits in RTop. In the course of doing so, we will show thatthe class of ∞-topoi is stable under various categorical constructions.
One of our goals in this book is to apply ideas from higher category theory
to study more classical mathematical objects such as topological spaces orordinary topoi. In order to do so, it is convenient to work in a setting where
all of these objects can be considered on the same footing. In §6.4, we will
introduce the definition of an n-topos for all 0 ≤ n ≤ ∞. When n = ∞,this will reduce to the theory introduced in §6.1. The case n = 1 will recover classical topos theory, and the case n = 0 is almost equivalent to thetheory of topological spaces. We will study the theory of n-topoi and introduce constructions which allow us to pass between n-topoi and ∞-topoi. In
particular, we associate an ∞-topos Shv(X) to every topological space X,
which will be the primary object of study in Chapter 7.
There are several different ways of thinking about what an ∞-topos X is.
On the one hand, we can view X as a generalized topological space; on thehighertopoi

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other hand, we can think of X as an alternative universe in which we cando homotopy theory. In §6.5, we will reinforce the second point of view by
studying the internal homotopy theory of an ∞-topos X. Just as in classicalhomotopy theory, one can define homotopy groups, Postnikov towers, Eilenberg MacLane spaces, and so forth. In Chapter 7, we will bring together
these two points of view by showing that classical geometric properties of a
topological space X are reflected in the internal homotopy of the ∞-topos
Shv(X) of sheaves on X.
There are several papers on higher topos theory in the literature. The
papers [75] and [11] both discuss a notion of 2-topos (the second from anelementary point of view). However, the basic model for these 2-topoi isthe 2-category of (small) categories rather than the 2-category of (small)groupoids. Jardine ([41]) has exhibited a model structure on the categoryof simplicial presheaves on a Grothendieck site, and the ∞-category associated to this model category is an ∞-topos in our sense. This construction is
generalized from ordinary categories with a Grothendieck topology to simplicial categories with a Grothendieck topology in [79] (and again produces∞-topoi). However, not every ∞-topos arises in this way: one can construct
only ∞-topoi which are hypercomplete (called t-complete in [79]); we willsummarize the situation in Section 6.5.2. Our notion of an ∞-topos is essentially equivalent to the notion of a Segal topos introduced in [79] and to
Charles Rezk’s notion of a model topos. We note also that the paper [79] hasconsiderable overlap with the ideas discussed here.

6.1 ∞-TOPOI: DEFINITIONS AND CHARACTERIZATIONSBefore we study the ∞-categorical version of topos theory, it seems appropriate to briefly review the classical theory. Recall that a topos is a category
C which behaves like the category of sets or (more generally) the category ofsheaves of sets on a topological space. There are several (equivalent) ways ofmaking this idea precise. The following result is proved (in a slightly differentform) in [2]:Proposition 6.1.0.1. Let C be a category. The following conditions areequivalent:(A) The category C is (equivalent to) the category of sheaves of sets onsome Grothendieck site.
(B) The category C is (equivalent to) a left exact localization of the categoryof presheaves of sets on some small category C0 .
(C) Giraud’s axioms are satisfied:(i) The category C is presentable (that is, C has small colimits and aset of small generators).

highertopoi

April 9, 2017∞-TOPOI

529(ii) Colimits in C are universal.
(iii) Coproducts in C are disjoint.
(iv) Equivalence relations in C are effective.
Definition 6.1.0.2. A category C is called a topos if it satisfies the equivalent conditions of Proposition 6.1.0.1.
Remark 6.1.0.3. A reader who is unfamiliar with some of the terminology
used in the statement of Proposition 6.1.0.1 should not worry: we will review the meaning of each condition in §6.1.1 as we search for ∞-categoricalgeneralizations of axioms (i) through (iv).
Our goal in this section is to introduce the ∞-categorical analogue ofDefinition 6.1.0.2. Proposition 6.1.0.1 suggests several possible approaches.
We begin with the simplest of these:
Definition 6.1.0.4. Let X be an ∞-category. We will say that X is an∞-topos if there exists a small ∞-category C and an accessible left exact
localization functor P(C) → X.
Remark 6.1.0.5. Definition 6.1.0.4 involves an accessibility condition which
was not mentioned in Proposition 6.1.0.1. This is because every left exact
localization of a category of set-valued presheaves is automatically accessible(see Proposition 6.4.1.6). We do not know if the corresponding result holds
for S-valued presheaves. However, it is true under a suitable hypercompleteness assumption: see [80].
Adopting Definition 6.1.0.4 amounts to selecting an extrinsic approach tohigher topos theory: the class of ∞-topoi is defined to be the smallest collection of ∞-categories which contains S and is stable under certain constructions (left exact localizations and the formation of functor categories). The
main objective of this section is to give several reformulations of Definition
6.1.0.2 which have a more intrinsic flavor. Our results may be summarized
in the following statement (all our our terminology will be explained laterin this section):
Theorem 6.1.0.6. Let X be an ∞-category. The following conditions areequivalent:(1) The ∞-category X is an ∞-topos.
(2) The ∞-category X is presentable, and for every small simplicial set K
and every natural transformation α : p → q of diagrams p, q : K . → X,
the following condition is satisfied:– If q is a colimit diagram and α = α|K is a Cartesian transformation, then p is a colimit diagram if and only if α is a Cartesian transformation.

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530

CHAPTER 6(3) The ∞-category X satisfies the following ∞-categorical analogues ofGiraud’s axioms:(i) The ∞-category X is presentable.
(ii) Colimits in X are universal.
(iii) Coproducts in X are disjoint.
(iv) Every groupoid object of X is effective.
We will review the meanings of conditions (i) through (iv) in §6.1.1 and§6.1.2. In §6.1.3, we will give several equivalent formulations of (2) and prove
the implications (1) ⇒ (2) ⇒ (3). The implication (3) ⇒ (1) is the most difficult; we will give the proof in §6.1.5 after establishing a crucial technicallemma in §6.1.4. Finally, in §6.1.6 we will establish yet another characterization of ∞-topoi based on the theory of classifying objects.
Remark 6.1.0.7. The characterization of the class of ∞-topoi given in part(2) of Theorem 6.1.0.6 is due to Rezk, as are many of the ideas presented in§6.1.3.
The equivalence (1) ⇔ (3) of Theorem 6.1.0.6 can be viewed as an ∞categorical analogue of the equivalence (B) ⇔ (C) in Proposition 6.1.0.1.
It is natural to ask if there is also some equivalent of the characterization(A). To put the question another way: given a small ∞-category C, does
there exist some natural description of the class of all left exact localizationsof C? Experience with classical topos theory suggests that we might try tocharacterize such localizations in terms of Grothendieck topologies on C. We
will introduce a theory of Grothendieck topologies on ∞-categories in §6.2.2
and show that every Grothendieck topology on C determines a left exact
localization of P(C). However, it turns out that not every ∞-topos arises
via this construction. This raises a natural question: is it possible to give an
explicit description of all left exact localizations of P(C), perhaps in terms ofsome more refined theory of Grothendieck topologies? We will give a partial
answer to this question in §6.5.
6.1.1 Giraud’s Axioms in the ∞-Categorical Setting
Our goal in this section is to formulate higher-categorical analogues of conditions (i) through (iv) of Proposition 6.1.0.1. We consider each axiom inturn. In each case, our objective is to find an analogous axiom which makes
sense in the setting of ∞-categories and is satisfied by the ∞-category S ofspaces.
(i) The category C is presentable.
The generalization to the case where C is a ∞-category is obvious: weshould merely require C to be a presentable ∞-category in the sense of
Definition 5.5.0.1. According to Example 5.5.1.8, this condition is satisfied
when C is the ∞-category of spaces.

highertopoi

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531∞-TOPOI(ii) Colimits in C are universal.
Let us first recall the meaning of this condition in classical category theory.
If the axiom (i) is satisfied, then C is presentable and therefore admits all(small) limits and colimits. In particular, every diagramf

X→S←T
has a limit XT = X ×S T . This construction determines a functorf ∗ : C/S → C/T
X 7→ XT ,which is a right adjoint to the functor given by composition with f .
We say that colimits in C are universal if the functor f ∗ is preserves
colimits for every map f : T → S in C. (In other words, colimits are universal
in C if any colimit in C remains a colimit in C after pulling back along amorphism T → S.)
Let us now attempt to make this notion precise in the setting of an arbitrary ∞-category C. Let OC = Fun(∆1 , C) and let p : OC → C be given byevaluation at {1} ⊆ ∆1 . Corollary 2.4.7.12 implies that p is a coCartesianfibration.
Lemma 6.1.1.1. Let X be an ∞-category and let p : OX → X be defined asabove. Let F be a morphism in OX corresponding to a diagram σ : ∆1 ×∆1 '(Λ22 )/ → X, which we will denote by
X0

X

f0

/ Y0
g

f


/ Y.

Then F is p-Cartesian if and only if the above diagram is a pullback in X. In
particular, p is a Cartesian fibration if and only if the ∞-category X admits
pullbacks.
Proof. For every simplicial set K, let K + denote the full simplicial subset of(K ? {x} ? {y}) × ∆1 spanned by all of the vertices except (x, 0) and definea simplicial set C by setting
Fun(K, C) = {m : K + → X : m|({x} ? {y}) × {1} = f, m|{y} × ∆1 = g}.
We observe that we have a commutative diagramC/ (OX )/g

X/f

/ X/Y 0which induces a map q : C → (OX )/g ×X/Y 0 X/f . We first claim that q
is a trivial fibration. Unwinding the definitions, we observe that the righthighertopoi

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CHAPTER 6

lifting property of q with respect to an inclusion ∂ ∆n ⊆ ∆n follows from theextension property of X with respect to Λn+2
n+1 , which follows in turn from
our assumption that X is an ∞-category.
The inclusion K + ⊆ K × ∆1 induces a projection q 0 : (OX )/F → C whichfits into a pullback diagram/C(OX )/F

g



X/σ

q

00



/ X/σ|Λ2 .
2

0

It follows that q is a right fibration and that q 0 is trivial if σ is a pullback
diagram. Conversely, we observe that (Λ22 )/ is a retract of (∆0 )+ , so thatthe map g is surjective on vertices. Consequently, if q 0 is a trivial fibration,then the fibers of q 00 are contractible, so that q 00 is a trivial fibration (Lemma
2.1.3.4) and σ is a pullback diagram.
By definition, F is p-Cartesian if and only if the compositionq ◦ q 0 : (OX )/F → (OX )/g ×X/Y 0 X/f
is a trivial fibration. Since q is a trivial fibration and q 0 is a right fibration,
this is also equivalent to the assertion that q 0 is a trivial fibration (Lemma
2.1.3.4).
Now suppose that X is an ∞-category which admits pullbacks, so thatthe projection p : OX → X is both a Cartesian fibration and a coCartesianfibration. Let f : S → T be a morphism in X. Taking the pullback of palong the corresponding map ∆1 → X, we obtain a correspondence fromp−1 (S) = X/S to p−1 (T ) = X/T associated to a pair of adjoint functors
X/X o

f!
f∗

/ X/T .

The functors f! and f ∗ are well-defined up to homotopy (in fact, up to acontractible space of choices). We may think of f! as the functor given bycomposition with f , and f ∗ as the functor given by pullback along f (in viewof Lemma 6.1.1.1).
We can now formulate the ∞-categorical analogue of (ii):
Definition 6.1.1.2. Let C be a presentable ∞-category. We will say thatcolimits in C are universal if, for any morphism f : T → S in C, the associatedpullback functorf ∗ : C/S → C/Tpreserves (small) colimits.
Assume that C is a presentable ∞-category and let f : T → S be amorphism in C. By the adjoint functor theorem, f ∗ : C/S → C/T preserves
all colimits if and only if it has a right adjoint f∗ . Since the existence ofhighertopoi

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533∞-TOPOI

adjoint functors can be tested inside the enriched homotopy category, thisgives a convenient criterion which allows us to test whether or not colimits
in C are universal.
Remark 6.1.1.3. Let X be an ∞-category. The assumption that colimits
in X are universal can be viewed as a kind of distributive law. We have thefollowing table of vague analogies:
Higher Category Theory

Algebra∞-Category

Set

Presentable ∞-category

Abelian group

Colimits

Sums

Limits

Products

lim(Xα ) ×S T ' lim(Xα ×S T )−→−→(x + y)z = xz + yz∞-Topos

Commutative ring

Definition 6.1.1.2 has a reformulation in the language of classifying functors (§3.3.2):
Proposition 6.1.1.4. Let X be an ∞-category which admits finite limits.
The following conditions are equivalent:(1) The ∞-category X is presentable, and colimits in X are universal.
(2) The Cartesian fibration p : OX → X is classified by a functor Xop →PrL .
Proof. We can restate condition (2) as follows: each fiber X/U of p is presentable, and each of the pullback functors f ∗ : X/V → X/U preserves small
colimits. It is clear that (1) ⇒ (2) and that (2) implies that colimits in X
are universal. Since X admits finite limits, it has a final object 1; condition(2) implies that X ' X/1 is presentable, which proves (1).

(iii) Coproducts in C are disjoint.

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If C is an ∞-category which admits finite coproducts, then we will saythat coproducts in C are disjoint if every coCartesian diagram
∅

X

{
#

Y

# ` |
X Y
is also Cartesian, provided that ∅ is an initial object of C. More informally,
to say that coproducts`are disjoint is to say that the intersection of X andY inside the union X Y is empty.
We now come to the most subtle and interesting of Giraud’s axioms:(iv) Every equivalence relation in C is effective.
Recall that if X is an object in an (ordinary) category C, then an equivalence relation R on X is an object of C equipped with a map p : R → X × X
such that for any S, the induced map
HomC (S, R) → HomC (S, X) × HomC (S, X)
exhibits HomC (S, R) as an equivalence relation on HomC (S, X).
If C admits finite limits, then it is easy to construct equivalence relations
in C: given any map X → Y in C, the induced map X ×Y X → X × X is anequivalence relation on X. If the category C admits finite colimits, then one
can attempt to invert this process: given an equivalence relation R on X, one
can form the coequalizer of the two projections R → X to obtain an object
which we will denote by X/R. In the category of sets, one can recover R as
the fiber product X ×X/R X. In general, this need not occur: one always has
R ⊆ X ×X/R X, but the inclusion may be strict (as subobjects of X × X).
If equality holds, then R is said to be an effective equivalence relation andthe map X → X/R is said to be an effective epimorphism.
Remark 6.1.1.5. Recall that a map f : X → Y in a category C is said to bea categorical epimorphism if the natural map HomC (Y, Z) → HomC (X, Z)
is injective for every object Z ∈ C, so that we may identify HomC (Y, Z)
with a subset of HomC (X, Z). To say that f is an effective epimorphism is
to say that we can characterize this subset: it is the collection of all maps
g : X → Z such that the diagram:Xg

X ×Y X
$

?Y
g

X

'/

7Z

highertopoi

April 9, 2017∞-TOPOI

535

commutes (which is obviously a necessary condition for the indicated dottedarrow to exist).
Using the terminology introduced above, we can neatly summarize someof the fundamental properties of the category of sets:
Fact 6.1.1.6. In the category of sets, every equivalence relation is effective
and the effective epimorphisms are precisely the surjective maps.
The first assertion of Fact 6.1.1.6 remains valid in any topos. According
to the axiomatic point of view, it is one of the defining features of a topos.
If C is a category with finite limits and colimits in which all equivalence
relations are effective, then we obtain a one-to-one correspondence between
equivalence relations on an object X and quotients of X (that is, isomorphismclasses of effective epimorphisms X → Y ). This correspondence is extremely
useful because it allows us to make elementary descent arguments: one candeduce statements about quotients of X from statements about X and about
equivalence relations on X (which live over X). We would like to formulate
an ∞-categorical analogue of this condition which will allow us to makesimilar arguments.
In the ∞-category S of spaces, the situation is more complicated. The
correct notion of surjection of spaces X → Y is a map which induces asurjection on path components π0 X → π0 Y . However, in this case, the(homotopy) fiber product R = X ×Y X does not give an equivalence relation
on X because the map R → X × X is not necessarily injective in any
reasonable sense. However, it does retain some of the pleasant features of anequivalence relation: instead of transitivity, we have a coherently associative
composition law R ×X R → R (this is perhaps most familiar in the situation
where X is a point: in this case, R can be identified with the based loop
space of Y , which is endowed with a multiplication given by concatenationof loops). In §6.1.2, we will make this idea precise and define groupoid objects
and effective groupoid objects in an arbitrary ∞-category. Granting thesenotions for the moment, we have a natural candidate for the ∞-categoricalgeneralization of condition (iv):(iv)0 Every groupoid object of C is effective.
6.1.2 Groupoid Objects
Let C be a category which admits finite limits. A groupoid object of C is afunctor F from C to the category Cat of (small) groupoids, which has thefollowing properties:(1) There exists an object X0 ∈ C and a (functorial) identification of
HomC (C, X0 ) with the set of objects in the groupoid F (C) for eachC ∈ C.
(2) There exists an object X1 ∈ C and a (functorial) identification of
HomC (C, X1 ) with the set of morphisms in groupoid F (C) for eachC ∈ C.

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Example 6.1.2.1. Let C be the category Set of sets. Then a groupoid objectof C is simply a (small) groupoid.
Giving a groupoid object of a category C is equivalent to giving a pairof objects X0 ∈ C (the “object classifier”) and X1 ∈ C (the “morphismclassifier”) together with a collection of maps which relate X0 to X1 andsatisfy appropriate identities, which imitate the usual axiomatics of category
theory. These identities can be very efficiently encoded using the formalismof simplicial objects. For every n ≥ 0, let [n] denote the category associatedto the linearly ordered set {0, . . . , n} and consider the functor Fn : C → Set
defined so that
Fn (C) = HomCat ([n], F (C)).
By assumption, F0 and F1 are representable by objects X0 , X1 ∈ C. Since C
is stable under finite limits, it follows that
Fn = F1 ×F0 · · · ×F0 F1
is representable by an object Xn = X1 ×X0 · · · ×X0 X1 . The objects Xn
can be assembled into a simplicial object X• of C. We can think of thisconstruction as a generalization of the process which associates to every
groupoid D its nerve N(D) (a simplicial set). Moreover, as in the classicalcase, the association F 7→ X• is fully faihtful. In other words, we can identify
groupoid objects of C with the corresponding simplicial objects. Of course,
not every simplicial object X• of C arises via this construction. This is true
if and only if certain additional conditions are met: for instance, the diagram
X2

d0

/ X1

d0


/ X0

d2


X1

d1

must be Cartesian.
The purpose of this section is to generalize the notion of a groupoid object
to the setting where C is an ∞-category. We begin by introducing the classof simplicial objects of C; we then define groupoid objects to be simplicial
objects which satisfy additional conditions.
Definition 6.1.2.2. Let ∆+ denote the category of finite (possibly empty)
linearly ordered sets. A simplicial object of an ∞-category C is a map of∞-categories
U• : N(∆)op → C .
An augmented simplicial object of C is a map
U•+ : N(∆+ )op → C .
We let C∆ denote the ∞-category Fun(N(∆)op , C); we will refer to C∆ as
the ∞-category of simplicial objects of C. Similarly, we will refer to the ∞category Fun(N(∆+ )op , C) as the ∞-category of augmented simplicial objectsof C, and we will denote it by C∆+ .
If U• is an (augmented) simplicial object of C and n ≥ 0 (n ≥ −1), we
will write Un for the object U ([n]) ∈ C.

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537∞-TOPOI

Remark 6.1.2.3. In the case where C is the nerve of an ordinary category
D, Definition 6.1.2.2 recovers the usual notion of a simplicial object of D.
More precisely, the ∞-category C∆ of simplicial objects of C is naturally
isomorphic to the nerve of the category of simplicial objects of D.
Lemma 6.1.2.4. Let f : X → Y be a map of simplicial sets satisfying thefollowing conditions:(1) The map f induces a bijection X0 → Y0 on vertex sets.
(2) The simplicial set Y is a Kan complex.
(3) The map f has the right lifting property with respect to every horn
inclusion Λni ⊆ ∆n for n ≥ 2.
(4) The map f is a weak homotopy equivalence.
Then f is a trivial Kan fibration.
Proof. In view of condition (4), it suffices to prove that f is a Kan fibration.
In other words, we must show that p has the right lifting property withrespect to every horn inclusion Λni ⊆ ∆n . If n > 1, this follows from (3).
We may therefore reduce to the case where n = 1; by symmetry, we may
suppose that i = 0.
Let e : y → y 0 be an edge of Y . Condition (1) implies that there is a
(unique) pair of vertices x, x0 ∈ X0 with y = f (x), y 0 = f (x0 ). Since f is a
homotopy equivalence, there is a path p from x to x0 in the topological space|X| such that the induced path |f |◦p in |Y | is homotopic to e via a homotopy
which keeps the endpoints fixed. By cellular approximation, we may supposethat this path is contained in the 1-skeleton of |X|. Consequently, there is apositive integer k, a sequence of vertices {z0 , . . . , zk } with z0 = x, zk = x0
such that each adjacent pair (zi , zi+1 ) is joined by an edge pi (running ineither direction), such that p is homotopic (relative to its boundary) to thepath obtained by concatenating the edges pi . Using conditions (2) and (3),
we note that X has the extension property with respect to the inclusionΛni ⊆ ∆n for each n ≥ 2. It follows that we may assume that pi runs fromzi to zi+1 : if it runs in the opposite direction, then we can extend the map(pi , s0 zi , •) : Λ22 → X
to a 2-simplex σ : ∆2 → X and then replace pi by d2 σ.
Without loss of generality, we may suppose that k > 0 is chosen as small
as possible. We claim that k = 1. Otherwise, we choose an extension τ :∆2 → X of the map(p1 , •, p0 ) : Λ21 → X.
We can then replace the initial segmentp0

p1

z0 → z1 → z2

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CHAPTER 6

by the edge d1 (τ ) : z0 → z2 and obtain a shorter path from x to x0 , contradicting the minimality of k.
The edges e and f (p0 ) are homotopic in Y relative to their endpoints.
Using (3), we see that p0 is homotopic (relative to its endpoints) to anedge e which satisfies f (e) = e. This completes the proof that f is a Kan
fibration.
Notation 6.1.2.5. Let K be a simplicial set. We let ∆/K denote the category of simplices of K defined in §4.2.3. The objects of ∆/K are pairs (J, η),
where J is an object of ∆ and η ∈ HomSet∆ (∆J , K). A morphism from (J, η)to (J 0 , η 0 ) is a commutative diagram/ ∆J 0

∆J
!

K.

}

Equivalently, we can describe ∆/K as the fiber product ∆ ×Set∆ (Set∆ )/K .
If C is an ∞-category, U : N(∆)op → C is a simplicial object of C, and K
is a simplicial set, then we let U [K] denote the composite map
N(∆/K )op → N(∆)op → C .
Proposition 6.1.2.6. Let C be an ∞-category and U : N(∆)op → C asimplicial object of C. The following conditions are equivalent:(1) For every weak homotopy equivalence f : K → K 0 of simplicial sets
which induces a bijection K0 → K00 on vertices, the induced map
C/U [K 0 ] → C/U [K] is a categorical equivalence.
(2) For every cofibration f : K → K 0 of simplicial sets which is a weakhomotopy equivalence and bijective on vertices, the induced map
C/U [K 0 ] → C/U [K]is a categorical equivalence.
(20 ) For every cofibration f : K → K 0 of simplicial sets which is a weakhomotopy equivalence and bijective on vertices, the induced map
C/U [K 0 ] → C/U [K]is a trivial fibration.
(3) For every n ≥ 2 and every 0 ≤ i ≤ n, the induced map
C/U [∆n ] → C/U [Λni ]is a categorical equivalence.
(30 ) For every n ≥ 2 and every 0 ≤ i ≤ n, the induced map
C/U [∆n ] → C/U [Λni ]is a trivial fibration.

highertopoi

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539∞-TOPOI(4) For every n ≥ 0 and every partition [n] = S ∪ S 0 such that S ∩ S 0
consists of a single element s, the induced map
C/U [∆n ] → C/U [K]`0
is a categorical equivalence, where K = ∆S {s} ∆S ⊆ ∆n .
(40 ) For every n ≥ 0 and every partition [n] = S ∪ S 0 such that S ∩ S 0
consists of a single element s, the induced map
C/U [∆n ] → C/U [K]`0
is a trivial fibration, where K = ∆S {s} ∆S ⊆ ∆n .
(400 ) For every n ≥ 0 and every partition [n] = S ∪ S 0 such that S ∩ S 0
consists of a single element s, the diagramU ([n])/ U (S)U (S 0 )
/ U ({s})is a pullback square in the ∞-category C.
Proof. The dual of Proposition 2.1.2.1 implies that any monomorphism K →K 0 of simplicial sets induces a right fibration C/U [K] → C/U [K] . By Corollary
2.4.4.6, a right fibration is a trivial fibration if and only if it is a categoricalequivalence. This proves that (2) ⇔ (20 ), (3) ⇔ (30 ), and (4) ⇔ (40 ). The
implications (1) ⇒ (2) ⇒ (3) are obvious.
We now prove that (3) implies (1). Let A denote the class of all morphisms
f : K 0 → K which induce a categorical equivalence C/U [K] → C/U [K 0 ] . Let
A0 denote the class of all cofibrations which have the same property; equivalently, A0 is the class of all cofibrations which induce a trivial fibration
C/U [K] → C/U [K 0 ] . From this characterization it is easy to see that A0 is
weakly saturated. Let A00 be the weakly saturated class of morphisms generated by the inclusions Λni ⊆ ∆n for n > 1. If we assume (3), then we havethe inclusions A00 ⊆ A0 ⊆ A.
Let f : K → K 0 be an arbitrary morphism of simplicial sets. By Proposition A.1.2.5, we can choose a map h0 : K 0 → M 0 which belongs to A00 ,where M 0 has the extension property with respect to Λni ⊆ ∆n for n > 1
and is therefore a Kan complex. Applying Proposition A.1.2.5 again, we canconstruct a commutative diagram
K

h

/M

h0


/ M 0,gf
K0

where the horizontal maps belong to A00 and g has the right lifting property
with respect to every morphism in A00 . If f is a weak homotopy equivalencehighertopoi

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CHAPTER 6

which is bijective on vertices, then g has the same properties, so that g isa trivial fibration by Lemma 6.1.2.4. It follows that g has the right lifting
property with respect to the cofibration g ◦ h : K → M 0 , so that g ◦ h is aretract of h and therefore belongs to A00 . Since g ◦ h = h0 ◦ f and h0 belong
to A00 ⊆ A, it follows that f belongs to A.
It is clear that (1) ⇒ (4). We next prove that (40 ) ⇒ (3). We must showthat if n > 1, then every inclusion Λni ⊆ ∆n belongs to the class A defined
above. The proof is by induction on n. Replacing i by n − i if necessary, we
may suppose that i < n. If (n, i) 6= (2, 0), we consider the composition
a
f0
f
∆n−1
∆{n−1,n} ,→ Λni ,→ ∆n .
{n−1}0Here f belongs to A by the inductive hypothesis and f 0 ◦ f belongs to A0 by
virtue of assumption (40 ); therefore f 0 also belongs to A. If n`= 2 and i = 0,0then we observe that the inclusion Λ21 ⊆ ∆2 is of the form ∆S {s} ∆S ⊆ ∆2 ,where S = {0, 1} and S 0 = {0, 2}.
To complete the proof, we show that (4) is equivalent to `(400 ). Fix n ≥ 0,0
0
S
S0
let S ∪ S = [n] be such that S ∩ S = {s}, and let K = ∆⊆ ∆n .
{s} ∆0
Let I denote the full subcategory of ∆/∆n spanned by the objects [n], S, S 0 ,and {s}. Let I ⊆ I0 be the full subcategory obtained by omitting the object[n]. Let p0 denote the composition
U

N(I0 )op → N(∆)op → C
and let p = p0 | N(I)op . Consider the diagram/ C/U [K]C/U [∆n ]
u

v





/ C/p .

C/p0

Condition (4) asserts that the upper horizontal map is a categorical equivalence, and condition (400 ) asserts that the lower horizontal map is a categorical equivalence. To prove that (4) ⇔ (400 ), it suffices to show that thevertical maps u and v are categorical equivalences.
We have a commutative diagram
C/U [∆n ]u/ C/p0%z
C/U [∆n ] .
Since ∆n is a final object of both ∆/∆n and I0 , the unlabelled maps are
trivial fibrations. It follows that u is a categorical equivalence.
To prove that v is a categorical equivalence, it suffices to show that theinclusion g : I ⊆ ∆/K induces a right anodyne map
N(g) : N(I) ⊆ N(∆/K )highertopoi

April 9, 2017

541∞-TOPOIof simplicial sets. We observe that the functor g has a left adjoint f , whichassociates to each simplex σ : ∆m → K the smallest simplex in I whichcontains the image of σ. The map N(g) is a section of N(f ), and there is a(fiberwise) simplicial homotopy from idN(∆/K ) to N(g)◦N(f ). We now invoke
Proposition 2.1.2.11 to deduce that N(g) is right anodyne, as desired.
Definition 6.1.2.7. Let C be an ∞-category. We will often denote simplicial
objects of C by U• and write Un for U• ([n]) ∈ C. We will say that a simplicial
object U• ∈ C∆ is a groupoid object of C if it satisfies the equivalent conditionsof Proposition 6.1.2.6. We will let Gpd(C) denote the full subcategory of C∆
spanned by the groupoid objects of C.
Remark 6.1.2.8. It follows from the proof of Proposition 6.1.2.6 that toverify that a simplicial object X• ∈ C∆ is a groupoid object, we need only
verify condition (400 ) in a small class of specific examples, but we will notneed this observation.
Proposition 6.1.2.9. Let C be a presentable ∞-category. The full subcategory Gpd(C) ⊆ C∆ is strongly reflective.
Proof. Let n ≥ 0 and [n] = S ∪ S 0 be as in statement (400 ) of Proposition 6.1.2.6. Let D(S, S 0 ) ⊆ C∆ be the full subcategory consisting of those
simplicial objects U ∈ C∆ for which the associated diagramU ([n])/ U (S)U (S 0 )
/ U ({s})is Cartesian. Lemmas 5.5.4.19 and 5.5.4.17 imply that D(S, S 0 ) is a strongly
reflective subcategory of C∆ . Let D denote the intersection of all these subcategories taken over all n ≥ 0 and all such decompositions [n] = S ∪ S 0 .
Lemma 5.5.4.18 implies that D ⊆ C∆ is strongly reflective, and Proposition
6.1.2.6 implies that D = Gpd(C).
Our next step is to exhibit a large class of examples of groupoid objects.
We first sketch the idea. Suppose that C is an ∞-category which admits
finite limits and let u : U → X be a morphism in C. Using this data, we
can construct a simplicial object U• of C, where Un is given by the (n + 1)fold fiber power of U over X. In order to describe this construction more
precisely, we need to introduce a bit of notation.
Notation 6.1.2.10. Let ∆≤n+ denote the full subcategory of ∆+ spannedby the objects {[k]}−1≤k≤n .
Proposition 6.1.2.11. Let C be an ∞-category and let U•+ : N(∆+ )op → C
be an augmented simplicial object of C. The following conditions are equivalent:highertopoi

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542

CHAPTER 6(1) The augmented simplicial object U•+ is a right Kan extension ofopU•+ | N(∆≤0+ ) .

(2) The underlying simplicial object U• is a groupoid object of C, and theopdiagram U•+ | N(∆≤1
is a pullback square
+ )
U1

/ U0


U0


/ U−1

in the ∞-category C.
Proof. Suppose first that (1) is satisfied. It follows immediately from thedefinition of right Kan extensions that the diagram
U1

/ U0


U0


/ U−1

is a pullback. To prove that U• is a groupoid, we show that U• satisfies
criterion (400 ) of Proposition 6.1.2.6. Let S and S 0 be sets with union [n]and intersection S ∩ S 0 = {s}. Let I be the nerve of the category (∆+ )/∆n .
For each subset J ⊆ [n], let I(J) denote the full subcategory of I spannedby the initial object together with the inclusions {j} → ∆n , j ∈ S. By
assumption, U•+ exhibits U• (S) as a limit of U•+ | N(I(S)), U• (S 0 ) as a limitof U•+ | N(I(S 0 )), U• ([n]) as a limit of U•+ | N(I([n])), and U• ({s}) as a limitof U•+ | N(I({s})). It follows from Corollary 4.2.3.10 that the diagramU• ([n])/ U• (S)U• (S 0 )/ U• ({s})

is a pullback.
We now prove that (2) implies (1). Using the above notation, we mustshow that for each n ≥ −1, U•+ exhibits U•+ ([n]) as a limit of U•+ | I([n]). Forn ≤ 0, this is obvious; for n = 1, it is equivalent to the assumption that
U1

/ U0


U0


/ U−1

is a pullback diagram. We prove the general case by induction on n. Using
the inductive hypothesis, we conclude that U• (∆S ) is a limit of U•+ | I(S)for all proper subsets S ⊂ [n]. Choose a decomposition {0, . . . , n} = S ∪ S 0 ,highertopoi

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543∞-TOPOI

where S ∩ S 0 = {s}. According to Proposition 4.4.2.2, the desired result is
equivalent to the assertion thatU• ([n])/ U• (S)U• (S 0 )/ U• ({s})

is a pullback diagram, which follows from our assumption that U• is agroupoid object of C.
We will say that an augmented simplicial object U•+ in an ∞-category C isa Čech nerve if it satisfies the equivalent conditions of Proposition 6.1.2.11.
In this case, U•+ is determined up to equivalence by the map u : U0 → U−1 ;we will also say that U•+ is the Čech nerve of u.
Notation 6.1.2.12. Let U• be a simplicial object in an ∞-category C.
We may regard U• as a diagram in C indexed by N(∆)op . We let |U• | :N(∆+ )op → C denote a colimit for U• (if such a colimit exists). We will refer
to any such colimit as a geometric realization of U• .
Remark 6.1.2.13. Note that we are regarding |U• | as a colimit diagramin C, not as an object of C. We also note that our notation is somewhat
abusive since |U• | is not uniquely determined by U• . However, if a colimit ofU• exists, then it is determined up to contractible ambiguity.
Definition 6.1.2.14. Let U• be a simplicial object of an ∞-category C. We
will say that U• is an effective groupoid if it can be extended to a colimit
diagram U•+ : N(∆+ )op → C such that U•+ is a Čech nerve.
Remark 6.1.2.15. It follows easily from characterization (2) of Proposition
6.1.2.11 that any effective groupoid U• is a groupoid.
We can now state the ∞-categorical counterpart of Fact 6.1.1.6: every
groupoid object in S is effective. This statement is somewhat less trivial thanits classical analogue. For example, a groupoid object U• in S with U0 = ∗can be thought of as a space U1 equipped with a coherently associative
multiplication operation. If U• is effective, then there exists a fiber diagram
U1

/∗


∗


/ U−1

so that U1 is homotopy equivalent to a loop space. This is a classical result(see, for example, [74]). We will give a somewhat indirect proof in the next
section.

highertopoi

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CHAPTER 6

6.1.3 ∞-Topoi and DescentIn this section, we will describe an elegant characterization of the notionof an ∞-topos based on the theory of descent. We begin by explaining theidea in informal terms. Let X be an ∞-category. To each object U of X
we can associate the overcategory X/U . If X admits finite limits, then thisd∞construction gives a contravariant functor from X to the ∞-category Catof (not necessarily small) ∞-categories. If X is an ∞-topos, then this functor
carries colimits in X to limits of ∞-categories. In other words, if an object
X ∈ X is obtained as the colimit of some diagram {Xα } in X, then giving amorphism Y → X is equivalent to giving a suitably compatible diagram ofmorphisms {Yα → Xα }. Moreover, we will eventually show that this property
characterizes the class of ∞-topoi. The ideas presented in this section are
due to Charles Rezk.
Definition 6.1.3.1. Let X be an ∞-category, K a simplicial set, and p, q :K → X two diagrams. We will say that a natural transformation α : p → q
is Cartesian if, for each edge φ : x → y in K, the associated diagram
p(x)

p(φ)

α(x)


q(x)

/ p(y)
α(y)

q(φ)


/ q(y)

is a pullback in X.
Lemma 6.1.3.2. Let X be an ∞-category, and let α : p → q be a naturaltransformation of diagrams p, q : K  ∆0 → X. Suppose that, for every vertex
x of K, the associated transformationp|{x}  ∆0 → q|{x}  ∆0
is Cartesian. Then α is Cartesian.
Proof. Let z be the cone point of K∆0 . We note that to every edge e : x → y
in K  ∆0 we can associate a diagram
/z

x
e


y

g
idz

/ z.

The transformation α restricts to a Cartesian transformation on the horizontal edges and the right vertical edge, either by assumption or because
they are degenerate. Applying Lemma 4.4.2.1, we deduce first that α(g) is a
Cartesian transformation, then that α(e) is a Cartesian transformation.
The condition that an ∞-category has universal colimits can be formulated
in the language of Cartesian transformations:highertopoi

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545∞-TOPOI

Lemma 6.1.3.3. Let X be a presentable ∞-category. The following conditions are equivalent:(1) Colimits in X are universal.
(2) Let p, q : (K .  ∆0 ) → X be diagrams which carry ∆0 to vertices
X, Y ∈ X and let α : p → q be a Cartesian transformation. If the map
q 0 : K . → X/Y associated to q is a colimit diagram, then the map
p0 : K . → X/X associated to p is a colimit diagram.
(3) Let p, q : K ?∆1 → X be diagrams which carry {1} to vertices X, Y ∈ X
and let α : p → q be a Cartesian transformation. If the map K . →X/Y associated to q is a colimit diagram, then the map K . → X/X
associated to p is a colimit diagram.
(4) Let p, q : K ?∆1 → X be diagrams which carry {1} to vertices X, Y ∈ X
and let α : p → q be a Cartesian transformation. If q|K?{0} is a colimit
diagram, then p|K ? {0} is a colimit diagram.
(5) Let α : p → q be a Cartesian transformation of diagrams K . → X. If
q is a colimit diagram, then p is a colimit diagram.
Proof. Assume that (1) is satisfied; we will prove (2). The transformation α
induces a map f : X → Y . Consider the map
φ : Fun(K . , OX ) → Fun(K . , X)
given by evaluation at the final vertex of ∆1 . Let δ(f ) denote the image of f
under the diagonal map δ : X → Fun(K . , X). Then we may identify α withan edge e of Fun(K . , OX ) which covers δ(f ). Since α is Cartesian, we canapply Lemma 6.1.1.1 and Proposition 3.1.2.1 to deduce that e is φ-Cartesian.
The composition f ∗ ◦ q 0 is the origin of a φ-Cartesian edge e0 : f ∗ ◦ q 0 → q 0 of
Fun(K . , OX ) covering δ(f ), so we conclude that f ∗ ◦ q 0 and p0 are equivalent
in Fun(K . , X/X ). Since q 0 is a colimit diagram and f ∗ preserves colimits,f ∗ ◦ q 0 is a colimit diagram. It follows that p0 is a colimit diagram, as desired.
We now prove that (2) ⇒ (1). Let f : X → Y be a morphism in X
and let q 0 : K . → X/Y be a colimit diagram. Choose a φ-Cartesian edge
e0 : f ∗ ◦ q 0 → q 0 as above, corresponding to a natural transformation α :p → q of diagrams p, q : (K .  ∆0 ) → X. Since e is φ-Cartesian, we may
invoke Proposition 3.1.2.1 and Lemma 6.1.1.1 to deduce that α restricts toa Cartesian transformation p|({x}  ∆0 ) → q|({x}  ∆0 ) for every vertex xof K . . It follows from Lemma 6.1.3.2 that α is Cartesian. Invoking (2), we
conclude that f ∗ ◦ q 0 is a colimit diagram, as desired.
The equivalence (2) ⇔ (3) follows from Proposition 4.2.1.2, and the equivalence (3) ⇔ (4) follows from Proposition 1.2.13.8. The implication (5) ⇒ (4)is obvious. The converse implication (4) ⇒ (5) follows from the observationthat K ? {0} is a retract of K ? ∆1 .

highertopoi

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CHAPTER 6

Notation 6.1.3.4. Let X be an ∞-category which admits pullbacks and let
S be a class of morphisms in X. We will say that S is stable under pullback
if for any pullback diagram
/X

X0
f0


Y0


/Y

f

in X such that f belongs to S, f 0 also belongs to S. We let OSX denote the(S)full subcategory of OX spanned by S, and OX the subcategory of OX whose
objects are elements of S and whose morphisms are pullback diagrams as
above. We observe that evaluation at {1} ⊆ ∆1 induces a Cartesian fibration(S)
OSX → X, which restricts to a right fibration OX → X (Corollary 2.4.2.5).
Lemma 6.1.3.5. Let X be a presentable ∞-category and suppose that colimits in X are universal. Let S be a class of morphisms of X which is stable
under pullback, K a small simplicial set, and q : K . → X a colimit diagram.
The following conditions are equivalent:op

d is a colimit diagram, where f :(1) The composition f ◦ q : K . → Cat∞
opd
X → Cat∞ classifies the Cartesian fibration OSX → X.
op(2) The composition f 0 ◦ q : K . → b
S is a colimit diagram, where f : X →op(S)b
S classifies the right fibration OX → X.
(3) For every natural transformation α : p → q of colimit diagrams K . →X, if α = α|K is a Cartesian transformation and α(x) ∈ S for eachvertex x ∈ K, then α is a Cartesian transformation and α(∞) ∈ S,
where ∞ denotes the cone point of K . .
0

Proof. Let C = Fun(K . , X)/q and C = Fun(K, X)/q . Let C denote the fullsubcategory of C spanned by Cartesian natural tranformations α : p → q
with the property that α(x) belongs to S for each vertex x ∈ K . and let C0
1
be defined similarly. Finally, let C denote the full subcategory of C spannedby those natural transformations α : p → q such that p is a colimit diagram,α = α|K is a Cartesian transformation, and α(x) belongs to S for each01vertex x ∈ K. Lemma 6.1.3.3 implies that C ⊆ C .
Let D denote the full subcategory of Fun(K . , X) spanned by the colimit diagrams. Proposition 4.3.2.15 asserts that the restriction map D → Fun(K, X)
is a trivial fibration. It follows that the associated map D/q → Fun(K, X)/q1
is also a trivial fibration and therefore restricts to a trivial fibration C → C0 .
According to Proposition 3.3.3.1, condition (1) is equivalent to the as0
sertion that the projection C → C0 is an equivalence of ∞-categories. Inview of the above argument, this is equivalent to the assertion that the fully0
10
faithful inclusion C ⊆ C is essentially surjective. Since C is clearly stablehighertopoi

April 9, 2017

547∞-TOPOI
0

1

under equivalence in C, (1) holds if and only if C = C , which is manifestly
equivalent to (3). The proof that (2) ⇔ (3) is similar but uses Proposition
3.3.3.3 in place of Proposition 3.3.3.1.
Lemma 6.1.3.6. Let X be a presentable category in which colimits are universal. Let f : X → ∅ be a morphism in X, where ∅ is an initial object of X.
Then X is also initial.
Proof. Observe that id∅ is both an initial object of X/∅ (Proposition 1.2.13.8)
and a final object of X/∅ . Let f ∗ : X/∅ → X/X be a pullback functor. Thenf ∗ preserves limits (since it is a right adjoint) and colimits (since colimits
in X are universal). Therefore f ∗ id∅ is both initial and final in X/X . It
follows that idX : X → X, being another final object of X/X , is also initial.
Applying Proposition 1.2.13.8, we deduce that X is an initial object of X,
as desired.
Lemma 6.1.3.7. Let X be a presentable ∞-category in which colimits areuniversal and let S be a class of morphisms in X which is stable underpullback. The following conditions are equivalent:(1) The Cartesian fibration OSX → X is classified by a colimit-preservingopd .
functor X → Cat∞(S)(2) The right fibration OX → X is classified by a colimit-preserving funcop
tor X → bS .
(3) The class S is stable under small coproducts, and for every pushout
diagram
f

α

/g

0


/ g0

β0

β


f0

α

in OX , if α and β are Cartesian transformations and f, f 0 , g ∈ S, then
α0 and β 0 are also Cartesian transformations and g 0 ∈ S.
Proof. The equivalence of (1) and (2) follows easily from Lemma 6.1.3.5. Letopd be a functor which classifies OX . Then (1) is equivalent tos : X → Cat∞
the assertion that s preserves small colimits. Supposing that (1) is satisfied,
we deduce (3) by applying Lemma 6.1.3.5 in the special cases of sums andcoproducts. For the converse, let us suppose that (3) is satisfied. Let ∅ denote
an initial object of X. Since colimits in X are universal, Lemma 6.1.3.6 implies
that X/∅ is equivalent to a final ∞-category ∆0 . Since the morphism id∅
belongs to S (since S is stable under empty coproducts), we conclude thats(∅) is a final ∞-category, so that s preserves initial objects. It follows from
Corollary 4.4.2.5 that s preserves finite coproducts. According to Proposition
4.4.2.6, it will suffice to prove that s preserves arbitrary coproducts. To

highertopoi

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548

CHAPTER 6

handle the case of infinite coproducts we apply Lemma 6.1.3.5 again: wemust `
show that if {fα }α∈A is a collection of elements of S having a coproductf = α∈A fα , then f ∈ S and each of the maps fα → f is a Cartesiantransformation. The first condition is true by assumption; for `the second we
let f 0 be a coproduct of the family {fβ }β∈A,β6=α , so that f ' f 0 fα and f 0 ∈S. Applying Lemma 6.1.3.5 (and the fact that s preserves finite coproducts),
we deduce that fα → f is a Cartesian transformation, as desired.
Definition 6.1.3.8. Let X be a presentable ∞-category in which colimits
are universal and let S be a class of morphisms in X. We will say that S is
local if it is stable under pullbacks and satisfies the equivalent conditions ofLemma 6.1.3.7.
Theorem 6.1.3.9. Let X be a presentable ∞-category. The following conditions are equivalent:(1) Colimits in X are universal, and for every pushout diagram
f

α

/g

0


/ g0

β0

β


f0

α

in OX , if α and β are Cartesian transformations, then α0 and β 0 are
also Cartesian transformations.
(2) Colimits in X are universal, and the class of all morphisms in X is
local.
(3) The Cartesian fibration OX → X is classified by a limit-preserving
functor Xop → PrL .
(4) Let K be a small simplicial set and α : p → q a natural transformationof diagrams p, q : K . → X. Suppose that q is a colimit diagram, andthat α = α|K is a Cartesian transformation. Then p is a colimit diagram if and only if α is a Cartesian transformation.
Proof. The equivalences (1) ⇔ (2) ⇔ (3) follow from Lemma 6.1.3.7 and
Proposition 6.1.1.4. The equivalence (3) ⇔ (4) follows from Lemmas 6.1.3.3
and 6.1.3.5.
We now have most of the tools required to establish the implication (1) ⇒(2) of Theorem 6.1.0.6. In view of Theorem 6.1.3.9, it will suffice to prove
the following:
Proposition 6.1.3.10. Let X be an ∞-topos. Then
(1) Colimits in X are universal.

highertopoi

April 9, 2017

549∞-TOPOI(2) For every pushout diagram
f

α

/g

α0


/ g0

β0

β


f0

in OX , if α and β are Cartesian transformations, then α0 and β 0 are
also Cartesian transformations.
Remark 6.1.3.11. Once we have established Theorem 6.1.0.6 in its entirety,
it will follow from Theorem 6.1.3.9 that the converse of Proposition 6.1.3.10
is also valid: a presentable ∞-category X is an ∞-topos if and only if itsatisfies conditions (1) and (2) as above. Condition (1) is equivalent to therequirement that for every morphism f : X → Y in X, the pullback functorf ∗ : X/Y → X/X has a right adjoint (in the case where Y is a final object ofX, this simply amounts to the requirement that every object Z ∈ X admits
an exponential Z X ; in other words, the requirement that X be Cartesianclosed), and condition (2) involves only finite diagrams in the ∞-category X.
One could conceivably obtain a theory of elementary ∞-topoi by dropping
the requirement that X be presentable (or replacing it by weaker conditions
which are also finite in nature). We will not pursue this idea further.
Before giving the proof of Proposition 6.1.3.10, we need to establish a few
easy lemmas.
Lemma 6.1.3.12. Let
p

φ

q0

q


φ0

/ψ

p0


/ ψ0

be a coCartesian square in the category of arrows of Set∆ . Suppose that p
and q are homotopy Cartesian and that q is a cofibration. Then
(1) The maps p0 and q 0 are homotopy Cartesian.
(2) Given any map of arrows r : ψ 0 → θ such that r ◦ p0 and r ◦ q 0 are
homotopy Cartesian, the map r is itself homotopy Cartesian.
Proof. Let r : ψ 0 → θ be as in (2). We must show that r is homotopy
Cartesian if and only if r ◦ p0 and r ◦ q 0 are homotopy Cartesian (taking
r = idψ0 , we will deduce (1)). Without loss of generality, we may replace φ,ψ, φ0 , and θ with minimal Kan fibrations. We now observe that r, r ◦ p0 , andr ◦ q 0 are homotopy Cartesian if and only if they are Cartesian; the desired
result now follows immediately.
Lemma 6.1.3.13. Let A be a simplicial model category containing an object Z which is both fibrant and cofibrant and let A/Z be endowed with theinduced model structure. Then the natural map θ : N(A◦/Z ) → N(A◦ )/Z is
an equivalence of ∞-categories.

highertopoi

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CHAPTER 6

Proof. Let φ : Z 0 → Z be an object of N(A◦ )/Z . Then we can choose afactorization
ψi

Z 0 → Z 00 → Z,
where i is a trivial cofibration and ψ is a fibration, corresponding to a fibrantcofibrant object of A/Z . The above diagram classifies an equivalence between
φ and ψ in N(A◦ )/Z , so that θ is essentially surjective.
Recall that for any simplicial category C containing a pair of objects X
and Y , there is a natural isomorphism of simplicial sets
HomR
N(C) (X, Y ) ' SingQ• (MapC (X, Y )),where Q• is the cosimplicial object of Set∆ introduced in §2.2.2. The same
calculation shows that if φ : X → Z, ψ : Y → Z are two morphisms in C,
then
HomR
N(C)/Z (φ, ψ) ' SingQ• (P ),where P denotes the path space
1

MapC (X, Y ) ×MapC (X,Z){0} MapC (X, Z)∆ ×MapC (X,Z){1} {φ}.
If C is fibrant, then we may identify M with the homotopy fiber of the map
ψ

f : MapC (X, Y ) → MapC (X, Z)
over the vertex φ. Consequently, we may identify the natural map
R
HomR
N(C/Z ) (φ, ψ) → HomN(C)/Z (φ, ψ)with SingQ• (θ), where θ denotes the inclusion of the fiber of f into thehomotopy fiber of f . Consequently, to show that SingQ• (θ) is a homotopy
equivalence, it suffices to prove that f is a Kan fibration. In the special casewhere C = A◦ and ψ is a fibration, this follows from the definition of asimplicial model category.
Lemma 6.1.3.14. Let S denote the ∞-category of spaces. Then
(1) Colimits in S are universal.
(2) For every pushout diagram
f

α

/g

α0


/ g0

β0

β


f0

in OS , if α and β are Cartesian transformations, then α0 and β 0 are
also Cartesian transformations.

highertopoi

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551∞-TOPOI

Proof. We first prove (1). Let f : X → Y be a morphism in S. Without lossof generality, we may suppose that f is a Kan fibration. We wish to showthat the projection
S/f → S/Y
has a right adjoint which preserves colimits. We obtain a commutative diagram of ∞-categories:
N((Set∆ )◦/X )F/ N((Set∆ )◦ )
/Y

φ

ψ





/ S/Y

S/f
φ0


S/X .
Lemma 6.1.3.13 asserts that ψ and φ0 ◦ φ are categorical equivalences, and
φ0 is a trivial fibration. It follows that φ is also a categorical equivalence.
Consequently, it will suffice to show that the functor F has a right adjoint
G which preserves colimits.
We observe that F is obtained by restricting the simplicial nerve of thefunctor f! : (Set∆ )/X → (Set∆ )/Y given by composition with f . The functor
f! is a left Quillen functor: it has a right adjoint f ∗ given by the formulaf ∗ (Y 0 ) = Y 0 ×Y X. According to Proposition 5.2.4.6, F admits a right adjoint
G, which is given by restricting the simplicial nerve of the functor f ∗ . To
prove that G preserves colimits, it will suffice to show that G itself admits
a right adjoint. Using Proposition 5.2.4.6 again, we are reduced to provingthat f ∗ is a left Quillen functor. We observe that f ∗ admits a right adjointf∗ given by the formula f∗ (X 0 ) = MapY (X, X 0 ). It is clear that f ∗ preserves
cofibrations; it also preserves weak equivalences since f is a fibration andSet∆ is a right proper model category (with its usual model structure).
To prove (2), we first apply Proposition 4.2.4.4 to reduce to the case where
the pushout diagram in question arises from a strictly commutative square
f

α

/g

α0


/ g0

β0

β


f0

of morphisms in the category Kan. We now complete the proof by applying
Lemma 6.1.3.12 and Theorem 4.2.4.1.
Lemma 6.1.3.15. Let X be a presentable ∞-category and let L : X → Y
be an accessible left exact localization. If colimits in X are universal, thencolimits in Y are universal.
Proof. We will use characterization (5) of Lemma 6.1.3.2. Let G be a rightadjoint to L and let α : p → q be a Cartesian transformation of diagrams

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K . → Y. Suppose that q is a colimit of q = q|K. Choose a colimit q 0 of G ◦ q,
so that there exists a morphism q 0 → G ◦ q in XG◦q/ which determines anatural transformation β : q 0 → q in Fun(K . , X). Form a pullback diagram
α0

p0

G◦p

/ q0
β


G◦α /
G◦q

.

in XK . Since G is left exact, G ◦ α is a Cartesian transformation. It follows
that α0 , being a pullback of G ◦ α, is also a Cartesian transformation. Since
colimits in X are universal, we conclude that p0 is a colimit diagram. Since
L is left exact, we obtain a pullback diagramL ◦ p0/ L ◦ q0


L◦G◦p


/ L ◦ G ◦ q.

L◦β

Since L preserves colimits, L◦q 0 and L◦p0 are colimit diagrams. The diagram
L ◦ G ◦ q is equivalent to q and therefore also a colimit diagram. We deducethat L ◦ β is an equivalence. Since the diagram is a pullback, the left verticalarrow is an equivalence as well, so that L ◦ G ◦ p is a colimit diagram. We
finally conclude that p is a colimit diagram, as desired.
We are now ready to give the proof of Proposition 6.1.3.10.
Proof of Proposition 6.1.3.10. Let us say that a presentable ∞-category X
is good if it satisfies conditions (1) and (2). Lemma 6.1.3.14 asserts that S isgood. Using Proposition 5.1.2.2, it is easy to see that if X is good, then so is
Fun(K, X) for every small simplicial set K. It follows that every ∞-category
P(C) of presheaves is good. To complete the proof, it will suffice to show that
if X is good and L : X → Y is an accessible left exact localization functor, thenY is good. Lemma 6.1.3.15 shows that colimits in Y are universal. Consider
a diagram σ : Λ20 → OY denoted byαβg ← f → h,
where α and β are Cartesian transformations. We wish to show that if σ isa colimit of σ in OY , then σ carries each edge to a Cartesian transformation.
Without loss of generality, we may suppose that σ = L ◦ σ 0 for some σ 0 :Λ20 → OX which is equivalent to G ◦ σ. Since G is left exact, G(α) and G(β)
are Cartesian transformations. Because X satisfies (2), there exists a colimit
σ 0 of σ 0 which carries each edge to a Cartesian transformation. Then L ◦ σ 0
is a colimit of σ. Since L is left exact, L ◦ σ 0 carries each edge to a Cartesiantransformation in OY .
Our final objective in this section is to prove the implication (2) ⇒ (3) of
Theorem 6.1.0.6 (Proposition 6.1.3.19 below).

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Lemma 6.1.3.16. Let X be an ∞-category and U•+ : N(∆+ )op → X anaugmented simplicial object of X. Let ∆∞ denote the category whose objects
are finite linearly ordered sets J, where Hom∆∞ (J, J 0 ) is the collection of allorder-preserving maps J ∪ {∞} → J 0 ∪ {∞} which carry ∞ to ∞ (here ∞is regarded as a maximal element of J ∪ {∞} and J 0 ∪ {∞}). Suppose thatU•+ extends to a functor F : N(∆∞ )op → X. Then U•+ is a colimit diagram
in X.
Proof. Let C denote the category whose objects are triples (J, J+ ), where
J is a finite linearly ordered set and J+ is an upward-closed subset of J.
0
We define HomC ((J, J+ ), (J 0 , J+)) to be the set of all order-preserving maps
00from J into J which carry J+ into J+. Observe that we have a functor
C → ∆∞ given by(J, J+ ) 7→ J − J+ .
Let F 0 denote the composite functor
N(C)op → N(∆∞ )op → X .
Let C be the full subcategory of C spanned by those pairs (J, J+ ) where0
J 6= ∅. Let C denote the full subcategory spanned by those pairs (J, J+ )00where J+ = ∅ and let C0 = C ∩ C. We observe that C can be identified with∆+ and that C0 can be identified with ∆ in such a way that U•+ is identified
0with F 0 | N(C )op .
Our first claim is that the inclusion N(C0 )op ⊆ N(C)op is cofinal. According
to Theorem 4.1.3.1, it will suffice to show that for every object X = (J, J+ )of C, the category C0/X has a contractible nerve. This is clear since the relevant category has a final object: namely, the map (J, ∅) → (J, J+ ). As aconsequence, we conclude that U•+ is a colimit diagram if and only if F 0 is acolimit diagram.
We now define C1 to be the full subcategory of C spanned by those pairs(J, J+ ) such that J+ is nonempty. We claim that F 0 | N(C)op is a left Kan
extension of F 0 | N(C1 )op . To prove this, we must show that for every (J, ∅) ∈C0 , the induced map(N(C1(J,∅)/ )op ). → N(C)op → X
is a colimit diagram. Let D denote the full subcategory of C1(J,∅)/ spanned by00
those morphisms (J, ∅) → (J 0 , J+
) which induce isomorphisms J ' J 0 − J+
.
1
op
op
We claim that the inclusion N(D) ⊆ N(C(J,∅)/ ) is cofinal. To prove this,
we once again invoke Theorem 4.1.3.1 to reduce to the following assertion:
00
00
for every morphism φ : (J, ∅) → (J 00 , J+), if J+
6= ∅, then the category D/φof all factorizations
000(J, ∅) → (J 0 , J+
) → (J 00 , J+)00such that J+
6= ∅ and such that J ' J 0 − J+hascontractible` a 000weakly
000
000
nerve. This is clear since D/φ has a final object (J J+, J+), where J+=highertopoi

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CHAPTER 600: (∀i ∈ J)[j ≥ φ(i)]}. Consequently, it will suffice to prove that the{j ∈ J+
induced functor

N(Dop ). → X
is a colimit diagram. This diagram can be identified with the constant diagram
N(∆+ )op → X
taking the value U• (J) and is a colimit diagram because the category ∆ hasa weakly contractible nerve (Corollary 4.4.4.10).
We now apply Lemma 4.3.2.7, which asserts that F 0 is a colimit diagram if and only if F 0 |(N(C1 )op ). is a colimit diagram. Let C2 ⊆ C1 be thefull subcategory spanned by those objects (J, J+ ) such that J = J+ . We
claim that the inclusion N(C2 )op ⊆ N(C1 )op is cofinal. According to Theorem 4.1.3.1, it will suffice to show that, for every object (J, J+ ) ∈ C1 , thecategory C2/(J,J+ ) has a weakly contractible nerve. This is clear since the map(2)(J+ , J+ ) → (J, J+ ) is a final object of the category C/(J,J+ ) . Consequently,
to prove that F 0 |(N(C1 )op ). is a colimit diagram, it will suffice to provethat F 0 |(N(C2 )op ). is a colimit diagram. But this diagram can be identified
with the constant map N(∆+ )op → X taking the value U• (∆−1 ), which isa colimit diagram because the simplicial set N(∆)op is weakly contractible(Corollary 4.4.4.10).
Lemma 6.1.3.17. Let X be an ∞-category and let U• : N(∆)op → X bea simplicial object of X. Let U•0 be the augmented simplicial object given bycomposing U• with the functor∆+ → ∆
a
J →J
{∞}.
Then
(1) The augmented simplicial object U•0 is a colimit diagram.
(2) If U• is a groupoid object of X, then the evident natural transformationof simplicial objects α : U•0 | N(∆)op → U• is Cartesian.
Proof. Assertion (1) follows immediately from Lemma 6.1.3.16. To prove (2),let us consider the collection S of all morphisms f : J → J 0 in ∆ such thatα(f ) is a pullback squareU•0 (J 0 )/ U• (J 0 )
U•0 (J)/ U• (J)in X. We wish to prove that every morphism of ∆ belongs to S. Using Lemma
4.4.2.1, we deduce that if f 0 ∈ S, then f ∈ S ⇔ (f ◦ f 0 ∈ S). Consequently,

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it will suffice to prove that every inclusion {j} ⊆ J belongs to S. Unwinding
the definition, this amounts the requirement that the diagram/ U• (J)U• (J ∪ {∞})
U• ({j, ∞})/ U• ({j})

be Cartesian, which follows immediately from criterion (400 ) of Proposition
6.1.2.6.
Remark 6.1.3.18. Assertion (2) of Lemma 6.1.3.17 has a converse: if α isa Cartesian transformation, then U• is a groupoid object of X. This can be
deduced easily by examining the proof of Proposition 6.1.2.6, but we will
not have need of it.
Proposition 6.1.3.19. Let X be an ∞-category satisfying the equivalentconditions of Theorem 6.1.3.9. Then X satisfies the ∞-categorical Giraud
axioms:(i) The ∞-category X is presentable.
(ii) Colimits in X are universal.
(iii) Coproducts in X are disjoint.
(iv) Every groupoid object of X is effective.
Proof. Axioms (i) and (ii) are obvious. To prove (iii), let us consider anarbitrary pair of objects X, Y ∈ X and let ∅ denote an initial object of X.
Let f : ∅ → X be a morphism (unique up to homotopy since ∅ is initial).
We observe that id∅ is an initial object of OX . Form a pushout diagram
id∅

f

/ idY

α

β0

β
α0

/ g

in OX . It is clear that α is a Cartesian transformation, and Lemma 6.1.3.6
implies that β is Cartesian as well. Invoking condition (2) of Theorem 6.1.3.9,
we deduce that α0 is a Cartesian transformation. But α0 can be identified
with a pushout diagram
/Y
∅

X

/X

`

Y.

It remains to prove that every groupoid object in X is effective. Let U• bea groupoid object of X and let U • : N(∆+ )op → X be a colimit of U• . LetU•0 : N(∆+ )op → X be the result of composing U • with the “shift” functor∆+ → ∆+

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CHAPTER 6a
J 7→ J{∞}.
(In other words, U•0 is the shifted simplicial object given by Un0 = Un+1 .)Lemma 6.1.3.17 implies that U•0 is a colimit diagram in X. We have a transformation α : U•0 → U • . Since U• is a groupoid, α = α| N(∆)op is a Cartesiantransformation (Lemma 6.1.3.17 again). Applying (4), we deduce that α isa Cartesian transformation. In particular, we conclude that
0
/ U−1
U00

U0


/ U −1is a pullback diagram in X. But this diagram can be identified with
/ U0
U1

U0


/ U −1 ,so that U• is effective by Proposition 6.1.2.11.
Corollary 6.1.3.20. Every groupoid object of S is effective.
6.1.4 Free Groupoids
Let X be an ∞-category which satisfies the ∞-categorical Giraud axioms (i)through (iv) of Theorem 6.1.0.6. We wish to prove that X is an ∞-topos. It
is clear that any proof will need to make use of the full strength of axioms (i)through (iv); in particular, we will need to apply (iv) to a class of groupoid
objects of X which are not obviously effective. The purpose of this section isto describe a construction which will yield nontrivial examples of groupoid
objects and to deduce a consequence (Proposition 6.1.4.2) which we will use
in the proof of Theorem 6.1.0.6.
Definition 6.1.4.1. Let f : X → Y be a functor between ∞-categories
which admit finite limits. Let Z be an object of X. We will say that f is left
exact at Z if, for every pullback square
/Y
W

X


/Z

in X, the induced square

is a pullback in Y.

f (W )/ f (Y )


f (X)


/ f (Z)highertopoi

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We can now state the main result of this section:
Proposition 6.1.4.2. Let X and Y be presentable ∞-categories and let f :X → Y be a functor which preserves small colimits. Suppose that groupoid
objects in X and Y are effective. Let
// U0

U1

/ U−1

s

be a coequalizer diagram in X and let
X

/ U0


U0



s

s

/ U−1

be a pullback diagram in X. Suppose that f is left exact at U0 . Then theassociated diagramf (X)/ f (U0 )
f (U0 )

/ f (U−1 )ssis a pullback square in Y.
Before giving the proof, we must establish some preliminary results.
Lemma 6.1.4.3. Let X and Y be ∞-categories which admit finite limits, let
f : X → Y be a functor, and let U• be a groupoid object of X. Suppose that f
is left exact at U0 . Then f ◦ U• is a groupoid object of Y.
Proof. This follows immediately from characterization (400 ) given in Proposition 6.1.2.6.
Let X be a presentable ∞-category. We define a simplicial resolution in
X to be an augmented simplicial object U•+ : N(∆+ )op → X which is acolimit of the underlying simplicial object U• = U•+ | N(∆)op . We let Res(X)denote the full subcategory of X∆+ spanned by the simplicial resolutions.
Note that since every simplicial object of X has a colimit, the restriction
functor Res(X) → X∆ is a trivial fibration and therefore an equivalenceof ∞-categories. We will say that a simplicial resolution U•+ is a groupoid
resolution if the underlying simplicial object U• is a groupoid object of X.
We will say that a map f : U•+ → V•+ of simplicial resolutions exhibits V•+
as the groupoid resolution generated by U•+ if V•+ is a groupoid resolution
and the induced map
MapRes(X) (V•+ , W•+ ) → MapRes(X) (U•+ , W•+ )is a homotopy equivalence for every groupoid resolution W•+ ∈ Res(X).

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Remark 6.1.4.4. Let X be a presentable ∞-category. Then for every simplicial resolution U•+ in X, there is a map f : U•+ → V•+ which exhibitsV•+ as the groupoid resolution generated by U•+ . In view of the equivalence
Res(X) → X∆ , this is equivalent to the assertion that Gpd(X) is a localization of X∆ . This follows from Proposition 6.1.2.6 together with Lemmas
5.5.4.18 and 5.5.4.19.
Lemma 6.1.4.5. Let X be a presentable ∞-category and let f : U•+ → V•+ bea map of simplicial resolutions which exhibits V•+ as the groupoid resolution
generated by U•+ . Let W•+ be an augmented simplicial object of X such thatthe underlying simplicial object W• ∈ X∆ is a groupoid. Composition with f
induces a homotopy equivalence
MapX∆ (V•+ , W•+ ) → MapX∆ (U•+ , W•+ ).
++Proof. Let |W• | be a colimit of W• . Then we have a commutative diagram
MapX∆ (V•+ , |W• |)/ MapX
MapX∆ (V•+ , W•+ )/ MapX

+

+

∆+

∆+

(U•+ , |W• |)
(U•+ , W•+ )where the vertical maps are homotopy equivalences (since U•+ and V•+ are
resolutions) and the upper horizontal map is a homotopy equivalence (since|W• | is a groupoid resolution).
Lemma 6.1.4.6. Let X be a presentable ∞-category. Suppose that f : U•+ →V•+ is a map in Res(X) which exhibits V•+ as the groupoid resolution generated by U•+ . Then f induces equivalences U−1 → V−1 and U0 → V0 .
−1
Proof. Let ∆≤0+ be the full subcategory of ∆+ spanned by the objects ∆≤0and ∆0 . Let j : ∆+ → ∆+ denote the inclusion functor and let j ∗ : X∆+ →OX be the associated restriction functor. We wish to show that j ∗ (f ) is anequivalence. Equivalently, we show that for every W ∈ OX , composition withj ∗ (f ) induces a homotopy equivalence

MapOX (j ∗ V•+ , W ) → MapOX (j ∗ U•+ , W ).
Let j∗ be a right adjoint to j ∗ (a right Kan extension functor). It will
suffice to prove that composition with f induces a homotopy equivalence
MapX∆ (V•+ , j∗ W ) → MapX∆ (U•+ , j∗ W ).
++The augmented simplicial object j∗ W is a Čech nerve, so that the underlying simplicial object of j∗ W is a groupoid by Proposition 6.1.2.11. We now
conclude by applying Lemma 6.1.4.5.
Let I denote the subcategory of ∆+ spanned by the objects ∅, [0], and [1],where the morphisms are given by injective maps of linearly ordered sets.
This category may be depicted as follows:/ [0]// [1].
∅

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We let I0 denote the full subcategory of I spanned by the objects [0] and[1]. We will say that a diagram N(I)op → X is a coequalizer diagram if it isa colimit of its restriction to N(I0 )op → X.
Let i denote the inclusion I ⊆ ∆+ and let i∗ denote the restriction functor
X∆+ → Fun(N(I)op , X). If X is a presentable ∞-category, then i∗ has a left
adjoint i! (a left Kan extension).
Lemma 6.1.4.7. Let X be a presentable ∞-category. The left Kan extension i! : Fun(N(I)op , X) → X∆+ carries coequalizer diagrams to simplicial
resolutions.
Proof. We have a commutative diagram of inclusions of subcategories
j0

I0
i0

/I
i


∆

j


/ ∆+

which gives rise to a homotopy commutative diagram of ∞-categories
Fun(N(I0 )op , X)

j!0

/ Fun(N(I)op , X)

i0!


X∆



i!

/ X∆+

j!

in which the morphisms are given by left Kan extensions. An object U ∈Fun(N(I)op X) is a coequalizer diagram if and only if it lies in the essential
image of j!0 . In this case, i! U lies in the essential image of i! ◦ j!0 ' j! ◦ i0! ,which is contained in the essential image of j! : namely, the resolutions.
Lemma 6.1.4.8. Let X be a presentable ∞-category and suppose we are
given a diagram U : N(I)op → C which we may depict as
U1

// U

/ U−1 .

0

Let V• = i! U ∈ X∆+ be a left Kan extension of U along i : N(J)op → ∆op+.
Then the augmentation maps V0 → V−1 and U0 → U−1 are equivalent in the∞-category OX .
Proof. This follows from Proposition 4.3.3.8 since
HomI (∆i , •) ' Hom∆+ (∆i , •)for i ≤ 0.
Proof of Proposition 6.1.4.2. Let U : N(I)op → C be a coequalizer diagramin X, which we denote by
U1/ / U0s/ U−1 ,

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and form a pullback square
X

/ U0


U0



s

s

/ U−1 .

Let V• = i! U ∈ X∆+ be a left Kan extension of U . According to Lemma
6.1.4.7, V• is a simplicial resolution. We may therefore choose a map V• →W• which exhibits W• as the groupoid resolution generated by V• (Remark
6.1.4.4). Since every groupoid object in X is effective, W• is a Čech nerve. It
follows from the characterization given in Proposition 6.1.2.11 that there isa pullback diagram
/ W0
W1

W0


/ W−1

in X. Using Lemma 6.1.4.8 and Lemma 6.1.4.6, we see that this diagram isequivalent to the pullback diagram
/ U0
X

U0

s



s

/ U−1 .

It therefore suffices to prove that the induced diagram/ f (W0 )f (W1 )f (W0 )

/ f (W−1 )is a pullback. We make a slightly stronger claim: the augmented simplicial
object f ◦ W• is a Čech nerve. Since every groupoid object in Y is effective,
it will suffice to prove that f ◦ W• is a groupoid resolution. Since f preserves
colimits, it is clear that f ◦ W• is a simplicial resolution. It follows from
Lemma 6.1.4.3 that the underlying simplicial object of f ◦ W• is a groupoid.

6.1.5 Giraud’s Theorem for ∞-Topoi
In this section, we will complete the proof of Theorem 6.1.0.6 by showingthat every ∞-category X which satisfies the ∞-categorical Giraud axioms (i)through (iv) arises as a left exact localization of an ∞-category of presheaves.
Our strategy is simple: we choose a small category C equipped with a functor
f : C → X. According to Theorem 5.1.5.6, we obtain a colimit-preserving
functor F : P(C) → X which extends f up to homotopy. We will apply
Proposition 6.1.4.2 to show that (under suitable hypotheses) F is a left
exact localization functor (Proposition 6.1.5.2).

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Lemma 6.1.5.1. Let X be a presentable ∞-category in which colimits areuniversal and coproducts are disjoint.
Let {φi : Zi → Z}i∈I be a family of morphisms in X which exhibit Z as acoproduct of the family of objects {Zi }i∈I . Let
W

Zj

α

/ Zi

φj


/Z

φi

be a square diagram in X. Then
(1) If i 6= j, the diagram is a pullback square if and only if W is an initial
object of X.
(2) If i = j, the diagram is a pullback square if and only if α is an equivalence.
Proof. Let Zi∨ be a coproduct for the objects {Zk }k∈I,k6=i and let ψ : Zi∨ →Z be a morphism such that each of the compositions
ψZk → Zi∨ → Z
is equivalent to Z. Then there is a pushout square
∅

Zi∨

β

/ Zi
φi

ψ


/ Z,

where ∅ denotes an initial object of X. Since coproducts in X are disjoint,
this pushout square is also a pullback.
Let φ∗i : X/Z → X/Zi denote a pullback functor. The above argumentshows that φ∗i (ψ) is an initial object of X/Zi . If j 6= i, then there is a mapof arrows φj → ψ in X/Z and therefore a map φ∗i (φj ) → φ∗i (ψ) in X/Zi .
Consequently, if α ' φ∗i (φj ), then W admits a map to an initial object of
X and is therefore itself initial by Lemma 6.1.3.6. This proves the “only if”
direction of (1). The converse follows from the uniqueness of initial objects.
Now suppose that i = j. We observe that idZ is a coproduct of φi and ψin the ∞-category X/Z . Since φ∗i preserves coproducts, we deduce that idZi
is a coproduct of φ∗ (φj ) : X → Zi and β : ∅ → Zi in X/Zi . Since β is an
initial object of X/Zi , we see that φ∗ (φj ) is an equivalence. The natural map
γ : α → φ∗i (φi ) corresponds to a commutative diagram
W
γ0

α

/ Zi
idZi

 φ∗i (φi ) / ZiXin the ∞-category X. Consequently, α is an equivalence if and only if γ0 is anequivalence, if and only if γ is an equivalence in X/Zi . This proves (2).

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Proposition 6.1.5.2. Let C be a small ∞-category which admits finite limits
and let X be an ∞-category which satisfies the ∞-categorical Giraud axioms(i) through (iv) of Theorem 6.1.0.6. Let F : P(C) → X be a colimit-preserving
functor. Suppose that the composition F ◦ j : C → X is left exact, where
j : C → P(C) denotes the Yoneda embedding. Then F is left exact.
Proof. According to Corollary 4.4.2.5, to prove that F is left exact, it will
suffice to prove that F preserves pullbacks and final objects. Since all final
objects are equivalent, to prove that F preserves final objects, it suffices toexhibit a single final object Z of P(C) such that F Y ∈ X is final. Let z bea final object of C (which exists by virtue of our assumption that C admits
finite limits). Then Z = j(z) is a final object of P(C) since j preserves limits
by Proposition 5.1.3.2. Consequently, F (Z) = f (z) is final since f is left
exact.
Let α : Y → Z be a morphism in P(C). We will say that α is good if forevery pullback square
W

/Y


X


/Z

α

in P(C), the induced square/ F (Y )F (W )

F (X)

F (α)

β


/ F (Z)is a pullback in X. Note that Lemma 4.4.2.1 implies that the class of goodmorphisms in P(C) is stable under composition.
We can rephrase the condition that a morphism α be good in terms of
the pullback functors α∗ : P(C)/Z → P(C)/Y , F (α)∗ : X/F (Z) → P(C)/F (Y ) .
Application of the functor F gives a map
t : F ◦ α∗ → F (α)∗ ◦ F
in the ∞-category of functors from P(C)/Z to X/F (Z) , and α is good if andonly if t is an equivalence. Note that t is a natural transformation of colimitpreserving functors. Since the image of the Yoneda embedding j : C → P(C)generates P(C) under colimits, it will suffice to prove that t is an equivalence
when evaluated on objects of the form β : j(x) → Z, where x is an object ofC.
Let us say that an object Z ∈ P(C) is good if every morphism α : Y → Z
is good. In other words, an object Z ∈ P(C) is good if F is left exact at Z in
the sense of Definition 6.1.4.1. By repeating the above argument, we deducethat Z is good if and only if every morphism of the form α : j(y) → Z isgood for y ∈ C.

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We next claim that for every object z ∈ C, the Yoneda image j(z) ∈ P(C)is good. In other words, we must show that for every pullback squareW/ j(y)j(x)
/ j(z)αβin P(C), the induced square
F (W )/ f (x)f (y)/ f (z)is a pullback in X. Since the Yoneda embedding is fully faithful, we may
suppose that α and β are the Yoneda images of morphisms x → z, y → z.
Since j preserves limits, we may reduce to the case where the first diagramis the Yoneda image of a pullback diagram in C. The desired result thenfollows from the assumption that f is left exact.
To complete the proof that F is left exact, it will suffice to prove thatevery object of P(C) is good. Because the Yoneda embedding j : C → P(C)generates P(C) under colimits, it will suffice to prove that the collectionof good objects of P(C) is stable under colimits. According to Proposition
4.4.3.3, it will suffice to prove that the collection of good objects of P(C) isstable under coequalizers and small coproducts.
We first consider the case of coproducts. Let {Zi }i∈I be a family of good
objects of P(C) indexed by a (small) set I and let {φi : Zi → Z}i∈I be afamily of morphisms which exhibit Z as a coproduct of the family {Zi }i∈I .
Suppose we are given a pullback diagramW/ j(y)


j(x)


/Z

α
β

in P(C). According to Proposition 5.1.2.2, evaluation at the object y induces
a colimit-preserving functor P(C) → S. Consequently, we have a homotopy
equivalenceaMapP(C) (j(y), Z) 'MapP(C) (j(y), Zi )i∈I

in the homotopy category H. Therefore we may assume that α factors as acomposition
α0

φij(y) → Zi → Z
for some i ∈ I. By assumption, the morphism α0 is good; it therefore suffices
to prove that φi is good. By a similar argument, we can replace β by a maphighertopoi

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φj : Zj → Z for some j ∈ I. We are now required to show that if
/ Zi

W0

φi


Zj

φj


/Z

is a pullback diagram in P(C), then
/ F (Zi )F (W 0 )F (φi )

F (φj )/ F (Z)F (Zj )

is a pullback diagram in X. Since F preserves initial objects, this follows
immediately from Lemma 6.1.5.1.
We now complete the proof by showing that the collection of good objectsof P(C) is stable under the formation of coequalizers. Let
// Z

Z1

/ Z−1

s
0

be a coequalizer diagram in P(C), and suppose that Z0 and Z1 are good. We
must show that any pullback diagramW/ j(y)


j(x)



α
β

/ Z−1

remains a pullback diagram after applying the functor F . The functor
P(C) → N(Set)
T 7→ HomhP(C) (j(x), T )can be written as a compositionπP(C) → S →0 N(Set),
where the first functor is given by evaluation at x. Both of these functors
commute with colimits. Consequently, we have a coequalizer diagram/ Hom// Hom(j(x), Z )(j(x), Z )Hom(j(x), Z )
hP(C)

1

0

hP(C)

hP(C)

−1

in the category of sets. In particular, the map α factors as a composition
α0sj(x) → Z0 → Z−1 .
Since we have already assumed that α0 is good, we can replace α by the map
s : Z0 → Z−1 in the above diagram. By a similar argument, we can replace
β : Y → Z−1 by the map s : Z0 → Z−1 . We now obtain the desired result
by applying Proposition 6.1.4.2.

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We are now ready to complete the proof of Theorem 6.1.0.6:
Proposition 6.1.5.3. Let X be an ∞-category. Suppose that X satisfies the∞-categorical Giraud axioms:(i) The ∞-category X is presentable.
(ii) Colimits in X are universal.
(iii) Coproducts in X are disjoint.
(iv) Every groupoid object of X is effective.
Then there exists a small ∞-category C which admits finite limits and anaccessible left exact localization functor P(C) → X. In particular, X is an∞-topos.
Proof. Let X be an ∞-topos. According to Proposition 5.4.7.4, there exists
a regular cardinal τ such that X is τ -accessible, and the full subcategoryXτ spanned by the τ -compact objects of X is stable under finite limits. Let
C be a minimal model for Xτ , so that there is an equivalence Indτ (C) →X. The proof of Theorem 5.5.1.1 shows that the inclusion Indτ (C) ⊆ P(C)
has a left adjoint L. The composition of L with the Yoneda embedding
C → P(C) can be identified with the Yoneda embedding C → Indτ (C) andtherefore preserves all limits which exist in C (Proposition 5.1.3.2). Applying
Proposition 6.1.5.2, we deduce that L is left exact, so that Indτ (C) is a left
exact localization (automatically accessible) of P(C). Since X is equivalent
to Indτ (C), we conclude that X is also an accessible left exact localization ofP(C).
6.1.6 ∞-Topoi and Classifying Objects
Let X be an ordinary category and let X be an object of X. Let Sub(X) denote the partially ordered collection of subobjects of X: an object of Sub(X)is an equivalence class of monomorphisms Y → X. If C is accessible, then
Sub(X) is actually a set. If X admits finite limits, then Sub(X) is contravariantly functorial in X: given a subobject Y → X and any map X 0 → X, thefiber product Y 0 = X 0 ×X Y is a subobject of X 0 . A subobject classifier isan object Ω of X which represents the functor Sub. In other words, Ω hasa universal subobject Ω0 ⊆ Ω such that every monomorphism Y → X fits
into a unique Cartesian diagram
/ Ω0
Y _
_

X


/ Ω.

(In this case, Ω0 is automatically a final object of C.)
Every topos has a subobject classifier. In fact, in the theory of elementary
topoi, the existence of a subobject classifier is taken as one of the axioms.

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Thus the existence of a subobject classifier is one of the defining characteristics of a topos. We would like to discuss the appropriate ∞-categoricalgeneralization of the theory of subobject classifiers. The ideas presented here
are due to Charles Rezk.
Definition 6.1.6.1. Let X be an ∞-category which admits pullbacks andlet S be a collection of morphisms of X which is stable under pullback. We
will say that a morphism f : X → Y classifies S if it is a final object of(S)
OX (see Notation 6.1.3.4). In this situation, we will also say that the object
Y ∈ X classifies S. A subobject classifier for X is an object which classifies
the collection of all monomorphisms in X.
Example 6.1.6.2. The ∞-category S of spaces has a subobject classifier:namely, the discrete space {0, 1} with two elements.
The following result provides a necessary and sufficient condition for the
existence of a classifying object for S:
Proposition 6.1.6.3. Let X be a presentable ∞-category in which colimits
are universal and let S be a class of morphisms in X which is stable underpullbacks. There exists a classifying object for S if and only if the following
conditions are satisfied:(1) The class S is local (Definition 6.1.3.8).
(2) For every object X ∈ X, the full subcategory of X/X spanned by theelements of S is essentially small.
Proof. Let s : Xop → b
S be a functor which classifies the right fibration(S)
OX → X. Then S has a classifying object if and only if s is a representablefunctor. According to the representability criterion of Proposition 5.5.2.2,
this is equivalent to the assertion that s preserves small limits, and theessential image of s consists of essentially small spaces. According to Lemma
6.1.3.7, s preserves small limits if and only if (1) is satisfied. It now suffices
to observe that for each X ∈ X, the space s(X) is essentially small if andonly if the full subcategory of X/X spanned by S is essentially small.
Using Proposition 6.1.6.3, one can show that every ∞-topos has a subobject classifier. However, in the ∞-categorical context, the emphasis on
subobjects misses the point. To see why, let us return to considering an ordinary category X with a subobject classifier Ω. By definition, for every object
X ∈ X, we may identify maps X → Ω with subobjects of X: that is, isomorphism classes of maps Y → X which happen to be monomorphisms. Evene such that an element
better would be an object classifier: that is, an object Ω
eof HomX (X, Ω) could be identified with an arbitrary map Y → X. But thisis an unreasonable demand: if Y → X is not a monomorphism, then there
may be automorphisms of Y as an object of X/X . It would be unnatural toignore these automorphisms. However, it is also not possible to take theme must be a set rather than a groupoid.
into account because HomX (X, Ω)highertopoi

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567∞-TOPOIIf we allow X to be an ∞-category, this objection loses its force. Informallyspeaking, we can consider the functor which associates to each X ∈ X themaximal ∞-groupoid contained in X/X (this is contravariantly functorial
in X provided that X has finite limits). We might hope that this functor isrepresentable by some Ω∞ ∈ X, which we would then call an object classifier.
Unfortunately, a new problem arises: it is generally unreasonable to ask
for the collection of all morphisms in X to be classified by an object of X
since this would require each slice X/X to be essentially small (Proposition
6.1.6.3). This is essentially a technical difficulty, which we will circumvent
by introducing a cardinality bound.
Definition 6.1.6.4. Let X be a presentable ∞-category. We will say that amorphism f : X → Y is relatively κ-compact if, for every pullback diagram
/X
X0
f0


Y0


/Y

f

such that Y 0 is κ-compact, X 0 is also κ-compact.
Lemma 6.1.6.5. Let X be a presentable ∞-category, κ a regular cardinal,J a κ-filtered ∞-category, and p : J. → X a colimit diagram. Let f : X → Y
be a morphism in X, where Y is the image under p of the cone point of J. .
For each α in J, let Yα = p(α) and form a pullback diagram
/X
Xα
f

fα


Yα

gα


/ Y.

Suppose that each fα is relatively κ-compact. Then f is relatively κ-compact.
Proof. Let Z be a κ-compact object of X and g : Z → Y a morphism. Since Z
is κ-compact and J is κ-filtered, there exists a 2-simplex of X corresponding
to a diagram
> Yα
Z

gα

/ Y.

g

We form a Cartesian rectangle ∆2 × ∆1 → X, which we will depict as
/X
/ Xα
Z0
f0


Z

fα


/ Yα

f


/ Y.

Since f 0 is a pullback of fα , we conclude that Z 0 is κ-compact. Lemma
4.4.2.1 implies that f 0 is also a pullback of f along g, so that f is relatively
κ-compact, as desired.

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Lemma 6.1.6.6. Let X be a presentable ∞-category in which colimits areuniversal. Let τ > κ be regular cardinals such that X is κ-accessible andthe full subcategory Xτ consisting of τ -compact objects of X is stable underpullbacks in X. Let α : σ → σ 0 be a Cartesian transformation between pushoutsquares σ, σ 0 : ∆1 × ∆1 → X, which we may view as a pushout square
f

α

/g

α0


/ g0

β0

β


f0

in Fun(∆1 , X). Suppose that f , g, and f 0 are relatively τ -compact. Then g 0
is relatively τ -compact.
Proof. Let C denote the full subcategory of Fun(∆1 × ∆1 , X) spanned by the
pushout squares and let Cτ = C ∩ Fun(∆1 × ∆1 , Xτ ). Since the class of τ compact objects of X is stable under pushouts (Corollary 5.3.4.15), we havea commutative diagramCτ/ Fun(Λ20 , Xτ )C/ Fun(Λ20 , X),

where the horizontal arrows are trivial fibrations (Proposition 4.3.2.15). The
proof of Proposition 5.4.4.3 shows that every object of Fun(Λ20 , X) can be
written as the colimit of a τ -filtered diagram in Fun(Λ20 , Xτ ). It follows thatσ 0 ∈ C can be obtained as the colimit of a τ -filtered diagram in Cτ . Since
colimits in X are universal, we conclude that the natural transformation α
can be obtained as a τ -filtered colimit of natural transformations αi : σi → σi0
in Cτ . Lemma 5.5.2.3 implies that the inclusion C ⊆ Fun(∆1 × ∆1 , X) is
colimit-preserving. Consequently, we deduce that g 0 can be written as a τ filtered colimit of morphisms {gi0 } determined by restricting {αi }. According
to Lemma 6.1.6.6, it will suffice to prove that each morphism gi0 is relatively
τ -compact. In other words, we may replace σ 0 by σi0 and thereby reduce tothe case where σ 0 belongs to Cτ . Since f, g, and f 0 are relatively τ -compact,
we conclude that σ|Λ20 takes values in Xτ . Since σ is a pushout diagram,
Corollary 5.3.4.15 implies that σ takes values in Xτ . Now we observe that g 0
is a morphism between τ -compact objects of X and therefore automaticallyrelatively τ -compact by virtue of our assumption that Xτ is stable underpullbacks in X.
Proposition 6.1.6.7. Let X be a presentable ∞-category in which colimits
are universal and let S be a local class of morphisms in X. For each regularcardinal κ, let Sκ denote the collection of all morphisms f which belong
to S and are relatively κ-compact. If κ is sufficiently large, then Sκ has aclassifying object.

highertopoi

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569∞-TOPOI

Proof. Choose κ0 such that X is κ0 -accessible. The restriction functor r :Fun((Λ22 )/ , X) → Fun(Λ22 , X) is accessible: in fact, it preserves all colimits(Proposition 5.1.2.2). Let g be a right adjoint to r (a limit functor); Proposition 5.4.7.7 implies that g is also accessible. Choose a regular cardinal
κ00 > κ0 such that g is κ00 -continuous and choose κ ≥ κ00 such that g carries
κ00 -compact objects of Fun(Λ22 , X) into Fun((Λ22 )/ , Xκ ). It follows that theclass of κ-compact objects of X is stable under pullbacks. We will show thatSκ has a classifying object.
We will verify the hypotheses of Proposition 6.1.6.3. First, we must showthat Sκ is local. For this, we will verify condition (3) of Lemma 6.1.3.7.
We begin by showing that Sκ is stable under small coproducts. Let {fα :Xα → Yα }α∈A be a small collectionof morphisms belonging to Sκ and let`
f : X → Y be a coproduct α∈A fα in Fun(∆1 , X). We wish to show thatf ∈ Sκ . Since S is local, we conclude that f ∈ S (using Lemma 6.1.3.7). It
therefore suffices to show that f is relatively κ-compact. Suppose we are givena κ-compact object Z ∈ X and a morphism g : Z → Y . Using Proposition
4.2.3.4 and Corollary 4.2.3.10, we `conclude that Y can be obtained as a
κ-filtered colimit of objects YA0 = α∈A0 Yα , where A0 ranges over the κsmall subsets of A. Since Z is κ-compact, we conclude that there exists afactorization
g0

g 00

Z → YA0 → Y
of g. Form a Cartesian rectangle ∆2 × ∆1 → X,
/X
/ XA
Z0
0


Z


/ Y.


/ YA0

`
Since S is local, we can identify XA0 with the coproduct α∈A0 Xα . Since
colimits are universal, we conclude that Z 0 is a coproduct of objects Zα0 =Xα ×Yα Z, where α ranges over A0 . Since each fα is relatively κ-compact,
we conclude that each Zα0 is κ-compact. Thus Z 0 , as a κ-small colimit of
κ-compact objects, is also κ-compact (Corollary 5.3.4.15).
We must now show that for every pushout diagram
f

α

/g

0


/ g0

β0

β


f0

α

in OX , if α and β are Cartesian transformations and f, f 0 , g ∈ Sκ , then α0
and β 0 are also Cartesian transformations and g 0 ∈ Sκ . The first assertion
follows immediately from Lemma 6.1.3.7 (since S is local), and we deduce
also that g 0 ∈ S. It therefore suffices to show that g is relatively κ-compact,which follows from Lemma 6.1.6.6.
It remains to show that, for each X ∈ X, the full subcategory of X/X
spanned by the elements of S is essentially small. Equivalently, we must

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CHAPTER 6(S)

show that the right fibration p : OX → X has essentially small fibers.
Let F : Xop → b
S classify p. Since S is local, F preserves limits. The fullsubcategory of b
S spanned by the essentially small Kan complexes is stable
under small limits, and X is generated by Xκ under small (κ-filtered) colimits.
Consequently, it will suffice to show that F (X) is essentially small when X
is κ-compact. In other words, we must show that there are only a bounded
number of equivalence classes of morphisms f : Y → X such that f ∈ Sκ .
We now observe that if f ∈ Sκ , then f is relatively κ-compact, so that Y
also belongs to Xκ . We now conclude by observing that the ∞-category Xκ
is essentially small.
We now give a characterization of ∞-topoi based on the existence of object
classifiers.
Theorem 6.1.6.8 (Rezk). Let X be a presentable ∞-category. Then X is
an ∞-topos if and only if the following conditions are satisfied:(1) Colimits in X are universal.
(2) For all sufficiently large regular cardinals κ, there exists a classifying
object for the class of all relatively κ-compact morphisms in X.
Proof. Assume that colimits in X are universal. According to Theorems
6.1.0.6 and 6.1.3.9, X is an ∞-topos if and only if the class S consistingof all morphisms of X is local. This clearly implies (2) in view of Proposition
6.1.6.7. Conversely, suppose that (2) is satisfied and let Sκ be defined as in
the statement of Proposition 6.1.6.7. Proposition 6.1.6.3 ensures that
S Sκ islocal for all sufficiently large regular cardinals κ. We note that S = Sκ . It
follows from Criterion (3) of Lemma 6.1.3.7 that S is also local, so that X is
an ∞-topos.

6.2 CONSTRUCTIONS OF ∞-TOPOI
According to Definition 6.1.0.4, an ∞-category X is an ∞-topos if and only
if X arises as an (accessible) left exact localization of a presheaf ∞-category
P(C). To complete the analogy with classical topos theory, we would like tohave some concrete description of the collection of left exact localizations ofP(C). In §6.2.1, we will study left exact localization functors in general andsingle out a special class which we call topological localizations. In §6.2.2, wewill study topological localizations of P(C) and show that they are in bijection with Grothendieck topologies on the ∞-category C by exact analogy withclassical topos theory. In particular, given a Grothendieck topology on C, one
can define an ∞-topos Shv(C) ⊆ P(C) of sheaves on C. In §6.2.3, we will characterize Shv(C) by a universal mapping property. Unfortunately, not every∞-topos X can be obtained as topological localization of an ∞-category ofpresheaves. Nevertheless, in §6.2.4 we will construct ∞-categories of sheaves

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571∞-TOPOI

which closely approximate X using the formalism of canonical topologies.
These ideas will be applied in §6.4 to obtain a classification theorem for
n-topoi.
6.2.1 Left Exact Localizations
Let X be an ∞-category. Up to equivalence, a localization L : X → Y isdetermined by the collection S of all morphisms f : X → Y in X such that
Lf is an equivalence in Y (Proposition 5.5.4.2). Our first result provides auseful criterion for testing the left exactness of L.
Proposition 6.2.1.1. Let L : X → Y be a localization of ∞-categories.
Suppose that X admits finite limits. The following conditions are equivalent:(1) The functor L is left exact.
(2) For every pullback diagram
X0
f0


Y0

/X

/Y

f

in X such that Lf is an equivalence in Y, Lf 0 is also an equivalence inY.
Proof. It is clear that (1) implies (2). Suppose that (2) is satisfied. We wish
to show that L is left exact. Let S be the collection of morphisms f in X
such that Lf is an equivalence. Without loss of generality, we may identify
Y with the full subcategory of X spanned by the S-local objects. Since thefinal object 1 ∈ X is obviously S-local, we have L1 ' 1. Thus it will suffice toshow that L commutes with pullbacks. We observe that given any diagram
X → Y ← Z, the pullback LX ×LY LZ is a limit of S-local objects of X
and therefore S-local. To complete the proof, it will suffice to show that thenatural map f : X ×Y Z → LX ×LY LZ belongs to S. We can write f as acomposition of maps
X ×Y Z → X ×LY Z → LX ×LY Z → LX ×LY LZ.
The last two maps are obtained from X → LX and Z → LZ by base change.
Assumption (2) implies that they belong to S. Thus it will suffice to showthat f 0 : X ×Y Z → X ×LY Z belongs to S. This map is a pullback of the
diagonal f 00 : Y → Y ×LY Y , so it will suffice to prove that f 00 ∈ S. Projection
to the first factor gives a left homotopy inverse g : Y ×LY Y → Y of f 00 ,so it suffices to prove that g ∈ S. But g is a base change of the morphism
Y → LY .
Proposition 6.2.1.2. Let X be a presentable ∞-category. Assume that colimits in X are universal and that the formation of pullbacks in X commuteswith filtered colimits (these conditions are satisfied, for example, if X is anhighertopoi

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CHAPTER 6∞-topos: see Example 7.3.4.7). Let S be a (small) set of morphisms in X,
and let S denote the smallest strongly saturated class of morphisms whichcontains S and is stable under pullbacks. Then S is generated (as a strongly
saturated class of morphisms) by a (small) set.
Proof. Choose a regular cardinal κ such that the domain and codomain ofeach morphism belonging to S are κ-compact. Enlarging κ if necessary, we
may assume that X is κ-accessible and that the full subcategory Xκ ⊆ X
is stable under the formation of finite limits (Proposition 5.4.7.4). Let S 0
be the smallest collection of morphisms of X which contains S and has thefollowing closure properties:(a) Given a pushout diagram
X

X0

f

f0

/Y

/ Y0

in Xκ such that f ∈ S 0 , the morphism f 0 also belongs to S 0 .
(b) Given a pullback diagram
X0

X

f0

f

/ Y0

/Y

in Xκ such that f ∈ S 0 , the morphism f 0 also belongs to S 0 .
(c) Given a morphism f : `X → Y in S 0 , the diagonal and codiagonal maps
X → X ×Y X and Y X Y → Y also belong to S 0 .
It is clear that the domain and codomain of every morphism in S 0 belongs
to Xκ , so that S 0 contains only a bounded number of morphisms up toequivalence. It is also clear that S 0 ⊆ S. To complete the proof, it will suffice
to show that S 0 generates S as a strongly saturated class of morphisms. Let
0
S denote the smallest strongly saturated class of morphisms containing S.
000
Then S ⊆ S . To prove that S ⊆ S , it will suffice to show that S is closed
under pullbacks. We first prove the following weaker assertion:(∗) Given a pullback diagram
X0

X
0

in X, if f ∈ S 0 then f 0 ∈ S .

f0

f

/ Y0

/Y

highertopoi

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573∞-TOPOITo prove this, we invoke the κ-accessibility of X to write Y 0 as a κ-filteredcolimit lim Yα0 , where each Yα0 is κ-compact. Since colimits in X are universal,−→
the map f 0 is a colimit of the induced maps Yα0 ×Y X → Yα0 , each of which
belongs to S 0 by virtue of (b).
(∗0 ) Suppose we are given a pullback diagram
f0

X0

X

/ Y0

/Y

f

0

in X, where f is a pushout of a morphism belonging to S 0 . Then f 0 ∈ S .
To prove (∗0 ), we choose a pushout diagram
X0

X

u

/ Y0

f


/Y

where u ∈ S 0 . Since colimits in X are universal, we deduce that f 0 is a0
pushout of the induced map X0 ×Y Y 0 → Y0 ×Y Y 0 , which belongs to S byassertion (∗).
Let L : X → S
following fact:

0 −1

X denote the localization functor. We will also need the(∗00 ) For every object X ∈ X, there exists an ordinal α and a transfinite
sequence of objects {Xβ }β≤α such that X0 = X, Xα = LX, Xγ 'limβ<γ Xβ when γ is a limit ordinal, and each of the maps Xβ → Xβ+1−→
is the pushout of a morphism belonging to S.
Assuming (∗00 ) for the moment, let us explain how to complete the proof.
Suppose we are given a pullback diagram
X0

f0


X

f

0

/ Y0

/ Y,
0

where f ∈ S . We wish to show that f 0 ∈ S . We can factor f 0 as a compositiong0g

X 0 ' X ×Y Y 0 → X ×LY Y 0 → Y 0 .
0

We will prove that both g and g 0 belong to S .
00Since f ∈ S , the map X → LY exhibits LY as an S -localization of X.
We may therefore choose a transfinite sequence
X0 → X1 → · · · → Xα = LYhighertopoi

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CHAPTER 6
0satisfying the requirements of (∗00 ). To prove that g 0 ∈ S , it will suffice to0show that each of the maps ρβ : X ×LY Y 0 → Xβ ×LY Y 0 belongs to S . We
will prove this using transfinite induction on β, the case β = 0 being trivial.
If β is a nonzero limit ordinal, this follows immediately from the inductive
hypothesis (since colimits in X are universal, we have ρβ ' limγ<β ργ , and−→0
S is closed under colimits). To carry out the inductive step, it suffices to0show that each of the maps Xβ ×LY Y 0 → Xβ+1 ×LY Y 0 belongs to S , which0follows from (∗ ).
We now prove that the morphism g : X ×Y Y 0 → X ×LY Y 0 belongs to0
S . Choose a transfinite sequence of morphisms
Y = Y0 → Y1 → · · · → Yα = LY
00

as in (∗ ). We will prove that each of the induced maps gβ : X ×Y Y 0 →0
X ×Yβ Y 0 belongs to S . We proceed by induction on β, the case β = 0 being
trivial. If β is a nonzero limit ordinal, then gβ is the colimit of the diagram{gγ }γ<β (since the formation of fiber products in X commutes with filteredcolimits), and the desired result follows from the inductive hypothesis. To
carry out the inductive step, we are reduced to proving that the morphism
q : X ×Yβ Y 0 → X ×Yβ+1 Y 00belongs to S . Using the fact that X is κ-accessible, we can write X and Y 0
as filtered colimits of κ-compact objects Xi and Yj0 . Since colimits in X are
universal, we conclude that q is a filtered colimit of morphisms
qij : Xi ×Yβ Yj0 → Xi ×Yβ+1 Yj0 .
0It will therefore suffice to show that each qij belongs to S .
Choose a pushout diagram
A

Yβ

u

/B

/ Yβ+1 ,where u ∈ S 0 . Since X is κ-compact, we can write Yβ as the colimit of adiagram {Yk }k∈K indexed by a κ-filtered partially ordered set K, where
each Yk is κ-compact. The object A is κ-compact, so that the map A → Yβ
factors through some Yk . Then the map` Yβ → Yβ+1 can be written as afiltered colimit of morphisms {Yl → Yl A B}l≥k . Enlarging k if necessary,
we may assume that the maps Xi → Yβ ← Yj0 also factor through Yk . Since
the formation of filtered colimits in X is left exact, we conclude that q is afiltered colimit of maps
qijk : Xi ×Yk Yj0 → Xi ×Yk `A B Yj0 .
0

We are therefore reduced to proving that each qijk belongs to S . Note thatqijk is a pullback of the diagonal map
δ : Yk → Yk ×Yk `A B Yk .

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575∞-TOPOI

Using (∗), we are reduced to proving that δ ∈ S 0 , which follows from (a) and(c).
It remains to prove (∗00 ). Let X ∈ X be an arbitrary object. For every
object X ∈ X, let U (X) denote the set of equivalence classes of pairs (u :A → B, v : A → X) where u is a morphism of X belonging to S 0 , and v isarbitrary. SetaaA(X) =A
B(X) =
B,(u:A→B,v)∈U (X)(u:A→B,v)∈U (X)`and let T (X) denote the pushout X A(X) B(X). Choosing a well-orderingof the set U (X), we see that the map X → T (X) can obtained as a transfinite
composition of pushouts of morphisms belonging to S 0 . We now define atransfinite sequence of objects {X α }α≤κ as follows: let X 0 = X, let X α+1 =T (X α ), and let X γ = limβ<γ X β when γ is a nonzero limit ordinal. The map−→
X = X 0 → X κ is a transfinite composition of morphisms of the form Y →T (Y ), hence a transfinite composition of pushouts of morphisms belonging
0
to S 0 . In particular, the morphism X → X κ belongs to S . To complete theproof of (∗00 ), it will suffice to show that X κ is S 0 -local (and therefore an0
S -localization of X). Let u : A → B be a morphism in S 0 ; we wish to showthat the induced map
θu : MapX (B, X κ ) → MapX (A, X κ )is a homotopy equivalence. We will prove that θu is n-connective for every
integer n ≥ −1; the proof proceeds by induction on n. If n ≥ 0, we can
invoke `the inductive hypothesis to deduce that θu is (n − 1)-connective. If
u0 : B A B → B is the codiagonal map, we are reduced to proving that
θu0 is (n − 1)-connective; this follows from the inductive hypothesis (sinceu0 ∈ S 0 by virtue of (c)). We are therefore reduced to the case n = −1: thatis, we must show that every map w : A → X κ factors through u. Since A isκ-compact, we may assume that w factors through a map w0 : A → X α forsome α < κ. By construction, we have a commutative diagram
/ Xα
A
u


B


/ X α+1 ,so that the composite map B → X α+1 → X κ gives the desired extension ofw.
Recall that a morphism f : Y → Z in an ∞-category X is a monomorphism
if it is a (−1)-truncated object of the ∞-category X/Z . Equivalently, f is amonomorphism if for every object X ∈ X, the induced map
MapX (X, Y ) → MapX (X, Z)
exhibits MapX (X, Y ) ∈ H as a summand of MapX (X, Z) in the homotopy
category H. If we fix Z ∈ X, then the collection of equivalence classes ofhighertopoi

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CHAPTER 6

monomorphisms Y → Z is partially ordered under inclusion. We will denote
this partially ordered collection by Sub(Z).
Proposition 6.2.1.3. Let X be a presentable ∞-category and let X be an
object of X. Then Sub(X) is a (small) partially ordered set.
Proof. By definition, the partially ordered set Sub(X) is characterized by
the existence of an equivalenceτ≤−1 X/X → N(Sub(X)).
Propositions 5.5.3.10 and 5.5.6.18 imply that N(Sub(X)) is presentable.
Consequently, there exists a small subset S ⊆ Sub(X) which generates
N(Sub(X)) under colimits. It follows that every element of Sub(X) can be
written as the supremum of a subset of S, so that Sub(X) is also small.
Definition 6.2.1.4. Let X be a presentable ∞-category and let S be astrongly saturated class of morphisms of X. We will say that S is topologicalif the following conditions are satisfied:(1) There exists S ⊆ S consisting of monomorphisms such that S generates
S as a strongly saturated class of morphisms.
(2) Given a pullback diagram
X0
f0


Y0

/X

/Y

f

in X for which f belongs to S, the morphism f 0 also belongs to S.
We will say that a localization L : X → Y is topological if the collection S of
all morphisms f : X → Y in X such that Lf is an equivalence is topological.
Proposition 6.2.1.5. Let X be a presentable ∞-category in which colimits
are universal and let S be a strongly saturated class of morphisms of X whichis topological. Then there exists a (small) subset S0 ⊆ S which consists ofmonomorphisms and generates S as a strongly saturated class of morphisms.
Proof. For every object U ∈ X, let Sub0 (U ) ⊆ Sub(U ) denote the collectionof equivalence classes of monomorphisms U 0 → U which belong to S. Choosea small collection of objects {Uα }α∈A which generates X under colimits.
For each α ∈ A and each element αe ∈ Sub0 (Uα ), choose a representative
monomorphism fαe : Vαe → Uα which belongs to S. Let
S0 = {fαe |α ∈ A, αe ∈ Sub0 (Uα )}.
It follows from Proposition 6.2.1.3 that S0 is a (small) set. Let S 0 denote thestrongly saturated class of morphisms generated by S0 . We will show thatS 0 = S.

highertopoi

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577∞-TOPOILet X0 be the full subcategory of X spanned by objects U with the following property: if f : V → U is a monomorphism and f ∈ S, then f ∈ S0 . By
construction, for each α ∈ A, Uα ∈ X0 . Since colimits in X are universal, it
is easy to see that X0 is stable under colimits in X. It follows that X0 = X,
so that every monomorphism which belongs to S also belongs to S0 . Since S
is generated by monomorphisms, we conclude that S = S0 , as desired.
Corollary 6.2.1.6. Let X be a presentable ∞-category. Every topologicallocalization L : X → Y is accessible and left exact.
6.2.2 Grothendieck Topologies and Sheaves in Higher Category
TheoryEvery ordinary topos is equivalent to a category of sheaves on some Grothendieck site. This can be deduced from the following pair of statements:(i) Every topos is equivalent to a left exact localization of some presheaf
opcategory SetC .
(ii) There is a bijective correspondence between left exact localizations of
op
SetC and Grothendieck topologies on C.
In §6.1, we proved the ∞-categorical analogue of assertion (i). Unfortunately, (ii) is not quite true in the ∞-categorical setting. In this section, we
will establish a slightly weaker statement: for every ∞-category C, there is abijective correspondence between Grothendieck topologies on C and topological localizations of P(C) (Proposition 6.2.2.17). Our first step is to introduce
the ∞-categorical analogue of a Grothendieck site. The following definition
is taken from [79]:Definition 6.2.2.1. Let C be an ∞-category. A sieve on C is a full subcategory of C(0) ⊆ C having the property that if f : C → D is a morphism in C
and D belongs to C(0) , then C also belongs to C(0) .
Observe that if f : C → D is a functor between ∞-categories and D(0) ⊆ D
is a sieve on D, then f −1 D(0) = D(0) ×D C is a sieve on C. Moreover, if f isan equivalence, then f −1 induces a bijection between sieves on D and sieves
on C.
If C ∈ C is an object, then a sieve on C is a sieve on the ∞-category C/C .
(0)

(0)

Given a morphism f : D → C and a sieve C/C on C, we let f ∗ C/C denote
(0)

(0)

the unique sieve on D such that f ∗ C/C ⊆ C/D and C/C determine the same
sieve on C/f .
A Grothendieck topology on an ∞-category C consists of a specification,for each object C of C, of a collection of sieves on C which we will refer toas covering sieves. The collections of covering sieves are required to possess
the following properties:(1) If C is an object of C, then the sieve C/C ⊆ C/C on C is a covering
sieve.

highertopoi

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CHAPTER 6(0)(2) If f : C → D is a morphism in C and C/C is a covering sieve on D,(0)then f ∗ C/C is a covering sieve on C.
(0)

(1)

(3) Let C be an object of C, C/C a covering sieve on C, and C/C anarbitrary sieve on C. Suppose that, for each f : D → C belonging to
(0)
(1)
(1)
the sieve C/C , the pullback f ∗ C/C is a covering sieve on D. Then C/C
is a covering sieve on C.
Example 6.2.2.2. Any ∞-category C may be equipped with the trivial(0)
topology in which a sieve C/C on an object C of C is covering if and only if(0)

C/C = C/C .
Remark 6.2.2.3. In the case where C is (the nerve of) an ordinary category,the definition given above reduces to the usual notion of a Grothendiecktopology on C. Even in the general case, a Grothendieck topology on C isjust a Grothendieck topology on the homotopy category hC. This is notcompletely obvious since for an object C of C, the functor
η : h(C/C ) → (hC)/C
is usually not an equivalence of categories. A morphism on the left hand sidecorresponds to a commutative triangle
/ D0

D


C

~

given by a specified 2-simplex σ : ∆2 → C (taken modulo homotopy), whileon the right hand side one requires only that the above diagram commute
up to homotopy: this amounts to requiring the existence of σ, but σ itself is
not taken as part of the data.
Although η need not be an equivalence of categories, η ∗ does induce abijection from the set of sieves on (hC)/C to the set of sieves on h(C/C ): for
this, it suffices to observe that η induces surjective maps
Homh( C/C ) (D, D0 ) → Hom(hC)/C (D, D0 )on morphism sets, which is obvious from the description given above.
The main objective of this section is to prove that for any (small) ∞category C, there is a bijective correspondence between Grothendieck topologies on C and (equivalence classes of) topological localizations of P(C). We begin by establishing a correspondence between sieves on C and (−1)-truncatedobjects of P(C). For each object U ∈ P(C), let C(0) (U ) ⊆ C be the full subcategory spanned by those objects C ∈ C such that U (C) 6= ∅. It is easy to
see that C(0) (U ) is a sieve on C. Conversely, given a sieve C(0) ⊆ C, there is aunique map C → ∆1 such that C(0) is the preimage of {0}. This construction

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579∞-TOPOI

determines a bijection between sieves on C and functors f : C → ∆1 , andwe may identify ∆1 with the full subcategory of Sop spanned by the objects∅, ∆0 ∈ Kan. Since every (−1)-truncated Kan complex is equivalent to either∅ or ∆0 , we conclude:
Lemma 6.2.2.4. For every small ∞-category C, the construction U 7→
C(0) (U ) determines a bijection between the set of equivalence classes of (−1)truncated objects of P(C) and the set of all sieves on C.
We now introduce a relative version of the above construction. Let C be asmall ∞-category as above and let j : C → P(C) be the Yoneda embedding.
Let C ∈ C be an object and let i : U → j(C) be a monomorphism in
P(C). Let C/C (U ) denote the full subcategory of C spanned by those objects
f : D → C of C/C such that there exists a commutative triangle
j(f )

j(D)

i

!

/ j(C)=

U.

It is easy to see that C/C (U ) is a sieve on C. Moreover, it is clear that if
i : U → j(C) and i0 : U 0 → j(C) are equivalent subobjects of j(C), thenC/C (U ) = C/C (U 0 ).
Proposition 6.2.2.5. Let C be a small ∞-category containing an object C
and let j : C → P(C) be the Yoneda embedding. The construction describedabove yields a bijection(i : U → j(C)) 7→ C/C (U )from Sub(j(C)) to the set of all sieves on C.
Proof. Use Corollary 5.1.6.12 to reduce to Lemma 6.2.2.4.
Definition 6.2.2.6. Let C be a (small) ∞-category equipped with a Grothendieck topology. Let S be the collection of all monomorphisms U → j(C)(0)
which correspond to covering sieves C/C ⊆ C/C . An object F ∈ P(C) isa sheaf if it is S-local. We let Shv(C) denote the full subcategory of P(C)spanned by S-local objects.
Proposition 6.2.2.7. Let C be a (small) ∞-category equipped with a Grothendieck topology. Then Shv(C) is a topological localization of P(C). In particular, Shv(C) is an ∞-topos.
In order to prove Proposition 6.2.2.7, we need to understand the sheafification construction P(C) → Shv(C) more precisely. This will require a few
explicit constructions.
Notation 6.2.2.8. Let C be an ordinary category equipped with a Grothendieck topology. We define a category Cov(C) as follows:highertopoi

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CHAPTER 6(0)• The objects of Cov(C) are pairs (C, C/C ), where C ∈ C is an object(0)and C/C ⊆ C/C is a covering sieve.
(0)

(0)

• Given a pair of objects (C, C/C ), (D, C/D ) ∈ Cov(C), a morphism from
(0)

(0)

(C, C/C ) to (D, C/D ) is a morphism u : C → D in the category C with(0)

property that, for every morphism C 0 → C belonging to the sieve C/C ,(0)

the composite map C 0 → C → D belongs to the sieve C/D .
If C is an ∞-category equipped with a Grothendieck topology, then theordinary category hC inherits a Grothendieck topology (Remark 6.2.2.3). We
let Cov(C) denote the fiber product
C ×N(hC) N(Cov(hC)).
Then Cov(C) is an ∞-category, whose objects can be identified with pairs(0)(0)(C, C/C ), where C ∈ C is an object and C/C is a covering sive on C. We
have an evident forgetful functor ρ : Cov(C) → C. This forgetful functor is a
Cartesian fibration, whose fiber over an object C ∈ C can be identified with
the nerve of the partially ordered set of covering sieves on C.
Construction 6.2.2.9. Let C be an ∞-category equipped with a Grothendieck topology. We let C+ denote the full subcategory of the fiber product(0)
Fun(∆1 , C) ×Fun({1},C) Cov(C) spanned by those pairs (u : C 0 → C, C/C )(0)where u belongs to the sieve C/C ⊆ C/C . We have evident forgetful functors
e : C+ → C and π : C+ → Cov(C), given on objects by
(0)

(0)

e(u : C 0 → C, C/C ) = C 0(0)π(u : C 0 → C, C/C ) = (C, C/C ).

Let ρ : Cov(C) → C be the forgetful functor of Notation 6.2.2.8. Then e, π,and ρ induce pullback functorse∗ : P(C) → P(C+ ) π ∗ : P(Cov(C)) → P(C+ ) ρ∗ : P(C) → P(Cov(C)).
We let π∗ denote a right adjoint to π (given by right Kan extension alongπ) and ρ! a left adjoint to ρ (given by left Kan extension along ρ). The
composition
e∗

π

π

P(C) → P(C+ ) →∗ P(Cov(C)) →! P(C)determines a functor from P(C) to itself, which we will denote by F 7→ F† .
Remark 6.2.2.10. In the situation of Construction 6.2.2.9, the functor
π : C+ → Cov(C) is a coCartesian fibration. It follows that, for every object(0)(C, C/C ) ∈ Cov(C), the inclusion
(0)

(0)

i : C/C ' π −1 {(C, C/C )} ,→ C+ ×Cov(C) Cov(C)/(C,C(0) )/C

is cofinal. We conclude that for every presheaf F ∈ P(C+ ), the presheafπ∗ F ∈ P(Cov(C)) is given on objects by the formula(0)(π∗ F)(C, C/C ) '(0)

lim F(C 0 → C, C/C ).
←−(0)

C 0 ∈C/Chighertopoi

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Remark 6.2.2.11. In the situation of Construction 6.2.2.9, the functor
ρ : Cov(C) → C is a Cartesian fibration. It follows that, for every object
C ∈ C, the inclusioni : (ρ−1 {C})op → (Cov(C) ×C CC/ )op
is cofinal. We conclude that for every presheaf F ∈ P(Cov(C)), the presheaf
ρ! F ∈ P(C) is given on objects by the formula(0)(ρ! F)(C) = lim F(C, C/C ),
−→
(0)
C/C

where the colimit is taken over the collection of covering sieves on C, partially ordered by reverse inclusion. Since the collection of covering sieves isclosed under intersection, it is filtered by reverse inclusion. Using Proposition 5.3.3.3, we deduce that the functor ρ! : P(Cov(C)) → P(C) is left exact.
Since the functors e∗ : P(C) → P(C+ ) and π∗ : P(C+ ) → P(Cov(C)) are
obviously left exact, we conclude that the construction F 7→ F† is left exact.
Remark 6.2.2.12. Let C be an ∞-category equipped with a Grothendiecktopology. Combining Remarks 6.2.2.10 and 6.2.2.11, we see that the construction F 7→ F† is given informally by the formulaF† (C) = lim lim F(C 0 ),
−→
←−(0)
(0)
C/C C 0 ∈C/C

where the colimit is taken over the collection of all covering sieves on C.
Remark 6.2.2.13. Let C be an ∞-category equipped with a Grothendiecktopology. The forgetful functor ρ : Cov(C) → C has an evident section s,
given by C 7→ (C, C/C ). Similarly, the composite functor ρ ◦ π : C+ → C isequipped with a section s+ , given by s+ (C) = (idC : C → C, C/C ). Notethat for each C ∈ C, the functor s+ carries C to a final object of the fiber
C+ ×Cov(C) {s(C)}. Combining this observation with Remark 6.2.2.10, we
deduce that the canonical map s∗ π∗ → s+∗ is an equivalence of functors
from P(C+ ) to P(C).
For every presheaf F ∈ P(C), we have a canonical map
F ' s+∗ e∗ F ' s∗ π∗ e∗ F ' ρ! s! s∗ π∗ e∗ F → ρ! π∗ e∗ F = F† .
This construction is functorial in F, and determines a natural transformationfrom the identity functor idP(C) to the functor F 7→ F† .
Lemma 6.2.2.14. Let C be an ∞-category equipped with a Grothendiecktopology, let F ∈ P(C) and let G ∈ Shv(C). Then composition with the canonical map F → F† of Remark 6.2.2.13 induces a homotopy equivalenceθ : MapP(C) (F† , G) → MapP(C) (F, G).
Proof. Let s : C → Cov(C) be the section of Remark 6.2.2.13, so that θ isgiven by the composition
MapP(C) (F† , G) ' MapP(Cov(C)) (π∗ e∗ F, ρ∗ G)→ MapP(C) (s∗ π∗ e∗ F, s∗ ρ∗ G)' MapP(C) (F, G).

highertopoi

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To prove that this map is a homotopy equivalence, it suffices to show thatρ∗ G is a right Kan extension of its restriction to the essential image of s.
Unwinding the definitions, we see that this condition is equivalent to ourhypothesis that G is a sheaf.
Remark 6.2.2.15. Let C be an ∞-category, let F : Cop → S be a functor,(0)
let C ∈ C be an object and let C/C ⊆ C/C be a sieve on C. For each map
(0)

(0)

f : D → C in C, we let f ∗ C/C denote the inverse image of C/C in C/D . The
construction(f : D → C) 7→

lim F(E)←− (0)

E∈f ∗ C/C

determines a functor F0 : C/C → S. Assume now that C is equipped with a(0)Grothendieck topology and that C/C is a covering sieve on C. Then we haveevident natural transformations
F | C/C → F0 → F† | C/C ,whose composition is induced by the natural transformation F → F† ofRemark 6.2.2.13. We have an evident commutative diagram
F(C)/ F0 (C)/ F† (C)





u



limD∈C(1) F(D)
←−
/C

/ lim
F0 (D)←−D∈C(0)
/C

u†

/ lim
F† (D)←−D∈C(1)/C

A simple cofinality argument shows that the vertical map in the middle is a
homotopy equivalence.
Proof of Proposition 6.2.2.7. By definition, Shv(C) = S −1 P(C), where S isthe collection of all monomorphisms i : U → j(C) which correspond tocovering sieves on C ∈ C. To prove that Shv(C) is a topological localizationof P(C), it will suffice to show that the localization functor P(C) → Shv(C)is left exact. Choose a regular cardinal κ with the following property:(0)(∗) For every object C ∈ C and every covering sieve C/C ⊆ C/C , theconstruction F 7→ limC 0 ∈C(0) F(C 0 ) determines a functor P(C) → S←−/Cwhich commutes with κ-filtered colimits.
Let [κ] denote the linearly ordered set of ordinals α ≤ κ. We construct a
diagram N[κ] → Fun(P(C), P(C)), which we will informally denote by α 7→Tα , using transfinite induction. Let T0 = idP(C) , and let Tγ = limβ<α Tβ−→when γ is a limit ordinal. If β < κ, we define Tβ+1 by the formula
Tβ+1 (F) = (Tβ F)† .
Using Proposition 5.3.3.3 and Remark 6.2.2.11, we deduce by induction on βthat each of the functors Tβ : P(C) → P(C) is left exact. We will complete thehighertopoi

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583∞-TOPOI

proof by showing that Tκ is left adjoint to the inclusion Shv(C) ,→ P(C). We
have an evident natural transformation idP(C) = T0 → Tκ . It will therefore
suffice to prove that for each F ∈ P(C), the induced map u : F → Tκ (F)exhibits Tκ (F) as an S-localization of F. We proceed in two steps:(a) The map u : F → Tκ (F) is an S-equivalence. That is, for every object
G ∈ Shv(C), composition with u induces a homotopy equivalence
MapP(C) (Tκ (F), G) → MapP(C) (F, G).
In fact, we prove more generally that each of the maps
MapP(C) (Tα (F), G) → MapP(C) (F, G)
is a homotopy equivalence. The proof proceeds by induction on α.
When α = 0 there is nothing to prove, and if α is a nonzero limit
ordinal we invoke our assumption that
Tα (F) ' lim Tβ (F).
−→β<αTo carry out the inductive step, it suffices to show that the maps
MapP(C) (Tα+1 (F), G) → MapP(C) (Tα (F), G)
are homotopy equivalences, which follows from Lemma 6.2.2.14.
(b) The object Tκ (F) is a sheaf. To prove this, suppose we are given an(0)
object C ∈ C and a covering sieve C/C ⊆ C/C . We wish to show thatthe canonical map θ : (Tκ F)(C) → limC 0 ∈C(0) (Tκ F)(C 0 ) is a homotopy←−/Cequivalence. Using (∗), we deduce that the map θ can be identified witha filtered colimit of maps
θα : (Tα F)(C) →lim (Tα F)(C 0 ).
←−(0)

C 0 ∈C/C

We will prove that, for each ordinal α < κ, the colimit of the sequenceof maps {θα+n }n≥0 is a homotopy equivalence. To prove this, it suffices
to show that each of the transition maps θα+n → θα+n+1 factors as acomposition
θα+n → φ → θα+n+1 ,where φ is an equivalence in S. This follows immediately from Remark
6.2.2.15.

The next lemma ensures us that we can recover a Grothendieck topology
on C from its ∞-category of sheaves Shv(C) ⊆ P(C).
Lemma 6.2.2.16. Let C be a (small ) ∞-category equipped with a Grothendieck topology and let L : P(C) → Shv(C) denote a left adjoint to the inclusion. Let j : C → P(C) denote the Yoneda embedding and let i : U → j(C) be(0)a monomorphism corresponding to a sieve C/C on C. Then Li is an equiv(0)alence if and only if C/C is a covering sieve.

highertopoi

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Proof. It is clear that if C/C is a covering sieve, then Li is an equivalence. Conversely, suppose that Li is an equivalence. Then τ≤0 (Li) is anequivalence. In view of Proposition 5.5.6.28, we can identify τ≤0 (Li) withL(τ≤0 i). The morphism τ≤0 i can be identified with a monomorphism η :F ⊆ HomhC (•, C) in the ordinary category of presheaves of sets on hC,
where
(0)

F(D) = {f ∈ HomhC (D, C) : f ∈ C/C }.
If η becomes an equivalence after sheafification, then the identity map idC :C → C belongs to F(C) locally; in other words, there exists a collection ofmorphisms {fα : Cα → C} which generate a covering sieve on C such that(0)(0)
each fα belongs to F(Cα ) and therefore to C/C . It follows that C/C contains
a covering sieve on C and is therefore itself covering.
We may summarize the results of this section as follows:
Proposition 6.2.2.17. Let C be a small ∞-category. There is a bijective correspondence between Grothendieck topologies on C and (equivalence classesof) topological localizations of P(C).
Proof. According to Proposition 6.2.2.7, every Grothendieck topology on C
determines a topological localization Shv(C) ⊆ P(C). Lemma 6.2.2.16 showsthat two Grothendieck topologies which determine the same ∞-categoriesof sheaves must coincide. To complete the proof, it will suffice to show thatevery topological localization of P(C) arises in this way. Let S be a strongly
saturated collection of morphisms in P(C) and suppose that S is topological.
Let S ⊆ S be the collection of all monomorphisms U → j(C) which belong
to S, where j : C → P(C) denotes the Yoneda embedding. Since the objects{j(C)}C∈C generate P(C) under colimits, and colimits in P(C) are universal,
we conclude that every monomorphism in S is a colimit of elements of S.
Since S is generated by monomorphisms, we conclude that S is generated
by S.
(0)
Let us say that a sieve C/C ⊆ C/C on an object C ∈ C is covering if thecorresponding monomorphism U → j(C) belongs to S. We will show thatthe collection of covering sieves determines a Grothendieck topology on C.
−1
Granting this, we observe that S P(C) is the ∞-category Shv(C) ⊆ P(C)of sheaves with respect to this Grothendieck topology, which will complete
the proof.
We now verify the axioms (1) through (3) of Definition 6.2.2.1:(1) Every sieve of the form C/C ⊆ C/C is covering since every identity map
idj(C) : j(C) → j(C) belongs to S.
(0)(2) Let f : C → D be a morphism in C and let C/D ⊆ C/D be a covering
sieve corresponding to a monomorphism i : U → j(D) which belongs to(0)
S. Then f ∗ C/C ⊆ C/C corresponds to a monomorphism u : U 0 → j(C)highertopoi

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which is a pullback of i along j(f ) and therefore belongs to S (since S
is stable under pullbacks).
(0)(3) Let C be an object of C, C/C a covering sieve on C corresponding to a(1)

monomorphism i : U → j(C) which belongs to S, and C/C an arbitrarysieve on C corresponding to a monomorphism v : U 0 → j(C). Suppose(0)that, for each f : D → C belonging to the sieve C/C , the pullback(1)

f ∗ C/C is a covering sieve on D. Since j 0 : C/C → P(C)/j(C) is a fully
faithful embedding which generates P(C)/j(C) under colimits (see theproof of Corollary 5.1.6.12), we conclude there is a diagram K → C/Csuch that j 0 ◦ K has colimit i0 . Since colimits in P(C) are universal, we
conclude that the map v 0 : U ×j(C) U 0 → U is a colimit of morphismsof the form j(D) ×j(C) U 0 → j(D), which belong to S by assumption.
Since S is stable under colimits, we conclude that i00 belongs to S. We
now have a pullback diagram
U ×j(C) U 0

v0

/U

v


/ j(C).

u0


U0

u

By assumption, u ∈ S. Thus v ◦u0 ∼ u◦v 0 ∈ S. Since u0 is a pullback of(1)u, we conclude that u0 ∈ S, so that v ∈ S. This implies that C/C ⊆ C/C
is a covering sieve, as we wished to prove.

For later use, we record the following characterization of initial objects in∞-categories of sheaves:
Proposition 6.2.2.18. Let C be a small ∞-category equipped with a Grothendieck topology and let C0 ⊆ C denote the full subcategory spanned by thoseobjects C ∈ C such that ∅ ⊆ C/C is a covering sieve on C. An object
F ∈ Shv(C) is initial if and only if it satisfies the following conditions:(1) If C ∈ C0 , then F(C) is contractible.
(2) If C ∈/ C0 , then F(C) is empty.
Proof. Let L : P(C) → Shv(C) be a left adjoint to the inclusion and let ∅be an initial object of P(C). Then L∅ is an initial object of Shv(C). Since
L is left exact, it preserves (−1)-truncated objects, as does the inclusion
Shv(C) ⊆ P(C). Thus L∅ is (−1)-truncated and corresponds to some sieve
C(0) ⊆ C (Lemma 6.2.2.4). As we saw in the proof of Lemma 6.2.2.16, asieve C(0) classifies an object of Shv(C) if and only if C(0) is saturated in
the following sense: if C ∈ C and the induced sieve C(0) ×C C/C is covering,highertopoi

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then C ∈ C(0) . An initial object of Shv(C) is an initial object of τ≤−1 Shv(C)and must therefore correspond to the smallest saturated sieve on C. An easyargument shows that this sieve is C0 and that F ∈ P(C) is a (−1)-truncatedobject classified by C0 if and only if conditions (1) and (2) are satisfied.
6.2.3 Effective Epimorphisms
In classical topos theory, the assumption that every equivalence relation iseffective leads to a bijective correspondence between equivalence relations on
an object X and effective epimorphisms X → Y . The purpose of this section
is to generalize the notion of an effective epimorphism to the ∞-categoricalsetting.
Our primary interest is studying the class of effective epimorphisms in an∞-topos X. However, we will later need to employ the same ideas when X isan n-topos for n < ∞. It is therefore convenient to work in a slightly moregeneral setting.
Definition 6.2.3.1. An ∞-category X is a semitopos if it satisfies the following conditions:(1) The ∞-category X is presentable.
(2) Colimits in X are universal.
(3) For every morphism f : U → X, the underlying groupoid of theČech nerve Č(f ) is effective (see §6.1.2).
Remark 6.2.3.2. Every ∞-topos is a semitopos; this follows immediatelyfrom Theorem 6.1.0.6.
Remark 6.2.3.3. If X is a semitopos, then so is X/X for every object X ∈ X.
Proposition 6.2.3.4. Let X be a semitopos. Let p : U → X be a morphismin X, let U• be the underlying simplicial object of the Čech nerve Č(p), andlet V ∈ X be a colimit of U• . The induced diagram
/V
U
p0

p

X

~

identifies p0 with a (−1)-truncation of p in X/X .
Proof. We first show that V is (−1)-truncated. It suffices to show that the
diagonal map V → V ×X V is an equivalence. We may identify V with
V ×V V . Since colimits in X are universal, it will suffice to prove that foreach m, n ≥ 0, the natural map
pm,n : Um ×V Un → Um ×X Un
is an equivalence. We next observe that each pm,n is a pullback of
p0,0 : U ×V U → U ×X U.

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Because U• is an effective groupoid, both sides may be identified with U1 .
To complete the proof, it suffices to show that the natural map
MapX/X (p0 , q) → MapX/X (p, q)
is an equivalence whenever q : E → X is a monomorphism. Note that bothsides are either empty or contractible. We must show that if MapX/X (p, q) is
nonempty, then so is MapX/X (p0 , q). We observe that the map X/q → X/X
is fully faithful, and that its essential image is a sieve on X/X . If that sieve
contains p, then it contains the entire groupoid U• (viewed as a groupoid in
X/X ). We conclude that there exists a groupoid object W• : N(∆)op → X/q
lifting U• . Let Ve ∈ X/q be a colimit of V• . According to Proposition 1.2.13.8,
the image of Ve in X/X can be identified with the map p0 : V → X. The
existence of Ve proves that MapX/X (p0 , q) is nonempty, as desired.
Corollary 6.2.3.5. Let X be a semitopos and let f : U → X be a morphism
in X. The following conditions are equivalent:(1) If we regard f as an object of the ∞-category X/X , then τ≤−1 (f ) is a
final object of X/X .
(2) The Čech nerve Č(f ) is a simplicial resolution of X.
We will say that a morphism f : U → X in a semitopos X is an effective epimorphism if it satisfies the equivalent conditions of Corollary 6.2.3.5.
There is a one-to-one correspondence between effective epimorphisms andeffective groupoids. More precisely, let ResEff (X) denote the full subcategory of the ∞-category X∆+ spanned by those augmented simplicial objects
U• which are both Čech nerves and simplicial resolutions. The restriction
functors
X∆+

X∆

}

%
Fun(∆1 , X)

induce equivalences of ∞-categories from ResEff (X) to the full subcategoryof X∆ spanned by the effective groupoids, and from ResEff (X) to the fullsubcategory of Fun(∆1 , X) spanned by the effective epimorphisms.
Remark 6.2.3.6. Let f∗ : X → Y be a geometric morphism of ∞-topoi
and let u : U → Y be an effective epimorphism in Y. Then f ∗ (u) is aneffective epimorphism in X. To see this, choose a Čech nerve U• of u. Sinceu is an effective epimorphism, U• is a colimit diagram. The left exactness off ∗ implies that f ∗ ◦ U• is a Čech nerve of f ∗ (u). Since f ∗ is a left adjoint,
we conclude that f ∗ ◦ U• is a colimit diagram, so that f ∗ (u) is an effective
epimorphism.

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The following result summarizes a few basic properties of effective epimorphisms:
Proposition 6.2.3.7. Let X be a semitopos.
(1) Any equivalence X → Y in X is an effective epimorphism.
(2) If f, g : X → Y are homotopic morphisms in X, then f is an effective
epimorphism if and only if g is an effective epimorphism.
(3) If F : Y → X is a left exact presentable functor between semitopoi andf : U → X is an effective epimorphism in X, then F (f ) is an effective
epimorphism in Y.
Proof. Assertions (1) and (2) are obvious. To prove (3), we observe that f isan effective epimorphism if and only if it can be extended to an augmented
simplicial object U• which is both a simplicial resolution and a Čech nerve.
Since F is left exact, it preserves the property of being a Čech nerve; since F
preserves colimits, it preserves the property of being a simplicial resolution.
Remark 6.2.3.8. Let X be a semitopos and let f : X → T be an effective
epimorphism in X. Applying part (3) of Proposition 6.2.3.7 to the geometricmorphism f : X/S → X/T induced by a morphism S → T in X, we deducethat any base change X ×T S → X of f is also an effective epimorphism.
In order to verify other basic properties of the class of effective epimorphisms, such as stability under composition, we will need to reformulate
the property of surjectivity in terms of subobjects. Let X be a presentable∞-category. For each X ∈ X, the ∞-category τ≤−1 X/X of subobjects of X
is equivalent to the nerve of a partially ordered set which we will denoteby Sub(X); we may identify Sub(X) with the set of equivalence classes ofmonomorphisms U → X. A morphism f : X → Y in X induces a left exact
pullback functor X/X → X/Y . This functor preserves (−1)-truncated objects
by Proposition 5.5.6.16 and therefore induces a map f ∗ : Sub(Y ) → Sub(X)of partially ordered sets.
Remark 6.2.3.9. Let X be a presentable ∞-category
in which colimits are`universal. Then any monomorphism u : U →Xα can be obtained as acoproduct of maps uα : Uα → Xα , where each uα is a pullback of u andtherefore also a monomorphism. It follows that the natural mapaYθ : Sub( Xα ) →Sub(Xα )is a monomorphism of partially ordered sets. If coproducts in X are disjoint,then θ is bijective.
Proposition 6.2.3.10. Let X be a semitopos. A morphism f : U → X in X
is an effective epimorphism if and only if f ∗ : Sub(X) → Sub(U ) is injective.

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Proof. Suppose first that f ∗ is injective. Let U• be the underlying groupoidof a Čech nerve of f , let V be a colimit of U• , let u : V → X be thecorresponding monomorphism, and let [V ] denote the corresponding elementof Sub(X). Since f factors through u, we conclude that f ∗ [V ] = f ∗ [X] =[U ] ∈ Sub(U ). Invoking the injectivity of f ∗ , we conclude that [V ] = [X], sothat u is an equivalence.
For the converse, let us suppose that f is an effective epimorphism. Let [V ]and [V 0 ] be elements of Sub(X), represented by monomorphisms u : V → X
and u0 : V 0 → X, and suppose that f ∗ [V ] = f ∗ [V 0 ]. We wish to provethat [V ] = [V 0 ]. Since f ∗ is a left exact functor, we have f ∗ ([V ] ∩ [V 0 ]) =f ∗ [V ×X V 0 ]. It will suffice to prove that [V 0 ] = [V ×X V 0 ]; the same argument
will then establish that [V ] = [V ×X V 0 ], and the proof will be complete. In
other words, we may assume without loss of generality that [V ] ⊆ [V 0 ], sothat there is a commutative diagram
/ V0

g

V0
u

X.

~

u0

We wish to show that g is an equivalence. The map g induces a naturaltransformation of augmented simplicial objects∗

α• : u∗ ◦ Č(f ) → u0 ◦ Č(f ).
We observe that g can be identified with α−1 . Since f is an effective epimorphism, Č(f ) is a colimit diagram. Since colimits in X are universal, we
conclude that α−1 is a colimit of α| N(∆)op . Consequently, to prove thatα−1 is an equivalence, it will suffice to prove that αn is an equivalence forn ≥ 0. Since each αn is a pullback of α0 , it will suffice to prove that α0
is an equivalence. But this is simply a reformulation of the condition thatf ∗ [V ] = f ∗ [V 0 ].
From this we immediately deduce some corollaries.
Corollary 6.2.3.11. Let X be a semitopos and let {fα : Xα → Yα } be a(small) collection of effective epimorphisms in X. Then the induced map
a
a
f:
Xα →
Yα
α

α

is an effective epimorphism.
Proof. Combine Proposition 6.2.3.10 with Remark 6.2.3.9.
Corollary 6.2.3.12. Let X be a semitopos containing a diagram
>Y
f

X

g
h

/ Z.

highertopoi

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CHAPTER 6(1) If f and g are effective epimorphisms, then so is h.
(2) If h is an effective epimorphism, then so is g.
Proof. This follows immediately from Proposition 6.2.3.10 and the observation that we have an equality f ∗ ◦ g ∗ = h∗ of functions Sub(Z) → Sub(X).
The theory of effective epimorphisms is a mechanism for proving theorems
by descent.
Lemma 6.2.3.13. Let X be a semitopos, let p : K . → X be a colimit diagram
and let ∞ denote the cone point of K . . Then the associated mapap(x) → p(∞)
x∈K0

(which is well-defined up to homotopy) is an effective epimorphism.
Proof. For each vertex x of K . , let Zx = p(x). If x belongs to K, we will
denote the corresponding map Zx → Z∞ by fx . Let E 00 ⊆ E 0 ∈ Sub(Z∞ )be such that fx∗ E 00 = fx∗ E 0 for each vertex x of K; we wish to show that
E 00 = E 0 . We can represent E 00 and E 0 by a 2-simplex σ∞ : ∆2 → X, which
we depict as
=

0
Z∞

!
/ Z∞ .

00
Z∞

Lift the above diagram to a 2-simplex σ : ∆2 → Fun(K . , X)
0

g

0

?p

g 00

/ p

g

p00

where g, g 0 , and g 00 are Cartesian transformations. Our assumption guarantees that the restriction of g 0 induces an equivalence p00 |K → p0 |K. Since
colimits in X are universal, g 0 is itself an equivalence, so that E 00 = E 0 , asdesired.
Proposition 6.2.3.14. Let X be an ∞-topos and let S be a collection ofmorphisms of X which is stable under pullbacks and coproducts. The following
conditions are equivalent:(1) The class S is local (Definition 6.1.3.8).
(2) Given a pullback diagram
/X

X0
f0


Y0

f

g


/ Y,

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where g is an effective epimorphism and f 0 ∈ S, we have f ∈ S.
Proof. We first show that (1) ⇒ (2). Let Y• : N(∆+ )op → X be a Čech nerveof the map g and choose a Cartesian transformation f• : X• → Y• of augmented simplicial objects which extends f . Then we can identify f 0 with
f0 : X0 → Y0 . Each fn is a pullback of f0 , and therefore belongs to S.
Applying Lemma 6.1.3.5, we deduce that f belongs to S as well.
Conversely, suppose that (2) is satisfied. We will show that S satisfies
criterion (3) of Lemma 6.1.3.7. Let
u

α

/v

0


/ v0

β0

β


u0

α

be a pushout diagram in OX , where α and β are Cartesian and u, v, u0 ∈ S.
Since X is an ∞-topos, we conclude that α0 and β 0 are also Cartesian. To
complete the proof, it will suffice to show that v 0 ∈ S. For this, we observethat there is a pullback diagram`/ X 00
X X0
v

Y

`

`

u0
g

Y0



v0

/ Y 00 ,where g is an effective epimorphism (Lemma 6.2.3.13), and apply hypothesis(2).
Proposition 6.2.3.15. Let X be a semitopos, and suppose we are given apullback square
X0

g0

f0


S0

g

/X

/S

f

in X. If f is an effective epimorphism, then so is f 0 . The converse holds if
g is an effective epimorphism.
0

Proof. Let g ∗ : X/S → X/S be a pullback functor. Without loss of generality,
we may suppose that f 0 = g ∗ f . Let U• : N(∆+ )op → X be a Čech nerve off . Since g ∗ is left exact (being a right adjoint), we conclude that g ∗ ◦ U•
is a Čech nerve of f 0 . If f is an effective epimorphism, then U• is a colimit
diagram. Because colimits in X are universal, g ∗ ◦U• is also a colimit diagram,so that f 0 is an effective epimorphism.
Conversely, suppose that f 0 and g are effective epimorphisms. Corollary
6.2.3.12 implies that g ◦ f 0 is an effective epimorphism. The commutativityof the diagram implies that f ◦ g 0 is an effective epimorphism, so that f isan effective epimorphism (Corollary 6.2.3.12 again).

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Lemma 6.2.3.16. Let X be a semitopos and suppose we are given a pullbacksquare
X0

g0

f0


S0

g

/X

/S

f

in X. Suppose that f 0 is an equivalence and that g is an effective epimorphism. Then f is an equivalence.
Proof. Let U• be a Čech nerve of g 0 and let V• be a Čech nerve of g. The
above diagram induces a transformation α• : U• → V• . The map α0 can be
identified with f 0 and is therefore an equivalence. For n ≥ 0, αn : Un → Vn isa pullback of α0 and is therefore also an equivalence. Since g is an effective
epimorphism, V• is a colimit diagram. Applying Proposition 6.2.3.15, weconclude that g 0 is also an effective epimorphism, so that U• is a colimit
diagram. It follows that f = α−1 is a colimit of equivalences and is therefore
an equivalence.

Proposition 6.2.3.17. Let X be a semitopos and suppose we are given apullback square
/X

X0
f0


S0

g


/S

f

in X. If f is n-truncated, then so is f 0 . The converse holds if g is an effective
epimorphism.
0

Proof. Let g ∗ : X/S → X/S be a pullback functor. The first part of (1)asserts that g ∗ carries n-truncated objects to n-truncated objects. This follows immediately from Proposition 5.5.6.16 since g ∗ is a right adjoint andtherefore left exact. We will prove the converse in a slightly stronger form: if
i : U → V is a morphism in X/S such that g ∗ (i) is an n-truncated morphism
0in X/S , then i is n-truncated. The proof is by induction on n. If n ≥ −1,
we can use Lemma 5.5.6.15 to reduce to the problem of showing that the
diagonal map δ : U → U ×V U is (n − 1)-truncated. Since g ∗ is left exact,
we can identify g ∗ (δ) with the diagonal map g ∗ U → g ∗ U ×g∗ V g ∗ U , which is(n − 1)-truncated according to Lemma 5.5.6.15; the desired result then follows from the inductive hypothesis. In the case n = −2, we have a pullback

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diagram
/U

g∗ U


g∗ i

g∗ V

S0

i

g0


/V

g


/ S.

Proposition 6.2.3.15 implies that g 0 is an effective epimorphism, and g ∗ i isan equivalence, so that i is also an equivalence by Lemma 6.2.3.16.
Let C be a small ∞-category equipped with a Grothendieck topology. Our
final goal in this section is to use the language of effective epimorphisms tocharacterize the ∞-topos Shv(C) by a universal property.
Lemma 6.2.3.18. Let C be a (small) ∞-category containing an object C,
let {fα : Cα → C}α∈A be a collection of morphisms indexed by a set A and(0)
let C/C ⊆ C/C be the sieve on C that they generate. Let j : C → P(C) denote
the Yoneda embedding and i : U → j(C) a monomorphism corresponding to(0)
the sieve C/C . Then i can be identified with a (−1)-truncation of the induced`
map α∈A j(Cα ) → j(C) in the ∞-topos P(C)/C .
Proof. Using Proposition 6.2.2.5, we can identify the equivalence classes of(0)(−1)-truncated object U ∈ P(C)/j(C) with sieves C/C ⊆ C/C . It is not dif(0)ficult to see that j(fα ) factors through U if and only if fα ∈ C/C . Con`
sequently, the (−1)-truncation of α∈A j(Cα ) → j(C) is associated to thesmallest sieve on C which contains each fα .
Lemma 6.2.3.19. Let X be an ∞-topos, C a small ∞-category equippedwith a Grothendieck topology, and f∗ : X → P(C) a functor with a left exact
left adjoint f ∗ : P(C) → X.
The following conditions are equivalent:(1) The functor f∗ factors through Shv(C) ⊆ P(C).
(2) For every collection of morphisms {vα : Cα → C} which generate acovering sieve in C, the induced mapaf ∗ (j(Cα )) → f ∗ (j(C))
is an effective epimorphism in X, where j : C → P(C) denotes the
Yoneda embedding.
Proof. Suppose first that (1) is satisfied and let {vα : Cα → C} be a collection of morphisms as in the statement of (2). Let L : P(C) → Shv(C)be a left adjoint to the inclusion. Then we have an equivalence of functors

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f ∗ ' (f ∗ | Shv(C)) ◦ L. Applying Remark 6.2.3.6, we are reduced to showingthat ifau:j(Cα ) → j(C)is the natural map, then Lu is an effective epimorphism in P(C). We factor
u as a compositionau0u00
j(Cα ) → U → j(C),where u0 is an effective epimorphism and u00 is a monomorphism. We wish
to show that Lu00 is an equivalence. Lemma 6.2.3.18 allows us to identify(0)
u00 with the monomorphism associated to the sieve C/C on C generated by
the maps vα . By assumption, this is a covering sieve, so that Lu00 is anequivalence in Shv(C) by construction.
(0)
Conversely, suppose that (2) is satisfied. Let C ∈ C and let C/C ⊆ C/C bea covering sieve on C associated to a monomorphism u00 : U → j(C). We
wish to show that f ∗ u00 is an equivalence. According to Lemma 6.2.3.18, we
have a factorizationau0
u00
j(Cα ) → U → j(C),α(0)

where the maps vα : Cα → C are chosen to generate the sieve C/C and
u0 is an effective epimorphism. Let u be a composition of u0 and u00 . Thenf ∗ u0 is an effective epimorphism (Remark 6.2.3.6), and f ∗ u is an effective
epimorphism by assumption (2). Corollary 6.2.3.12 now shows that f ∗ u00 isan effective epimorphism. Since f ∗ u00 is also a monomorphism, we conclude
that f ∗ u00 is an equivalence, as desired.
Proposition 6.2.3.20. Let X be an ∞-topos and let C be a small ∞-category
equipped with a Grothendieck topology. Let L : P(C) → Shv(C) denote aleft adjoint to the inclusion and j : C → P(C) the Yoneda embedding. Let
Fun∗ (Shv(C), X) denote the ∞-category of left exact colimit-preserving functors from Shv(C) to X (Definition 6.3.1.10). The composition
Lj

J : Fun∗ (Shv(C), X) → Fun∗ (P(C), X) → Fun(C, X)
is fully faithful. Suppose furthermore that C admits finite limits. Then afunctor f : C → X belongs to the essential image of J if and only if thefollowing conditions are satisfied:(1) The functor f is left exact.
(2) For every collection of morphisms {Cα → C}α∈A which generates acovering sieve on C, the associated morphismaf (Cα ) → f (C)α∈A

is an effective epimorphism in X.

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Proof. If the topology on C is trivial, then Theorem 5.1.5.6 implies that J
is fully faithful, and the description of the essential image of J follows from
Proposition 6.1.5.2. In the general case, Proposition 5.5.4.20 implies thatcomposition with L induces a fully faithful embedding
J 0 : Fun∗ (Shv(C), X) → Fun∗ (P(C), X),
so that J is a composition of J 0 with a fully faithful functor
J 00 : Fun∗ (P(C), X) → Fun(C, X).
Suppose that C admits finite limits and that f satisfies (1), so that f isequivalent to J 00 (u∗ ) for some left exact colimit-preserving u∗ : P(C) → X.
The functor u∗ is unique up to equivalence, and Lemma 6.2.3.19 ensuresthat u∗ belongs to the essential image of J 0 if and only if condition (2) issatisfied.
Remark 6.2.3.21. It is possible to formulate a generalization of Proposition
6.2.3.20 which describes the essential image of J even when C does not admit
finite limits. The present version will be sufficient for the applications in thisbook.
6.2.4 Canonical Topologies
Let X be an ∞-topos. Suppose that we wish to identify X with an ∞-categoryof sheaves. The first step is to choose a pair of adjoint functors
P(C) oF
G/X

where F is left exact. According to Theorem 5.1.5.6, F is determined up toequivalence by the compositionjF

f : C → P(C) → X .
We might then try to choose a topology on C such that G factors as acomposition
G0

X → Shv(C) ⊆ P(C).
Though it is not always possible to guarantee that G0 is an equivalence, wewill show that for an appropriately chosen topology (Definition 6.2.4.1), the∞-topos Shv(C) is a close approximation to X (Proposition 6.2.4.6).
Definition 6.2.4.1. Let X be a semitopos, C a small ∞-category which
admits finite limits, and f : C → X a left exact functor. We will say that a(0)sieve C/C ⊆ C/C on an object C ∈ C is a canonical covering relative to f if
(0)

there exists a collection of morphisms {uα : Cα → C} belonging to C/C such`that the induced map f (Cα ) → f (C) is an effective epimorphism in X.
Our first goal is to verify that the canonical topology is actually a Grothendieck topology on C.

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Proposition 6.2.4.2. Let f : C → X be as in Definition 6.2.4.1. The collection of canonical coverings relative to f determines a Grothendieck topology
on C.
Proof. Since any identity map idf (C) : f (C) → f (C) is an effective epimorphism, it is clear that the sieve C/C is a canonical covering of C for every(0)

C ∈ C. Suppose that C/C ⊆ C/C is a canonical covering of C and thatg : D → C is a morphism in C. We wish to prove that the induced sieve(0)g ∗ C/C is a canonical covering. Choose a collection of objects uα : Cα → C`(0)of C/C such that the induced map α f (Cα ) → f (C) is an effective epimorphism and form pullback diagrams
Dα

vα

/D

uα


/C

g


Cα

in C. Using the fact that f is left exact and that colimits in X are universal,
we conclude that the diagram`/ f (D)f (Dα )` f (Cα )

/ f (C)is a pullback, so that the upper horizontal map is an effective epimorphism by(0)(0)
Proposition 6.2.3.15. Since each vα belongs to g ∗ C/C , it follows that g ∗ C/C
is a canonical covering.
(0)
(1)
(0)
Now suppose that C/C and C/C are sieves on C ∈ C, where C/C is a
(0)

(1)

canonical covering, and for each g : D → C in C/C , the covering g ∗ C/C isa canonical covering of D. Choose a collection of morphisms gα : Dα → C`(0)belonging to C/C with the property thatf (Dα ) → f (C) is an effective
epimorphism. For each Dα , choose a collection of morphisms hα,β : Eα,β →`(1)Dα belonging to gα∗ C/C such that the map β f (Eα,β ) → f (Dα ) is aneffective epimorphism. Using Corollary 6.2.3.11, we conclude that the mapaaf (Eα,β ) →f (Dα )α,βα

is an effective epimorphism. Since effective epimorphisms are stable undercomposition (Corollary 6.2.3.12), we have an effective epimorphismaf (Eα,β ) → f (C)α,βinduced by the collection of compositions gα ◦hα,β : Eα,β → C. Each of these(1)(1)
compositions belongs to C/C , so that C/C is a canonical covering of C.

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For later use, we record a few features of the canonical topology:
Lemma 6.2.4.3. Let f : C → X be as in Definition 6.2.4.1 and regard C as
endowed with the canonical topology relative to f . Let j : C → P(C) denote
the Yoneda embedding and let L : P(C) → Shv(C) be a left adjoint to theinclusion. Suppose that C ∈ C is such that f (C) is an initial object of X.
Then Lj(C) is an initial object of Shv(C).
Proof. If f (C) is an initial object of X, then the empty sieve ∅ ⊆ C/C is acovering sieve with respect to the canonical topology. By construction, theassociated monomorphism ∅ → j(C) becomes an equivalence after applying
L, so that Lj(C) is initial in Shv(C).
Lemma 6.2.4.4. Let f : C → X be as in Definition 6.2.4.1. Suppose that f is
fully faithful and that coproducts in X are disjoint and let {uα : Cα → C} bea small collection of morphisms in C such that the morphisms f (uα ) exhibitf (C) as a coproduct of the family {f (Cα )}. Let F : Cop → S be a sheaf on C
(with respect to the canonical topology induced by f ). Then the morphisms{F(uα )} exhibit F(C) as a product of {F(Cα )} in S.
Q
Proof. We wish to show that the natural map F(C) → F(Cα ) is an isomorphism in the homotopy category H. We may identify the left
` hand sidewith MapP(C) (j(C), F) and the right hand side with MapP(C) ( j(Cα ), F).
Consequently, it will suffice to show that the natural mapav:j(Cα ) → j(C)becomes an equivalence after applying the localization functor L : P(C) →Shv(C). Choose a factorization of v as a composite
a
v0
v 00
j(Cα ) → U → j(C),where v 0 is an effective epimorphism and v 00 is a monomorphism. We observe(0)that v 00 is the monomorphism associated to the sieve C/C → C generated
by the morphisms uα . This is clearly a covering sieve with respect to thecanonical topology, so that Lv 00 is an equivalence in Shv(C). It follows that
Lv is equivalent to Lv 0 and is therefore an effective epimorphism (Remark
6.2.3.6). Form a pullback diagramV` j(Cα )v/`

j(Cβ )vv

/ j(C).

We wish to prove that Lv is an equivalence. According to Lemma 6.2.3.16,
it will suffice to show that Lv is an equivalence. Since colimits in P(C) areuniversal, we may identify v with a coproduct of morphisms
v β : Vβ → j(Cβ ),highertopoi

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where Vβ can be written as a coproduct α j(Cα ×C Cβ ). Using Lemma
6.1.5.1, we can identify the summand j(Cβ ×C Cβ ) of Vβ with j(Cβ ), and therestriction of v β to this summand is an equivalence. To complete the proof,
it will suffice to show that for every other summand Dα,β = j(Cα ×C Cβ ),the localization LD is an initial object of Shv(C). To prove this, we observethat Lemma 6.1.5.1 implies that f (Cα ×C Cβ ) is an initial object of X andapply Lemma 6.2.4.3.
Lemma 6.2.4.5. Let C be a small ∞-category equipped with a Grothendiecktopology and let u : F0 → F be a morphism in Shv(C). Suppose that, for eachC ∈ C and each η ∈ π0 F(C), there exists a collection of morphisms {Cα →C} which generates a covering sieve on C and a collection of ηα ∈ π0 F0 (Cα )such that η and ηα have the same image in π0 F(Cα ). Then u is an effective
epimorphism.
Proof. Replacing F by its image in F0 if necessary, we may suppose that u isa monomorphism. Let L : P(C) → Shv(C) be a left adjoint to the inclusion
and let D be the full subcategory of P(C) spanned by those objects G suchthat, for every pullback diagram
G0

F0

u0

u

/G

/F

in P(C), Lu0 is an equivalence in Shv(C). To prove that u is an equivalence,
it will suffice to show that the equivalent morphism Lu is an equivalence.
For this, it will suffice to prove that F ∈ D. We will in fact prove that
D = P(C). We first observe that since colimits in P(C) are universal and
L commutes with colimits, D is stable under colimits in P(C). Since P(C)is generated under colimits by the image of the Yoneda embedding, it will
suffice to prove that j(C) ∈ D for each C ∈ C. Choose a map j(C) → F,classified up to homotopy by η ∈ π0 F(C), and form a pullback diagram
U

F0

u0

/ j(C)u/F

as above. Then u0 is a monomorphism; according to Proposition 6.2.2.5, it is(0)(0)
classified by a sieve C/C on C. Our hypothesis guarantees that C/C contains
a collection of morphisms {Cα → C} which generate a covering sieve, so(0)that C/C is itself covering. It follows immediately from the construction ofShv(C) that Lu0 is an equivalence.
We close with the following result, which implies that any ∞-topos is
closely approximated by an ∞-category of sheaves.

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Proposition 6.2.4.6. Let X be a semitopos, C a small ∞-category which
admits finite limits, andF

P(C) oG/Xa pair of adjoint functors. Suppose that the compositionjF

f : C → P(C) → X
is left exact and regard C as endowed with the canonical topology relative tof . Then:(1) The functor G factors through Shv(C).
(2) Suppose that f is fully faithful and generates X under colimits. Then
G carries effective epimorphisms in X to effective epimorphisms in
Shv(C).
Proof. In view of the definition of the canonical topology, (1) is equivalent
to the following assertion: given a`collection of morphisms {uα : Cα → C} in
C such that the induced map u : α Cα → C is an effective epimorphism in
X, if i : U → j(C) is the monomorphism in P(C) corresponding to the sieve(0)
C/C ⊆ C/C generated by the collection {uα }, then F (i) is an equivalence in`
X. Let u0 : α j(Cα ) → j(C) be the coproduct of the family {j(uα )} and0let V• : ∆op+ → P(C) be a Čech nerve of u . Then i can be identified with
the induced map from the colimit of V• | N(∆)op to V−1 . Since F preserves
colimits, to show that F (i) is an equivalence, it will suffice to show thatF ◦ V• is a colimit diagram. Since u is an effective epimorphism, it suffices
to observe that F ◦ V• is equivalent to the Čech nerve of u.
We now prove (2). Suppose that u : Y → Z is an effective epimorphism
in X. We wish to prove that Gu is an effective epimorphism in Shv(C). We
will show that the criterion of Lemma 6.2.4.5 is satisfied. Choose an object
C ∈ C and a point η ∈ π0 MapP(C) (j(C), GZ) ' π0 MapX (f (C), Z). Form apullback diagram
Y0

Y

u0

/ f (C)

u


/Z

s

so that u0 is an effective epimorphism. Since`f (C) generates X under colimits,
there exists an effective epimorphism u00 : α f (Cα ) → Y . The compositionu0 ◦ u00 is an effective epimorphism and corresponds to a family of maps
wα : f (Cα ) → f (C) in X. Since f is fully faithful, we may suppose that eachwα = f vα for some map vα : Cα → C in C. It follows that the collectionof maps {vα } generates a covering sieve on C with respect to the canonicaltopology. Moreover, each of the compositionsaf (Cα ) →f (Cα ) → Yαhighertopoi

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gives rise to a point ηα ∈ π0 MapX (f (Cα ), Y ) ' π0 MapP(C) (j(Cα ), G(Y ))with the desired properties.

6.3 THE ∞-CATEGORY OF ∞-TOPOIIn this section, we will show that the collection of all ∞-topoi can be organized into an ∞-category RTop. The objects of RTop are ∞-topoi, andthe morphisms are called geometric morphisms; we will give a definition in§6.3.1. In §6.3.2, we will show that RTop admits (small) colimits. In §6.3.3,
we will show that RTop admits (small) filtered limits; we will treat the caseof general limits in §6.3.4.
Let X be an ∞-topos containing an object U . In §6.3.5, we will show thatthe ∞-category X/U is an ∞-topos. Moreover, this ∞-topos is equipped witha canonical geometric morphism X/U → X. Geometric morphisms whicharise via this construction are said to be étale. In §6.3.6, we will define amore general notion of algebraic morphism between ∞-topoi. We will also
prove a structure theorem which implies that every ∞-topos X satisfying
some mild hypotheses admits an algebraic morphism to an ∞-category ofsheaves on a 2-category.
6.3.1 Geometric Morphisms
In classical topos theory, the correct notion of morphism between two topoi
is that of an adjunction
Xo

f∗
f∗

/ Y,

where the functor f ∗ is left exact. We will introduce the same ideas in the∞-categorical setting.
Definition 6.3.1.1. Let X and Y be ∞-topoi. A geometric morphism fromX to Y is a functor f∗ : X → Y which admits a left exact left adjoint (which
we will typically denote by f ∗ ).
Remark 6.3.1.2. Let f∗ : X → Y be a geometric morphism from an ∞topos X to another ∞-topos Y, so that f∗ admits a left adjoint f ∗ . Either ofthe functors f∗ and f ∗ determines the other up to equivalence (in fact, up tocontractible ambiguity). We will often abuse terminology by referring to f ∗ asa geometric morphism from X to Y. We will always indicate in our notation
whether the left or the right adjoint is being considered: a superscripted
asterisk indicates a left adjoint (pullback functor), and a subscripted asterisk
indicates a right adjoint (pushforward functor).
Remark 6.3.1.3. Any equivalence of ∞-topoi is a geometric morphism. If
f∗ , g∗ : X → Y are homotopic, then f∗ is a geometric morphism if and only ifg∗ is a geometric morphism (because we can identify left adjoints of f∗ withleft adjoints of g∗ ).

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Remark 6.3.1.4. Let f∗ : X → Y and g∗ : Y → Z be geometric morphisms.
Then f∗ and g∗ admit left exact left adjoints, which we will denote by f ∗and g ∗ , respectively. The composite functor f ∗ ◦ g ∗ is left exact and is aleft adjoint to g∗ ◦ f∗ by Proposition 5.2.2.6. We conclude that g∗ ◦ f∗ is ageometric morphism, so the class of geometric morphisms is stable undercomposition.
d∞ denote the ∞-category of (not necessarily
Definition 6.3.1.5. Let Cat
d∞ as follows:
small) ∞-categories. We define subcategories LTop, RTop ⊆ Cat(1) The objects of LTop and RTop are the ∞-topoi.
(2) A functor f ∗ : X → Y between ∞-topoi belongs to LTop if and only iff ∗ preserves small colimits and finite limits.
(3) A functor f∗ : X → Y between ∞-topoi belongs to RTop if and only iff∗ has a left adjoint which is left exact.
The ∞-categories LTop and RTop are canonically antiequivalent. To provethis, we will use the argument of Corollary 5.5.3.4. First, we need a definition.
Definition 6.3.1.6. A map p : X → S of simplicial sets is a topos fibration
if the following conditions are satisfied:(1) The map p is both a Cartesian fibration and a coCartesian fibration.
(2) For every vertex s of S, the corresponding fiber Xs = X ×S {s} is an∞-topos.
(3) For every edge e : s → s0 in S, the associated functor Xs → Xs0 is left
exact.
The following analogue of Proposition 5.5.3.3 follows immediately fromthe definitions:
Proposition 6.3.1.7. (1) Let p : X → S be a Cartesian fibration of simd∞ . Then p is a topos
plicial sets classified by a map χ : S op → Catd∞ .
fibration if and only if χ factors through RTop ⊆ Cat(2) Let p : X → S be a coCartesian fibration of simplicial sets classified
d∞ . Then p is a topos fibration if and only if χ
by a map χ : S → Catd∞ .
factors through LTop ⊆ Cat
Corollary 6.3.1.8. For every simplicial set S, there is a canonical bijection[S, LTop] ' [S op , RTop],
where [K, C] denotes the collection of equivalence classes of objects of the∞-category Fun(K, C). In particular, LTop and RTopop are canonically isomorphic in the homotopy category of ∞-categories.

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Proof. According to Proposition 6.3.1.7, both [S, LTop] and [S op , RTop] canbe identified with the collection of equivalence classes of topos fibrations
X → S.
The following proposition is a simple reformulation of some of the resultsof §5.5.6.
Proposition 6.3.1.9. Let f∗ : X → Y be a geometric morphism between ∞topoi having a left adjoint f ∗ . Then f ∗ and f∗ carry k-truncated objects to
k-truncated objects and k-truncated morphisms to k-truncated morphisms,
for any integer k ≥ −2. Moreover, there is a (canonical) equivalence ofYX ∗functors f ∗ τ≤k
' τ≤k
f .
Proof. The first assertion follows immediately from Lemma 5.5.6.15 since f∗
and f ∗ are both left exact. The second follows from Proposition 5.5.6.28.
Definition 6.3.1.10. Let X and Y be ∞-topoi. We let Fun∗ (X, Y) denotethe full subcategory of Fun(X, Y) spanned by geometric morphisms f∗ : X →Y, and Fun∗ (Y, X) the full subcategory of Fun(Y, X) spanned by their left
adjoints.
Remark 6.3.1.11. It follows from Proposition 5.2.6.2 that the ∞-categories
Fun∗ (X, Y) and Fun∗ (Y, X) are canonically antiequivalent to one another.
Warning 6.3.1.12. If X and Y are ∞-topoi, then the ∞-category Fun∗ (X, Y)
of geometric morphisms from X to Y is not necessarily small or even equivalent to a small ∞-category. This phenomenon is familiar in classical topos
theory. For example, there is a classifying topos A for abelian groups having
the property that for any topos X, the category C of geometric morphisms
X → A is equivalent to the category of abelian group objects of X. This
category is almost never small (for example, when X is the topos of sets, C
is equivalent to the category of abelian groups).
In spite of Warning 6.3.1.12, the ∞-category of geometric morphisms between two ∞-topoi can be reasonably controlled:
Proposition 6.3.1.13. Let X and Y be ∞-topoi. Then the ∞-category
Fun∗ (Y, X) of geometric morphisms from X to Y is accessible.
Proof. For each regular cardinal κ, let Yκ denote the full subcategory of Y
spanned by κ-compact objects. Choose a regular cardinal κ such that Y is κaccessible and Yκ is stable under finite limits in Y. We may therefore identify
Y with Indκ (C), where C is a minimal model for Yκ . According to Proposition
5.3.5.10, composition with the Yoneda embedding j : C → Y induces anequivalence from the ∞-category of κ-continuous functors Funκ (Y, X) to the∞-category Fun(C, X). We now make the following observations:(1) A functor F : Y → X preserves all small colimits if and only if F ◦ j :C → X preserves κ-small colimits (Proposition 5.5.1.9).

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603∞-TOPOI(2) A colimit-preserving functor F : Y → X is left exact if and only if thecomposition F ◦ j : C → X is left exact (Proposition 6.1.5.2).
Invoking Proposition 5.2.6.2, we deduce that the ∞-category Fun∗ (Y, X)
is equivalent to the full subcategory M ⊆ XC consisting of functors which
preserve κ-small colimits and finite limits. Proposition 5.4.4.3 implies that
Fun(C, X) is accessible. For every κ-small (finite) diagram p : K → C, the fullsubcategory of Fun(C, X) spanned by those functors which preserve colimits(limits) of p is an accessible subcategory of Fun(C, X) (Example 5.4.7.9).
Up to isomorphism, there are only a bounded number of κ-small (finite)
diagrams in C. Consequently, M is an intersection of a bounded number ofaccessible subcategories of Fun(C, X) and therefore accessible (Proposition
5.4.7.10).
6.3.2 Colimits of ∞-Topoi
Our goal in this section is to construct colimits in the ∞-category RTop of∞-topoi. According to Corollary 6.3.1.8, it will suffice to construct limits inthe ∞-category LTop.
Proposition 6.3.2.1. Let {Xα }α∈A be aQcollection of ∞-topoi parametrized
by a (small) set A. Then the product X = α∈A Xα is an ∞-topos. Moreover,each projection πα∗ : X → Xα is left exact and colimit-preserving. The corresponding geometric morphisms exhibit X as a product of the family {Xα }α∈A
in the ∞-category LTop.
Proof. Proposition 5.5.3.5 implies that X is presentable. It is clear that adiagram p : K . → X is a colimit if and only if each composition πα∗ ◦ p :K . → Xα is a colimit diagram in Xα . Similarly, a diagram q : K / → X is alimit if and only if each composition πα∗ ◦ q : K / → Xα is a limit diagram in
Xα . Using criterion (2) of Theorem 6.1.0.6, we deduce that X is an ∞-topos,
and that each πα∗ preserves all limits and colimits that exist in X. Choose aright adjoint π∗α : Xα → X to each πα∗ .
According to Theorem 4.2.4.1, the ∞-category X is a product of the familyd∞ . Since LTop is a subcategory of Cat
d∞ , it{Xα }α∈A in the ∞-category Cat
will suffice to prove the following assertion:• For every ∞-topos Y and every functor f ∗ : Y → X such that each of
the composite functors Y → Xα is left exact and colimit-preserving, f ∗is itself left exact and colimit-preserving.
This follows immediately from the fact that limits and colimits are computed pointwise.
Proposition 6.3.2.2. Let
X0

q0 ∗

p0 ∗


Y0

q

∗

/X

/Y

p∗

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be a diagram of ∞-categories which is homotopy Cartesian (with respect to
the Joyal model structure). Suppose further that X, Y, and Y0 are ∞-topoi
and that p∗ and q ∗ are left exact and colimit-preserving. Then X0 is an ∞topos. Moreover, for any ∞-topos Z and any functor f ∗ : Z → X, f ∗ is∗left exact and colimit-preserving if and only if the compositions p0 ◦ f ∗ and0∗∗
q ◦ f are left exact and colimit-preserving. In particular (taking f ∗ = idX ),∗∗the functors p0 and q 0 are left exact and colimit-preserving.
Proof. The second claim follows immediately from Lemma 5.4.5.5 and thedual result concerning limits. To prove the first, we observe that X0 is presentable by Proposition 5.5.3.12. To show that X is an ∞-topos, it will suffice
to show that it satisfies criterion (2) of Theorem 6.1.0.6. This follows immediately from Lemma 5.4.5.5 since X and Y0 satisfy criterion (2) of Theorem
6.1.0.6.
Proposition 6.3.2.3. The ∞-category LTop admits small limits, and the
d∞ preserves small limits.
inclusion functor LTop ⊆ Cat
Proof. According to Proposition 4.4.2.6, it suffices to prove this result for
pullbacks and small products. In the case of products, we apply Proposition
6.3.2.1. For pullbacks, we use Proposition 6.3.2.2 and Theorem 4.2.4.1.
6.3.3 Filtered Limits of ∞-Topoi
We now consider the problem of computing limits in the ∞-category RTopof ∞-topoi. This is quite a bit more difficult than the analogous problem ford∞ does not commute
colimits because the inclusion functor i : RTop ⊆ Catwith limits in general. However, the inclusion i does commute with filteredlimits:
Theorem 6.3.3.1. The ∞-category RTop admits small filtered limits (thatis, limits indexed by diagrams Cop → RTop where C is a small filtered ∞d∞ preserves small filteredcategory). Moreover, the inclusion RTop ⊆ Cat
limits.
The remainder of this section is devoted to the proof of Theorem 6.3.3.1.
Our basic strategy is to mimic the proof of Theorem 5.5.3.18. Our first stepd∞ ) of a filtered diagram of ∞-topoi is itself
is to show that the limit (in Cat
an ∞-topos. This is equivalent to a more concrete assertion: if p : X → S isa topos fibration and S op is a small filtered ∞-category, then the ∞-category
C of Cartesian sections of p is an ∞-topos. We saw in Proposition 5.5.3.17
that C is an accessible localization of the ∞-category MapS (S, X) spannedby all sections of p. Our first step will be to show that MapS (S, X) is an∞-topos. For this, the hypothesis that S op is filtered is irrelevant.
Lemma 6.3.3.2. Let p : X → S be a topos fibration, where S is a small
simplicial set. The ∞-category MapS (S, X) of sections of p is an ∞-topos.

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Proof. This is a special case of Proposition 5.4.7.11.
Proposition 6.3.3.3. Let A be a (small) filtered partially ordered set andlet p : X → N(A) be a topos fibration. Let C = MapN(A) (N(A), X) be the ∞category of sections of p and let C0 ⊆ C be the full subcategory of C spannedby the Cartesian sections of p. Then C0 is an accessible left exact localizationof C.
Lemma 6.3.3.4. Let X be an ∞-topos. Suppose we are given a set of accessible
T left exact localization functors {Lα : X → X}α∈A . Then the intersectionα∈A Lα X is an accessible left exact localization of X (and, in particular,
an ∞-topos).
Proof. Without loss of generality, we may assume that A is an initial segmentof the ordinals: that is, A = {α : α < β}, for some ordinal β. Choosea regular cardinal κ such that each of the functors Lα commutes with κfiltered colimits. Let βκ denote the ordinal product of α with κ, and let [βκ]denote the linearly ordered set of ordinals ≤ βκ. We will construct a map
N[βκ] → Fun(X, X), which we denote informally as a transfinite sequence offunctors {Fγ }γ≤βκ . Let F0 = idX and Fγ = limδ<γ Fδ when γ is a nonzero−→
limit ordinal. To complete the construction, it will suffice to define Fγ+1
assuming that Fγ has already been defined. Writing γ = βδ + α for α < β,we set Fγ+1 = Lα ◦ Fγ .
Since each Lα is an accessible left exact functor and filtered colimits in X
are left exact (see Example 7.3.4.7), it follows by induction on γ that eachof the functors Fγ is left exact and accessible. Set F =TFβκ . We claim that
F is a localization functor with essential image X0 = α∈A Lα X. To provethis, it suffices to verify the following:(a) For each X ∈ X and each Y ∈ X0 , the canonical map X → F (X) induces a homotopy equivalence MapX (F (X), Y ) → MapX (X, Y ). Moregenerally, we claim that each of the maps
MapX (Fγ (X), Y ) → MapX (F0 (X), Y ) = MapX (X, Y )is a homotopy equivalence. The proof proceeds by induction on γ, thecase γ = 0 being trivial. If γ is a nonzero limit ordinal, we invoke thefact that Fγ ' limδ<γ Fδ . To handle the case of successor ordinals,−→
it suffices to show that MapX (Fγ+1 (X), Y ) → MapX (Fγ (X), Y ) is a
homotopy equivalence for each ordinal γ. Writing γ = βδ + α, we haveFγ+1 (X) = Lα Fγ (X). The desired result now follows from the factthat Y ∈ X0 is Lα -local.
(b) For each X ∈ X, the object F (X) belongs to X0 . Fix α ∈ A; we wish
to show that F (X) ∈ Lα X. Note that the collection of ordinals of theform βδ + α + 1, where δ < κ, is cofinal in the set {γ : γ < βκ}. It
follows that
F (X) ' lim Fβδ+α+1 (X) = lim Lα Fβδ+α (X)
−→
−→
δ<κ

δ<κ

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is a κ-filtered colimit of objects belonging to Lα X. Since Lα commuteswith κ-filtered colimits, we conclude that F (X) ∈ Lα X.

Proof of Proposition 6.3.3.3. We say that a subset B ⊆ A is cofinal if, forevery element α ∈ A, there exists β ∈ B such that α ≤ β (this is equivalent
to the requirement that the inclusion N(B) ,→ N(A) is a cofinal map ofsimplicial sets). For every cofinal subset B ⊆ A, let CB denote the fullsubcategory of C spanned by those sections s of p which are p-right Kan
extensions of s| N(B). By virtue of Lemma 6.3.3.4, it will suffice to prove thefollowing pair of assertions:(a) For every cofinal subset B ⊆ A, the full subcategory CB is an accessible
left exact localization of C.
T(b) Let P be the set of all cofinal subsets of A. Then C0 = B∈P CB .
To prove (a), consider the restriction functor φ : C → MapN(A) (N(B), X).
Using Proposition 4.3.2.15, we deduce that φ restricts to a trivial Kan fibration φ0 : CB → MapN(A) (N(B), X). Any choice of section of φ0 determines
a fully faithful embedding MapN(A) (N(B), X) → C which is right adjoint toφ, having essential image CB . This proves that CB is a localization of C. The
accessibility and left exactness follow from the fact that MapN(A) (N(B), X)
is an ∞-topos (Lemma 6.3.3.2) and that the restriction functor φ is left
exact.
We now prove (b). Suppose first that E : N(A) → X is a section of p which
belongs to C0 . We claim that E ∈ CB for every cofinal subset B ⊆ A. To provethis, it suffices to show that for each element α ∈ A, the functor E exhibits
E(α) as a p-limit of the diagram E| N(Bα ), where Bα = {β ∈ B : β ≥ α}. By
assumption, E carries each edge of N(A) to a p-Cartesian morphism in X.
Using Propositions 4.3.1.9, 4.3.1.10, and Corollary 4.4.4.10, we are reduced
to proving that the simplicial set N(Bα ) is weakly contractible. This is clear,since the cofinality of B in A guarantees that N(Bα ) is filtered.
it will suffice to show that every object E ∈T To complete the proof,
0C
belongs
toC.
Fix
a pair of elements α ≤ β in A; we wish
B∈P B
to show that the map E(α) → E(β) is p-Cartesian. To prove this, set
B = {γ ∈ A : γ ≥ β}. Then B is cofinal in A, so E is a p-right Kan extension
of E| N(B). It follows that E(α) is a p-limit of the diagram E| N(B). Since
β is an initial object of N(B), we conclude that the map E(α) → E(β) is
p-Cartesian, as desired.

Proposition 6.3.3.5. Let A be a (small) filtered partially ordered set, let
p : X → N(A)op , and let Y ⊆ MapN(A) (N(A), X) be the full subcategory
spanned by the Cartesian sections of p. For each α ∈ A, the evaluation map
π∗ : Y → Xα is a geometric morphism of ∞-topoi.

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Proof. Let A0 = {β ∈ A : α ≤ β}. Using Theorem 4.1.3.1, we conclude
that the inclusion N(A0 ) ⊆ N(A) is cofinal. Corollary 3.3.3.2 implies that the
restriction map
MapN(A) (N(A), X) → MapN(A) (N(A0 ), X)
induces an equivalence on the full subcategories spanned by Cartesian sections. Consequently, we are free to replace A by A0 and thereby assume that
α is a least element of A.
The functor π∗ factors as a composition
ψ∗φ∗

Y → MapN(A) (N(A), X) → Xα ,where φ∗ denotes the inclusion functor and ψ∗ the evaluation functor. Proposition 6.3.3.3 implies that φ∗ is a geometric morphism; it therefore suffices
to show that ψ∗ is a geometric morphism as well.
Let ψ ∗ be a left adjoint to ψ∗ (the existence of ψ ∗ follows from Proposition
4.3.2.17 as indicated below). We wish to show that ψ ∗ is left exact. According
to Proposition 5.1.2.2, it will suffice to show that the composition
ψ∗eβθ : Xα → MapN(A) (N(A), X) → Xβ
is left exact, where eβ denotes the functor given by evaluation at β.
Let f : ∆1 → N(A) be the edge joining α to β, let C be the ∞-categoryof coCartesian sections of p, and let C0 be the ∞-category of coCartesiansections of the induced map p0 : X ×N(A) ∆1 → ∆1 . We observe that C
consists precisely of those sections s : N(A) → X of p which are p-left Kan
extensions of s|{α}. Applying Proposition 4.3.2.15, we conclude that theevaluation map eα : C → Xα is a trivial fibration and that (by Proposition
4.3.2.17) we may identify ψ ∗ with the compositionq

Xα → C ⊆ MapN(A) (N(A), X),
where q is a section of eα | C. Let q 0 : Xα → C0 be the composition of q withthe restriction map C → C0 . Then θ can be identified with the compositionq0eβ

Xα → C0 → Xβ ,which is the functor Xα → Xβ associated to f : α → β by the coCartesianfibration p. Since p is a topos fibration, θ is left exact, as desired.
Let G be a profinite group and let X be a set with a continuous actionof G. Then we can recover X as the direct limit of the fixed-point sets X U ,where U ranges over the collection of open subgroups of G. Our next result
is an ∞-categorical analogue of this observation.
Lemma 6.3.3.6. Let p : X → S . be a Cartesian fibration of simplicial sets,.
which is classified by a colimit diagram S . → Catop∞ , and let s : S → X bea Cartesian section of p. Then s is a p-colimit diagram.

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Proof. By virtue of Corollary 3.3.1.2, we may suppose that S is an ∞category. Unwinding the definitions, we must show that the map Xs/ → Xs/
induces an equivalence of ∞-categories when restricted to the inverse imageof the cone point of S . . Fix an object x ∈ X lying over the cone point of
S . . Let f : S . → X be the constant map with value x and let f = f |S. To
complete the proof, it will suffice to show that the restriction map
θ : MapFun(S . ,X) (s, f ) → MapFun(S,X) (s, f )is a homotopy equivalence. To prove this, we choose a p-Cartesian transformation α : g → f , where g : S . → X is a section of p (automatically
Cartesian). Let g = g|S and let α : g → f be the associated transformation.
Let C be the full subcategory of MapS . (S . , X) spanned by the Cartesiansections of p and let C ⊆ MapS . (S, X) be defined similarly. We have a commutative diagram in the homotopy category H
MapC (s, g)θ0/ MapC (s, g)αα

MapFun(S . ,X) (s, f )θ/ MapFun(S,X) (s, f ).

Proposition 2.4.4.2 implies that the vertical maps are homotopy equivalences,and Proposition 3.3.3.1 implies that θ0 is a homotopy equivalence (since therestriction map C → C is an equivalence of ∞-categories). It follows that θ
is a homotopy equivalence as well.
Lemma 6.3.3.7. Let p : X → S be a presentable fibration and let C be thefull subcategory of MapS (S, X) spanned by the Cartesian sections of p. For
each vertex s of S, let ψ(s)∗ : C → Xs be the functor given by evaluation ats and let ψ(s)∗ be a left adjoint to ψ(s)∗ . There exists a diagram θ : S →Fun(C, C) with the following properties:(1) For each vertex s of S, θ(s) is equivalent to the composition ψ(s)∗ ◦ψ(s)∗ .
(2) The identity functor idC is a colimit of θ in the ∞-category of functors
Fun(C, C).
Proof. Without loss of generality, we may suppose that p extends to apresentable fibration p : X → S . , which is classified by colimit diagram
S . → PrL (and therefore by a colimit diagram S . → Catop∞ by virtue ofTheorem 5.5.3.18). Let C be the ∞-category of Cartesian sections of p, sothat we have trivial fibrations
C ← C → X∞ ,where X∞ = X ×S . {∞} and ∞ denotes the cone point of S . . For eachvertex s of S . , we let ψ(s)∗ : C → X s be the functor given by evaluation ats and ψ(s)∗ a left adjoint to ψ(s)∗ . To complete the proof, it will suffice toconstruct a map θ0 : S → Fun(C, X∞ ) with the following properties:

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609∞-TOPOI(10 ) For each vertex s of S, θ0 (s) is equivalent to the composition ψ(∞)∗ ◦ψ(s)∗ ◦ ψ(s)∗ .
(20 ) The functor ψ(∞)∗ is a colimit of θ0 .
Let e : C×S . → X be the evaluation map. Choose a p-coCartesian naturaltransformation e → e0 , where e0 is a map from C × S . to X∞ . Lemma 6.3.3.6
implies that for each object X ∈ C, the restriction e|{X} × S . is a p-colimit
diagram in X. Applying Proposition 4.3.1.9, we deduce that e0 |{X} × S . isa colimit diagram in X∞ . According to Proposition 5.1.2.2, e0 determines acolimit diagram S . → Fun(C, X∞ ). Let θ0 be the restriction of this diagram
to S. Then the colimit of θ0 can be identified with e0 |C × {∞}, which isequivalent to e|C×{∞} = ψ(∞)∗ . This proves (20 ). To verify (10 ), we observethat e0 |C × {s} can be identified with the composition of ψ(s)∗ = e|C × {s}with the functor Xs → X∞ associated to the coCartesian fibration p, which
can in turn be identified with ψ(∞)∗ ◦ ψ(s)∗ (both are left adjoints to thepullback functor X∞ → Xs associated to p).
Proposition 6.3.3.8. Let A be a (small) filtered partially ordered set, let p :X → N(A), and let Y ⊆ MapN(A) (N(A), X) be the full subcategory spannedby the Cartesian sections of p. Let Z be an ∞-topos and let π∗ : Z → Y be
an arbitrary functor. Suppose that, for each α ∈ A, the compositionπZ →∗ Y → Xα
is a geometric morphism of ∞-topoi. Then π∗ is a geometric morphism of∞-topoi.
Proof. Let π ∗ denote a left adjoint to π∗ . Since π ∗ commutes with colimits,
Lemma 6.3.3.7 implies that π ∗ can be written as the colimit of a diagramq : N(A) → ZY having the property that for each α ∈ A, q(α) is equivalent toπ ∗ ψ(α)∗ ψ(α)∗ , where ψ(α)∗ denotes the evaluation functor at α and ψ(α)∗
is its left adjoint. Each composition π ∗ ψ(α)∗ is left adjoint to the geometricmorphism ψ(α)∗ π∗ and therefore left exact. It follows that q(α) is left exact.
Since filtered colimits in Z are left exact (Example 7.3.4.7), we conclude thatthe functor π ∗ is left exact, as desired.
Proof of Theorem 6.3.3.1. Let C be a small filtered ∞-category and let q :d∞ ofCop → RTop be an arbitrary diagram. Choose a limit q : (C. )op → Cat
d∞ . We must show that q factors through RTop andq in the ∞-category Cat
is a limit diagram in RTop.
Using Proposition 5.3.1.18, we may assume without loss of generality that
C is the nerve of a filtered partially ordered set A. Let p : X → N(A)op be thetopos fibration classified by q (Proposition 6.3.1.7). Then the image of thecone point of (C. )op under q is equivalent to the ∞-category X of Cartesiansections of p (Corollary 3.3.3.2). It follows from Proposition 6.3.3.3 that X
is an ∞-topos. Moreover, Proposition 6.3.3.5 ensures that for each α ∈ A,
the evaluation map X → Xα is a geometric morphism. This proves that q

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factors through RTop. To complete the proof, we must show that q is a limit
d∞ and q is a limit
diagram in RTop. Since RTop is a subcategory of Catddiagram in Cat∞ , this reduces immediately to the statement of Proposition
6.3.3.8.
6.3.4 General Limits of ∞-Topoi
Our goal in this section is to construct general limits in the ∞-category RTop.
Our strategy is necessarily rather different from that of §6.3.3 because thed does not preserve limits in general. In fact, i does
inclusion i : RTop → Cat
not even preserve the final object:
Proposition 6.3.4.1. Let X be an ∞-topos. Then Fun∗ (S, X) is a contractible Kan complex. In particular, S is a final object in the ∞-category
RTop of ∞-topoi.
Proof. We observe that S ' Shv(∆0 ), where the ∞-category ∆0 is endowedwith the “discrete” topology (so that the empty sieve does not constitutea cover of the unique object). According to Proposition 6.2.3.20, the ∞category Fun∗ (S, X) is equivalent to the full subcategory of X ' Fun(∆0 , X)
spanned by those objects X ∈ X which correspond to left exact functors∆0 → X. It is clear that these are precisely the final objects of X andtherefore form a contractible Kan complex (Proposition 1.2.12.9).
To construct limits in general, we first develop some tools for describing∞-topoi via “generators and relations.” This will allow us to reduce theconstruction of limits in RTop to the problem of constructing colimits in
Cat∞ .
Lemma 6.3.4.2. Let C be a small ∞-category, let κ be a regular cardinal,and suppose we are given a (small) collection of κ-small diagrams {f α :Kα. → C}α∈A . Then there exists a functor F : C → D with the following
properties:(1) The ∞-category D is small and admits κ-small colimits.
(2) For each α ∈ A, the induced map F ◦f α : Kα. → C is a colimit diagram.
(3) Let E be an arbitrary ∞-category which admits κ-small colimits. Let
Fun0 (D, E) denote the full subcategory of Fun(D, E) spanned by thosefunctors which preserve κ-small colimits. Then composition with F
induces a fully faithful embeddingθ : Fun0 (D, E) → Fun(C, E).
The essential image of θ consists of those functors F 0 : C → E suchthat each F 0 ◦ f α is a colimit diagram in E.
Proof. Let j : C → P(C) denote the Yoneda embedding. For each α ∈ A,
let fα = f α |Kα and let Cα ∈ C denote the image of the cone point underhighertopoi

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611∞-TOPOIf α . Let Dα ∈ P(C) denote a colimit of the induced diagram j ◦ fα , so thatj ◦ f α induces a map sα = Dα → j(Cα ). Let S = {sα }α∈A , let X denotethe localization S −1 P(C), and let L : P(C) → X denote a left adjoint to
the inclusion. Let D0 be the smallest full subcategory of X that contains theessential image of the functor L ◦ j and is stable under κ-small colimits, let
D be a minimal model for D0 , and let F : C → D denote the compositionof L ◦ j with a retraction of D0 onto D. It follows immediately from theconstruction that D satisfies conditions (1) and (2).
We observe that for each α ∈ A, the domain and codomain of sα are bothκ-compact objects of P(C). It follows that X is stable under κ-filtered colimitsin P(C). Corollary 5.5.7.3 implies that L carries κ-compact objects of P(C)to κ-compact objects of X. Since the collection of κ-compact objects of X
is stable under κ-small colimits, we conclude that D0 consists of κ-compact
objects of X. Invoking Proposition 5.3.5.11, we deduce that the inclusion
D ⊆ X determines an equivalence Indκ (D) ' X.
We now prove (3). We observe that there exists a fully faithful embeddingi : E → E0 which preserves κ-small colimits, where E0 admits arbitrary small
colimits (for example, we can take E0 = Fun(E, bS)op and i to be the Yoneda
0
embedding). Replacing E by E if necessary, we may assume that E itself
admits arbitrary small colimits. We have a homotopy commutative diagram
FunL (X, E)
Fun0 (D, E)θ0θ/ FunL (P(C), E)/ Fun(C, E),

where FunL (Y, E) denotes the full subcategory of Fun(Y, E) spanned by thosefunctors which preserve small colimits. Propositions 5.3.5.10 and 5.5.1.9 imply that the left vertical arrow is an equivalence, while Theorem 5.1.5.6
implies that the right vertical arrow is an equivalence. It will therefore suffice to show that θ0 is fully faithful and that the essential image of θ0 consistsof those colimit-preserving functors F 0 from P(C) to E such that F 0 ◦ j ◦ f α isa colimit diagram for each α ∈ A. This follows immediately from Proposition
5.5.4.20.
Definition 6.3.4.3. Let Catlex∞ denote the subcategory of Cat∞ defined asfollows:(1) A small ∞-category C belongs to Catlex∞ if and only if C admits finite
limits.
(2) Let f : C → D be a functor between small ∞-categories which admit
finite limits. Then f is a morphism in Catlex∞ if and only if f preserves
finite limits.
Lemma 6.3.4.4. The ∞-category Catlex∞ admits small colimits.

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Proof. Let p : J → Catlex∞ be a small diagram which carries each vertex j ∈ J
to an ∞-category Cj . Let C be a colimit of the diagram p in Cat∞ , and foreach j ∈ J let φj : Cj → C be the associated functor. Consider the collectionof all isomorphism classes of diagrams {f : K / → C}, where K is a finite
simplicial set and the map f admits a factorization
f0

φj

K / → Cj → C,
where f0 is a limit diagram in Cj . Invoking the dual of Lemma 6.3.4.2, wededuce the existence of a functor F : C → D with the following properties:(1) The ∞-category D is small and admits finite limits.
(2) Each of the compositions F ◦ φj is left exact.
(3) For every ∞-category E which admits finite limits, composition with F
induces an equivalence from the full subcategory of Fun(D, E) spannedby the left exact functors to the full subcategory of Fun(C, E) spannedby those functors F 0 : C → E such that each F 0 ◦ φj is left exact.
It follows that that D can be identified with a colimit of the diagram p in
the ∞-category Catlex∞.
Lemma 6.3.4.5. Let C be a small ∞-category which admits finite limits andlet f∗ : X → P(C) be a geometric morphism of ∞-topoi. Then there exists
a small ∞-category D which admits finite limits and a left exact functor
f 00 : C → D such that f∗ is equivalent to the composition
f0

f 00

∗
∗
X→
P(D) →
P(C),where f∗0 is a fully faithful geometric morphism and f∗00 is given by composition with f 00 .
Proof. Without loss of generality, we may assume that X is minimal. Let f ∗be a left adjoint to f∗ . Choose a regular cardinal κ large enough that thecomposition
jC

f∗

C → P(C) → X
carries each object C ∈ C to a κ-compact object of X. Enlarging κ if necessary, we may assume that X is κ-accessible and that the collection of
κ-compact objects is stable under finite limits (Proposition 5.4.7.4). Let D
be the collection of κ-compact objects of X. The proof of Proposition 6.1.5.3
shows that the inclusion D ⊆ X can be extended to a left exact localization∗
functor f 0 : P(D) → X.
Using Theorem 5.1.5.6, we conclude that the composition jD ◦f ∗ ◦jC : C →∗P(D) can be extended to a colimit-preserving functor f 00 : P(C) → P(D)
∗
0∗
00 ∗
∗
and that f ◦ f is homotopic to f . Proposition 6.1.5.2 implies that f 00
00 ∗
0
00
0∗
is left exact. It follows that f and f admit right adjoints f∗ and f∗ withthe desired properties.

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Proposition 6.3.4.6. The ∞-category RTop of ∞-topoi admits pullbacks.
Proof. Suppose first that we are given a pullback square
f∗0

W

/X

g∗0

g∗



f∗
/Z
Y
in the ∞-category of RTop. We make the following observations:(a) Suppose that Z is a left exact localization of another ∞-topos Z0 . Thenthe induced diagram
/X
W

/ Z0


Y
is also a pullback square.

(b) Let S −1 X and T −1 Y be left exact localizations of X and Y, respectively.
Let U be the smallest strongly saturated collection of morphisms in W∗∗
which contains f 0 S and g 0 T and is closed under pullbacks. Using
Proposition 6.2.1.2, we deduce that U is generated by a (small) set ofmorphisms in W. It follows that the diagram/ S −1 X

U −1 WT −1 Y
is again a pullback in RTop.

/Z

Now suppose we are given an arbitrary diagram
g∗

f∗

X→Z←Y
in RTop. We wish to prove that there exists a fiber product X ×Z Y in RTop.
The proof of Proposition 6.1.5.3 implies that there exists a small ∞-category
C which admits finite limits, such that Z is a left exact localization of P(C).
Using (a), we can reduce to the case where Z = P(C). Using (b) and Lemma
6.3.4.5, we can reduce to the case where X = P(D) for some small ∞-category
D which admits finite limits, and g∗ is induced by composition with a left
exact functor g : C → D. Similarly, we can assume that f∗ is determinedby a left exact functor f : C → D0 . Using Lemma 6.3.4.4, we can form apushout diagram
EO o
DO
g

D0 o

f

C

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in the ∞-category Catlex∞ . Using Proposition 6.1.5.2 and Theorem 5.1.5.6, it
is not difficult to see that the induced diagram/ P(D)P(E)
P(D0 )g∗f∗
/ P(C)

is the desired pullback square in RTop.
Corollary 6.3.4.7. The ∞-category RTop admits small limits.
Proof. Using Corollaries 4.2.3.11 and 4.4.2.4, it suffices to show that RTop
admits filtered limits, a final object, and pullbacks. The existence of filteredlimits follows from Theorem 6.3.3.1, the existence of a final object follows
from Proposition 6.3.4.1, and the existence of pullbacks follows from Proposition 6.3.4.6.
Remark 6.3.4.8. Our construction of fiber products in RTop is somewhat
inexplicit. We will later give a more concrete construction in the case ofordinary products; see §7.3.3.
We conclude this section by proving a companion result to Corollary
6.3.4.7. First, a few general remarks. The ∞-category RTop is most naturally viewed as an ∞-bicategory since we can also consider noninvertible
natural transformations between geometric morphisms. Correspondingly, we
can consider a more general theory of ∞-bicategorical limits in RTop. While
we do not want to give any precise definitions, we would like to point out that
Corollary 6.3.4.7 can be generalized to show that RTop admits all (small)∞-bicategorical limits. In more concrete terms, this just means that RTopis cotensored over Cat∞ in the following sense:
Proposition 6.3.4.9. Let X be an ∞-topos and let C be a small ∞-category.
Then there exists an ∞-topos Mor(C, X) and a functor
e : C → Fun∗ (Mor(C, X), X)
with the following universal property:(∗) For every ∞-topos Y, composition with e induces an equivalence of∞-categories
Fun∗ (Y, Mor(C, X)) → Fun(C, Fun∗ (Y, X)).
Proof. We first treat the case where X = P(D), where D is a small ∞category which admits finite limits. Using Lemma 6.3.4.2, we conclude that
there exists a functor e0 : Cop × D → D0 with the following properties:(1) The ∞-category D0 is small and admits finite limits.

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615∞-TOPOI(2) For each object C ∈ C, the induced functore0D0
D ' {C} × D ⊆ Cop × D →is left exact.
(3) Let E be an arbitrary ∞-category which admits finite limits. Thencomposition with e0 induces an equivalence from the full subcategoryof Fun(D0 , E) spanned by the left exact functors to the full subcategoryof Fun(Cop × D, E) spanned by those functors which restrict to left
exact functors {C} × D → E for each C ∈ C.
In this case, we can define Mor(C, X) to be P(D0 ) ande : C → Fun∗ (P(D0 ), P(D))
to be given by composition with e0 ; the universal property (∗) follows immediately from Theorem 5.1.5.6 and Proposition 6.1.5.2.
In the general case, we invoke Proposition 6.1.5.3 to reduce to the casewhere X = S −1 X0 is an accessible left exact localizaton of an ∞-topos X0 ,where X0 ' P(D) is as above so that we can construct an ∞-topos Mor(C, X0 )and a map e0 : C → Fun∗ (Mor(C, X0 ), X0 ) satisfying the condition (∗). For
each C ∈ C, let e0 (C)∗ denote the corresponding geometric morphism from
Mor(C, X0 ) to X0 , let e0 (C)∗ denote a left adjoint to e0 (C)∗ , and let S(C) =e0 (C)∗ S be the image of S in the collection of morphisms of Mor(C, X0 ). Since
each e0 (C)∗ is a colimit-preserving functor, each of the sets S(C) is generated
under colimits by a small collection of morphisms in Mor(C, X0 ). Let T be thesmallest collection of morphisms in Mor(C, X0 ) which is strongly saturated, isstable under pullbacks, and contains each of the sets SC . Using Proposition
6.2.1.2, we conclude that T is generated (as a strongly saturated class ofmorphisms) by a small collection of morphisms in Mor(C, X0 ). It follows
that Mor(C, X) = T −1 Mor(C, X0 ) is an ∞-topos. By construction, the map
e0 restricts to give a functor e : C → Fun∗ (Mor(C, X), X). Unwinding thedefinitions, we see that e has the desired properties.
Remark 6.3.4.10. Let X be an ∞-topos and let RTop∆ denote the simpli∆
d corresponding to the subcategory RTop ⊆ Catd∞ ,cial subcategory of Cat∞∆
so that RTop ' N(RTop ). The construction Y 7→ Fun∗ (X, Y) determines a∆d , which in turn induces a functor
simplicial functor from RTop∆ to Cat∞

d∞ .
θX : RTop → Cat
We claim that θX preserves small limits (this translates into the condition
that limits in RTop really give ∞-bicategorical limits in the ∞-bicategory of∞-topoi).
d∞ → b
To prove this, fix an arbitrary ∞-category C and let eC : Cat
S be
the functor corepresented by C. It will suffice to show that eC ◦ θX preserves
small limits. The collection of all ∞-categories C which satisfy this condition
is stable under all colimits, so we may assume without loss of generality that
C is small. It now suffices to observe that eC ◦ θX is equivalent to the functor
corepresented by the ∞-topos Fun(C, X).

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6.3.5 Étale Morphisms of ∞-Topoi
Let f : X → Y be a continuous map of topological spaces. We say that f isétale (or a local homeomorphism) if, for every point x ∈ X, there exist open
sets U ⊆ X containing x and V ⊆ Y containing f (x) such that f induces
a homeomorphism U → V . Let F denote the sheaf of sections of f : that is,
F is a sheaf of sets on Y such that for every open set V ⊆ Y , F(V ) is theset of all continuous maps s : V → X such that f ◦ s = id : V → Y . The
construction (f : X → Y ) 7→ F determines an equivalence of categories, fromthe category of topological spaces which are étale over Y to the category ofsheaves (of sets) on Y . In particular, we can recover the topological space
X (up to homeomorphism) from the sheaf of sets F on Y . For example, we
can reconstruct the category ShvSet (X) of sheaves on X as the overcategory
ShvSet (Y )/ F .
Our goal in this section is to develop an analogous theory of étale morphisms in the setting of ∞-topoi. Suppose we are given a geometric morphismf∗ : X → Y. Under what circumstances should we say that f∗ is étale? Byanalogy with the case of topological spaces, we should expect that an étalemorphism determines a “sheaf” on Y: that is, an object U of the ∞-category
Y. Moreover, we should then be able to recover the ∞-category X as an
overcategory Y/U . The following result guarantees that this expectation issomewhat reasonable:
Proposition 6.3.5.1. Let X be an ∞-topos and let U be an object of X.
(1) The ∞-category X/U is an ∞-topos.
(2) The projection π! : X/U → X has a right adjoint π ∗ which commuteswith colimits. Consequently, π ∗ itself has a right adjoint π∗ : X/U → X
which is a geometric morphism of ∞-topoi.
Proof. The existence of a right adjoint π ∗ to the projection π! : X/U →X follows from the assumption that X admits finite limits. Moreover, theassertion that π ∗ preserves colimits is a special case of the assumption thatcolimits in X are universal. This proves (2).
To prove (1), we will show that X/U satisfies criterion (2) of Theorem
6.1.0.6. We first observe that X/U is presentable (Proposition 5.5.3.10). Let
K be a small simplicial set and let α : p → q be a natural transformationof diagrams p, q : K . → X/U . Suppose that q is a colimit diagram andthat α = α|K is a Cartesian transformation. The projection π! preserves all
colimits (since it is a left adjoint), so that π! ◦ q is a colimit diagram in X.
Since π! preserves pullback squares (Proposition 4.4.2.9), π! ◦α is a Cartesiantransformation. By assumption, X is an ∞-topos, so that Theorem 6.1.0.6
implies that π! ◦ p is a colimit diagram if and only if π! ◦ α is a Cartesiantransformation. Using Propositions 4.4.2.9 and 1.2.13.8, we conclude thatp is a colimit diagram if and only if α is a Cartesian transformation, as
desired.

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A geometric morphism f∗ : X → Y of ∞-topoi is said to be étale if it arises
via the construction of Proposition 6.3.5.1; that is, if f admits a factorization
f0

f 00

∗
∗
X→
Y/U →
Y,

where U is an object of Y, f∗0 is a categorical equivalence, and f∗00 is a right∗adjoint to the pullback functor f 00 : Y → Y/U . We note that in this case, f ∗00
0
has a left adjoint f! = f! ◦ f∗ . Consequently, f ∗ preserves all limits, not justfinite limits.
Remark 6.3.5.2. Given an étale geometric morphism f : X/U → X of∞-topoi, the description of the pushforward functor f∗ is slightly more complicated than that of f! (which is merely the forgetful functor) or f ∗ (which
is given by taking products with U ). Given an object p : X → U of X/U , thepushforward f∗ X is an object of X which represents the functor “sections ofp.”
The collection of étale geometric morphisms contains all equivalences andis stable under composition. Consequently, we can consider the subcategory
RTopét ⊆ RTop containing all objects of RTop whose morphisms are precisely the étale geometric morphisms. Our goal in this section is to study the∞-category RTopét . Our main results are the following:(a) If X is an ∞-topos containing an object U , then the associated étale
geometric morphism π∗ : X/U → X can be described by a universal
property. Namely, X/U is universal among ∞-topoi Y with a geometricmorphism φ∗ : Y → X such that φ∗ U admits a global section (Proposition 6.3.5.5).
(b) There is a simple criterion for testing whether a geometric morphismπ∗ : X → Y is étale. Namely, π∗ is étale if and only if the functor π ∗admits a left adjoint π! , the functor π! is conservative, and an appropriate push-pull formula holds in the the ∞-category Y (Proposition
6.3.5.11).
(c) Given a pair of topological spaces X0 and X1 and a homeomorphism φ :U0 ' U1 between open subsets U0 ⊆ X0 and U1 ⊆ X1 , we can`“glue”
X0 to X1 along φ to obtain a new topological space X = X0 U0 X1 .
Moreover, the topological space X contains open subsets homeomorphic to X0 and X1 . In the setting of ∞-topoi, it is possible to makemuch more general “gluing” constructions of the same type. We canformulate this idea more precisely as follows: given any diagram {Xα }in the ∞-category RTopét having a colimit X in RTop, each of theassociated geometric morphisms Xα → X is étale (Theorem 6.3.5.13).
Using this fact, we will show that the ∞-category RTopét admits small
colimits.
Remark 6.3.5.3. We will say that a geometric morphism of ∞-topoi f ∗ :Y → X is étale if and only if its right adjoint f∗ : X → Y is étale. Wehighertopoi

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CHAPTER 6

let LTopét denote the subcategory of LTop spanned by the étale geometricopmorphisms, so that there is a canonical equivalence RTopét ' LTopét.
Our first step is to obtain a more precise formulation of the universal
property described in (a):Definition 6.3.5.4. Let f ∗ : X → Y be a geometric morphism of ∞-topoi.
Let U be an object of X and α : 1Y → f ∗ U a morphism in Y, where 1Y
denotes a final object of Y. We will say that α exhibits Y as a classifying∞-topos for sections of U if, for every ∞-topos Z, the diagram
Fun∗ (Y, Z)
φ


Z∗

◦f ∗

/ Fun∗ (X, Z)
φ0


/Z

is a homotopy pullback square of ∞-categories. Here Z∗ denotes the ∞category of pointed objects of Z (that is, the full subcategory of Fun(∆1 , Z)
spanned by morphisms f : Z → Z 0 , where Z is a final object of Z), and themorphisms φ and φ0 are given by evaluation on α and U , respectively.
Let X be an ∞-topos containing an object U . It follows immediately fromthe definition that a classifying ∞-topos for sections of U is uniquely determined up to equivalence provided that it exists. For the existence, we havethe following result:
Proposition 6.3.5.5. Let X be an ∞-topos containing an object U , let
π! : X/U → X be the projection map and let π ∗ : X → X/U be a right adjoint
to π! . Let 1U denote the identity map from U to itself, regarded as a (final)
object of X/U , and let α : 1U → π ∗ U be adjoint to the identity map π! 1U ' U .
Then α exhibits X/U as a classifying ∞-topos for sections of U .
Before giving the proof of Proposition 6.3.5.5, we summarize some of thepleasant consequences.
Corollary 6.3.5.6. Let X be an ∞-topos containing an object U , and let
π ∗ : X → X/U be the corresponding étale geometric morphism. For every∞-topos Z, composition with π ∗ induces a left fibration
Fun∗ (X/U , Z) → Fun∗ (X, Z).
Moreover, the fiber over a geometric morphism φ∗ : X → Z is homotopy
equivalent to the mapping space MapZ (1Z , φ∗ U ).
Remark 6.3.5.7. Corollary 6.3.5.6 implies, in particular, the existence of
homotopy fiber sequences
MapZ (1Z , φ∗ U ) → MapLTop (X/U , Z) → MapLTop (X, Z)(where the fiber is taken over a geometric morphism φ∗ ∈ MapLTop (X, Z)).
Suppose that Z = X/V and that φ∗ is a right adjoint to the projection
X/V → Z. We then deduce the existence of a canonical homotopy equivalence
MapX (V, U ) ' MapZ (1Z , φ∗ U ) ' MapLTopX / (X/U , X/V ).

highertopoi

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619∞-TOPOI

Remark 6.3.5.8. It follows from Remark 6.3.5.7 that if f ∗ : X → Y is ageometric morphism of ∞-topoi and U ∈ X is an object, then the induced
diagram
X

/Y


X/U


/ Y/f ∗ U

is a pushout square in LTop.
Corollary 6.3.5.9. Suppose we are given a commutative diagram
?Y
f∗

g∗



/Z

h∗

X

in LTopop , where g∗ is étale. Then f∗ is étale if and only if h∗ is étale.
Proof. The “only if” direction is obvious. To prove the converse, let us suppose that g∗ and h∗ are both étale, so that we have equivalences X ' Z/U
and U ' Z/V for some pair of objects U, V ∈ Z. Using Remark 6.3.5.7, wededuce that the morphism f∗ is determined by a map U → V in Z, which
we can identify with an object V ∈ Y such that X ' Y/V .
Remark 6.3.5.10. Let X be an ∞-topos. The projection map
p : Fun(∆1 , X) → Fun({1}, X) ' X
is a Cartesian fibration. Moreover, for every morphism α : U → V in X,
the associated functor α∗ : X/V → X/U is an étale geometric morphism of∞-topoi, so that p is classified by a functor χ0 : Xop → LTopét . The functor
χ0 carries the final object of X to an ∞-topos equivalent to X and therefore
factors as a compositionχ

Xop → (LTopét )X / → LTopét .
The argument of Remark 6.3.5.7 shows that χ is fully faithful, and it follows
immediately from the definitions that χ is essentially surjective. Corollary
6.3.5.9 allows us to identify (LTopét )X / with the full subcategory of LTopX /spanned by the étale geometric morphisms f ∗ : X → Y. Consequently, we canregard χ as a fully faithful embedding of X into the ∞-category (LTopop )/ X
of ∞-topoi over X, whose essential image consists of those ∞-topoi which
are étale over X.
Proof of Proposition 6.3.5.5. Let p : M → ∆1 be a correspondence from
X/U ' M ×∆1 {0} to X ' M ×∆1 {1} associated to the adjoint functors
X/U o

π!
π∗

/X .

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CHAPTER 6

Let α0 be a morphism from 1U ∈ X/U to 1X ∈ X in M (so that α0 isdetermined uniquely up to homotopy). We observe that there is a retractionr : M → X/U which restricts to π ∗ on X ⊆ M, and we can identify α withr(α0 ).
Let Z be an arbitrary ∞-topos. Let C be the full subcategory of Fun(M, Z)
spanned by those functors F : M → Z with the following properties:(a) The restriction F | X/U : X/U → Z preserves small colimits and finite
limits.
(b) The functor F is a left Kan extension of F | X/U . In other words, F
carries p-Cartesian morphisms in M to equivalences in Z.
Proposition 4.3.2.15 implies that the restriction map C → Fun∗ (X/U , Z) is atrivial Kan fibration. Moreover, this trivial Kan fibration has a section given
by composition with r. It will therefore suffice to show that the diagramC/ Fun∗ (X, Z)


Z∗


/Z

is a homotopy pullback square. In other words, we wish to show that restriction along α0 and the inclusion X ⊆ M induce a categorical equivalence
C → Z∗ ×Z Fun∗ (X, Z).
We define simplicial subsets M00 ⊆ M0 ⊆ M as follows:`(i) Let M00 ' X {1} ∆1 be the union of X with the 1-simplex of M
corresponding to the morphism α0 .
(ii) Let M0 be the full subcategory of M spanned by X together with theobject 1U .
We can identify Z∗ ×Z Fun∗ (X, Z) with the full subcategory
C00 ⊆ Fun(M00 , Z)
spanned by those functors F satisfying the following conditions:(a0 ) The restriction F | X preserves small colimits and finite limits.
(b0 ) The object F (1U ) is final in Z.
Let C0 be the full subcategory of Fun(M0 , Z) spanned by those functors which
satisfy (a0 ) and (b0 ). To complete the proof, it will suffice to show that the
restriction mapsuv

C → C0 → C00
are trivial Kan fibrations.
We first show that u is a trivial Kan fibration. In view of Proposition
4.3.2.15, it will suffice to prove the following:highertopoi

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621∞-TOPOI(∗) A functor F : M → Z satisfies (a) and (b) if and only if it satisfies (a0 )and (b0 ) and F is a right Kan extension of F | M0 .
To prove the “only if” direction, let us suppose that F satisfies (a) and (b).
Without loss of generality, we may suppose F = F0 ◦ r, where F0 = F | X/U .
Then F | X = F0 ◦ π ∗ . Since F0 and π ∗ both preserve small colimits andfinite limits, we deduce (a0 ). Condition (b0 ) is an immediate consequence of(a). We must show that F is a right Kan extension of F | X. Unwinding thedefinitions (and applying Corollary 4.1.3.1), we are reduced to showing thatfor every object V ∈ X/U corresponding to a morphism V → U in X, thediagramF (V )/ F (V )F (1U )


/ F (U )is a pullback square in Z. Since F = F0 ◦ r and F0 preserves finite limits, itsuffices to show that the squareV/ π∗ V
1U/ π∗ Uis a pullback square in X/U . In view of Proposition 1.2.13.8, it suffices toobserve that the squareV/ V ×UU/ U ×Uis a pullback in X.
Now let us suppose that F is a right Kan extension of F1 = F | M0 andthat F1 satisfies conditions (a0 ) and (b0 ). We first claim that F satisfies (b).
In other words, we claim that for every object V ∈ X, the canonical map
F (π ∗ V ) → F (V ) is an equivalence. Consider the diagram
F (π ∗ V )/ F (V × U )/ F (V )F (1U )


/ F (U )
/ F (1X ).

Since F is a right Kan extension of F1 , the left square is a pullback. Since F1satisfies (a), the right square is a pullback. Therefore the outer square is apullback. Condition (b0 ) implies that the lower horizontal composition is anequivalence, so the upper horizontal composition is an equivalence as well.

highertopoi

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CHAPTER 6

We now prove that F satisfies (a). Condition (b0 ) implies that the functor
F0 = F | X/U preserves final objects. It will therefore suffice to show that
F0 preserves small colimits and pullback squares. Since F is a right Kan
extension of F1 , the functor F0 can be described by the formula
V 7→ F (π! V ) ×F (U ) F (1U ).
It therefore suffices to show that the functors π! , F | X, and • ×F (U ) F (1U )preserve small colimits and pullback squares. For π! , this follows from Propositions 1.2.13.8 and 4.4.2.9. For F | X, we invoke assumption (a0 ). For thefunctor • ×F (U ) F (1U ), we invoke our assumption that Z is an ∞-topos (sothat colimits in Z are universal). This completes the verification that u is atrivial Kan fibration.
To complete the proof, we must show that the functor v is a trivial Kan
fibration. We note that v fits into a pullback diagram
v
/ C00
C0

Fun(M0 , Z)v0
/ Fun(M00 , Z).

It will therefore suffice to show that v 0 is a trivial Kan fibration. Since Z is an∞-category, we need only show that the inclusion M00 ⊆ M0 is a categoricalequivalence of simplicial sets. This is a special case of Proposition 3.2.2.7.
We next establish the recognition principle promised in (b):Proposition 6.3.5.11. Let f ∗ : X → Y be a geometric morphism of ∞topoi. Then f ∗ is étale if and only if the following conditions are satisfied:(1) The functor f ∗ admits a left adjoint f! (in view of Corollary 5.5.2.9,
this is equivalent to the assumption that f ∗ preserves small limits).
(2) The functor f! is conservative. That is, if α is a morphism in Y suchthat f! α is an equivalence in X, then α is an equivalence in Y.
(3) For every morphism X → Y in X, every object Z ∈ Y, and every
morphism f! Z → Y , the induced diagram/ f! Zf! (f ∗ X ×f ∗ Y Z)

X


/Y

is a pullback square in X.
Remark 6.3.5.12. Condition (3) of Proposition 6.3.5.11 can be regarded
as a push-pull formula: it provides a canonical equivalencef! (f ∗ X ×f ∗ Y Z) ' X ×Y f! Z.
In particular, when Y is final in X, we have an equivalence f! (f ∗ X × Z) 'X × f! Z: in other words, the functor f! is “linear” with respect to the actionof X on Y.

highertopoi

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623∞-TOPOI

Proof of Proposition 6.3.5.11. Suppose first that f ∗ is an étale geometricmorphism. Without loss of generality, we may suppose that Y = X/U andthat f ∗ is right adjoint to the forgetful functor f! : X/U → X. Assertions (1)and (2) are obvious, and assertion (3) follows from the observation that, forevery diagram
X → Y ← Z → U,
the induced map (X × U ) ×Y ×U Z → X ×Y Z is an equivalence in X.
For the converse, let us suppose that (1), (2), and (3) are satisfied. We
wish to show that f ∗ is étale. Let U = f! 1Y . Let F denote the compositionf!
Y ' Y/1Y → X/U . To complete the proof, it will suffice to show that F is
an equivalence of ∞-categories. Proposition 5.2.5.1 implies that F admits aright adjoint G given by the formula(X → U ) 7→ f ∗ X ×f ∗ U 1Y .
Assumption (3) guarantees that the counit map v : F G → idX/U is anequivalence. To complete the proof, it suffices to show that for each Y ∈ Y,
the unit map uY : Y → GF Y is an equivalence. The map F uY : F Y →F GF Y has a left homotopy inverse (given by vF Y ) which is an equivalence,
so that F uY is an equivalence. It follows that f! uY is an equivalence, so thatuY is an equivalence by virtue of assumption (2). Thus G is a homotopy
inverse to F , so that F is an equivalence of ∞-categories, as desired.
Our final goal in this section is to prove the following result:
Theorem 6.3.5.13. The ∞-category RTopét admits small colimits, and theinclusion RTopét ⊆ RTop preserves small colimits.
The proof of Theorem 6.3.5.13 is rather technical and will occupy ourattention for the remainder of this section. However, the analogous result iselementary if we work with ∞-topoi which are assumed to be étale over a
fixed base X. In this case, Theorem 6.3.5.13 can be reduced (with the aid ofRemark 6.3.5.10) to the following assertion:
Proposition 6.3.5.14. Let X be an ∞-topos and let χ : X → LTopop/ X bethe functor of Remark 6.3.5.10. Then χ preserves small colimits.
Proof. Combine Propositions 1.2.13.8 and 6.3.2.3 with Theorem 6.1.3.9.
Proof of Theorem 6.3.5.13. As a first step, we establish the following:(∗) Suppose we are given a small diagram p : K → LTopop and a geometricmorphism of ∞-topoi φ∗ : lim(p) → Y. Suppose further that for each−→vertex v in K, the induced geometric morphism φ(v)∗ : p(v) → Y isétale. Then φ∗ is étale.
To prove (∗), we note that φ∗ determines a functor p : K → LTopop/ Y liftingp. Since each φ(v)∗ is étale, Remark 6.3.5.10 implies that p factors as acompositionqχ

K → Y → LTopop/Y .

highertopoi

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CHAPTER 6

Let U ∈ Y be a colimit of the diagram q. Then Corollary 6.3.5.9 implies thatlim(p) ' Y/U , so that φ∗ is étale, as desired.
−→
We now return to the proof of Theorem 6.3.5.13. Using Proposition 4.4.3.2
and its proof, we can reduce the proof to the following special cases:
op(a) The ∞-category LTopétadmits small coproducts, and the inclusion
op
op
LTopét ⊆ LTop preserves small coproducts.
op(b) The ∞-category LTopétadmits coequalizers, and the inclusion
op
LTopét⊆ LTopoppreserves coequalizer diagrams.
We first prove (a). In view of (∗), it will suffice to prove the following:(a0 ) Let {Xα } be a small collection of ∞-topoi and let X be their coopproduct
Q in LTop (so that we have an equivalence of ∞-categories
X ' α Xα ). Then each of the associated geometric morphisms φ∗α :X → Xα is étale.
Q
To prove (a0 ), we may assume without loss of generality that X = α Xα∗and that φα is given by projection onto the corresponding factor. The desired result then follows from the criterion of Proposition 6.3.5.11 (moreconcretely: let let U ∈ X be an object whose image in Xα is a final object
Uα ∈ Xα andQ whose image in Xβ is an initial object Uβ ∈ Xβ for β 6= α;then X/U ' β (Xβ )/Uβ ' Xα .)To prove (b), we can again invoke (∗) to reduce to the following assertion:(b0 ) Suppose we are given a diagram
Y

// X0

op
in LTopéthaving colimit X in LTopop . Then the induced geometricmorphism φ∗ : X0 → X is étale.

To prove (b0 ), we identify the diagram in question with a functor p :I → LTopop and I with the subcategory of N(∆)op spanned by the objects{[0], [1]} and injective maps of linearly ordered sets. Let X• be the simplicial
object of LTopop given by left Kan extension along the inclusion I ⊆ N(∆)op ,so that each Xn is equivalent to a coproduct (in LTopop ) of X0 with n copiesof Y. Using (∗) and Corollary 6.3.5.9, we deduce that X• is a simplicial
opobject in LTopét. Consequently, assertion (b0 ) is an immediate consequenceof Lemma 4.3.2.7 and the following:
op(b00 ) Let X• be a simplicial object of LTopét
and let X be its geometric realop
ization in LTop . Then the induced geometric morphism φ∗ : X0 → X
is étale.

The proof of (b00 ) is based on the following lemma whose proof we defer
until the end of this section:

highertopoi

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625∞-TOPOI

op
Lemma 6.3.5.15. Suppose we are given a simplicial object X• in LTopét.
op0
Then there exists a morphism of simplicial objects X• → X• of LTopét withthe following properties:(1) The induced map X0 → X00 is an equivalence of ∞-topoi.
(2) The simplicial object X0• is a groupoid object in LTopop .
(3) The induced map of geometric realizations (in LTopop ) is an equivalence | X• | → | X0• |.
Using Lemma 6.3.5.15, we can reduce the proof of (b00 ) to the special casewhere X• is a groupoid object of LTopop . The diagram
op

X
op
d
⊆ Cat
N(∆)op →• LTopét∞

is classified by a Cartesian fibration q : Z → N(∆)op . Here we can identify
Xn with the fiber Z[n] = Z ×N(∆)op {[n]}, and every map of linearly orderedsets α : [m] → [n] induces a geometric morphism α∗ : Z[m] → Z[n] . Since thegeometric morphism α∗ is étale, it admits a left adjoint α! , so that q is alsoa coCartesian fibration (Corollary 5.2.2.4).
It follows from Propositions 6.3.2.3 and 3.3.3.1 that we can identify X
with the full subcategory of FunN(∆)op (N(∆)op , Z) spanned by the Cartesiansections of q; under this identification, the pullback functor φ∗ corresponds
to the functor X → Z[0] ' X0 given by evaluation at [0].
Let 1X denote a final object of X, which we regard as a section of q. Let
T : N(∆)op → N(∆)op denote the shift functor [n] 7→ [n]?[0] and let β0 : T →idN(∆)op denote the evident natural transformation. Let β : (1X ◦T ) → U• bea natural transformation in Fun(N(∆)op , Z) lifting β which is q-coCartesian.
op
Since X• is a groupoid object of LTopét, we deduce that U• is a Cartesiansection of q, which we can identify with an object of X.
Let S = N(∆)op × ∆1 , so that β0 defines a map S → N(∆)op . Let Z0 =Z ×N(∆)op S and let βS = β, regarded as a section of the projection q 0 :/β

Z0 → S. Let Z00 = Z0 S (see §4.2.2 for an explanation of this notation). Let
q 00 : Z00 → S. The fibers of q 00 can be described as follows:
/1Z

[n+1]
• The fiber of q 00 over ([n], 0) can be identified with Z[n+1]' Z[n+1] .

/U• The fiber of q 00 over ([n], 1) can be identified with Z[n]n ' Z[n+1] .
Proposition 4.2.2.4 implies that the projection q 00 : Z00 → S is a coCarted∞ . The above description
sian fibration classified by a map χ : S → Catshows that χ can be regarded as an equivalence from χ0 = χ| N(∆)op × {0}d∞ .
to χ1 = χ| N(∆)op × {1} in the ∞-category of simplicial objects of Cat
Moreover, the functor χ0 classifies the pullback of the coCartesian fibration
q by the translation map T : N(∆)op → N(∆op ), so that χ0 and χ1 factor
opthrough LTopét. Lemma 6.1.3.17 implies that the colimit of χ0 (hence alsohighertopoi

April 9, 2017626

CHAPTER 6of χ1 ) in LTopop is canonically equivalent to X0 . On the other hand, Propositions 3.3.3.1 and 6.3.2.3 allow us to identify lim(χ1 ) with the ∞-category−→of Cartesian sections of the projection Z00 ×S (N(∆)op × {1}) → N(∆)op ,which is isomorphic to X/U• as a simplicial set. We now complete the proof
by observing that the resulting identification X0 ' X/U• is compatible withthe projection φ∗ : X0 → X.
The remainder of this section is devoted to the proof of Lemma 6.3.5.15.
We first need to introduce a bit of notation. We begin with a few remarks
about the behavior of ∞-topoi under change of universe.
Notation 6.3.5.16. Let X be an ∞-topos and C an arbitrary ∞-category.
We let ShvC (X) denote the full subcategory of Fun(Xop , C) spanned by thosefunctors which preserve small limits. We will refer to ShvC (X) as the ∞category of C-valued sheaves on X.
Remark 6.3.5.17. Let X be an ∞-topos. Proposition 5.5.2.2 implies that
the Yoneda embedding X → ShvS (X) is an equivalence; in other words, we
can identify X with the ∞-category of sheaves of (small) spaces on itself. Letb
S denote the ∞-category of spaces which belong to some larger universe U.
We claim the following:(a) The ∞-category ShvSb (X) can be regarded as an ∞-topos in U.
(b) The inclusion ShvS (X) ⊆ ShvSb (X) preserves small colimits.
To prove (a), let us suppose that X = S −1 P(C), where C is a small ∞category and S is a strongly saturated class of morphisms in P(C) which isstable under pullbacks and of small generation. Theorem 5.1.5.6 and Proposibbtion 5.5.4.20 allow us to identify ShvSb (X) with S −1 P(C),where P(C)denotesop bb
the presheaf ∞-category Fun(C , S). Let S denote the strongly saturatedbclass of morphisms of P(C)generated by S. Then Sb is of small generation(and therefore of U-small generation); to complete the proof of (a) it will
suffice to show that Sb is stable under pullbacks.
b 0 denote the full subcategory of P(C)b
Let P(C)spanned by those objects
X with the following property:(∗) Let
/ Y0

Y
f


X

f0


/ X0

b
be a pullback diagram in P(C).
If f 0 ∈ S then f ∈ Sb0 .
bb 0 is stable under
Since colimits in P(C)are universal, the subcategory P(C)b
U-small colimits in P(C).
Moreover, since S is stable under pullbacks inhighertopoi

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627∞-TOPOIb
P(C) (and since the inclusion P(C) ⊆ P(C)is fully faithful), the ∞-category
0
b
b
P(C)
contains P(C). Since P(C)is generated (under U-small colimits) by
the essential image of the Yoneda embedding C → P(C), we conclude thatb 0 = P(C).
b
P(C)
bWe now let S 0 denote the collection of all morphisms in P(C)such that,for every pullback diagram
/ Y0

Y

f0

f


/ X0


X

b The above argument shows that S ⊆ S 0 .
in P(C), if f 0 ∈ S 0 then f ∈ S.
0
Since S is strongly saturated, we conclude that Sb ⊆ S 0 , so that Sb is stable
under pullbacks, as desired. This completes the proof of (a).
To prove (b), it will suffice to show that the composite mapb
P(C) → S −1 P(C) → Sb−1 P(C)preserves small colimits. We can rewrite this as the composition of a pair offunctorsi b
Lb
P(C) → P(C)→ Sb−1 P(C).

bb
The functor L is left adjoint to the inclusion of Sb−1 P(C)into P(C)andtherefore preserves all U-small colimits. It therefore suffices to show that theinclusion i : Fun(Cop , S) → Fun(Cop , bS) preserves small colimits. In view of
Proposition 5.1.2.2, it will suffice to prove the inclusion i0 : S → b
S preserves
small colimits. We note that i0 is an equivalence from S to the full sub0
category b
S ⊆b
S spanned by the essentially small spaces. It now suffices toobserve that the collection of essentially small spaces is stable under small
colimits (this follows from Corollaries 5.4.1.5 and 5.3.4.15).
Remark 6.3.5.18. Let U be a universe as in Example 6.3.5.17, let f ∗ : X →Y be a geometric morphism of ∞-topoi, and let fb∗ : ShvSb (X) → ShvSb (Y) be
given by composition with f ∗ . Then fb∗ can be identified with a geometricmorphism in the universe U. To prove this, let κ denote the regular cardinalin the universe U such that small sets (in our original universe) can be
identified with κ-small sets in U. It follows from Corollary 5.3.5.4 that wedκ (X) and Ind
dκ (Y), respectively.
can identify ShvSb (X) and ShvSb (Y) with Ind
Proposition 5.2.6.3 implies that fb∗ admits a left adjoint fb∗ which fits into acommutative diagram
X
ShvSb (X)

f∗

fb∗

/Y

/ Shvb (Y).
S

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To complete the proof, it will suffice to show that fb∗ is left exact. Since f ∗preserves final objects, the functor fb∗ preserves final objects as well. It therefore suffices to show that fb∗ preserves pullback diagrams. Using Proposition
5.3.5.15 and Example 7.3.4.7, we conclude that every pullback diagram in
ShvSb (X) can be obtained as a U-small κ-filtered colimit of pullback diagramsin X. The desired result now follows from the assumption that f ∗ is left exact
and the observation that the class of pullback diagrams in ShvSb (Y) is stable
under U-small filtered colimits (Example 7.3.4.7).
For the remainder of this section, we fix a larger universe U. Let b
S denotethe ∞-category of U-small spaces.
Notation 6.3.5.19. Let F : LTop → b
S be a functor. For every ∞-topos X,
we let FX : Xop → b
S denote the compositionFX/
Xop ' LTopét → LTop → b
S.

We will say that FX is a sheaf if, for every ∞-topos X, the functor FX
opdpreserves small limits. We let Shv(LTop) denote the full subcategory ofbFun(LTop, S) spanned by those functors which are sheaves.
Example 6.3.5.20. Let X be an ∞-topos and let eX : LTop → b
S be thefunctor represented by X. Proposition 6.3.5.14 implies that eX belongs to
op
op
d
d
Shv(LTop). We will say that a sheaf F ∈ Shv(LTop) is representable if
F ' eX for some ∞-topos X.
opdLemma 6.3.5.21. The ∞-category Shv(LTop) is an ∞-topos in the universe U. Moreover, for every ∞-topos X, the restriction functor F 7→ FX
opddetermines a functor Shv(LTop) → ShvSb (X) which preserves U-small colimits and finite limits.

Proof. Let Funét (∆1 , LTop) be the full subcategory Fun(∆1 , LTop) spannedby the étale morphisms and let e : Funét (∆1 , LTop) → LTop be given byevaluation at the vertex {0} ∈ ∆1 . Since the collection of étale morphisms in
LTop is stable under pushouts (Remark 6.3.5.8), the map e is a coCartesianfibration.
We define a simplicial set K equipped with a projection p : K → LTop sothat the following universal property is satisfied: for every simplicial set K,
we have a natural bijection
HomInd(Gop ) (K, K) = HomSet∆ (K ×LTop Funét (∆1 , LTop), b
S).
Then K is an ∞-category whose objects can be identified with pairs (X, FX ),X/
where X is an ∞-topos and FX : LTopét → b
S is a functor. It follows from
Corollary 3.2.2.13 that the projection p is a Cartesian fibration and that amorphism (X, FX ) → (Y, FY ) is p-Cartesian if and only if, for every object
U ∈ X, the canonical map FX (X/U ) → FY (Y/f ∗ U ) is a homotopy equivalence,
where f ∗ denotes the underlying geometric morphism from X to Y.

highertopoi

April 9, 2017

629∞-TOPOILet K0 denote the full subcategory of K spanned by pairs (X, FX ), where
the functor FX preserves small limits. It follows from the above that the
Cartesian fibration p restricts to a Cartesian fibration p0 : K0 → LTop (withthe same class of Cartesian morphisms). The fiber of K0 over an object X ∈LTop can be identified with ShvSb (X), which is an ∞-topos in the universe
U (Remark 6.3.5.17). Moreover, to every geometric morphism f ∗ : X → Y
in LTop, the Cartesian fibration p0 associates the pushforward functor fb∗ :ShvSb (Y) → ShvSb (X) given by composition with f ∗ . It follows from Remark
6.3.5.18 that fb∗ admits a left adjoint fb∗ and that fb∗ is left exact. We may
summarize the situation by saying that p0 is a topos fibration (see Definition
6.3.1.6); in particular, p0 is a coCartesian fibration.
Let Y = FunLTop (LTop, K0 ) denote the ∞-category of sections of p0 .
Unwinding the definitions, we obtain an identification
Y ' Fun(Funét (∆1 , LTop), b
S).
Let LTop0 denote the essential image of the (fully faithful) diagonal embedding LTop → Fun(∆1 , LTop). Consider the following conditions on a sections : LTop → K0 of p0 :(a) The functor s carries étale morphisms in LTop to p-coCartesian morphisms in X.
(b) Let S : Funét (∆1 , LTop) → b
S be the functor corresponding to s. Then,for every commutative diagram
?Y

/Z
X
of étale morphisms in LTop, the induced map S(X → Z) → S(Y → Z)
is an equivalence in b
S.
(c) For every étale morphism f ∗ : X → Y in LTop, the canonical map
S(idX ) → S(f ∗ ) is an equivalence in bS.
(d) The functor S is a left Kan extension of S| LTop0 .
Unwinding the definitions, we see that (a) ⇔ (b) ⇒ (c) ⇔ (d). Moreover,the implication (c) ⇒ (b) follows by a two-out-of-three argument. Let Y0denote the full subcategory of Y spanned by those sections which satisfy theequivalent conditions (a) through (d); it follows from Proposition 4.3.2.15
that composition with the diagonal embedding LTop → Funét (∆1 , LTop)opdinduces an equivalence θ : Y0 → Shv(LTop). The desired result now follows
from Proposition 5.4.7.11.
To prove Lemma 6.3.5.15, we need a criterion which will allow us to detect étale geometric morphisms of ∞-topoi in terms of the functors thatthey represent. To formulate this criterion, we introduce a bit of temporary
terminology.

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opdDefinition 6.3.5.22. Let α : F → G be a morphism in Shv(LTop). Wewill say that α is universal if, for every geometric morphism of ∞-topoi
f ∗ : X → Y, the induced diagramfb∗ FX/ fb∗ GX


FY


/ GY

is a pullback square in ShvSb (Y) (here fb∗ denotes the geometric morphism
described in Remark 6.3.5.18).
opdRemark 6.3.5.23. The collection of universal morphisms in Shv(LTop)
is stable under pullbacks and composition and contains every equivalence inopdShv(LTop).
opdRemark 6.3.5.24. Let p : K → Shv(LTop) be a small diagram having
a colimit F . Assume that for every edge v → v 0 of K, the induced map
p(v) → p(v 0 ) is universal. Then each of the induced maps p(v) → F isuniversal. This follows immediately from Theorem 6.1.3.9.
opdLemma 6.3.5.25. Let α : F → G be a morphism in Shv(LTop) andassume that G is representable by an ∞-topos X. Then F is representable by
an ∞-topos étale over X if the following conditions are satisfied:(1) The morphism α is universal (in the sense of Definition 6.3.5.22).
(2) For every ∞-topos Y, the homotopy fibers of the induced map F (Y) →G(Y) are essentially small.
Remark 6.3.5.26. In fact, the converse to Lemma 6.3.5.25 is true as well,
but we will not need this fact.
Proof. Choose a point η ∈ G(X) which induces an equivalence eX → G.
The map η induces a global section η of GX in the ∞-topos ShvSb (X). Let
F0 denote the fiber product FX ×GX 1ShvSb (X) . Assumption (2) implies thatthe functor F0 : Xop → b
S takes values which are essentially small. It follows
from Proposition 5.5.2.2 that F0 is representable by an object U ∈ X. In
particular, we have a tautological point η 0 ∈ F0 (U ), which determines acommutative diagram
eX/U

/F




/G

α

eX

op
d
in Shv(LTop). To complete the proof, it will suffice to show that the upperhorizontal map is an equivalence.

highertopoi

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631∞-TOPOIopd
Fix an ∞-topos Y and let R : Shv(LTop) → ShvSb (Y) be the restriction
map; we will show that the induced map R(eX/U ) → R(F ) is an equivalence in ShvSb (Y). It will suffice to show that for every V ∈ Y, the map
R(eX/U (V ) → R(F )(V ) induces a homotopy equivalence after passing to
the homotopy fibers over any point η 0 ∈ R(G)(V ). Replacing Y by Y/V , we
may assume that η 0 is induced by a geometric morphism f ∗ : X → Y which
determines a map 1 → R(G), where 1 denotes the final object of ShvSb (Y).
Let F 0 = R(eX/U ) ×R(G) 1 and F 00 = R(F ) ×R(G) 1; to complete the proof it
will suffice to show that the induced map F 0 → F 00 is an equivalence.
We have a commutative diagram

F 00

1

fb∗ η

/ fb∗ FX

/ R(F )

/ fb∗ GX/ R(G)

in the ∞-category ShvSb (Y). Here the right square is a pullback since α isuniversal, and the outer square is a pullback by construction. It follows thatthe left square is also a pullback, so that
F 00 ' fb∗ (1X ×G FX ) ' fb∗ F0 .
X

We note that fb∗ F0 can be identified with the functor represented by theobject f ∗ U ∈ Y, which (by virtue of Remark 6.3.5.7) is equivalent to F 0 , asdesired.
Lemma 6.3.5.27. Let κ be an uncountable regular cardinal and let X• be asimplicial object of S with the following properties:(a) For each n ≥ 0, the connected components of X• are essentially κsmall.
(b) For every morphism [m] → [n] in ∆, the induced map Xn → Xm hasessentially κ-small homotopy fibers.
Let X be the geometric realization of X• . Then the induced map X0 → X
has essentially κ-small homotopy fibers.
Proof. Replacing X by one of its connected components X 0 (and each Xnby the inverse image X 0 ×X Xn ), we may suppose that X is connected.
Let R ⊆ π0 X0 × π0 X0 denote the image of π0 X1 and let ∼ denote theequivalence relation on π0 X0 generated by R. It follows from assumption (b)that for every κ-small subset A ⊆ π0 X0 , the intersections R ∩ (A × π0 X0 )and R ∩ (π0 X0 × A) are again κ-small. Since κ is uncountable, it follows
that the every ∼-equivalence class is κ-small. Since (π0 X0 )/ ∼ is isomorphic
to π0 X ' ∗, we conclude that π0 X is itself κ-small. Combining this with(a), we conclude that X0 is essentially κ-small. Invoking (b), we deducethat each Xn is essentially κ-small, so that X is essentially κ-small. The
desired conclusion now follows from the long exact sequences associated to
the fibration sequences X0 ×X {∗} → X0 → X.

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Lemma 6.3.5.28. Let X be an ∞-topos. Then
(1) The inclusion ShvSb (X) ⊆ Fun(Xop , bS) admits a left exact left adjoint
L.
(2) Let F ∈ Fun(Xop , bS) be a functor such that each of the spaces F (X) isessentially small. Then each of the spaces LF (X) is essentially small.
(3) Let α : F → G be a morphism in Fun(Xop , bS) such that, for eachX ∈ X, the homotopy fibers of the induced map F (X) → G(X) areessentially small. Then for each X ∈ X, the homotopy fibers of themap LF (X) → LG(X) are also essentially small.
Proof. The existence of the left adjoint L follows from Lemma 5.5.4.19. Since
ShvSb (X) contains the essential image of the Yoneda embedding j : X →Fun(Xop , bS), we can identify L ◦ j with j. Since j is left exact, Proposition
6.1.5.2 implies that L is also left exact. This proves (1).
We now prove (2). Choose a (small) regular cardinal κ such that X is
κ-accessible and let Xκ denote the full subcategory of X spanned by the
κ-compact objects. Let T denote the composition
T0

T 00

Fun(Xop , bS) → Fun((Xκ )op , bS) → Fun(Xop , b
S),
where T 0 is the restriction functor and T 00 is given by the right Kan extension.
We have an evident natural transformation id → T which exhibits T as alocalization functor on Fun(Xop , b
S). Proposition 6.1.3.6 implies that everyb
S-valued sheaf on X is T -local. It follows that the canonical map L → LT
is an equivalence of functors. In particular, to prove that LF (X) is locallysmall, we may assume without loss of generality that F is T -local.
Let Fun0 (Xop , bS) denote the full subcategory of Fun(Xop , bS) spanned by
the T -local functors (in other words, those functors which are right Kan
extensions of their restriction to (Xκ )op ; by Proposition 4.3.2.15 this ∞category is equivalent to Fun((Xκ )op , bS)). We can identify Fun0 (Xop , bS) withκb
the ∞-category of S-valued sheaves ShvSb (P(X )) on the ∞-topos P(Xκ ).
Let F 0 be the image of F under this identification; we observe that thefunctor F 0 : P(Xκ )op → b
S takes essentially small values. In Remark 6.3.5.17,
we saw that this ∞-category contains ShvSb (X) as a left exact localizationand that the localization functor L0 : ShvSb (P(Xκ )) → ShvSb (X) is equivalent
to L when restricted to ShvS (P(Xκ )) ' P(Xκ ). Since F 0 belongs to theessential image of the inclusion ShvS (P(Xκ )) ⊆ ShvSb (P(Xκ )), the argument
given there proves that L0 F 0 belongs to the essential image of the inclusion
ShvS (X) ⊆ ShvSb (X), so that LF (X) is essentially small, as desired.
To prove (3), let us fix a point η ∈ LG(X). We wish to prove the following
stronger version of (3):(30 ) For every map U → X in X, the homotopy fiber of the induced map
LF → LG is essentially small (here the homotopy fiber is taken overthe point determined by η).

highertopoi

April 9, 2017

633∞-TOPOILet X0/X denote the full subcategory of X/X spanned by those morphisms
U → X for which condition (20 ) is satisfied. Since LF and LG belong to
ShvSb (X) (and since the collection of essentially small spaces is stable undersmall limits), we conclude that X0/X is stable under small colimits in X/X .
Let X1/X be the largest sieve contained in X0/X (in other words, a morphism
U → X belongs to X1/X if and only if, for every morphism V → U in X, thecomposite map V → X belongs to X0/X ). Since colimits in X are universal,
we conclude that X1/X is stable under small colimits in X/X . It follows that
X1/X ' X/X0 for some monomorphism i : X0 → X in X. We wish to showthat i is an equivalence.
Since L is left exact, we have L(G ×LG j(X)) ' Lj(X) ' j(X). In particular, the map G ×LG j(X) → j(X0 ) cannot factor through j(X0 ) unless i isan equivalence. It will therefore suffice to show that G ×j(X) j(X0 ) ' G. In
other words, it will suffice to show that if U ∈ X/X and η 0 ∈ G(U ) is a pointsuch that the images of η and η 0 lie in the same connected component ofLG(U ), then U ∈ X1/X . Since the existence of η 0 is stable under the processof replacing U by some further refinement V → U , it will suffice to showthat U ∈ X0/X . Replacing X by U , we obtain the following reformulation of(30 ):(300 ) Let η 0 ∈ G(X). Then the homotopy fiber Z of the induced map
LF (X) → LG(X) (over the point determined by η) is essentially small.
Since L is left exact, we can identify Z with LF0 (X), where F0 = F ×Gj(X). Since the homotopy fibers of the maps F (Y ) → G(Y ) are essentiallysmall, we may assume without loss of generality that F0 ∈ Fun(Xop , S).
Invoking (2), we deduce that the values of LF0 are essentially small, as
desired.
op
Proof of Lemma 6.3.5.15. Let X• be a simplicial object of LTopét
and letopdF• be its image under the Yoneda embedding j : LTop → Shv(LTopop ).
Let F be a geometric realization of |F• |. We will prove the following:(∗) The map β : F0 → F satisfies conditions (1) and (2) of Lemma 6.3.5.25.
Assuming (∗) for the moment, we will complete the proof of Lemma 6.3.5.15.
Let F•0 be a Čech nerve of the induced map F0 → F (so that Fn0 ' F0 ×F F0 ×
· · ·×F F0 ; in particular F00 ' F0 ). We first claim that each Fn0 is representableby an ∞-topos X0n and that each inclusion [0] ,→ [n] induces an étale map
opdof ∞-topos X0n → X00 ' X0 . Since F•0 is a groupoid object of Shv(LTop)(and F00 ' F0 is representable by the ∞-topos X0 ), it will suffice to prove
this result when n = 1. Consider the pullback diagram
F10

F0

β0

β

/ F00

/ F.

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It follows from condition (∗) that β 0 satisfies conditions (1) and (2) of Lemma
6.3.5.25, so that F10 is representable by an ∞-topos étale over X0 , as desired.
Since the Yoneda embedding j is fully faithful, we may assume without lossof generality that F•0 is the image under j of a groupoid object X0• of LTopop .
Using Corollary 6.3.5.9, we deduce that X0• defines a simplicial object of theopsubcategory LTopét. The evident natural transformation F• → F•0 induces
a map of simplicial objects α : X• → X0• ; we claim that α has the desired
properties. The only nontrivial point is to verify that the induced map ofgeometric realizations | X• | → | X0• | is an equivalence of ∞-topoi. For this,it suffices to show that for every ∞-topos Y, the upper horizontal map in
the diagram
MapLTopop (| X0• |, Y)/ MapLTopop (| X• |, Y)/ lim MapLTopop (Xn , Y)lim MapLTopop (X0n , Y)←−←−
is a homotopy equivalence. Since the vertical maps are homotopy equivalences, it suffices to show that the lower horizontal map is a homotopy
equivalence. Since j is fully faithful, it suffices to show that the lower horizontal map in the analogous diagram
0
MapShv(LTopop ) (|F• |, eY )d0lim MapShv(LTopop ) (Fn , eY )d←−

/ Map dShv(LTopop )(|F• |, eY )/ lim Map d(Fn , eY )Shv(LTopop )←−is a homotopy equivalence. Again, the vertical maps are homotopy equivalences, so we are reduced to showing that the upper horizontal map is a
homotopy equivalence. This follows from the fact that we have an equivaopopdd) since groupoid objects in Shv(LTop)
lence |F• | ' F ' |F•0 | in Shv(LTopare effective (Lemma 6.3.5.21).
It remains to prove (∗). Remark 6.3.5.24 implies that β : F0 → F isuniversal. To complete the proof, we must show that for every ∞-topos Y,
the homotopy fibers of the induced map F0 (Y) → F (Y) are essentially small.
opd
Let R : Shv(LTop) → ShvSb (Y) denote the restriction map, let G• denotethe image of F• under R, and let G = |G• | ' R(F ). Let G0 denote thegeometric realization of G• in the larger ∞-category Fun(Yop , bS) and let
op b
L : Fun(Y , S) → ShvSb (Y) be a left adjoint to the inclusion (see Lemma
6.3.5.28). Then we can identify the map G0 → G with the image under L ofthe map u : G0 → G0 . In view of Lemma 6.3.5.28, it will suffice to show thatfor every object U ∈ Y, the induced map G0 (U ) → G0 (U ) has essentiallysmall homotopy fibers.
For each n ≥ 0 and each object U ∈ Y, we can identify Gn (U ) with the
maximal Kan complex contained in Fun∗ (Xn , Y/U ). Since the ∞-category
Fun∗ (Xn , Y/U ) is locally small (Proposition 6.3.1.13), we conclude that eachhighertopoi

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635∞-TOPOI

connected component of Gn (U ) is essentially small. Moreover, for every morphism [m] → [n] in ∆, the induced map β : Gn (U ) → Gm (U ) is induced bycomposition with an étale geometric morphism g ∗ : Xm → Xn , so that thehomotopy fibers of β are essentially small by Remark 6.3.5.7. The desired
result now follows from Lemma 6.3.5.27.
6.3.6 Structure Theory for ∞-Topoi
In this section we will analyze the following question: given a geometricmorphism f : X → Y of ∞-topoi, when is f an equivalence? Clearly, this istrue if and only if the pullback functor f ∗ is both fully faithful and essentiallysurjective. It is useful to isolate and study these conditions individually.
Definition 6.3.6.1. Let f : X → Y be a geometric morphism of ∞-topoi.
The image of f is defined to be the smallest full subcategory of X whichcontains f ∗ Y and is stable under small colimits and finite limits. We will saythat f is algebraic if the image of f coincides with X.
Our first goal is to prove that the image of a geometric morphism is itself
an ∞-topos.
Proposition 6.3.6.2. Let f : X → Z be a geometric morphism of ∞-topoi
and let Y be the image of f . Then Y is an ∞-topos. Moreover, the inclusion
Y ⊆ X is left exact and colimit-preserving, so we obtain a factorization of f
as a composition of geometric morphismsgh

X → Y → Z,
where h is algebraic and g ∗ is fully faithful.
Proof. We will show that Y satisfies the ∞-categorical versions of Giraud’s
axioms (see Theorem 6.1.0.6). Axioms (ii), (iii), and (iv) are concernedwith the interaction between colimits and finite limits. Since X satisfies theseaxioms and Y ⊆ X is stable under the relevant constructions, Y automaticallysatisfies these axioms as well. The only nontrivial point is to verify (i), whichasserts that Y is presentable.
Choose a small collection of objects {Zα } which generate Z under colimits.
Now choose an uncountable regular cardinal τ with the following properties:(1) Each f ∗ (Zα ) is a τ -compact object of X.
(2) The final object 1X is τ -compact.
(3) The limits functor Fun(Λ22 , X) → X (a right adjoint to the diagonal
functor) is τ -continuous and preserves τ -compact objects.
Let Y0 be the collection of all objects of Y which are τ -compact when
considered as objects of X. Clearly, each object of Y0 is also τ -compact whenregarded as an object of Y. Moreover, because X is accessible, Y0 is essentiallysmall. It will therefore suffice to prove that Y0 generates Y under colimits.

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Choose a minimal model Y00 for Y. Since X is accessible, the full subcategory Xκ spanned by the κ-compact objects is essentially small, so that Y00 issmall. According to Proposition 5.3.5.10, there exists a τ -continuous functor
F : Indτ (Y00 ) → X whose composition with the Yoneda embedding is equivalent to the inclusion Y00 ⊆ X. Since Y00 admits τ -small colimits, Indτ (Y00 ) ispresentable. Proposition 5.3.5.11 implies that F is fully faithful; let Y00 be its
essential image. To complete the proof, it will suffice to show that Y00 = Y.
Since Y is stable under colimits in X, we have Y00 ⊆ Y. According to
Proposition 5.5.1.9, F preserves small colimits, so that Y00 is stable undersmall colimits in X. By construction, Y00 contains each f ∗ (Zα ). Since f ∗preserves colimits, we conclude that Y00 contains f ∗ Z. By definition Y is thesmallest full subcategory of X which contains f ∗ Z and is stable under small
colimits and finite limits. It remains only to show that Y00 is stable underfinite limits. Assumption (2) guarantees that Y00 contains the final objectof X, so we need only show that Y00 is stable under pullbacks. Consider adiagram p : Λ22 → Y00 . The proof of Proposition 5.4.4.3 (applied with K = Λ22
and κ = ω) shows that p can be written as a τ -filtered colimit of diagrams
pα : Λ22 → Y00 . Since filtered colimits in X are left exact (Example 7.3.4.7),
we conclude that the limit of p can be obtained as a τ -filtered colimit oflimits of the diagrams pβ . In view of assumption (3), each of these limits liesin Y0 , so that the limit of p lies in Y00 , as desired.
Remark 6.3.6.3. The factorization of Proposition 6.3.6.2 is unique up to(canonical) equivalence.
The terminology of Definition 6.3.6.1 is partially justified by the following
observations:
Proposition 6.3.6.4. (1) Every étale geometric morphism between ∞topoi is algebraic.
(2) The collection of algebraic geometric morphisms of ∞-topoi is stable
under filtered limits (in RTop).
Proof. We first prove (1). Let X be an ∞-topos, let U be an object of X, let
π! : X/U → X be the projection functor and let π ∗ be a left adjoint to π! . Let
f : X → U be an object of X/U , and let F : f → idU be a morphism in X/U(uniquely determined up to equivalence; for example, we can take F to be
the composition of f with a retraction ∆1 × ∆1 → ∆1 ). Let g : F → π ∗ π! F
be the unit map for the adjunction between π ∗ and π! . We claim that g isa pullback square in X/U . According to Proposition 4.4.2.9, it will suffice toverify that the image of g under π! is a pullback square in X. But this square
can be identified with
/ X ×UXUδ/ U × U,highertopoi

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637∞-TOPOI

which is easily shown to be Cartesian. It follows that, in X/U , f can beobtained as a fiber product of the final object with objects that lie in theessential image of π ∗ . It follows that π ∗ X generates X/U under finite limits,so that π is algebraic.
To prove (2), we consider a geometric morphism f : X → Y which is afiltered limit of algebraic geometric morphisms {fα : Xα → Yα } in the ∞category Fun(∆1 , RTop). Let X0 ⊆ X be a full subcategory which is stable
under finite limits, stable under small colimits, and contains f ∗ Y. We wish
to prove that X0 = X. For each α, we have a diagram of ∞-topoi
X

f

ψ(α)


Xα

fα

/Y

/ Yα .

Let X0α be the preimage of X0 under ψ(α)∗ . Then X0α ⊆ Xα is stable underfinite limits, stable under small colimits, and contains the essential imageof fα∗ . Since fα is algebraic, we conclude that X0α = Xα . In other words,
X0 contains the essential image of each ψ(α)∗ . Lemma 6.3.3.7 implies thatevery object of X can be realized as a filtered colimit of objects, each ofwhich belongs to the essential image of fα∗ for α appropriately chosen. Since
X0 is stable under small colimits, we conclude that X0 = X. It follows that f
is algebraic, as desired.
Remark 6.3.6.5. It is possible to formulate a converse to Proposition
6.3.6.4. Namely, one can characterize the class of algebraic morphisms as thesmallest class of geometric morphisms which contains all étale morphisms
and is stable under certain kinds of filtered limits. However, it is necessary
to allow limits parametrized not only by filtered ∞-categories but also byfiltered stacks over ∞-topoi. The precise statement requires ideas which lie
outside the scope of this book.
Having achieved a rudimentary understanding of the class of algebraic
geometric morphisms, we now turn our attention to the opposite extreme:
namely, geometric morphisms f : X → Y, where f ∗ is fully faithful.
Proposition 6.3.6.6. Let f : X → Y be a geometric morphism of ∞-topoi.
Suppose that f ∗ is fully faithful and essentially surjective on 1-truncatedobjects. Then f ∗ is essentially surjective on n-truncated objects for all n.
The proof uses ideas which will be introduced in §6.5.1 and §7.2.2.
Proof. Without loss of generality, we may identify Y with the essential imageof f ∗ . We use induction on n. The result is obvious for n = 1. Assume thatn > 1 and let X be an n-truncated object of X. By the inductive hypothesis,
U = τ≤n−1 X belongs to Y. Replacing X and Y by X/U and Y/U , we may
suppose that X is n-connective.

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We observe that πn X is an abelian group object of the ordinary topos
Disc(X/X ). Since X is 2-connective, Proposition 7.2.1.13 implies that the
pullback functor Disc(X) → Disc(X/X ) is an equivalence of categories. We
may therefore identify πn X with an abelian group object A ∈ Disc(X). Since
A is discrete, it belongs to Y. It follows that the Eilenberg-MacLane object
K(A, n + 1) belongs to Y. Since X is an n-gerb banded by A, Theorem
7.2.2.26 implies the existence of a pullback diagram
/ 1X
X

1X


/ K(A, n + 1).

Since Y is stable under pullbacks in X, we conclude that X ∈ Y, as desired.
Corollary 6.3.6.7. Let f : X → Y be a geometric morphism of ∞-topoi.
Suppose that f ∗ is fully faithful and essentially surjective on 1-truncatedobjects and that X is n-localic (see §6.4.5). Then f is an equivalence of ∞topoi.
Remark 6.3.6.8. In the situation of Corollary 6.3.6.7, one can eliminate thehypothesis that X is n-localic in the presence of suitable finite-dimensionality
assumptions on X and Y; see §7.2.1.
Remark 6.3.6.9. Let X be an n-localic ∞-topos and let Y be the 2-localic∞-topos associated to the 2-topos τ≤1 X, so that we have a geometric morphism f : X → Y. It follows from Corollary 6.3.6.7 that f is algebraic.
Roughly speaking, this tells us that there is only a very superficial interaction between the theory of k-categories and “topology,” for k > 2. On
the other hand, this statement fails dramatically if k = 1: the relationship between an ordinary topos and its underlying locale is typically very
complicated and not algebraic in any reasonable sense. It is natural to ask
what happens when k = 2. In other words, does Proposition 6.3.6.6 remain
valid if f ∗ is only assumed to be essentially surjective on discrete objects?
An affirmative answer would indicate that our theory of ∞-topoi is a relatively modest extension of classical topos theory. A counterexample could
be equally interesting if it were to illustrate a nontrivial interaction between
higher category theory and geometry.

6.4 N -TOPOI
Roughly speaking, an ordinary topos is a category which resembles the category of sheaves of sets on a topological space X. In §6.1, we introduced
the definition of an ∞-topos. In the same rough terms, we can think of an∞-topos as an ∞-category which resembles the ∞-category of sheaves of∞-groupoids on a topological space X. Phrased in this way, it is natural tohighertopoi

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639∞-TOPOI

guess that these two notions have a common generalization. In §6.4.1, we will
introduce the notion of an n-topos for every 0 ≤ n ≤ ∞. The idea is that an
n-topos should be an n-category which resembles the n-category of sheavesof (n − 1)-groupoids on a topological space X. Of course, there are many
approaches to making this idea precise. Our main result, Theorem 6.4.1.5,
asserts that several candidate definitions are equivalent to one another. The
proof of Theorem 6.4.1.5 will occupy our attention for most of this section.
In §6.4.3, we study an axiomatization of the class of n-topoi in the spirit of
Giraud’s theorem, and in §6.4.4 we will give a characterization of n-topoi
based on their descent properties. The case of n = 0 is somewhat exceptional and merits special treatement. In §6.4.2, we will show that a 0-topos
is essentially the same thing as a locale (a mild generalization of the notionof a topological space).
Our main motivation for introducing the definition of an n-topos is thatit allows us to study ∞-topoi and topological spaces (or more generally, 0topoi) in the same setting. In §6.4.5, we will introduce constructions whichallow us to pass back and forth between m-topoi and n-topoi for any 0 ≤m ≤ n ≤ ∞. We introduce an ∞-category TopR
n of n-topoi for each n ≤ 0
and show that each TopRn can be regarded as a localization of the ∞-category
RTop. In other words, the study of n-topoi for n < ∞ can be regarded as a
special case of the theory of ∞-topoi.
6.4.1 Characterizations of n-Topoi
In this section, we will introduce the definition of n-topos for 0 ≤ n < ∞.
In view of Theorem 6.1.0.6, there are several reasonable approaches to thesubject. We will begin with an extrinsic approach.
Definition 6.4.1.1. Let 0 ≤ n < ∞. An ∞-category X is an n-topos if there
exists a small ∞-category C and an (accessible) left exact localization
L : P≤n−1 (C) → X,
where P≤n−1 (C) denotes the full subcategory of P(C) spanned by the (n−1)truncated objects.
Remark 6.4.1.2. The accessibility condition on the localization functor
L : P≤n−1 (C) → X of Definition 6.4.1.1 is superfluous: we will show that sucha left exact localization of P≤n−1 (C) is automatically accessible (combine
Proposition 6.4.1.6 with Corollary 6.2.1.6).
Remark 6.4.1.3. An ∞-category X is a 1-topos if and only if it is equivalent
to the nerve of an ordinary (Grothendieck) topos; this follows immediatelyfrom characterization (B) of Proposition 6.1.0.1.
Remark 6.4.1.4. Definition 6.4.1.1 also makes sense in the case n = −1 but
is not very interesting. Up to equivalence, there is precisely one (−1)-topos:
the final ∞-category ∗.
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Theorem 6.4.1.5. Let X be a presentable ∞-category and let 0 ≤ n < ∞.
The following conditions are equivalent:(1) There exists a small n-category C which admits finite limits, a Grothendieck topology on C, and an equivalence of X with the full subcategory of
Shv≤n−1 (C) ⊆ Shv(C) consisting of (n−1)-truncated objects of Shv(C).
(2) There exists an ∞-topos Y and an equivalence X → τ≤n−1 Y.
(3) The ∞-category X is an n-topos.
(4) Colimits in X are universal, X is equivalent to an n-category, and theclass of (n − 2)-truncated morphisms in X is local (see §6.1.3).
(5) Colimits in X are universal, X is equivalent to an n-category, and forall sufficiently large regular cardinals κ, there exists an object of X
which classifies (n − 2)-truncated relatively κ-compact morphisms inX.
(6) The ∞-category X satisfies the following n-categorical versions of Giraud’s axioms:(i) The ∞-category X is equivalent to a presentable n-category.
(ii) Colimits in X are universal.
(iii) If n > 0, then coproducts in X are disjoint.
(iv) Every n-efficient (see §6.4.3) groupoid object of X is effective.
The proof of Theorem 6.4.1.5 will require a preliminary result.
Proposition 6.4.1.6. Let X be an n-topos, where 0 ≤ n < ∞, and let
L : X → Y be a left-exact localization. Then L is a topological localization.
Proof. Choose a left-exact localization functor F : P≤n (C) → X, for somesmall ∞-category C. Replacing C by a truncation if necessary, we may suppose that C is equivalent to an n-category. Let S be the collection of Lequivalences in X. If we can prove that F −1 (S) is generated (as a strongly
saturated class of morphisms) by monomorphisms, it will follow that S =F (F −1 (S)) is generated by monomorphisms (since F is left exact, it carries
monomorphisms to monomorphisms). We may therefore reduce to the casewhere X = P≤n−1 (C), for some small n-category C.
Let j : C → P≤n−1 (C) denote the Yoneda embedding. For C ∈ C, let(0)
us say that a sieve C/C ⊆ C/C is covering if the induced monomorphism
U → j(C) belongs to S. Arguing as in Proposition 6.2.2.17, we see that
this collection of covering sieves determines a Grothendieck topology on C.
Let Shv≤n−1 (C) = Shv(C) ∩ P≤n−1 (C). Then L factors as a composition of
localization functors
L0L00

P≤n−1 (C) → Shv≤n−1 (C) → Y,highertopoi∞-TOPOI

April 9, 2017641

where L0 is a left adjoint to the inclusion. We will complete the proof byshowing that L00 is an equivalence of ∞-categories. This is equivalent to thefollowing assertion:(∗) If f : X → Y is a morphism in P≤n (C) such that Lf is an equivalence,then L0 f is an equivalence.
Since P≤n (C) is equivalent to an (n + 1)-category, we may assume thatthe morphism f is k-truncated for some integer k. The proof now proceeds
by induction on k, beginning with the case k = −1. Assume that f is amonomorphism. Write Y as a colimit of objects of the form j(C), where C ∈C. Then f is a colimit of the induced maps fC : X ×Y j(C) → j(C). Since L isleft exact, it carries each fC to a pullback of Lf and therefore an equivalencein Y. By construction, each X ×Y j(C) → j(C) is a monomorphism andtherefore corresponds to a covering sieve on C, from which we conclude that
L0 fC is an equivalence. Since L0 commutes with colimits, we deduce that
L0 (f ) is an equivalence.
Now suppose that k ≥ 0. Let δ : X → X ×Y X be the diagonal map(which is well-defined up to equivalence). According to Lemma 5.5.6.15, δis (k − 1)-truncated. Since L is left exact, L(δ) can be identified with adiagonal map LX → LX ×LY LX which is therefore an equivalence. The
inductive hypothesis implies that L0 (δ) is an equivalence. It follows that
L0 (f ) is a monomorphism. Using the inductive hypothesis, we conclude that
L0 (L0 (f )) ' L0 (f ) is an equivalence, as desired.
Proof of Theorem 6.4.1.5. The case n = 0 will be analyzed very explicitlyin §6.4.2; let us therefore restrict our attention to the case n > 0. The
implication (1) ⇒ (2) is obvious (take Y = Shv(C)). Suppose that (2) issatisfied. Without loss of generality, we may suppose that Y is an (accessible)
left exact localization of P(C) for some small ∞-category C. Then X is a left
exact localization of P≤n−1 (C), which proves (3). If (3) is satisfied, then the
proof of Proposition 6.4.1.6 shows that X ' Shv≤n−1 (C) for some n-category
C equipped with a Grothendieck topology, and can therefore be describedas the ∞-category of (n − 1)-truncated objects of the ∞-topos Shv(C); we
conclude that (3) ⇒ (2).
The implication (2) ⇒ (4) will be established in §6.4.4 (Propositions
6.4.4.6 and 6.4.4.7). The proof of Theorem 6.1.6.8 adapts without change
to show that (4) ⇔ (5). The implication (4) ⇒ (6) will be proven in §6.4.4(Proposition 6.4.4.9). Finally, the “difficult” implication (6) ⇒ (1) will be
proven in §6.4.3 (Proposition 6.4.3.6) using an inductive argument quite
similar to the proof of Giraud’s original result.
Remark 6.4.1.7. Theorem 6.4.1.5 is slightly stronger than its ∞-categoricalanalogue, Theorem 6.1.0.6: it asserts that every n-topos arises as an ncategory of sheaves on some n-category C equipped with a Grothendiecktopology.

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Remark 6.4.1.8. Let X be a presentable ∞-category in which colimits areuniversal. Then there exists a regular cardinal κ such that every monomorphism is relatively κ-compact. In this case, characterization (5) of Theorem
6.4.1.5 recovers a classical description of ordinary topos theory: a category
X is a topos if and only if it is presentable, colimits in X are universal, andX has a subobject classifier.
6.4.2 0-Topoi and Locales
Our goal in this section is to prove Theorem 6.4.1.5 in the special case n = 0.
A byproduct of our proof is a classification result (Corollary 6.4.2.6) which
identifies the theory of 0-topoi with the classical theory of locales (Definition
6.4.2.3).
We begin by observing that when n = 0, a morphism in an ∞-category
X is (n − 2)-truncated if and only if it is an equivalence. Consequently,
any final object of X is an (n − 2)-truncated morphism classifier, and theclass of (n − 2)-truncated morphisms is automatically local (in the senseof Definition 6.1.3.8). Moreover, if X is a 0-category, then every groupoid
object in X is equivalent to a constant groupoid and therefore automaticallyeffective. Consequently, characterizations (4) through (6) in Theorem 6.4.1.5
all reduce to the same condition on X and we may restate the desired result
as follows:
Theorem 6.4.2.1. Let X be a presentable 0-category. The following conditions are equivalent:(1) There exists a small 0-category C which admits finite limits, a Grothendieck topology on C, and an equivalence X → Shv≤−1 (C).
(2) There exists an ∞-topos Y and an equivalence X → τ≤−1 Y.
(3) The ∞-category X is a 0-topos.
(4) Colimits in X are universal.
Before giving a proof of Theorem 6.4.2.1, it is convenient to reformulatecondition (4). Recall that any 0-category X is equivalent to N(U), where U
is a partially ordered set which is well-defined up to canonical isomorphism(see Example 2.3.4.3). The presentability of X is equivalent to the assertion
that U is a complete lattice: that is, every subset of U has a least upperbound in U (this condition formally implies the existence of greater lower
bounds as well).
Remark 6.4.2.2. If n = 0, then every presentable n-category is essentiallysmall. This is typically not true for n > 0.
We note that the condition that colimits in X be universal can also be
formulated in terms of the partially ordered set U: it is equivalent to theassertion that meets in U commute with infinite joins in the following sense:highertopoi

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643

Definition 6.4.2.3. Let U be a partially ordered set. We will say that U isa locale if the following conditions are satisfied:S(1) Every subset {Uα } of elements of U has a least upper bound α Uα inU.
(2) The formation of least upper bounds commutes with meets in the sensethat[[(Uα ∩ V ) = ( Uα ) ∩ V.
(Here (U ∩ V ) denotes the greatest lower bound of U and V , which
exists by virtue of assumption (1).)
Example 6.4.2.4. For every topological space X, the collection U(X) of
open subsets of X forms a locale. Conversely, if U is a locale, then there isa natural topology on the collection of prime filters of U which allows us to
extract a topological space from U. These two constructions are adjoint to
one another, and in good cases they are actually inverse equivalences. Moreprecisely, the adjunction gives rise to an equivalence between the categoryof spatial locales and the category of sober topological spaces. In general, alocale can be regarded as a sort of generalized topological space in which one
may speak of open sets but one does not generally have a sufficient supplyof points. We refer the reader to [42] for details.
We can summarize the above discussion as follows:
Proposition 6.4.2.5. Let X be a presentable 0-category. Then colimits in
X are universal if and only if X is equivalent to N(U), where U is a locale.
We are now ready to give the proof of Theorem 6.4.2.1.
Proof. The implications (1) ⇒ (2) ⇒ (3) are easy. Suppose that (3) is satisfied, so that X is a left exact localization of P≤−1 (C) for some small ∞category C. Up to equivalence, there are precisely two (−1)-truncated spaces:∅ and ∗. Consequently, τ≤−1 S is equivalent to the two-object ∞-category ∆1 .
It follows that P≤−1 (C) is equivalent to Fun(Cop , ∆1 ).
e denote the collection of sieves on C ordered by inclusion. Then,
Let X
identifying a functor f : C → ∆1 with the sieve f −1 {0} ⊆ C, we deduce thateFun(C, ∆1 ) is isomorphic to the nerve N(X).
Without loss of generality, we may identify X with the essential image ofe → N(X).
e The map L may be identified witha localization functor L : N(X)e to itself. Unwinding the definitions,a map of partially ordered sets from X
we find that the condition that L be a left exact localization is equivalent to
the following three properties:
e →X
e is idempotent.
(A) The map L : X
e U ⊆ L(U ).
(B) For each U ∈ X,

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e → X
e preserves finite intersections (since X is a left(C) The map L : X
eexact localization of N(X).)
e : LU = U }. Then it is easy to see that X is equivalent
Let U = {U ∈ X
to the nerve N(U) and that the partially ordered set X satisfies conditions(1) and (2) of Definition 6.4.2.3. Therefore U is a locale, so that colimits in
N(U) are universal by Proposition 6.4.2.5. This proves that (3) ⇒ (4).
Now suppose that (4) is satisfied. Using Proposition 6.4.2.5, we may suppose without loss of generality that X = N(U), where U is a locale. We
observe that X is itself small. Let us say that a sieve {Uα → U } on an object
U ∈ X is covering if
[
U=
Uα
α

in U. Using the assumption that U is a locale, it is easy to see that thecollection of covering sieves determines a Grothendieck topology on X. The∞-category P≤−1 (X) can be identified with the nerve of the partially ordered
set of all downward-closed subsets U0 ⊆ U. Moreover, an object of P≤−1 (X)belongs to Shv≤−1 (X) if and only if the corresponding subset U0 ⊆ U isstable under joins. Every such subset U0 ⊆ U has a largest element U ∈ U,
and we then have an identification U0 = {V ∈ U : V ≤ U }. It follows thatShv≤−1 (X) is equivalent to the nerve of the partially ordered set U, which
is X. This proves (1) and concludes the argument.
We may summarize the results of this section as follows:
Corollary 6.4.2.6. An ∞-category X is a 0-topos if and only if it is equivalent to N(U), where U is a locale.
Remark 6.4.2.7. Coproducts in a 0-topos are typically not disjoint.
In classical topos theory, there are functorial constructions for passing backand forth between topoi and locales. Given a locale U (such as the locale
U(X) of open subsets of a topological space X), one may define a topos X ofsheaves (of sets) on U. The original locale U may then be recovered as thepartially ordered set of subobjects of the final object of X. In fact, for any
topos X, the partially ordered set U of subobjects of the final object forms
a locale. In general, X cannot be recovered as the category of sheaves on
U; this is true if and only if X is a localic topos: that is, if and only if X isgenerated under colimits by the collection of subobjects of the final object
1X . In §6.3, we will discuss a generalization of this picture which will allow
us to pass between m-topoi and n-topoi for any m ≤ n.
6.4.3 Giraud’s Axioms for n-Topoi
In §6.1.1, we sketched an axiomatic approach to the theory of ∞-topoi. The
axioms we introduced were closely parallel to Giraud’s axioms for ordinary
topoi with one important difference. If X is an ∞-topos, then every groupoid

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object of X is effective. If X is an ordinary topos, then a groupoid U• iseffective only if the diagram// U0
U1
exhibits U1 as an equivalence relation on U0 . Our first goal in this section is
to formulate an analogue of this condition, which will lead us to an axiomatic
description of n-topoi for all 0 ≤ n ≤ ∞.
Definition 6.4.3.1. Let X be an ∞-category and U• a groupoid object of
X. We will say that U• is n-efficient if the natural map
U1 → U0 × U0
(which is well-defined up to equivalence) is (n − 2)-truncated.
Remark 6.4.3.2. By convention, we regard every groupoid object as ∞efficient.
Example 6.4.3.3. If C is (the nerve of) an ordinary category, then giving
a 1-efficient groupoid object U• of C is equivalent to giving an object U0 of
C and an equivalence relation U1 on U0 .
Proposition 6.4.3.4. An ∞-category X is equivalent to an n-category if
and only if every effective groupoid in X is n-efficient.
Proof. Suppose first that C is equivalent to an n-category. Let U• be aneffective groupoid in X. Then U• has a colimit U−1 . The existence of apullback diagram
/ U0
U1

U0


/ U−1

implies that the map f 0 : U1 → U0 × U0 is a pullback of the diagonal map
f : U−1 → U−1 × U−1 . We wish to show that f 0 is (n − 2)-truncated. By
Lemma 5.5.6.12, it suffices to show that f is (n − 2)-truncated. By Lemma
5.5.6.15, this is equivalent to the assertion that U−1 is (n − 1)-truncated.
Since C is equivalent to an n-category, every object of C is (n − 1)-truncated.
Now suppose that every effective groupoid in X is n-efficient. Let U ∈ X
be an object; we wish to show that U is (n − 1)-truncated. The constant
simplicial object U• taking the value U is an effective groupoid and therefore
n-efficient. It follows that the diagonal map U → U × U is (n − 2)-truncated.
Lemma 5.5.6.15 implies that U is (n − 1)-truncated, as desired.
We are now almost ready to supply the “hard” step in the proof of Theorem 6.4.1.5 (namely, the implication (6) ⇒ (1)). We first need a slightly
technical lemma whose proof requires routine cardinality estimates.
Lemma 6.4.3.5. Let X be a presentable ∞-category in which colimits areuniversal. There exists a regular cardinal τ such that X is τ -accessible, andthe full subcategory of Xτ ⊆ X spanned by the τ -compact objects is stable
under the formation of subobjects and finite limits.

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Proof. Choose a regular cardinal κ such that X is κ-accessible. We observe
that, up to equivalence, there are a bounded number of κ-compact objectsof X and therefore a bounded number of subobjects of κ-compact objects of
X. Now choose an uncountable regular cardinal τ  κ such that:(1) The ∞-category Xκ is essentially τ -small.
(2) For each X ∈ Xκ and each monomorphism i : U → X, U is τ -compact.
It is clear that X is τ -accessible, and Xτ is stable under finite limits (in fact,κ-small limits) by Proposition 5.4.7.4. To complete the proof, we must showthat Xτ is stable under the formation of subobjects. Let i : U → X be amonomorphism, where X is τ -compact. Since X is κ-accessible, we can write
X as the colimit of a κ-filtered diagram p : J → Xκ . Since X is τ -compact,
it is a retract of the colimit X 0 of some τ -small subdiagram p| J0 . Since τ is
uncountable, we can use Proposition 4.4.5.12 to write X as the colimit of a
τ -small diagram Idem → X which carries the unique object of Idem to X 0 .
Since colimits in X are universal, it follows that U can be written as a τ -small
colimit of a diagram Idem → X which takes the value U 0 = U ×X X 0 . It will
therefore suffice to prove that U 0 is τ -compact. Invoking the universality ofcolimits once more, we observe that U 0 is a τ -small colimit of objects of theform U 00 = U 0 ×X 0 p(J), where J is an object of J0 . We now observe that U 00
is a subobject of p(J) ∈ Xκ and is therefore τ -compact by assumption (2).
It follows that U 0 , being a τ -small colimit of τ -compact objects of X, is also
τ -compact.
Proposition 6.4.3.6. Let 0 < n < ∞ and let X be an ∞-category satisfying
the following conditions:(i) The ∞-category X is presentable.
(ii) Colimits in X are universal.
(iii) Coproducts in X are disjoint.
(iv) The effective groupoid objects of X are precisely the n-efficient groupoids.
Then there exists a small n-category C which admits finite limits, a Grothendieck topology on C, and an equivalence X → Shv≤n−1 (C).
Proof. Without loss of generality, we may suppose that X is minimal. Choosea regular cardinal κ such that X is κ-accessible, and the full subcategory
C ⊆ X spanned by the κ-compact objects of X is stable under the formation ofsubobjects and finite limits (Lemma 6.4.3.5). We endow C with the canonicaltopology induced by the inclusion C ⊆ X. According to Theorem 5.1.5.6,
there is an (essentially unique) colimit-preserving functor F : P(C) → X
such that F ◦ j is equivalent to the inclusion C ⊆ X, where j : C → P(C)denotes the Yoneda embedding. The proof of Theorem 5.5.1.1 shows that Fhighertopoi

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has a fully faithful right adjoint G : X → P(C). We will complete the proof
by showing that the essential image of G is precisely Shv≤n−1 (C).
Since X is equivalent to an n-category (Proposition 6.4.3.4) and G is left
exact, we conclude that G factors through P≤n−1 (C). It follows from Proposition 6.2.4.6 that G factors through Shv≤n−1 (C). Let X0 ⊆ Shv≤n−1 (C) denotethe essential image of G. To complete the proof, it will suffice to show that
X0 = Shv≤n−1 (C). Let ∅ be an initial object of X. The space MapX (X, ∅)is contractible if X is an initial object of X and empty otherwise (Lemma
6.1.3.6). It follows from Proposition 6.2.2.18 that G(∅) is an initial object of
Shv≤n−1 (C).
We next claim that X0 is stable under small coproducts in Shv≤n−1 (C).
It will suffice to show that the map G preserves coproducts. Let {Uα } be asmall collection of objects of X and U their coproduct in X. According toLemma 6.1.5.1, we have a pullback diagram
Vα,β
φ0


Uβ

φ

/ Uα

/ U,

where Vα,β is an initial object of X if α 6= β, while φ and φ0 are equivalences
if α = β. The functor G preserves all limits, so that the diagram
G(Vα,β )/ G(Uα )G(Uβ )/ G(U )

is a pullback in Shv≤n−1 (C). Let U 0 denote a coproduct of the objects i(Uα )in Shv≤n−1 (C) and let g : U 0 → G(U ) be the induced map. Since colimits in
X are universal, we obtain a natural identification of U 0 ×G(U ) U 0 with thecoproductaa(G(Uα ) ×G(U ) G(Uβ )) 'Uα ' U 0 ,α,βα

where the second equivalence follows from our observation that G preserves
initial objects. Applying Lemma 5.5.6.15, we deduce that g is a monomorphism.
To prove that g is an equivalence, it will suffice to show that the map
π0 U 0 (C) → π0 G(U )(C) = π0 MapX (C, U )is surjective for every object C ∈ C. Since colimits in X are universal, every
map h : C → U can be written as a coproduct of maps hα : Cα → Uα .
Each Cα is a subobject of C (Lemma 6.4.4.8) and therefore belongs to C.
hα
Let h0α ∈ π0 U 0 (Cα ) denote the homotopy class of the composition G(Cα ) →G(Uα ) → U 0 . Since the topology on C is canonical, Lemma 6.2.4.4 implies

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that π0 U 0 (C) ' α π0 U 0 (Cα ) contains an element h0 which restricts to eachh0α . It is now clear that h is the image of h0 under the map π0 U 0 (C) →π0 MapX (C, U ).
We will prove the following result by induction on k: if there exists a ktruncated morphism f : X → Y , where Y ∈ X0 and X ∈ Shv≤n−1 (C), thenX ∈ X0 . Taking k = n − 1 and Y to be a final object of Shv≤n−1 (C) (which
belongs to X0 because C contains a final object), we conclude that every
object of Shv≤n−1 (C) belongs to X0 , which completes the proof.
If k = −2, then f is an equivalence so that X ∈ X0 , as desired. Assume now
that k ≥ −1. Since X0 contains the essential image of the Yoneda embedding
and is stable under coproducts, there exists an effective epimorphism p :U → X in Shv≤n−1 (C), where U ∈ X0 . Let U • be a Čech nerve of p in
Shv≤n−1 (C) and U• be the associated groupoid object. We claim that U•
is a groupoid object of X0 . Since X0 is stable under limits in Shv≤n−1 (C),it suffices to prove that U0 = U and U1 = U ×X U belong to X0 . We now
observe that there exists a pullback diagramQ

δ0

U ×X UX/ U ×Y U/ X ×Y X.

δSince f is k-truncated, δ is (k − 1)-truncated (Lemma 5.5.6.15), so that δ 0 is
(k − 1)-truncated. Since U ×Y U belongs to X0 (because X0 is stable underlimits), our inductive hypothesis allows us to conclude that U ×X U ∈ X0 ,as desired.
We observe that U• is an n-efficient groupoid object of X0 . Invoking assumption (iv), we conclude that U• is effective in X0 . Let X 0 ∈ X0 be a colimitof U• in X0 , so that we have a morphism u : X → X 0 in Shv≤n−1 (C)U• / .
To complete the proof that X ∈ X0 , it will suffice to show that u is anequivalence. Since u induces an equivalence
U ×X U → U ×X 0 U,
it is a monomorphism (Lemma 5.5.6.15). It will therefore suffice to show thatu is an effective epimorphism. We have a commutative diagram
p0

U
p

/ X0
>
u

X,
where p is an effective epimorphism; it therefore suffices to show that p0
is an effective epimorphism, which follows immediately from Proposition
6.2.4.6.
Remark 6.4.3.7. Proposition 6.4.3.6 is valid also for n = 0 but is almost
vacuous: coproducts in a 0-topos X are never disjoint unless X is trivial(equivalent to the final ∞-category ∗).

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Remark 6.4.3.8. In a certain respect, the theory of ∞-topoi is simpler
than the theory of ordinary topoi: in an ∞-topos, every groupoid object
is effective; it is not necessary to impose any additional conditions like nefficiency. The absense of this condition gives the theory of ∞-topoi a slightly
different flavor than ordinary topos theory. In an ∞-topos, we are free to form
quotients of objects not only by equivalence relations but also by arbitrarygroupoid actions. In geometry, this extra flexibility allows the constructionof useful objects such as orbifolds and algebraic stacks, which are useful ina variety of mathematical situations.
One can imagine weakening the gluing conditions even further and considering axioms having the form “every category object is effective.” This
seems like a natural approach to a theory of topos-like (∞, ∞)-categories.
However, we will not pursue the matter any further here.
6.4.4 n-Topoi and Descent
Let X be an ∞-category which admits finite limits and let OX denote thefunctor ∞-category Fun(∆1 , X) equipped with the Cartesian fibration e :d∞
OX → X (given by evaluation at {1} ⊆ ∆1 ), as in §6.1.1. Let F : Xop → Cat
be a functor which classifies e; informally, F associates to each object U ∈ X
the ∞-category X/U . According to Theorem 6.1.3.9, X is an ∞-topos if andd∞ . The
only if the functor F preserves limits and factors through PrL ⊆ Cat
assumption that F preserves limits can be viewed as a descent condition: it
asserts that if X → U is a morphism of X and U is decomposed into “pieces”
Uα , then X can be canonically reconstructed from the “pieces” X ×U Uα .
The goal of this section is to obtain a similar characterization of the class of
n-topoi for 0 ≤ n < ∞.
We begin by considering the case where X is the (nerve of) the categoryof sets. In this case, we can think of F as a contravariant functor from sets
to categories, which carries a set U to the category Set/U . This functor does
not preserve pullbacks: given a pushout square
X

Y

z
$

Y

Z

#` {
XZ

in the category Set, there is an associated functor
θ : Set/Y `X Z → Set/Y ×Set/X Set/Z(here the right hand side indicates a homotopy fiber product of categories).
The functor θ is generally not an equivalence of categories: for example, θfails to be an equivalence if Y = Z = ∗, provided that X has cardinality of at

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least 2. However, θ is always fully faithful. Moreover, we have the following
result:Fact 6.4.4.1. The functor θ induces an isomorphism of partially ordered
sets
a
Sub(Y
Z) → Sub(Y ) ×Sub(X) Sub(Z),X

where Sub(M ) denotes the partially ordered set of subsets of M .
In this section, we will show that an appropriate generalization of Fact
6.4.4.1 can be used to characterize the class of n-topoi for all 0 ≤ n ≤ ∞.
First, we need to introduce some terminology.
Notation 6.4.4.2. Let X be an ∞-category which admits pullbacks andlet 0 ≤ n ≤ ∞. We let OnX denote the full subcategory of OX spanned by(n)morphisms f : U → X which are (n − 2)-truncated, and OX ⊆ OnX thesubcategory whose objects are (n − 2)-truncated morphisms in X and whosemorphisms are Cartesian transformations (see Notation 6.1.3.4).
Example 6.4.4.3. Let X be an ∞-category which admits pullbacks. ThenO0X is the full subcategory of OX spanned by the final objects in each fiberof the morphism p : OX → X. Since p is a coCartesian fibration (Corollary
2.4.7.12), Proposition 2.4.4.9 asserts that the restriction p| O0X is a trivial
fibration of simplicial sets.
Lemma 6.4.4.4. Let X be a presentable ∞-category in which colimits areuniversal and coproducts are disjoint and let n ≥ −2. Then the class ofn-truncated morphisms in X is stable under small coproducts.
Proof. The proof is by induction on n, where the case n = −2 is obvious.
Suppose that {fα : Xα → Yα } is a family of n-truncated morphisms in X
having coproduct f : X → Y . Since colimits in X are universal, we conclude
that X ×Y X can be written as a coproductaa(Xα ×Y Xβ ) '(Xα ×Yα (Yα ×Y Yβ ) ×Yβ Xβ ).
α,β

α,β

Applying Lemma 6.1.5.1, we can rewrite this coproduct asa(Xα ×Yα Xα ).
αConsequently, the diagonal map δ : X → X ×Y X is a coproduct of diagonalmaps {δα : Xα → Xα ×Yα Xα }. Applying Lemma 5.5.6.15, we deduce thateach δα is (n − 1)-truncated, so that δ is (n − 1)-truncated by the inductive
hypothesis. We now apply Lemma 5.5.6.15 again to deduce that f is ntruncated, as desired.
Combining Lemmas 6.1.3.3, 6.1.3.5, 6.1.3.7, and 6.4.4.4, we deduce thefollowing analogue of Theorem 6.1.3.9.

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Theorem 6.4.4.5. Let X be a presentable ∞-category in which colimits areuniversal and coproducts are disjoint. The following conditions are equivalent:(1) For every pushout diagram
f

α

/g

0


/ g0

β0

β


f0

α

in OnX , if α and β are Cartesian transformations, then α0 and β 0 are
also Cartesian transformations.
(2) The class of (n − 2)-truncated morphisms in X is local.
(3) The Cartesian fibration OnX → X is classified by a limit-preservingd∞ .
functor Xop → Cat(n)(4) The right fibration OX → X is classified by a limit-preserving functor
Xop → bS.
(5) Let K be a small simplicial set and α : p → q a natural transformationof colimit diagrams p, q : K . → X. Suppose that α = α|K is a Cartesian transformation and that α(x) is (n − 2)-truncated for every vertexx ∈ K. Then α is a Cartesian transformation and α(∞) is (n − 2)truncated, where ∞ denotes the cone point of K . .
Our next goal is to establish the implication (2) ⇒ (4) of Theorem 6.4.1.5.
We will deduce this from the equivalence (2) ⇔ (3) (which we have alreadyestablished) together with Propositions 6.4.4.6 and 6.4.4.7 below.
Proposition 6.4.4.6. Let X be an n-topos, 0 ≤ n ≤ ∞. Then colimits in X
are universal.
Proof. Using Lemma 6.1.3.15, we may reduce to the case X = P≤n−1 (C) forsome small ∞-category C. Using Proposition 5.1.2.2, we may further reduceto the case where X = τ≤n−1 S.
Let f : X → Y be a map of (n − 1)-truncated spaces and let f ∗ : S/Y →/XSbe a pullback functor. Since X is stable under limits in S, f ∗ restricts
to give a functor X/Y → X/X ; we wish to prove that this restricted functor
commutes with colimits. We observe that X/X and X/Y can be identified
with the full subcategories of S/X and S/Y spanned by the (n − 1)-truncatedobjects, by Lemma 5.5.6.14. Let τX : S/X → X/X and τY : S/Y → X/Ydenote left adjoints to the inclusions. The functor f ∗ preserves all colimits(Lemma 6.1.3.14) and all limits (since f ∗ has a left adjoint). Consequently,
Proposition 5.5.6.28 implies that τX ◦ f ∗ ' f ∗ ◦ τY .
Let p : K . → X/Y be a colimit diagram. We wish to show that f ∗ ◦ p
is a colimit diagram. According to Remark 5.2.7.5, we may assume thathighertopoi

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p = τY ◦ p0 for some colimit diagram p0 : K . → S/Y . Since colimits in S
are universal (Lemma 6.1.3.14), the composition f ∗ ◦ p0 : K . → S/X is acolimit diagram. Since τX preserves colimits, we conclude that τX ◦ f ∗ ◦ p0 :K . → X/X is a colimit diagram, so that f ∗ ◦ τY ◦ p0 = f ∗ ◦ p is also a colimit
diagram, as desired.
Proposition 6.4.4.7. Let Y be an ∞-topos and let X = τ≤n Y, 0 ≤ n ≤ ∞.
Then the class of (n − 2)-truncated morphisms in X is local.
Proof. Combining Propositions 6.2.3.17 and 6.2.3.14 with Lemma 6.4.4.4,
we conclude that the class of (n − 2)-truncated morphisms in Y is local.
Consequently, the Cartesian fibration OnY → Y is classified by a colimitop
d . It follows that O(n) → X is classified bypreserving functor F : Y → Cat∞
X
F | X. To prove that F | X is colimit-preserving, it will suffice to show that F
is equivalent to F ◦ τ≤n . In other words, we must show that F carries eachopd . Replacing Y by Y/τ Y ,n-truncation Y → τ≤n Y to an equivalence in Cat∞≤nwe reduce to Lemma 7.2.1.13.
We conclude this section by proving the following generalization of Proposition 6.1.3.19, which also establishes the implication (4) ⇒ (6) of Theorem
6.4.1.5. We will assume n > 0; the case n = 0 was analyzed in §6.4.2.
Lemma 6.4.4.8. Let X be a presentable ∞-category in which colimits areuniversal and let f : ∅ → X be a morphism in X, where ∅ is an initial objectof X. Then f is a monomorphism.
Proof. Let Y be an arbitrary object of X. We wish to show that composition
with f induces a (−1)-truncated map
MapX (Y, ∅) → MapX (Y, X).
If Y is an initial object of X, then both sides are contractible; otherwise theleft side is empty (Lemma 6.1.3.6).
Proposition 6.4.4.9. Let 1 ≤ n ≤ ∞ and let X be a presentable n-category.
Suppose that colimits in X are universal and that the class of (n−2)-truncatedmorphisms in X is local. Then X satisfies the n-categorical Giraud axioms:(i) The ∞-category X is equivalent to a presentable n-category.
(ii) Colimits in X are universal.
(iii) Coproducts in X are disjoint.
(iv) Every n-efficient groupoid object of X is effective.
Proof. Axioms (i) and (ii) hold by assumption. To show that coproductsin X are disjoint, let us consider an arbitrary pair of objects X, Y ∈ X andlet ∅ denote an initial object of X. Let f : ∅ → X be a morphism (unique
up to homotopy since ∅ is initial). Since colimits in X are universal, f is ahighertopoi

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653∞-TOPOI

monomorphism (Lemma 6.4.4.8) and therefore belongs to OnX since n ≥ 1.
We observe that id∅ is an initial object of OX , so we can form a pushout
diagram
α /
idY
id∅
β

β0


α0
/ g
f
in OnX . It is clear that α is a Cartesian transformation, and Lemma 6.1.3.6
implies that β is Cartesian as well. Invoking Theorem 6.4.4.5, we deduce that
α0 is a Cartesian transformation. But α0 can be identified with a pushout
diagram/Y∅X/ X ` Y.

This proves (iii).
Now suppose that U• is an n-efficient groupoid object of X; we wish toprove that U• is effective. Let U • : N(∆+ )op → X be a colimit of U• . LetU•0 : N(∆+ )op → X be the result of composing U • with the shift functor∆+ → ∆+a
J 7→ J{∞}.
(In other words, U•0 is the shifted simplicial object given by Un0 = Un+1 .)Lemma 6.1.3.17 asserts that U•0 is a colimit diagram in X. We have a transformation α : U•0 → U • . Let V • denote the constant augmented simplicial
object of X taking the value U0 , so that we have a natural transformation
β : U•0 → V • . Let W • denote a product of U • and V • in the ∞-category X∆+
of augmented simplicial objects and let γ : U•0 → W • be the induced map.
We observe that for each n ≥ 0, the map γ(∆n ) : Un+1 → W n is a pullbackof U1 → U0 × U0 and therefore (n − 2)-truncated (since U• is assumed tobe n-efficient). Since U• is a groupoid, we conclude that γ = γ| N(∆)op is
a Cartesian transformation. Invoking Theorem 6.4.4.5, we deduce that γ isalso a Cartesian transformation, so that the diagram
/ U0
U1


/ U0 × W −1
W0
is Cartesian. Combining this with the Cartesian diagram/ W −1W0/ U −1 ,U0
we deduce that U is effective, as desired.

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6.4.5 Localic ∞-Topoi
The standard example of an ordinary topos is the category Shv(X; Set) ofsheaves (of sets) on a topological space X. Of course, not every topos is of
this form: the category Shv(X; Set) is generated under colimits by subobjectsof its final object (which can be identified with open subsets of X). A topos
X with this property is said to be localic and is determined up to equivalenceby the locale Sub(1X ) which we may view as a 0-topos. The objective of thissection is to obtain an ∞-categorical analogue of this picture, which will allow
us to relate the theory of n-topoi to that of m-topoi for all 0 ≤ m ≤ n ≤ ∞.
Definition 6.4.5.1. Let X and Y be n-topoi for 0 ≤ n ≤ ∞. A geometricmorphism from X to Y is a functor f∗ : X → Y which admits a left exact left
adjoint (which we will typically denote by f ∗ ).
We let Fun∗ (X, Y) denote the full subcategory of the ∞-category Fun(X, Y)
spanned by the geometric morphisms and let TopRn denote the subcategorydof Cat∞ whose objects are n-topoi and whose morphisms are geometric morphisms.
Remark 6.4.5.2. In the case where n = 1, the ∞-category of geometricmorphisms Fun∗ (X, Y) between two 1-topoi is equivalent to (the nerve of)the category of geometric morphisms between the ordinary topoi hX and hY.
Remark 6.4.5.3. In the case where n = 0, the ∞-category of geometricmorphisms Fun∗ (X, Y) between two 0-topoi is equivalent to the nerve of thepartially ordered set of homomorphisms from the underlying locale of Y to
the underlying locale of X. (A homomorphism between locales is a map ofpartially ordered sets which preserve finite meets and arbitrary joins.) In thecase where X and Y are associated to (sober) topological spaces X and Y ,this is simply the set of continuous maps from X to Y partially ordered by
specialization.
R
If m ≤ n, then the ∞-categories TopRm and Topn are related by the following observation:

Proposition 6.4.5.4. Let X be an n-topos and let 0 ≤ m ≤ n. Then the fullsubcategory τ≤m−1 X spanned by the (m − 1)-truncated objects is an m-topos.
Proof. If m = n = ∞, the result is obvious. Otherwise, it follows immediatelyfrom (2) of Theorem 6.4.1.5.
Lemma 6.4.5.5. Let C be a small n-category which admits finite limits andlet Y be an ∞-topos. Then the restriction map
Fun∗ (Y, P(C)) → Fun∗ (τ≤n−1 Y, P≤n−1 (C))
is an equivalence of ∞-categories.
Proof. Let M ⊆ Fun(P(C), Y) and M0 ⊆ Fun(P≤n−1 (C), τ≤n−1 Y) denotethe full subcategories spanned by left exact colimit-preserving functors. Inhighertopoi

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655∞-TOPOI

view of Proposition 5.2.6.2, it will suffice to prove that the restriction map
θ : M → M0 is an equivalence of ∞-categories.
Let M00 denote the full subcategory of Fun(P(C), τ≤n−1 Y) spanned bycolimit-preserving functors whose restriction to P≤n−1 (C) is left exact. Corollary 5.5.6.22 implies that the restriction map θ0 : M00 → M0 is an equivalenceof ∞-categories.
Let j : C → P≤n−1 (C) ⊆ P(C) denote the Yoneda embedding. Composition
with j yields a commutative diagram
/ M0

θ

M

ψ0

ψ


Fun(C, τ≤n−1 Y).

Fun(C, τ≤n−1 Y)

Theorem 5.1.5.6 implies that ψ and ψ 0 ◦ θ0 are fully faithful. Since θ0 is anequivalence of ∞-categories, we deduce that ψ 0 is fully faithful. Thus θ is fully
faithful; to complete the proof, we must show that ψ and ψ 0 have the sameessential image. Suppose that f : C → τ≤n−1 Y belongs to the essential imageof ψ 0 . Without loss of generality, we may suppose that f is a compositiong∗j

C → P≤n−1 (C) → τ≤n−1 Y .
As a composition of left exact functors, f is left exact. We may now invoke
Proposition 6.1.5.2 to deduce that f belongs to the essential image of ψ.
Lemma 6.4.5.6. Let C be a small n-category which admits finite limits andis equipped with a Grothendieck topology and let Y be an ∞-topos. Then the
restriction map
θ : Fun∗ (Y, Shv(C)) → Fun∗ (τ≤n−1 Y, Shv≤n−1 (C))
is an equivalence of ∞-categories.
Proof. We have a commutative diagram
Fun∗ (Y, Shv(C))
Fun∗ (Y, P(C))θ/ Fun∗ (τ≤n−1 Y, Shv≤n−1 (C))θ0

/ Fun∗ (τ≤n−1 Y, P≤n−1 (C)),

where the vertical arrows are inclusions of full subcategories and θ0 is anequivalence of ∞-categories (Lemma 6.4.5.5). To complete the proof, it will
suffice to show that if f∗ : Y → P(C) is a geometric morphism such thatf∗ |τ≤n−1 Y factors through Shv≤n−1 (C), then f∗ factors through Shv(C).
Let f ∗ be a left adjoint to f∗ and let S denote the collection of all morphisms in P(C) which localize to equivalences in Shv(C). We must show thatf ∗ S consists of equivalences in Y. Let S ⊆ S be the collection of monomorphisms which belong to S. Since Shv(C) is a topological localization of P(C),it will suffice to show that f ∗ S consists of equivalences in Y. Let g : X → Yhighertopoi

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belong to S. Since P(C) is generated under colimits by the essential image of the Yoneda embedding, we can write Y as a colimit of a diagram
K → P≤n−1 (C). Since colimits in P(C) are universal, we obtain a corresponding expression of g as a colimit of morphisms {gα : Xα → Yα } which are
pullbacks of g, where Yα ∈ P≤n−1 (C). In this case, gα is again a monomorphism, so that Xα is also (n − 1)-truncated. Since f ∗ commutes with colimits, it will suffice to show that each f ∗ (gα ) is an equivalence. But thisfollows immediately from our assumption that f∗ |τ≤n−1 Y factors throughShv≤n−1 (Y).
Proposition 6.4.5.7. Let 0 ≤ m ≤ n ≤ ∞ and let Y be an m-topos. Thereexists an n-topos X and a categorical equivalence f∗ : τ≤m−1 X → Y with thefollowing universal property: for any n-topos Z, composition with f∗ induces
an equivalence of ∞-categories
θ : Fun∗ (Z, X) → Fun∗ (τ≤m−1 Z, Y).
Proof. If m = ∞, then n = ∞, and we may take X = Y. Otherwise, we may
apply Theorem 6.4.1.5 to reduce to the case where Y = Shv≤m−1 (C), where
C is a small m-category which admits finite limits and is equipped with aGrothendieck topology. In this case, we let X = Shv≤n−1 (C) and define f∗
to be the identity. Let Z be an arbitrary n-topos. According to Theorem
6.4.1.5, we may assume without loss of generality that Z = τ≤n−1 Z0 , where
Z0 is an ∞-topos. We have a commutative diagramθ

0

Fun∗ (Z, X)
6
θ
θ 00

Fun∗ (Z0 , Shv(C))(/ Fun∗ (τ≤m−1 Z, Y).

Lemma 6.4.5.6 implies that θ0 and θ00 are equivalences of ∞-categories, sothat θ is also an equivalence of ∞-categories.
Definition 6.4.5.8. Let 0 ≤ m ≤ n ≤ ∞ and let X be an n-topos. We will
say that X is m-localic if, for any n-topos Y, the natural map
Fun∗ (Y, X) → Fun∗ (τ≤m−1 Y, τ≤m−1 X)
is an equivalence of ∞-categories.
According to Proposition 6.4.5.7, every m-topos X is equivalent to thesubcategory of (m − 1)-truncated objects in an m-localic n-topos X0 , and X0
is determined up to equivalence. More precisely, the truncation functor
TopR
n

τ≤m−1

/ TopRm

R
induces an equivalence C → TopR
m , where C ⊆ Topn is the full subcategory spanned by the m-localic n-topoi. In other words, we may view the∞-category of m-topoi as a localization of the ∞-category of n-topoi. In
particular, the theory of m-topoi for m < ∞ can be regarded as a special
case of the theory of ∞-topoi. For this reason, we will focus our attention
on the case n = ∞ for most of the remainder of this book.

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657

Proposition 6.4.5.9. Let X be an n-localic ∞-topos. Then any topologicallocalization of X is also n-localic.
Proof. The proof of Proposition 6.4.5.7 shows that X is n-localic if and only
if there exists a small n-category C which admits finite limits, a Grothendiecktopology on C, and an equivalence X → Shv(C). In other words, X is n-localic
if and only if it is equivalent to a topological localization of P(C), where C isa small n-category which admits finite limits. It is clear that any topologicallocalization of X has the same property.
Let X be an ∞-topos. One should think of the ∞-categories τ≤n−1 X as
“Postnikov sections” of X. The classical 1-truncation τ≤1 X of a homotopy
type X remembers only the fundamental groupoid of X. It therefore knows
all about local systems of sets on X but nothing about fibrations over X
with nondiscrete fibers. The relationship between X and τ≤0 X is analogous:τ≤0 X knows about the sheaves of sets on X but has forgotten about sheaves
with nondiscrete stalks.
Remark 6.4.5.10. In view of the above discussion, the notation τ≤0 X isunfortunate because the analogous notation for the 1-truncation of a homotopy type X is τ≤1 X. We caution the reader not to regard τ≤0 X as the resultof applying an operation τ≤0 to X; it instead denotes the essential image ofthe truncation functor τ≤0 : X → X.

6.5 HOMOTOPY THEORY IN AN ∞-TOPOS
In classical homotopy theory, the most important invariants of a (pointed)
space X are its homotopy groups πi (X, x). Our first objective in this section
is to define analogous invariants in the case where X is an object of anarbitrary ∞-topos X. In this setting, the homotopy groups are not ordinarygroups but are instead sheaves of groups on the underlying topos Disc(X). In§6.5.1, we will study these homotopy groups and the closely related theoryof n-connectivity. The main theme is that the internal homotopy theory ofa general ∞-topos X behaves much like the classical case X = S.
One important classical fact which does not hold in general for an ∞topos is Whitehead’s theorem. If f : X → Y is a map of CW complexes,
then f is a homotopy equivalence if and only if f induces bijective maps
πi (X, x) → πi (Y, f (x)) for any i ≥ 0 and any base point x ∈ X. If f : X → Y
is a map in an arbitrary ∞-topos X satisfying an analogous condition on(sheaves of) homotopy groups, then we say that f is ∞-connective. We will
say that an ∞-topos X is hypercomplete if every ∞-connective morphismin X is an equivalence. Whitehead’s theorem may be interpreted as sayingthat the ∞-topos S is hypercomplete. An arbitrary ∞-topos X need not
be hypercomplete. We will survey the situation in §6.5.2, where we also givesome reformulations of the notion of hypercompleteness and show that every
topos X has a hypercompletion X∧ . In §6.5.3, we will show that an ∞-topos

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X is hypercomplete if and only if X satisfies a descent condition with respect
to hypercoverings (other versions of this result can be found in [20] and [79]).
Remark 6.5.0.1. The Brown-Joyal-Jardine theory of (pre)sheaves of simplicial sets on a topological space X is a model for the hypercomplete ∞topos Shv(X)∧ . In many respects, the ∞-topos Shv(X) of sheaves of spaces
on X is better behaved before hypercompletion. We will outline some of theadvantages of Shv(X) in §6.5.4 and in Chapter 7.
6.5.1 Homotopy Groups
Let X be an ∞-topos and let X be an object of X. We will refer to a discrete
object of X/X as a sheaf of sets on X. Since X is presentable, it is automatically cotensored over spaces as explained in Remark 5.5.2.6. Consequently,
for any object X of X and any simplicial set K, there exists an object X Kof X equipped with natural isomorphisms
MapX (Y, X K ) → MapH (K, MapX (Y, X))
in the homotopy category H of spaces.
Definition 6.5.1.1. Let S n = ∂ ∆n+1 ∈ H denote the (simplicial) n-sphereand fix a base point ∗ ∈ S n . Then evaluation at ∗ induces a morphismns : X S → X in X. We may regard s as an object of X/X , and we define
πn (X) = τ≤0 s ∈ X/X to be the associated discrete object of X/X .
We will generally identify πn (X) with its image in the underlying topos
Disc(X/X ) (where it is well-defined up to canonical isomorphism). The conn
stant map S n → ∗ induces a map X → X S which determines a base pointof πn (X).
Suppose that K`and K 0 are pointed simplicial sets and let K ∨ K 0 denotethe coproduct K ∗ K 0 . There is a pullback diagram
X K∨K

XK

0

$
{

$

X

XK

0

z
0

0

in X, so that X K∨K may be identified with a product of X K and X K in
the ∞-topos X/X . We now make the following general observation:
Lemma 6.5.1.2. Let X be an ∞-topos. The truncation functor τ≤n : X → X
preserves finite products.
Proof. We must show that for any finite collection of objects {Xα }α∈A having product X, the induced mapYτ≤n X →
τ≤n Xαα∈Ahighertopoi

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659∞-TOPOI

is an equivalence. If X is the ∞-category of spaces, then this follows fromWhitehead’s theorem: simply compute homotopy groups (and sets) on bothsides. If X = P(C), then to prove that a map in X is an equivalence, it suffices
to show that it remains an equivalence after evaluation at any object C ∈ C;
thus we may reduce to the case where X = S considered above. In the generalcase, X is equivalent to the essential image of a left exact localization functor
L : P(C) → P(C) for some small ∞-category C. Without loss of generality,
we may identify X with a full subcategory of P(C). Then X ⊆ X0 = P(C) isstable under limits, so that X may be identified with a product of the family{Xα }α∈A in X0 . It follows from the case treated above that the natural map
Y
X0
X0
τ≤n
X→
τ≤n
Xα
α∈A
0

X
is an equivalence. Proposition 5.5.6.28 implies that L ◦ τ≤n| X is an ntruncation functor for X. The desired result now follows by applying thefunctor L to both sides of the above equivalence and invoking the assumption that L is left exact (here we must require the finiteness of A).

It follows from Lemma 6.5.1.2 that there is a canonical isomorphism
X

X

0

X

0

τ≤0/X (X K∨K ) ' τ≤0/X (X K ) × τ≤0/X (X K )in the topos Disc(X/X ). In particular, for n > 0, the usual comultiplication
S n → S n ∨ S n (a well-defined map in the homotopy category H) induces
a multiplication map πn (X) × πn (X) → πn (X). As in ordinary homotopy
theory, we conclude that πn (X) is a group object of Disc(X/X ) for n > 0,
which is commutative for n > 1.
In order to work effectively with homotopy sets, it is convenient to define
the homotopy sets πn (f ) of a morphism f : X → Y to be the homotopy setsof f considered as an object of the ∞-topos X/Y . In view of the equivalences
X/f → X/X , we may identify πn (f ) with an object of Disc(X/X ), which isagain a sheaf of groups if n ≥ 1, and abelian groups if n ≥ 2. The intuition isthat the stalk of these sheaves at a point p of X is the nth homotopy groupof the homotopy fiber of f taken with respect to the base point p.
Remark 6.5.1.3. It is useful to have the following recursive definition
for homotopy groups. Let f : X → Y be a morphism in an ∞-topos X.
Regarding f as an object of the topos X/Y , we may take its 0th truncaXtion τ≤0/Y f . This is a discrete object of X/Y , and by definition we haveXXX

π0 (f ) ' f ∗ τ≤0/Y (X) ' X ×Y τ≤0/Y (f ). The natural map X → τ≤0/Y (f ) gives
a global section of π0 (f ). Note that in this case, π0 (f ) is the pullback of adiscrete object of X/Y : this is because the definition of π0 does not requirea base point.
If n > 0, then we have a natural isomorphism πn (f ) ' πn−1 (δ) in the
topos Disc(X/X ), where δ : X → X ×Y X is the associated diagonal map.

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Remark 6.5.1.4. Let f : X → Y be a geometric morphism of ∞-topoi
and let g : Y → Y 0 be a morphism in Y. Then there is a canonical isomorphism f ∗ (πn (g)) ' πn (f ∗ (g)) in Disc(X/f ∗ Y ). This follows immediately from
Proposition 5.5.6.28.
fg

Remark 6.5.1.5. Given a pair of composable morphisms X → Y → Z,
there is an associated sequence of pointed objects
δn+1

δ

n
· · · → f ∗ πn+1 (g) → πn (f ) → πn (g ◦ f ) → f ∗ πn (g) →πn−1 (f ) → · · ·in the ordinary topos Disc(X/X ), with the usual exactness properties. To
construct the boundary map δn , we observethat the n-sphere S n can be`−+
written as a (homotopy) pushout Dof two hemispheres along
S n−1 D∗the equator. By construction, f πn (g) can be identified with the 0-truncation
of
−

n

n

X ×Y Y S ×Z Sn Z ' X D ×Y D− Y S ×Z Sn Z,
which maps by restriction to
XS

n−1

+

×Y Sn−1 Y D ' X S

n−1

×Y Sn−1 Y.

We now observe that the 0-truncation of the latter object is naturally isomorphic to πn−1 (f ) ∈ Disc(X/X ).
To prove the exactness of the above sequence in an ∞-topos X, we firstchoose an accessible left exact localization L : P(C) → X. Without loss offg

generality, we may suppose that the diagram X → Y → Z is the image
under L of a diagram in P(C). Using Remark 6.5.1.4, we conclude that thesequence constructed above is equivalent to the image under L of an analogous sequence in the ∞-topos P(C). Since L is left exact, it will suffice toprove that this second sequence is exact; in other words, we may reduce tothe case X = P(C). Working componentwise, we can reduce further to thecase where X = S. The desired result now follows from classical homotopy
theory. (Special care should be taken regarding the exactness of the above
sequence at π0 (f ): this should really be interpreted in terms of an action of
the group f ∗ π1 (g) on π0 (f ). We leave the details of the construction of thisaction to the reader.)Remark 6.5.1.6. If X = S and η : ∗ → X is a pointed space, then η ∗ πn (X)can be identified with the nth homotopy group of X with base point η.
We now study the implications of the vanishing of homotopy groups.
Proposition 6.5.1.7. Let f : X → Y be an n-truncated morphism in an∞-topos X. Then πk (f ) ' ∗ for all k > n. If n ≥ 0 and πn (f ) ' ∗, then fis (n − 1)-truncated.
Proof. The proof goes by induction on n. If n = −2, then f is an equivalence
and there is nothing to prove. Otherwise, the diagonal map δ : X → X ×Y X
is (n−1)-truncated (Lemma 5.5.6.15). The inductive hypothesis and Remark

highertopoi

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661∞-TOPOI

6.5.1.3 allow us to deduce that πk (f ) ' πk−1 (δ) ' ∗ whenever k > n andk > 0. Similarly, if n ≥ 1 and πn (f ) ' πn−1 (δ) ' ∗, then δ is (n − 2)truncated by the inductive hypothesis, so that f is (n−1)-truncated (Lemma
5.5.6.15).
The case of small k and n requires special attention: we must show that if f
is 0-truncated, then f is (−1)-truncated if and only if π0 (f ) ' ∗. Because f isX
0-truncated, we have an equivalence τ≤0/Y (f ) ' f , so that π0 (f ) ' X ×Y X.
To say π0 (f ) ' ∗ is to assert that the diagonal map δ : X → X ×Y X isan equivalence, which is equivalent to the assertion that f is (−1)-truncated(Lemma 5.5.6.15).
Remark 6.5.1.8. Proposition 6.5.1.7 implies that if f is n-truncated forsome n  0, then we can test whether or not f is m-truncated for any
particular value of m by computing the homotopy groups of f . In contrast
to the classical situation, it is not possible to drop the assumption that f isn-truncated for n  0.
Lemma 6.5.1.9. Let X be an object in an ∞-topos X and let p : X → Y be
an n-truncation of X. Then p induces isomorphisms πk (X) ' p∗ πk (Y ) for
all k ≤ n.
Proof. Let φ : X → Y be a geometric morphism such that φ∗ is fully faithful.
By Proposition 5.5.6.28 and Remark 6.5.1.4, it will suffice to prove the lemmain the case where X = Y. We may therefore assume that Y is an ∞-categoryof presheaves. In this case, homotopy groups and truncations are computed
pointwise. Thus we may reduce to the case X = S, where the conclusion
follows from classical homotopy theory.
Definition 6.5.1.10. Let f : X → Y be a morphism in an ∞-topos X
and let 0 ≤ n ≤ ∞. We will say that f is n-connective if it is an effective
epimorphism and πk (f ) = ∗ for 0 ≤ k < n. We shall say that the object
X is n-connective if f : X → 1X is n-connective, where 1X denotes the
final object of X. By convention, we will say that every morphism f in X is(−1)-connective.
Definition 6.5.1.11. Let X be an object of an ∞-topos X. We will say that
X is connected if it is 1-connective: that is, if the truncation τ≤0 X is a final
object in X.
Proposition 6.5.1.12. Let X be an object in an ∞-topos X and let n ≥ −1.
Then X is n-connective if and only if τ≤n−1 X is a final object of X.
Proof. The case n = −1 is trivial. The proof in general proceeds by induction
on n ≥ 0. If n = 0, then the conclusion follows from Proposition 6.2.3.4.
Suppose n > 0. Let p : X → τn−1 X be an (n − 1)-truncation of X. Ifτ≤n−1 X is a final object of X, thenπk X ' p∗ πk (τn−1 X) ' ∗highertopoi

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CHAPTER 6

for k < n by Lemma 6.5.1.9. Since the map p : X → τ≤n−1 X ' 1X is an effective epimorphism (Proposition 7.2.1.14), it follows that X is n-connective.
Conversely, suppose that X is n-connective. Then p∗ πn−1 (τ≤n−1 X) ' ∗.
Since p is an effective epimorphism, Lemma 6.2.3.16 implies that
πn−1 (τ≤n−1 X) ' ∗.
Using Proposition 6.5.1.7, we conclude that τ≤n−1 X is (n − 2)-truncated, sothat τ≤n−1 X ' τ≤n−2 X. Repeating this argument, we reduce to the casewhere n = 0 which was handled above.
Corollary 6.5.1.13. The class of n-connective objects of an ∞-topos X isstable under finite products.
Proof. Combine Proposition 6.5.1.12 with Lemma 6.5.1.2.
Let X be an ∞-topos and X an object of X. Since
MapX (X, Y ) ' MapX (τ≤n X, Y )whenever Y is n-truncated, we deduce that X is (n + 1)-connective if andonly if the natural map MapX (1X , Y ) → MapX (X, Y ) is an equivalence for all
n-truncated Y . From this, we can immediately deduce the following relative
version of Proposition 6.5.1.12:
Corollary 6.5.1.14. Let f : X → X 0 be a morphism in an ∞-topos X.
Then f is (n + 1)-connective if and only if composition with f induces ahomotopy equivalence
MapX/X 0 (idX 0 , Y ) → MapX/X 0 (f, Y )for every n-truncated object Y ∈ X/X 0 .
Remark 6.5.1.15. Let L : X → Y be a left exact localization of ∞-topoi
and let f : Y → Y 0 be an n-connective morphism in Y. Then f is equivalent(in Fun(∆1 , Y)) to Lf0 , where f0 is an n-connective morphism in X. To see
this, we choose a (fully faithful) right adjoint G to L and a factorization
f0

G(Y )

<X
G(f0 )f 00#/ G(Y 0 ),where f 0 is n-connective and f 00 is (n−1)-truncated. Then Lf 00 ◦Lf 0 is equivalent to f and is therefore n-connective. It follows that Lf 00 is an equivalence,
so that Lf 0 is equivalent to f .
We conclude by noting the following stability properties of the class ofn-connective morphisms:
Proposition 6.5.1.16. Let X be an ∞-topos.

highertopoi

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663∞-TOPOI(1) Let f : X → Y be a morphism in X. If f is n-connective, then it ism-connective for all m ≤ n. Conversely, if f is n-connective for alln < ∞, then f is ∞-connective.
(2) Any equivalence in X is ∞-connective.
(3) Let f, g : X → Y be homotopic morphisms in X. Then f is n-connective
if and only if g is n-connective.
(4) Let π ∗ : X → Y be left adjoint to a geometric morphism from π∗ : Y →X and let f : X → X 0 be an n-connective morphism in X. Then π ∗ fis an n-connective morphism in Y.
(5) Suppose we are given a diagram
>Y
f

g


h
/Z
X
in X, where f is n-connective. Then g is n-connective if and only if h
is n-connective.
(6) Suppose we are given a pullback diagram
X0
f0

q0

/X
f



q
/Y
Y0
in X. If f is n-connective, then so is f 0 . The converse holds if q is aneffective epimorphism.
Proof. The first three assertions are obvious. Claim (4) follows from Propositions 6.5.1.12 and 5.5.6.28. To prove (5), we first observe that Corollary
6.2.3.12 implies that g is an effective epimorphism if and only if h is aneffective epimorphism. According to Remark 6.5.1.5, we have a long exact
sequence· · · → f ∗ πi+1 (g)→πi (f ) → πi (h) → f ∗ πi (g) → πi−1 (f ) → · · ·of pointed objects in the topos Disc(X/X ). It is then clear that if f and g
are n-connective, then so is h. Conversely, if f and h are n-connective, thenf ∗ πi (g) ' ∗ for i ≤ n. Since f is an effective epimorphism, Lemma 6.2.3.16
implies that πi (g) ' ∗ for i ≤ n, so that g is also n-connective.
The first assertion of (6) follows from (4) since a pullback functor q ∗ :X/Y → X/Y 0 is left adjoint to a geometric morphism. For the converse, let
us suppose that q is an effective epimorphism and that f 0 is n-connective.
According to Lemma 6.2.3.15, the maps f and q 0 are effective epimorphisms.
Applying Remark 6.5.1.4, we conclude that there are canonical isomorphisms∗∗
q 0 πk (f ) ' πk (f 0 ) in the topos Disc(X/X 0 ), so that q 0 πk (f ) ' ∗ for k < n.
Applying Lemma 6.2.3.16, we conclude that πk (f ) ' ∗ for k < n, so that f
is n-connective, as desired.

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CHAPTER 6

Corollary 6.5.1.17. Let
X0

/X

g

f0

f



/Y
Y0
be a pushout diagram in an ∞-topos X. Suppose that f 0 is n-connective.
Then f is n-connective.
Proof. Choose an accessible left exact localization functor L : P(C) → X.
Using Remark 6.5.1.15, we can assume without loss of generality that f 0 =Lf00 , where f00 : X00 → Y00 is a morphism in P(C). Similarly, we may assume
g = Lg0 for some morphism g0 : X00 → X0 . Form a pushout diagram
X00

/ X0

g0

f00

f0


Y00


/ Y0

in P(C). Then the original diagram is equivalent to the image (under L) of
the diagram above. In view of Proposition 6.5.1.16, it will suffice to showthat f0 is n-connective. Using Propositions 6.5.1.12 and 5.5.6.28, we see thatf0 is n-connective if and only if its image under the evaluation map P(C) → S
associated to any object C ∈ C is n-connective. In other words, we can reduceto the case where X = S, and the result now follows from classical homotopy
theory.
We conclude by establishing a few results which will be needed in §7.2:
Proposition 6.5.1.18. Let f : X → Y be a morphism in an ∞-topos X,
δ : X → X ×Y X the associated diagonal morphism, and n ≥ 0. The following
conditions are equivalent:(1) The morphism f is n-connective.
(2) The diagonal map δ : X → X ×Y X is (n − 1)-connective and f is aneffective epimorphism.
Proof. The proof is immediate from Definition 6.5.1.10 and Remark 6.5.1.3.
Proposition 6.5.1.19. Let X be an ∞-topos containing an object X andlet σ : ∆2 → X be a 2-simplex corresponding to a diagram
Y

/Z

f

~

g

X.
Then f is an n-connective morphism in X if and only if σ is an n-connective
morphism in X/X .

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665∞-TOPOI

Proof. We observe that X/g → X/Z is a trivial fibration, so that an objectof X/g is n-connective if and only if its image in X/Z is n-connective.
Proposition 6.5.1.20. Let f : X → Y be a morphism in an ∞-topos X,
let s : Y → X be a section of f (so that f ◦ s is homotopic to idY ), and letn ≥ 0. Then f is n-connective if and only if s is (n − 1)-connective.
Proof. We have a 2-simplex σ : ∆2 → X which we may depict as follows:
>X
s

f

idY
/ Y.
Y
Corollary 6.2.3.12 implies that f is an effective epimorphism; this completes
the proof in the case n = 0. Suppose that n > 0 and that s is (n − 1)connective. In particular, s is an effective epimorphism. The long exact sequence of Remark 6.5.1.5 gives an isomorphism πi (s) ' s∗ πi+1 (f ), so thats∗ πk (f ) vanishes for 1 ≤ k < n. Applying Lemma 6.2.3.16, we conclude thatπk (f ) ' ∗ for 1 ≤ k < n. Moreover, since s is an effective epimorphism,
it induces an effective epimorphism π0 (idY ) → π0 (f ) in the ordinary topos
Disc(X/Y ), so that π0 (f ) ' ∗ as well. This proves that f is n-connective.
Conversely, if f is n-connective, then πi (s) ' ∗ for i < n − 1; the only
nontrivial point is to verify that s is an effective epimorphism. According to
Proposition 6.5.1.19, it will suffice to prove that σ is an effective epimorphism
when viewed as a morphism in X/Y . Using Proposition 7.2.1.14, we may
X

reduce to proving that σ 0 = τ≤0/Y (σ) is an equivalence in X/Y . This is clear
since the source and target of σ 0 are both final objects of X/Y (by virtue of
our assumption that f is 1-connective).
6.5.2 ∞-Connectedness
Let C be an ordinary
category equipped with a Grothendieck topology andop
let A = SetC
∆ be the category of simplicial presheaves on C.
Proposition 6.5.2.1 (Jardine [41]). There exists a left proper, combinatorial, simplicial model structure on the category A which admits the following
description:(C) A map f : F• → G• of simplicial presheaves on C is a local cofibration
if it is an injective cofibration: that is, if and only if the induced map
F• (C) → G• (C) is a cofibration of simplicial sets for each object C ∈ C.
(W ) A map f : F• → G• of simplicial presheaves on C is a local equivalence
if and only if, for any object C ∈ C and any commutative diagram of
topological spaces/ |F• (C)|
S n−1
_

Dn


/ |G• (C)|,highertopoi

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CHAPTER 6

there exists a collection of morphisms {Cα → C} which generates acovering sieve on C, such that in each of the induced diagrams
S n−1
_

/ |F• (Cα )|
9


Dn


/ |G• (Cα )|,one can produce a dotted arrow so that the upper triangle commutes
and the lower triangle commutes up to a homotopy which is fixed on
S n−1 .
We refer the reader to [41] for a proof (one can also deduce Proposition
6.5.2.1 from Proposition A.2.6.15). We will refer to the model structure of
Proposition 6.5.2.1 as the local model structure on A.
Remark 6.5.2.2. In the case where the topos X of sheaves of sets on C
has enough points, there is a simpler description of the class (W ) of local
equivalences: a map F → G of simplicial presheaves is a local equivalence
if and only if it induces weak homotopy equivalences Fx → Gx of simplicial
sets after passing to the stalk at any point x of X. We refer the reader to[41] for details.
Let A◦ denote the full subcategory of A consisting of fibrant-cofibrantobjects (with respect to the local model structure) and let X = N(A◦ ) be
the associated ∞-category. We observe that the local model structure on A
is a localization of the injective model structure on A. Consequently, the∞-category X is a localization of the ∞-category associated to the injectivemodel structure on A, which (in view of Proposition 5.1.1.1) is equivalentto P(N(C)). It is tempting to guess that X is equivalent to the left exact
localization Shv(N(C)) constructed in §6.2.2. This is not true in general;
however, as we will explain below, we can always recover X as an accessible
left exact localization of Shv(N(C)). In particular, X is itself an ∞-topos.
In general, the difference between X and Shv(N(C)) is measured by the
failure of Whitehead’s theorem. Essentially by construction, the equivalencesin A are those maps which induce isomorphisms on homotopy sheaves. In
general, this assumption is not strong enough to guarantee that a morphismin Shv(N(C)) is an equivalence. However, this is the only difference: the∞-category X can be obtained from Shv(N(C)) by inverting the class of ∞connective morphisms (Proposition 6.5.2.14). Before proving this, we study
the class of ∞-connective morphisms in an arbitrary ∞-topos.
Lemma 6.5.2.3. Let p : C → D be a Cartesian fibration of ∞-categories,
let C0 be a full subcategory of C, and suppose that for every p-Cartesianmorphism f : C → C 0 in C, if C 0 ∈ C0 , then C ∈ C0 . Let D be an object of D
and let f : C → C 0 be a morphism in the fiber CD = C ×D {D} which exhibits
C 0 as a C0D -localization of C (see Definition 5.2.7.6). Then f exhibits C 0 as
a C-localization of C.

highertopoi

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667∞-TOPOI

Proof. According to Proposition 2.4.3.2, p induces a Cartesian fibration
CC/ → DD/ , which restricts to give a Cartesian fibration p0 : C0C/ → DD/ .
We observe that f is an object of C0C/ which is an initial object of (p0 )−1 {idD }(Remark 5.2.7.7), and that idD is an initial object of DD/ . Lemma 2.4.4.7 implies that f is an initial object of C0C/ , so that f exhibits C 0 as C0 -localizationof C (Remark 5.2.7.7), as desired.
Lemma 6.5.2.4. Let p : C → D be a Cartesian fibration of ∞-categories,
let C0 be a full subcategory of C, and suppose that for every p-Cartesianmorphism f : C → C 0 in C, if C 0 ∈ C0 , then C ∈ C0 . Suppose that foreach object D ∈ D, the fiber C0D = C0 ×D {D} is a reflective subcategory of
CD = C ×D {D} (see Remark 5.2.7.9). Then C0 is a reflective subcategory ofC.
Proof. Combine Lemma 6.5.2.3 with Proposition 5.2.7.8.
Lemma 6.5.2.5. Let X be a presentable ∞-category, let C be an accessible∞-category, and let α : F → G be a natural transformation between accessible functors F, G : C → X. Let C(n) be the full subcategory of C spanned bythose objects C such that α(C) : F (C) → G(C) is n-truncated. Then C(n) is
an accessible subcategory of C (see Definition 5.4.7.8).
Proof. We will work by induction on n. If n = −2, then we have a (homotopy) pullback diagram
C(n)

/C


E


/ Fun(∆1 , X),

αwhere E is the full subcategory of Fun(∆1 , X) spanned by equivalences.
The inclusion of E into Fun(∆1 , X) is equivalent to the diagonal map X →Fun(∆1 , X) and therefore accessible. Proposition 5.4.6.6 implies that C(n) is
an accessible subcategory of C as desired.
If n ≥ −1, we apply the the inductive hypothesis to the diagonal functor
δ : F → F ×G F using Lemma 5.5.6.15.
Lemma 6.5.2.6. Let X be a presentable ∞-category and let −2 ≤ n <∞. Let C be the full subcategory of Fun(∆1 , X) spanned by the n-truncatedmorphisms. Then C is a strongly reflective subcategory of Fun(∆1 , X).
Proof. Applying Lemma 6.5.2.4 to the restriction functor Fun(∆1 , X) →Fun({1}, X), we conclude that C is a reflective subcategory of Fun(∆1 , X).
The accessibility of C follows from Lemma 6.5.2.5.
Lemma 6.5.2.7. Let X be an ∞-topos, let 0 ≤ n ≤ ∞, and let D(n) be thefull subcategory of Fun(∆1 , X) spanned by the n-connective morphisms of X.
Then D(n) is an accessible subcategory of X and is stable under colimits inX.

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Proof. Suppose first that n < ∞. Let C(n) ⊆ Fun(∆1 , X) be the full subcategory spanned by the n-truncated morphisms in X. According to Lemma
6.5.2.6, the inclusion C(n) ⊆ Fun(∆1 , X) has a left adjoint L : Fun(∆1 , X) →C(n). Moreover, the proof of Lemma 6.5.2.3 shows that f is n-connective if
and only if Lf is an equivalence. It is easy to see that the full subcategory
E ⊆ C(n) spanned by the equivalences is stable under colimits in C(n), sothat D(n) is stable under colimits in Fun(∆1 , X). The accessibility of D(n)follows from the existence of the (homotopy) pullback diagram
D(n)/ Fun(∆1 , X)E
/ C(n)

Land Proposition 5.4.6.6.
If n = ∞, we observe that D(n) = ∪m<∞ D(m), which is manifestly
1stable under colimits and is an accessible subcategory of X∆ by Proposition
5.4.7.10.
Proposition 6.5.2.8. Let X be an ∞-topos and let S denote the collectionof ∞-connective morphisms of X. Then S is strongly saturated and of small
generation (see Definition 5.5.4.5).
Proof. Lemma 6.5.2.7 implies that S is stable under colimits in Fun(∆1 , X),
and Corollary 6.5.1.17 shows that S is stable under pushouts. To prove that
S has the two-out-of-three property, we consider a diagram σ : ∆2 → X,
which we depict as
>Y
f

X

g
h

/ Z.

If f is ∞-connective, then Proposition 6.5.1.16 implies that g is ∞-connective
if and only if h is ∞-connective. Suppose that g and h are ∞-connective.
The long exact sequence· · · → f ∗ πn+1 (g) → πn (f ) → πn (h) → f ∗ πn (g) → πn−1 (f ) → · · ·of Remark 6.5.1.5 shows that πn (f ) ' ∗ for all n ≥ 0. It will therefore suffice to prove that f is an effective epimorphism. According to Proposition
6.5.1.19, it will suffice to show that σ is an effective epimorphism in X/Z . AcXX

cording to Proposition 7.2.1.14, it suffices to show that τ≤0/Z (h) and τ≤0/Z (g)are both final objects of X/Z , which follows from the 0-connectivity of g andh (Proposition 6.5.1.12).
To show that S is of small generation, it suffices (in view of Lemma
5.5.4.14) to show that the full subcategory of Fun(∆1 , X) spanned by S
is accessible. This follows from Lemma 6.5.2.7.

highertopoi

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669∞-TOPOILet X be an ∞-topos. We will say that an object X of X is hypercomplete
if it is local with respect to the class of ∞-connective morphisms. Let X∧
denote the full subcategory of X spanned by the hypercomplete objects of
X. Combining Propositions 6.5.2.8 and 5.5.4.15, we deduce that X∧ is anaccessible localization of X. Moreover, since Proposition 6.5.1.16 implies thatthe class of ∞-connective morphisms is stable under pullback, we deducefrom Proposition 6.2.1.1 that X∧ is a left exact localization of X. It follows
that X∧ is itself an ∞-topos. We will show in a moment that X∧ can be
described by a universal property.
Lemma 6.5.2.9. Let X be an ∞-topos and let n < ∞. Then τ≤n X ⊆ X∧ .
Proof. Corollary 6.5.1.14 implies that an n-truncated object of X is local withrespect to every n-connective morphism of X and therefore with respect toevery ∞-connective morphism of X.
Lemma 6.5.2.10. Let X be an ∞-topos, let L : X → X∧ be a left adjoint to
the inclusion, and let X ∈ X be such that LX is an ∞-connective object ofX∧ . Then LX is a final object of X∧ .
Proof. For each n < ∞, we have equivalences∧

XXX
1X ' τ≤n
LX ' Lτ≤n
X ' τ≤nX,

where the first is because of our hypothesis that LX is ∞-connective, thesecond is given by Proposition 5.5.6.28, and the third is given by Lemma
6.5.2.9. It follows that X is an ∞-connective object of X, so that LX is a
final object of X∧ by construction.
We will say that an ∞-topos X is hypercomplete if X∧ = X; in other words,
X is hypercomplete if every ∞-connective morphism of X is an equivalence,
so that Whitehead’s theorem holds in X.
Remark 6.5.2.11. In [79], the authors use the term t-completeness to referto the property that we have called hypercompleteness.
Lemma 6.5.2.12. Let X be an ∞-topos. Then the ∞-topos X∧ is hypercomplete.
Proof. Let f : X → Y be an ∞-connective morphism in X∧ . Applying
Lemma 6.5.2.10 to the ∞-topos (X∧ )/Y ' (X/Y )∧ , we deduce that f is anequivalence.
We are now prepared to characterize X∧ by a universal property:
Proposition 6.5.2.13. Let X and Y be ∞-topoi. Suppose that Y is hypercomplete. Then composition with the inclusion X∧ ⊆ X induces an isomorphism
Fun∗ (Y, X∧ ) → Fun∗ (Y, X).

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Proof. Let f∗ : Y → X be a geometric morphism; we wish to prove that f∗factors through X∧ . Let f ∗ denote a left adjoint to f∗ ; it will suffice to showthat f ∗ carries each ∞-connective morphism u of X to an equivalence in Y.
Proposition 6.5.1.16 implies that f ∗ (u) is ∞-connective, and the hypothesisthat Y is hypercomplete guarantees that u is an equivalence.
The following result establishes the relationship between our notion of
hypercompleteness and the Brown-Joyal-Jardine theory of simplicial presheaves.
Proposition 6.5.2.14. Let C be a small category equipped with a Grothendieck topology and let A denote the category of simplicial presheaves on C endowed with the local model structure (see Proposition 6.5.2.1). Let A◦ denotethe full subcategory consisting of fibrant-cofibrant objects and let A = N(A◦ )be the corresponding ∞-category. Then A is equivalent to Shv(C)∧ ; in particular, it is a hypercomplete ∞-topos.
Proof. Let P(C) denote the ∞-category P(N(C)) of presheaves on N(C) andlet A0 denote the model category of simplicial presheaves on C endowed with
the injective model structure of §A.3.3. According to Proposition 4.2.4.4, thesimplicial nerve functor induces an equivalence◦θ : N(A0 ) → P(C).
◦We may identify N(A◦ ) with the full subcategory of N(A0 ) spanned by
the S-local objects, where S is the class of local equivalences (Proposition
A.3.7.3).
We first claim that θ| N(A◦ ) factors through Shv(C). Consider an ob(0)ject C ∈ C and a sieve C/C ⊆ C/C . Let χC : C → Set be the functor(0)

D 7→ HomC (D, C) represented by C, let χC be the subfunctor of χC deter(0)(0)
mined by the sieve C/C , and let i : χC → χC be the inclusion. We regard(0)χC and χC as simplicial presheaves on C which take values in the full subcategory of Set∆ spanned by the constant simplicial sets. We observe thatevery simplicial presheaf on C which is valued in constant simplicial sets is
automatically fibrant and every object of A0 is cofibrant. Consequently, we◦
may regard i as a morphism in the ∞-category N(A0 ). It is easy to see that(0)
θ(i) represents the monomorphism U → j(C) classified by the sieve C/C . If
(0)

C/C is a covering sieve on C, then i is a local equivalence. Consequently, every object X ∈ N(A◦ ) is i-local, so that θ(X) is θ(i)-local. By construction,
Shv(C) is the full subcategory of P(C) spanned by those objects which are(0)θ(i)-local for every covering sieve C/C on every object C ∈ C. We conclude
that θ| N(A◦ ) factors through Shv(C).
Let X = θ−1 Shv(C), so that N(A◦ ) can be identified with the collectionof S 0 -local objects of X, where S 0 is the collection of all morphisms in X
which belong to S. Then θ induces an equivalence N(A◦ ) → θ(S 0 )−1 Shv(C).
We now observe that a morphism f in X belongs to S 0 if and only ifhighertopoi

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671∞-TOPOI

θ(f ) is an ∞-connective morphism in Shv(C) (since the condition of being a local equivalence can be tested on homotopy sheaves). It follows thatθ(S 0 )−1 Shv(C) = Shv(C)∧ , as desired.
Remark 6.5.2.15. In [79], the authors discuss a generalization of Jardine’s
construction in which the category C is replaced by a simplicial category.
Proposition 6.5.2.14 holds in this more general situation as well.
We conclude this section with a few remarks about localizations of an∞-topos X. In §6.2.1, we introduced the class of topological localizations ofX, which consists of those left exact localizations which can be obtained byinverting monomorphisms in X. The hypercompletion X∧ is, in some sense,at the other extreme: it is obtained by inverting the ∞-connective morphisms
in X, which are never monomorphisms unless they are already equivalences.
In fact, X∧ is the maximal (left exact) localization of X which can be obtained
without inverting monomorphisms:
Proposition 6.5.2.16. Let X and Y be ∞-topoi and let f ∗ : X → Y be a left
exact colimit-preserving functor. The following conditions are equivalent:(1) For every monomorphism u in X, if f ∗ u is an equivalence in Y, thenu is an equivalence in X.
(2) For every morphism u ∈ X, if f ∗ u is an equivalence in Y, then f is∞-connective.
Proof. Suppose first that (2) is satisfied. If u is a monomorphism and f ∗ u
is an equivalence in Y, then u is ∞-connective. In particular, u is both amonomorphism and an effective epimorphism and therefore an equivalence in
X. This proves (1). Conversely, suppose that (1) is satisfied and let u : X → Z
be an arbitrary morphism in X such that f ∗ (u) is an equivalence. We will
prove by induction on n that u is n-connective.
We first consider the case n = 0. Choose a factorization
?Y
u0

X

u00
u


/ Z,

where u0 is an effective epimorphism and u00 is a monomorphism. Since f ∗ u
is an equivalence, Corollary 6.2.3.12 implies that f ∗ u00 is an effective epimorphism. Since f ∗ u00 is also a monomorphism (by virtue of our assumptionthat f is left exact), we conclude that f ∗ u00 is an equivalence. Applying (1),
we deduce that u00 is an equivalence, so that u is an effective epimorphism,
as desired.
Now suppose n > 0. According to Proposition 6.5.1.18, it will suffice toshow that the diagonal map δ : X → X ×Z X is (n − 1)-connective. By the
inductive hypothesis, it will suffice to prove that f ∗ (δ) is an equivalence in
Y. We conclude by observing that f ∗ is left exact, so we can identify δ withthe diagonal map associated to the equivalence f ∗ (u) : f ∗ X → f ∗ Z.

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Definition 6.5.2.17. Let X be an ∞-topos and let Y ⊆ X be an accessible
left exact localization of X. We will say that Y is an cotopological localizationof X if the left adjoint L : X → Y to the inclusion of Y in X satisfies theequivalent conditions of Proposition 6.5.2.16.
Remark 6.5.2.18. Let f ∗ : X → Y be the left adjoint of a geometricmorphism between ∞-topoi and suppose that the equivalent conditions of
Proposition 6.5.2.16 are satisfied. Let u : X → Z be a morphism in X andchoose a factorization
?Y
u0

X

u00
u


/ Z,

where u0 is an effective epimorphism and u00 is a monomorphism. Then u00
is an equivalence if and only if f ∗ (u00 ) is an equivalence. Applying Corollary
6.2.3.12, we conclude that u is an effective epimorphism if and only if f ∗ (u)is an effective epimorphism.
The hypercompletion X∧ of an ∞-topos X can be characterized as the maximal cotopological localization of X (that is, the cotopological localization
which is obtained by inverting as many morphisms as possible). According toour next result, every localization can be obtained by combining topologicaland cotopological localizations:
Proposition 6.5.2.19. Let X be an ∞-topos and let X00 ⊆ X be an accessible
left exact localization of X. Then there exists a topological localization X0 ⊆ Xsuch that X00 ⊆ X0 is a cotopological localization of X0 .
Proof. Let L : X → X00 be a left adjoint to the inclusion, let S be thecollection of all monomorphisms u in X such that Lu is an equivalence, and
let X0 = S −1 X be the collection of S-local objects of X. Since L is left exact, S
is stable under pullbacks and therefore determines a topological localizationof X. By construction, we have X00 ⊆ X0 . The restriction L| X0 exhibits X00
as an accessible left exact localization of X0 . Let u be a monomorphism
in X0 such that Lu is an equivalence. Then u is a monomorphism in X,
so that u ∈ S. Since X0 consists of S-local objects, we conclude that u isan equivalence. It follows that X00 is a cotopological localization of X0 , asdesired.
Remark 6.5.2.20. It is easy to see that the factorization of Proposition
6.5.2.19 is essentially uniquely determined: more precisely, X0 is unique provided we assume that it is stable under equivalences in X.
Combining Proposition 6.5.2.19 with Remark 7.2.1.16, we see that every∞-topos X can be obtained in following way:
(1) Begin with the ∞-category P(C) of presheaves on some small ∞category C.

highertopoi

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673∞-TOPOI(2) Choose a Grothendieck topology on C: this is equivalent to choosing a
opleft exact localization of the underlying topos Disc(P(C)) = SethC .
(3) Form the associated topological localization Shv(C) ⊆ P(C), which canbe described as the pullback
P(C) ×P(N(hC)) Shv(N(hC))
in RTop.
(4) Form a cotopological localization of Shv(C) by inverting some class of∞-connective morphisms of Shv(C).
Remark 6.5.2.21. Let X be an ∞-topos. The collection of all ∞-connective
morphisms in X is saturated. It follows from Proposition 5.5.5.7 that there
exists a factorization system (SL , SR ) on X, where SL is the collection of
all ∞-connective morphisms in X. We will say that a morphism in X is
hypercomplete if it belongs to SR . Unwinding the definitions (and using thefact that a morphism in X/Y is ∞-connective if and only if its image in X is∞-connective), we conclude that a morphism f : X → Y is hypercomplete
if and only if it is hypercomplete when viewed as an object of the ∞-topos
X/Y (see §6.5.2).
Using Proposition 5.2.8.6, we deduce that the collection of hypercomplete
morphisms in X is stable under limits and the formation of pullback squares.
Remark 6.5.2.22. Let X be an ∞-topos. The condition that a morphism
f : X → Y be hypercomplete is local: that is, if {Yα → Y`
} is a collectionof morphisms which determine an effective epimorphism
Yα → Y , andeach of the induced maps fα : X ×Y Yα → Y`α is hypercomplete, thenQ f ishypercomplete. To prove this, we set Y0 = α Yα ; then X/Y0 ' α X/Yα(since coproducts in X are disjoint), so it is easy to see that the induced
map f 0 : X ×Y Y0 → Y0 is hypercomplete. Let Y• be the simplicial objectof X given by the Čech nerve of the effective epimorphism Y0 → Y . For
every map Z → Y , let Z• be the simplicial object described by the formula
Zn = Yn ×Y Z (equivalently, Z• is the Čech nerve of the effective epimorphism
Z ×Y Y0 → Z). Using Remark 6.5.2.21, we conclude that each of the maps
Xn → Yn is hypercomplete.
For every map A → Y , the mapping space MapX/Y (A, X) can be obtained
as the totalization of a cosimplicial spacen 7→ MapX/Yn (An , Xn ).
If g : A → B is an ∞-connective morphism in X/Y , then each of the induced
maps An → Bn is ∞-connective, so the induced map
MapX/Yn (Bn , Xn ) → MapX/Yn (An , Xn )is a homotopy equivalence. Passing to the totalization, we obtain a homotopy
equivalence MapX/Y (B, X) → MapX/Y (A, X). Thus f is hypercomplete, as
desired.

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6.5.3 Hypercoverings
Let X be an ∞-topos. In §6.5.2, we defined the hypercompletion X∧ ⊆ X to
be the left exact localization of X obtained by inverting the ∞-connective
morphisms. In this section, we will give an alternative description of the hypercomplete objects X ∈ X∧ : they are precisely those objects of X which satisfy a descent condition with respect to hypercoverings (Theorem 6.5.3.12).
We begin by reviewing the definition of a hypercovering.
Let X be a topological space and let F be a presheaf of sets on X. To
construct the sheaf associated to F, it is natural to consider the presheaf F†
defined byF† = lim lim F(V ).
−→ ←−U V ∈U

Here the direct limit is taken over all sieves U which cover U (see Construction 6.2.2.9). There is an obvious map F → F† which is an isomorphism
whenever F is a sheaf. Moreover, F† is “closer” to being a sheaf than F is.
More precisely, F† is always a separated presheaf: two sections of F† which
agree locally automatically coincide. If F is itself a separated presheaf, thenF† is a sheaf.
For a general presheaf F, we need to apply the above construction twice
to construct the associated sheaf (F† )† . To understand the problem, let us
try to prove that F† is a sheaf (to see where
S the argument breaks down).
Suppose we are given an open covering X = Uα and a collection of sectionssα ∈ F† (Uα ) such thatsα |Uα ∩ Uβ = sβ |Uα ∩ Uβ .
Refining the covering Uα if necessary, we may assume that each sα is theimage of some section tα ∈ F(Uα ). However, the equationtα |Uα ∩ Uβ = tβ |Uα ∩ Uβ
holds only locally on Uα ∩Uβ , so the sections tα do not necessarily determine
a global section of F† . To summarize: the freedom to consider arbitrarily fineopen covers U = {Uα } is not enough; we also need to be able to refine theintersections Uα ∩ Uβ . This leads very naturally to the notion of a hypercovering. Roughly speaking, a hypercovering of X consists of an open covering{Uα } of X, an open covering {Vαβγ } of each intersection Uα ∩ Uβ , and analogous data associated to more complicated intersections (see Definition 6.5.3.2
for a more precise formulation).
In classical sheaf theory, there are two ways to construct the sheaf associated to a presheaf F:(1) One can apply the construction F 7→ F† twice.
(2) Using the theory of hypercoverings, one can proceed directly by defining
F+ (U ) = lim lim F(V ),−→ ←−Uwhere the direct limit is now taken over arbitrary hypercoverings U.

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In higher category theory, the difference between these two approaches
becomes more prominent. For example, suppose that F is not a presheaf ofsets but a presheaf of groupoids on X. In this case, one can construct theassociated sheaf of groupoids using either approach. However, in the case of
approach (1), it is necessary to apply the construction F 7→ F† three times:
the first application guarantees that the automorphism groups of sectionsof F are separated presheaves, the second guarantees that they are sheaves,and the third guarantees that F itself satisfies descent. More generally, if F
is a sheaf of n-truncated spaces, then the sheafification of F via approach(1) takes place in (n + 2)-stages.
When we pass to the case n = ∞, the situation becomes more complicated.
If F is a presheaf of spaces on X, then it is not reasonable to expect toobtain a sheaf by applying the construction F 7→ F† any finite number oftimes. In fact, it is not obvious that F† is any closer than F to being a sheaf.
Nevertheless, this is true: our proof of Proposition 6.2.2.7 shows that thesheafification of F can be obtained by transfinite iteration of the construction
F 7→ F† .
Alternatively, one can use the construction F 7→ F+ to construct a sheafof spaces from F in a single step. The universal property asserted above
guarantees the existence of a morphism of sheaves θ : F(α) → F+ . However,
the morphism θ is generally not an equivalence. Instead, θ realizes F+ as
the hypercompletion of F(α) in the ∞-topos Shv(X). We will not prove
this statement directly but will instead establish a reformulation (Corollary
6.5.3.13) which does not make reference to the sheafification constructions
outlined above.
Before we can introduce the definition of a hypercovering, we need toreview some simplicial terminology.
Notation 6.5.3.1. For each n ≥ 0, let ∆≤n denote the full subcategoryof ∆ spanned by the set of objects {[0], . . . , [n]}. If X is a presentable ∞category, the restriction functor
skn : X∆ → Fun(N(∆≤n )op , X)
has a right adjoint given by right Kan extension along the inclusion functorN(∆≤n )op ⊆ N(∆)op . Let coskn : X∆ → X∆ be the composition of skn with
its right adjoint. We will refer to coskn as the n-coskeleton functor.
Definition 6.5.3.2. Let X be an ∞-topos. A simplicial object U• ∈ X∆ isa hypercovering of X if, for each n ≥ 0, the unit map
Un → (coskn−1 U• )nis an effective epimorphism. We will say that U• is an effective hypercoveringof X if the colimit of U• is a final object of X.
Remark 6.5.3.3. More informally, a simplicial object U• ∈ X∆ is a hypercovering of X if each of the associated maps
U0 → 1X

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U1 → U0 × U0
U2 → · · ·is an effective epimorphism.
Lemma 6.5.3.4. Let X be an ∞-topos and let U• be a simplicial object in X.
Let L : X → X∧ be a left adjoint to the inclusion. The following conditions
are equivalent:(1) The simplicial object U• is a hypercovering of X.
(2) The simplicial object L ◦ U• is a hypercovering of X∧ .
Proof. Since L is left exact, we can identify L ◦ coskn U• with coskn (L ◦ U• ).
The desired result now follows from Remark 6.5.2.18.
Lemma 6.5.3.5. Let X be an ∞-topos and let U be an ∞-connective objectof X. Let U• be the constant simplicial object with value U . Then U• is a
hypercovering of X.
Proof. Using Lemma 6.5.3.4, we can reduce to the case where X is hypercomplete. Then U ' 1X , so that U• is equivalent to the constant functor withvalue 1X and is therefore a final object of X∆ . For each n ≥ 0, the coskeleton
functor coskn−1 preserves small limits, so coskn−1 U• is also a final object ofU• . It follows that the unit map U• → coskn−1 U• is an equivalence.
Notation 6.5.3.6. Let ∆s be the subcategory of ∆ with the same objects
but where the morphisms are given by injective order-preserving maps between nonempty linearly ordered sets. If X is an ∞-category, we will refer toa diagram N(∆s )op → X as a semisimplicial object of X.
op
Lemma 6.5.3.7. The inclusion N(∆ops ) ⊆ N(∆ ) is cofinal.

Proof. According to Theorem 4.1.3.1, it will suffice to prove that for everyn ≥ 0, the category C = ∆s ×∆ ∆/[n] has a weakly contractible nerve. To
prove this, we let F : C → C be the constant functor taking value givenby the inclusion [0] ⊆ [n] and G : C → C be the `functor which carries
an arbitrary map [m] → [n] to the induced map [0] [m] → [n]. We havenatural transformations of functors
F → G ← idC .
Let X be the topological space | N(C)|. The natural transformations aboveshow that the identity map idX is homotopic to a constant, so that X iscontractible, as desired.
Consequently, if U• is a simplicial object in an ∞-category X and U•s =U• | N(∆ops ) is the associated semisimplicial object, then we can identify colimits of U• with colimits of U•s .
We will say that a simplicial object U• in an ∞-category X is n-coskeletal
if it is a right Kan extension of its restriction to N(∆op≤n ). Similarly, we will
say that a semisimplicial object U• of X is n-coskeletal if it is a right Kan
extension of its restriction to N(∆ops,≤n ), where ∆s,≤n = ∆s ×∆ ∆≤n .

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Lemma 6.5.3.8. Let X be an ∞-category, let U• be a simplicial object ofX, and let U•s = U• | N(∆ops ) the associated semisimplicial object. Then U• isn-coskeletal if and only if U•s is n-coskeletal.
Proof. It will suffice to show that, for each ∆m ∈ ∆, the nerve of the inclusion(∆s )/[m] ×∆s ∆s,≤n ⊆ ∆/[m] ×∆ ∆≤n
is cofinal. Let θ : [m0 ] → [m] be an object of ∆/[m] ×∆ ∆≤n . We let C denotethe category of all factorizations
θ0

θ 00

[m0 ] → [m00 ] → [m]for θ such that θ00 is a monomorphism and m00 ≤ n. According to Theorem
4.1.3.1, it will suffice to prove that N(C) is weakly contractible (for every
choice of θ). We now simply observe that C has an initial object (given by
the unique factorization where θ0 is an epimorphism).
Lemma 6.5.3.9 ([20]). Let X be an ∞-topos and let U• be an n-coskeletal
hypercovering of X. Then U• is effective.
Proof. We will prove this result by induction on n. If n = 0, then U• canbe identified with the underlying groupoid of the Čech nerve of the map
θ : U0 → 1X , where 1X is a final object of X. Since U• is a hypercovering,θ is an effective epimorphism, so the Čech nerve of θ is a colimit diagram
and the desired result follows. Let us therefore assume that n > 0. Let
V• = coskn−1 U• and let f• : U• → V• be the adjunction map. For eachm ≥ 0, the map fm : Um → Vm is a composition of finitely many pullbacksof fn . Since U• is a hypercovering, fn is an effective epimorphism, so each fm
is also an effective epimorphism. We also observe that fm is an equivalence
for m < n.
Let W+ : N(∆+ × ∆)op → X be a Čech nerve of f• (formed in the ∞category X∆ of simplicial objects of X). We observe that W+ | N({∅} × ∆)op
can be identified with V• . Since V• is an (n − 1)-coskeletal hypercovering ofX, the inductive hypothesis implies that any colimit |V• | is a final object of
X. The inclusion N({∅} × ∆)op ⊆ N(∆+ × ∆)op is cofinal (being a productof N(∆)op and the inclusion of a final object into N(∆+ )op ), so we may
identify colimits of W+ with colimits of V• . It follows that any colimit of
W+ is a final object of X. We next observe that each of the augmented
simplicial objects W+ | N(∆+ ×{[m]})op is a Čech nerve of fm and therefore
a colimit diagram (since fm is an effective epimorphism). Applying Lemma
4.3.3.9, we conclude that W+ is a left Kan extension of the bisimplicial
object W = W+ | N(∆ × ∆)op . According to Lemma 4.3.2.7, we can identify
colimits of W+ with colimits of W , so any colimit of W is a final object ofX.
Let D• : N(∆op ) → X be the simplicial object of X obtained by composing
W with the diagonal map δ : N(∆op ) → N(∆ × ∆)op . According to Lemmahighertopoi

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5.5.8.4, δ is cofinal. We may therefore identify colimits of W with colimitsof D• , so that any colimit |D• | of D• is a final object of X.
opss
Let U•s = U• | N(∆ops ) and let D• = D• | N(∆s ). We will prove that U• issa retract of D• in the ∞-category of semisimplicial objects of X. According
to Lemma 6.5.3.7, we can identify colimits of D•s with colimits of D• . It will
follow that any colimit of U•s is a retract of a final object of X and therefore
itself final. Applying Lemma 6.5.3.7 again, we will conclude that any colimitof U• is a final object of X, and the proof will be complete.
We observe that D•s is the result of composing W with the (opposite of
the nerve of the) diagonal functor
δ s : ∆s → ∆ × ∆ .
Similarly, the semisimplicial object U•s is obtained from W via the composition : ∆s ⊆ ∆ ' {[0]} × ∆ ⊆ ∆ × ∆ .
There is an obvious natural transformation of functors δ s →  which yields
a map of semisimplicial objects θ : U•s → D•s . To complete the proof, it will
suffice to show that there exists a map
θ0 : D•s → U•ssuch that θ0 ◦ θ is homotopic to the identity on U•s .
According to Lemma 6.5.3.8, U•s is n-coskeletal as a semisimplicial objectssof X. Let D≤n
and U≤ndenote restrictions of D•s and U•s to N(∆ops,≤n )ss
and θ≤n : U≤n → D≤n be the morphism induced by θ. We have canonicalhomotopy equivalencesss
MapFun(N(∆op(D•s , U•s ) ' MapFun(N(∆op),X) (D≤n , U≤n )s ),X)s,≤nss
MapFun(N(∆op(U•s , U•s ) ' MapFun(N(∆op),X) (U≤n , U≤n ).
s ),X)s,≤n

It will therefore suffice to prove that there exists a map
0
s
s
θ≤n
: D≤n
→ U≤n
0
s
such that θ≤n◦ θ≤n is homotopic to the identity on U≤n.
Consider the functorssδ : ∆s,≤n → ∆+ × ∆ : ∆s,≤n → ∆+ × ∆defined as follows:sδ ([m]) =((∅, [m]) if m < n([n], [n]) if m = n((∅, [m])([m]) =([0], [n])if m < nif m = n.

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We have a commutative diagram of natural transformations
s

/ δs




/ 

δ

which gives rise to a diagram
D≤n o
O
s

ψ≤n

s
D≤n
O
θ≤n

U ≤n ossU≤nin the ∞-category Fun(N(∆ops,≤n ), X). The vertical arrows are equivalences.
Consequently, it will suffice to produce a (homotopy) left inverse to ψ≤n .
sss
For m ≥ 0, let V≤m= V• | ∆s,≤m . We can identify D≤n and U ≤n withs
objects X, Y ∈ X/V≤n−1and ψ≤n with a morphism f : X → Y . To complete
the proof, it will suffice to produce a left inverse to f in the ∞-categorys
X/V≤n−1. We observe that because V• is (n−1)-coskeletal, we have a diagramof trivial fibrationsss→ X/V≤n−1.
X/Vn ← X/V≤n

Using this diagram (and the construction of W ), we conclude that Y can besidentified with a product of (n + 1) copies of X in X/V≤n−1
and that f can be
identified with the identity map. The existence of a left homotopy inverse to
f is now obvious (choose any of the (n + 1)-projections from Y onto X).
Lemma 6.5.3.10. Let X be an ∞-topos and let f• : U• → V• be a naturaltransformation between simplicial objects of X. Suppose that, for each k ≤ n,
the map fk : Uk → Vk is an equivalence. Then the induced map |f• | : |U• | →|V• | of colimits is n-connective.
Proof. Choose a left exact localization functor L : P(C) → X. Without lossof generality, we may suppose that f• = L ◦ f • , where f • : U • → V • is atransformation between simplicial objects of P(C), where f k is an equivalence for k ≤ n. Since L preserves colimits and n-connectivity (Proposition
6.5.1.16), it will suffice to prove that |f• | is n-connective. Using Propositions
6.5.1.12 and 5.5.6.28, we see that |f• | is n-connective if and only if, for eachobject C ∈ C, the induced morphism in S is n-connective. In other words,
we may assume without loss of generality that X = S.
According to Proposition 4.2.4.4, we may assume that f• is obtained bytaking the simplicial nerve of a map f•0 : U•0 → V•0 between simplicial objects
in the ordinary category Kan. Without loss of generality, we may supposethat U•0 and V•0 are projectively cofibrant (as diagrams in the model category
Set∆ ). According to Theorem 4.2.4.1, it will suffice to prove that the induced
map from the (homotopy) colimit of U•0 to the (homotopy) colimit of V•0highertopoi

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has n-connective homotopy fibers, which follows from classical homotopy
theory.
Lemma 6.5.3.11. Let X be an ∞-topos and let U• be a hypercovering of X.
Then the colimit |U• | is ∞-connective.
Proof. We will prove that θ is n-connective for every n ≥ 0. Let V• =coskn+1 U• and let u : U• → V• be the adjunction map. Lemma 6.5.3.10
asserts that the induced map |U• | → |V• | is n-connective, and Lemma 6.5.3.9
asserts that |V• | is a final object of X. It follows that |U• | ∈ X is n-connective,
as desired.
The preceding results lead to an easy characterization of the class of hypercomplete ∞-topoi:
Theorem 6.5.3.12. Let X be an ∞-topos. The following conditions areequivalent:(1) For every X ∈ X, every hypercovering U• of X/X is effective.
(2) The ∞-topos X is hypercomplete.
Proof. Suppose that (1) is satisfied. Let f : U → X be an ∞-connective
morphism in X and let f• be the constant simplicial object of X/X withvalue f . According to Lemma 6.5.3.5, f is a hypercovering of X/X . Invoking(1), we conclude that f ' |f• | is a final object of X/X ; in other words, f isan equivalence. This proves that (1) ⇒ (2).
Conversely, suppose that X is hypercomplete. Let X ∈ X be an object
and U• a hypercovering of X/X . Then Lemma 6.5.3.11 implies that |U• | isan ∞-connective object of X/X . Since X is hypercomplete, we conclude that|U• | is a final object of X/X , so that U• is effective.
Corollary 6.5.3.13 (Dugger-Hollander-Isaksen [20], Toën-Vezzosi [79]). Let
X be an ∞-topos. For each X ∈ X and each hypercovering U• of X/X , let|U• | be the associated morphism of X (which has target X). Let S denote thecollection of all such morphisms |U• |. Then X∧ = S −1 X. In other words, anobject of X is hypercomplete if and only if it is S-local.
Remark 6.5.3.14. One can generalize Corollary 6.5.3.13 as follows: let
L : X → Y be an arbitrary left exact localization of ∞-topoi and let S be
the collection of all morphisms of the form |U• |, where U• is a simplicial
object of X/X such that L ◦ U• is an effective hypercovering of Y/LX . Then
L induces an equivalence S −1 X → Y.
It follows that every ∞-topos can be obtained by starting with an ∞category of presheaves P(C), selecting a collection of augmented simplicial
objects U•+ , and inverting the corresponding maps |U• | → U−1 . The specification of the desired class of augmented simplicial objects can be viewed
as a kind of “generalized topology” on C in which one specifies not only thecovering sieves but also the collection of hypercoverings which are to becomehighertopoi

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effective after localization. It seems plausible that this notion of topology canbe described more directly in terms of the ∞-category C, but we will not
pursue the matter further.
6.5.4 Descent versus Hyperdescent
Let X be a topological space and let U(X) denote the category of open
subsets of X. The category U(X) is equipped with a Grothendieck topology in
S which the covering sieves on U are those sieves {Uα ⊆ U } such thatU = α Uα . We may therefore consider the ∞-topos Shv(N(U(X))), which
we will call the ∞-topos of sheaves on X and denote by Shv(X). In §6.5.2,
we discussed an alternative theory of sheaves on X, which can be obtained
either through Jardine’s local model structure on the category of simplicial presheaves or by passing to the hypercompletion Shv(X)∧ of Shv(X).
According to Theorem 6.5.3.12, Shv(X)∧ is distinguished from Shv(X) inthat objects of Shv(X)∧ are required to satisfy a descent condition for arbitrary hypercoverings of X, while objects of Shv(X) are required to satisfy adescent condition only for ordinary coverings.
Warning 6.5.4.1. We will always use the notation Shv(X) to indicate the∞-category of S-valued sheaves on X rather than the ordinary category of
set-valued sheaves. If we need to indicate the latter, we will denote it by
ShvSet (X).
The ∞-topos Shv(X)∧ seems to have received more attention than Shv(X)in the literature (though there is some discussion of Shv(X) in [20] and [79]).
We would like to make the case that for most purposes, Shv(X) has better
properties. A large part of Chapter 7 will be devoted to justifying some of
the claims made below.
(1) In §6.4.5, we saw that the construction
X 7→ Shv(X)could be interpreted as a right adjoint to the functor which associates toevery ∞-topos Y the underlying locale of subobjects of the final objectof Y. In other words, Shv(X) occupies a universal position among ∞topoi which are related to the original space X.
(2) Suppose we are given a Cartesian square
X0

ψ0

π0


S0

ψ

/X

/S

π

in the category of locally compact topological spaces. In classical sheaf
theory, there is a base change transformationψ ∗ π∗ → π∗0 ψ 0∗

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of functors between the derived categories of (left bounded) complexesof (abelian) sheaves on X and on S 0 . The proper base change theorem
asserts that this transformation is an equivalence whenever the map π
is proper.
∗

The functors ψ ∗ , ψ 0 , π∗ , and π∗0 can be defined on the ∞-topoi Shv(X),
Shv(X 0 ), Shv(S), and Shv(S 0 ), and on their hypercompletions. Moreover, one has a base change mapψ ∗ π∗ → π 0∗ ψ 0∗

in this nonabelian situation as well.
It is natural to ask if the base change transformation is an equivalence when π is proper. It turns out that this is false if we work withhypercomplete ∞-topoi. Let us sketch a counterexample:
Counterexample 6.5.4.2. Let Q denote the Hilbert cube [0, 1] ×
[0, 1] × · · · . For each i, we let Qi ' Q denote “all but the first i”
factors of Q, so that Q = [0, 1]i × Qi .
We construct a sheaf of spaces F on X = Q × [0, 1] as follows. Begin
with the empty stack. Adjoin to it two sections defined over the open
sets [0, 1) × Q1 × [0, 1) and (0, 1] × Q1 × [0, 1). These sections both
restrict to give sections of F over the open set (0, 1) × Q1 × [0, 1).
We next adjoin paths between these sections defined over the smaller
open sets (0, 1) × [0, 1) × Q2 × [0, 21 ) and (0, 1) × (0, 1] × Q2 × [0, 12 ).
These paths are both defined on the smaller open set (0, 1) × (0, 1) ×
Q2 × [0, 21 ), so we next adjoin two homotopies between these paths
over the open sets (0, 1) × (0, 1) × [0, 1) × Q3 × [0, 31 ) and (0, 1) ×
(0, 1) × (0, 1] × Q3 × [0, 13 ). Continuing in this way, we obtain a sheaf
F. On the closed subset Q × {0} ⊂ X, the sheaf F is ∞-connective
by construction, and therefore its hypercompletion admits a global
section. However, the hypercompletion of F does not admit a global
section in any neighborhood of Q×{0} since such a neighborhood must
contain Q × [0, n1 ) for n  0 and the higher homotopies required forthe construction of a section are eventually not globally defined.
However, in the case where π is a proper map, the base change mapψ ∗ π∗ → π∗0 ψ 0∗
is an equivalence of functors from Shv(X) to Shv(S 0 ). One may regard
this fact as a nonabelian generalization of the classical proper basechange theorem. We refer the reader to §7.3 for a precise statement
and proof.
Remark 6.5.4.3. A similar issue arises in classical sheaf theory if
one chooses to work with unbounded complexes. In [73], Spaltenstein
defines a derived category of unbounded complexes of sheaves on X,

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where X is a topological space. Spaltenstein’s definition forces all quasiisomorphisms to become invertible, which is analogous to the procedure of obtaining X∧ from X by inverting the ∞-connective morphisms.
Spaltenstein’s work shows that one can extend the definitions of all of
the basic objects and functors. However, it turns out that the theorems do not all extend: in particular, one does not have the properbase change theorem in Spaltenstein’s setting (Counterexample 6.5.4.2
can be adapted to the setting of complexes of abelian sheaves). The
problem may be rectified by imposing weaker descent conditions which
do not invert all quasi-isomorphisms; we will give a more detailed discussion in [51].
(3) The ∞-topos Shv(X) often has better finiteness properties than the∞-topos Shv(X)∧ . Recall that a topological space X is coherent if thecollection of compact open subsets of X is stable under finite intersections and forms a basis for the topology of X.
Proposition 6.5.4.4. Let X be a coherent topological space. Then the∞-category Shv(X) is compactly generated: that is, Shv(X) is generated under filtered colimits by its compact objects.
Proof. Let Uc (X) be the partially ordered set of compact open subsetsof X, let Pc (X) = P(N(Uc (X))), and let Shvc (X) be the full subcategory of Pc (X) spanned by those presheaves F with the following
properties:(1) The object F(∅) ∈ C is final.
(2) For every pair of compact open sets U, V ⊆ X, the associateddiagram/ F(U )
F(U ∩ V )

F(V )


/ F(U ∪ V )is a pullback.
In §7.3.5, we will prove that the restriction functor Shv(X) → Shvc (X)is an equivalence of ∞-categories (Theorem 7.3.5.2). It will therefore
suffice to prove that Shvc (X) is compactly generated.
Using Lemmas 5.5.4.19, 5.5.4.17, and 5.5.4.18, we deduce that Shvc (X)is an accessible localization of Pc (X). Let X be a compact object of
Pc (X). We observe that X and LX corepresent the same functor on
Shvc (X). Proposition 5.3.3.3 implies that the subcategory Shvc (X) ⊆Pc (X) is stable under filtered colimits in Pc (X). It follows that LX is
a compact object of Shv0 (X). Since Pc (X) is generated under filteredcolimits by its compact objects (Proposition 5.3.5.12), we conclude
that Shvc (X) has the same property.

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It is not possible replace Shv(X) by Shv(X)∧ in the statement of
Proposition 6.5.4.4.
Counterexample 6.5.4.5. Let S = {x, y, z} be a topological spaceconsisting of three points, with topology generated by the open subsets
S + = {x, y} ⊂ S and S − = {x, z} ⊂ S. Let X = S × S × · · · be a
product of infinitely many copies of S. Then X is a coherent topologicalspace. We will show that the global sections functor Γ : Shv(X)∧ →S does not commute with filtered colimits, so that the final objectof Shv(X)∧ is not compact. A more elaborate version of the same
argument shows that Shv(X)∧ contains no compact objects other thanits initial object.
To show that Γ does not commute with filtered colimits, we use avariant on the construction of Counterexample 6.5.4.2. We define asequence of sheaves
F0 → F1 → · · ·as follows. Let F0 be generated by sections
0η+∈ F(S + × S × · · · )0η−∈ F(S − × S × · · · ).

Let F1 be the sheaf obtained from F0 by adjoining paths1η+: ∆1 → F({x} × S + × S × · · · )1
η−: ∆1 → F({x} × S − × S × · · · )00from η+to η−. Similarly, let F2 be obtained from F1 by adjoining
homotopies2η+: (∆1 )2 → F({x} × {x} × S + × S × · · · )2
η−: (∆1 )2 → F({x} × {x} × S − × S × · · · )1
1. Continuing this procedure, we obtain a sequence ofto η−
from η+sheaves

F0 → F1 → F2 → · · ·whose colimit F∞ ∈ Shv(X)∧ admits a section (since we allow descent with respect to hypercoverings). However, none of the individual
sheaves Fn admits a global section.
Remark 6.5.4.6. The analogue of Proposition 6.5.4.4 fails, in general,
if we replace the coherent topological space X by a coherent topos.
For example, we cannot take X to be the topos of étale sheaves on an
algebraic variety. However, it turns out that the analogue Proposition
6.5.4.4 is true for the topos of Nisnevich sheaves on an algebraic variety;
we refer the reader to [51] for details.

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Remark 6.5.4.7. A point of an ∞-topos X is a geometric morphismp∗ : S → X, where S denotes the ∞-category of spaces (which is a
final object of RTop by virtue of Proposition 6.3.4.1). We say that
X has enough points if, for every morphism f : X → Y in X having
the property that p∗ (f ) is an equivalence for every point p of X, f
is itself an equivalence in X. If f is ∞-connective, then every stalkp∗ (f ) is ∞-connective, hence an equivalence by Whitehead’s theorem.
Consequently, if X has enough points, then it is hypercomplete.
In classical topos theory, Deligne’s version of the Gödel completenesstheorem (see [54]) asserts that every coherent topos has enough points.
Counterexample 6.5.4.5 shows that there exist coherent topologicalspaces with Shv(X)∧ 6= Shv(X), so that Shv(X) does not necessarily
have enough points. Consequently, Deligne’s theorem does not hold in
the ∞-categorical context.
(4) Let k be a field and let C denote the category of chain complexes of
k-vector spaces. Via the Dold-Kan correspondence, we may regard C
as a simplicial category. We let Mod(k) = N(C) denote the simplicial
nerve. We will refer to Mod(k) as the ∞-category of k-modules; it is apresentable ∞-category which we will discuss at greater length in [51].
Let X be a compact topological space and choose a functorial injective
resolution
F → I 0 (F) → I 1 (F) → · · ·on the category of sheaves F of k-vector spaces on X. For every open
subset U on X, we let kU denote the constant sheaf on U with valuek, extended by zero to X. Let HBM (U ) = Γ(X, I • (kU ))∨ , the dualof the complex of global sections of the injective resolution I • (kU ).
Then HBM (U ) is a complex of k-vector spaces whose homologies are
precisely the Borel-Moore homology of U with coefficients in k (in otherwords, they are the dual spaces of the compactly supported cohomology
groups of U ). The assignment
U 7→ HBM (U )determines a presheaf on X with values in the ∞-category Mod(k).
In view of the existence of excision exact sequences for Borel-Moore
homology, it is natural to suppose that HBM (U ) is actually a sheaf
on X with values in Mod(k). This is true provided that the notion ofsheaf is suitably interpreted: namely, HBM extends (in an essentiallyunique fashion) to a colimit-preserving functor
φ : Shv(X) → Mod(k)op .
(In other words, the functor U 7→ HBM (U ) determines a Mod(k)valued sheaf on X in the sense of Definition 7.3.3.1.) However, thesheaf HBM is not necessarily hypercomplete in the sense that φ does
not necessarily factor through Shv(X)∧ .

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Counterexample 6.5.4.8. There exists a compact Hausdorff space
X and a hypercovering U• of X such that the natural map HBM (X) →lim HBM (U• ) is not an equivalence. Let X be the Hilbert cube Q =←−[0, 1]×[0, 1]×· · · (more generally, we could take X to be any nonempty
Hilbert cube manifold). It is proven in [15] that every point of X
has arbitrarily small neighborhoods which are homeomorphic to Q ×[0, 1). Consequently, there exists a hypercovering U• of X, where each
Un is a disjoint union of open subsets of X homeomorphic to Q ×[0, 1). The Borel-Moore homology of every Un vanishes; consequently,
lim HBM (U• ) is zero. However, the (degree zero) Borel-Moore homol←−ogy of X itself does not vanish since X is nonempty and compact.
Borel-Moore homology is a very useful tool in the study of a locallycompact space X, and its descent properties (in other words, the existence of various Mayer-Vietoris sequences) are very naturally encodedin the statement that HBM is a k-module in the ∞-topos Shv(X) (inother words, a sheaf on X with values in Mod(k)); however, this kmodule generally does not lie in Shv(X)∧ . We see from this example
that nonhypercomplete sheaves (with values in Mod(k) in this case)on X often arise naturally in the study of infinite-dimensional spaces.
(5) Let X be a topological space and f : Shv(X) → Shv(∗) ' S be thegeometric morphism induced by the projection X → ∗. Let K be a
Kan complex regarded as an object of S. Then π0 f∗ f ∗ K is a naturaldefinition of the sheaf cohomology of X with coefficients in K. If X
is paracompact, then the cohomology set defined above is naturally
isomorphic to the set [X, |K|] of homotopy classes of maps from X into
the geometric realization |K|; we will give a proof of this statement in§7.1. The analogous statement fails if we replace Shv(X) by Shv(X)∧ .
(6) Let X be a topological space. Combining Remark 6.5.2.2 with Proposition 6.5.2.14, we deduce that Shv(X)∧ has enough points and that
Shv(X)∧ = Shv(X) if and only if Shv(X) has enough points. The possible failure of Whitehead’s theorem in Shv(X) may be viewed either
as a bug or a feature. The existence of enough points for Shv(X) is extremely convenient; it allows us to reduce many statements about the∞-topos Shv(X) to statements about the ∞-topos S of spaces where
we can apply classical homotopy theory. On the other hand, if Shv(X)does not have enough points, then there is the possibility that it detects certain global phenomena which cannot be properly understoodby restricting to points. Let us consider an example from geometric
topology. A map f : X → Y of compact metric spaces is called celllike if each fiber Xy = X ×Y {y} has trivial shape (see [18]). This notion
has good formal properties provided that we restrict our attention tometric spaces which are absolute neighborhood retracts. In the generalcase, the theory of cell-like maps can be badly behaved: for example,
a composition of cell-like maps need not be cell-like.

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The language of ∞-topoi provides a convenient formalism for discussing the problem. In §7.3.6, we will introduce the notion of a cell-likemorphism p∗ : X → Y between ∞-topoi. By definition, p∗ is cell-like
if it is proper and if the unit map u : F → p∗ p∗ F is an equivalence
for each F ∈ Y. A cell-like map p : X → Y of compact metric spaces
need not give rise to a cell-like morphism p∗ : Shv(X) → Shv(Y ). The
hypothesis that each fiber Xy has trivial shape ensures that the unitu : F → p∗ p∗ F is an equivalence after passing to stalks at each pointy ∈ Y . This implies only that u is ∞-connective, and in general u neednot be an equivalence.
Remark 6.5.4.9. It is tempting to try to evade the problem describedabove by working instead with the hypercomplete ∞-topoi Shv(X)∧and Shv(Y )∧ . In this case, we can test whether or not u : F → p∗ p∗ F
is an equivalence by passing to stalks. However, since the proper basechange theorem does not hold in the hypercomplete context, the stalk(p∗ p∗ F)y is not generally equivalent to the global sections of p∗ F |Xy .
Thus, we still encounter difficulties if we want to deduce global consequences from information about the individual fibers Xy .
(7) The counterexamples described in this section have one feature in common: the underlying space X is infinite-dimensional. In fact, this isnecessary: if the space X is finite-dimensional (in a suitable sense),
then the ∞-topos Shv(X) is hypercomplete (Corollary 7.2.1.12). This
finite-dimensionality condition on X is satisfied in many of the situations to which the theory of simplicial presheaves is commonly applied,such as the Nisnevich topology on a scheme of finite Krull dimension.

highertopoi

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Chapter Seven
Higher Topos Theory in Topology
In this chapter, we will sketch three applications of the theory of ∞-topoi tothe study of classical topology. We begin in §7.1 by showing that if X is aparacompact topological space, then the ∞-topos Shv(X) of sheaves on X
can be interpreted as a homotopy theory of topological spaces Y equippedwith a map to X. We will deduce, as an application, that if p∗ : Shv(X) →Shv(∗) is the geometric morphism induced by the projection X → ∗, thenthe composition p∗ p∗ is equivalent to the functor
K 7→ K X
from (compactly generated) topological spaces to itself.
Our second application is to the dimension theory of topological spaces.
There are many different notions of dimension for a topological space X,
including the notion of covering dimension (when X is paracompact), Krull
dimension (when X is Noetherian), and cohomological dimension. We willdefine the homotopy dimension of an ∞-topos X, which specializes to thecovering dimension when X = Shv(X) for a paracompact space X and isclosely related to both cohomological dimension and Krull dimension. We
will show that any ∞-topos which is (locally) finite-dimensional is hypercomplete, thereby justifying assertion (7) of §6.5.4. We will conclude by proving
a bound on the homotopy dimension of Shv(X), where X is a Heyting space(see §7.2.4 for a definition); this may be regarded as a generalization ofGrothendieck’s vanishing theorem, which applies to nonabelian cohomology
and to (certain) non-Noetherian spaces X.
Our third application is a generalization of the proper base change theorem.
Suppose we are given a Cartesian diagram
X0
q0

p0

/X
q



p
/Y
Y0
of locally compact topological spaces. There is a natural transformation∗η : p∗ q∗ → q∗0 p0
of functors from the derived category of abelian sheaves on X to the derived
category of abelian sheaves on Y 0 . The proper base change theorem assertsthat η is an isomorphism whenever q is a proper map. In §7.3, we will generalize this statement to allow nonabelian coefficient systems. To give theproof, we will develop a theory of proper morphisms between ∞-topoi, which
is of some interest in itself.

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7.1 PARACOMPACT SPACES
Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group Hn (X; G); we will single out threeof them for discussion here. First of all, we have the singular cohomology
groups Hnsing (X; G), which are defined to be cohomology of a chain complexof G-valued singular cochains on X. An alternative is to regard Hn (•, G) asa representable functor on the homotopy category of topological spaces, sothat Hnrep (X; G) can be identified with the set of homotopy classes of maps
from X into an Eilenberg-MacLane space K(G, n). A third possibility is to
use the sheaf cohomology Hnsheaf (X; G) of X with coefficients in the constantsheaf G on X.
If X is a sufficiently nice space (for example, a CW complex), then thesethree definitions give the same result. In general, however, all three givedifferent answers. The singular cohomology of X is defined using continuous
maps from ∆k into X and is useful only when there is a good supply ofsuch maps. Similarly, the cohomology group Hnrep (X; G) is defined using
continuous maps from X to a simplicial complex and is useful only when
there is a good supply of real-valued functions on X. However, the sheaf
cohomology of X seems to be a good invariant for arbitrary spaces: it has
excellent formal properties and gives sensible answers in situations where
the other definitions break down (such as the étale topology of algebraic
varieties).
We will take the position that the sheaf cohomology of a space X is thecorrect answer in all cases. It is then natural to ask for conditions underwhich the other definitions of cohomology give the same answer. We should
expect this to be true for singular cohomology when there are many continuous functions into X and for Eilenberg-MacLane cohomology when there
are many continuous functions out of X. It seems that the latter class ofspaces is much larger than the former: it includes, for example, all paracompact spaces, and consequently for paracompact spaces one can show thatthe sheaf cohomology Hnsheaf (X; G) coincides with the Eilenberg-MacLane
cohomology Hnrep (X; G). Our goal in this section is to prove a generalizationof the preceding statement to the setting of nonabelian cohomology (Theorem 7.1.0.1 below; see also Theorem 7.1.4.3 for the case where the coefficientsystem G it not assumed to be constant).
As we saw in §6.5.4, we can associate to every topological space X an∞-topos Shv(X) of sheaves (of spaces) on X. Moreover, given a continuous
map p : X → Y of topological spaces, p−1 induces a map from the categoryof open subsets of Y to the category of open subsets of X. Composition with
p−1 induces a geometric morphism p∗ : Shv(X) → Shv(Y ).
Fix a topological space X and let p : X → ∗ denote the projection from X
to a point. Let K be a Kan complex which we may identify with an objectof S ' Shv(∗). Then p∗ K ∈ Shv(X) may be regarded as the constant sheaf
on X having value K and p∗ p∗ K ∈ S as the space of global sections of p∗ K.
Let |K| denote the geometric realization of K (a topological space) and let

highertopoi

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CHAPTER 7[X, |K|] denote the set of homotopy classes of maps from X into |K|. The
main goal of this section is to prove the following:
Theorem 7.1.0.1. If X is paracompact, then there is a canonical bijection
φ : [X, |K|] → π0 (p∗ p∗ K).
Remark 7.1.0.2. In fact, the map φ exists without the assumption that
X is paracompact: the construction in general can be formally reduced tothe paracompact case since the universal example X = |K| is paracompact.
However, in the case where X is not paracompact, the map φ is not necessarily bijective.
Our first step in proving Theorem 7.1.0.1 is to realize the space of maps
from X into |K| as a mapping space in an appropriate simplicial categoryof spaces over X. In §7.1.2, we define this category and endow it with a(simplicial) model structure. We may therefore extract an underlying ∞category N(Top◦/X ).
Our next goal is to construct an equivalence between N(Top◦/X ) and the∞-topos Shv(X) of sheaves of spaces on X (a very similar comparison result
has been obtained by Toën; see [78]). To prove this, we will attempt torealize N(Top◦/X ) as a localization of a certain ∞-category of presheaves.
We will give an explicit description of the relevant localization in §7.1.3 andshow that it is equivalent to N(Top◦/X ) in §7.1.4. In §7.1.5, we will deduce
Theorem 7.1.0.1 as a corollary of this more general comparison result. We
conclude with §7.1.6, in which we apply our results to obtain a reformulationof classical shape theory in the language of ∞-topoi.
7.1.1 Some Point-Set Topology
Let X be a paracompact topological space. In order to prove Theorem 7.1.0.1,
we will need to understand the homotopy theory of presheaves on X. We
then encounter the following technical obstacle: an open subset of a paracompact space need not be paracompact. Because we wish to deal only withparacompact spaces, it will be convenient to restrict our attention to presheaves which are defined only with respect to a particular basis B for X
consisting of paracompact open sets. The existence of a well-behaved basis
is guaranteed by the following result:
Proposition 7.1.1.1. Let X be a paracompact topological space and U an
open subset of X. The following conditions are equivalent:(i) There exists a continuous function f : X → [0, 1] such that U = {x ∈X : f (x) > 0}.
(ii) There exists a sequence of closed subsets {Kn ⊆ X}n≥0
S such that each
Kn+1 contains an open neighborhood of Kn and U = n≥0 Kn .
(iii) S
There exists a sequence of closed subsets {Kn ⊆ X}n≥0 such that U =n≥0 Kn .

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691

HIGHER TOPOS THEORY IN TOPOLOGYLet B denote the collection of all open subsets of X which satisfy these conditions. Then
(1) The elements of B form a basis for the topology of X.
(2) Each element of B is paracompact.
(3) The collection B is stable under finite intersections (in particular, X ∈B).
(4) The empty set ∅ belongs to B.
Remark 7.1.1.2. A subset of X which can be written as a countable unionof closed subsets of X is called an Fσ -subset of X. Consequently, the basis B
for the topology of X appearing in Proposition 7.1.1.1 can be characterized
as the collection of open Fσ -subsets of X.
Remark 7.1.1.3. If the topological space X admits a metric d, then every
open subset U ⊆ X belongs to the basis B of Proposition 7.1.1.1. Indeed,
we may assume without loss of generality that the diameter of X is at most
1 (adjusting the metric if necessary), in which case the function
f (x) = d(x, X − U ) = inf d(x, y)y ∈U/satisfies condition (i).
Proof. We first show that (i) and (ii) are equivalent. If (i) is satisfied, thenthe closed subsets Kn = {x ∈ X : f (x) ≥ n1 } satisfy the demands of (ii).
Suppose next that (ii) is satisfied. For each n ≥ 0, let Gn denote the closureof X − Kn+1 , so that Gn ∩ Kn = ∅. It follows that there exists a continuousfunction fn : X → [0, 1] such that that fn vanishes on Gn and the restrictionof f P
to Kn is the constant function taking the value 1. Then the function
f = n>0 2fnn has the property required by (i).
We now prove that (ii) ⇔ (iii). The
S implication (ii) ⇒ (iii) is obvious.
For the converse, suppose that U = n Kn , where the Kn are closed subsetsof X. We define a new sequence of closed subsets {Kn0 }n≥0 by induction as
follows. Let K00 = K0 . Assuming that Kn0 has already been defined, let V
and W be disjoint open neighborhoods of the closed sets Kn0 ∪ Kn+1 andX − U , respectively (the existence of such neighborhoods follows from theassumption that X is paracompact; in fact, it would suffice to assume that0
X is normal) and define Kn+1
to be the closure of V . It is then easy to seethat the sequence of closed sets {Kn0 }n≥0 satisfies the requirements of (ii).
We now verify properties (1) through (4) of the collection of open sets
B. Assertions (3) and (4) are obvious. To prove (1), consider an arbitrary
point x ∈ X and an open set U containing x. Then the closed sets {x} and
X − U are disjoint, so there exists a continuous function f : X → [0, 1]
supported on U such that f (x) = 1. Then U 0 = {y ∈ X : f (y) > 0} is an
open neighborhood of x contained in U , and U 0 ∈ B.

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It remains to proveS (2). Let U ∈ B; we wish to prove that U is paracompact. Write U = n≥0 Kn , where each Kn is a closed subset of X containing a neighborhood of Kn−1 (by convention, we set Kn = ∅ for n < 0).
Let {Uα } be an open covering of X. Since each Kn is paracompact, we
can choose a locally finite covering {Vα,n } of Kn which refines {Uα ∩ Kn }.
0
Let Vα,ndenote the intersection of Vα,n with the interior of Kn and let
0
Wα,n = Vα,n∩ (X − Kn−2 ). Then {Wα,n } is a locally finite open covering of
X which refines {Uα }.
Let X be a paracompact topological space and let B be the basis constructed in Proposition 7.1.1.1. Then B can be viewed as a category withfinite limits and is equipped with a natural Grothendieck topology. To simplify the notation, we will let Shv(B) denote the ∞-topos Shv(N(B)). Notethat because N(B) is the nerve of a partially ordered set, the ∞-topos Shv(B)is 0-localic. Moreover, the corresponding locale Sub(1) of subobjects of the
final object 1 ∈ Shv(B) is isomorphic to the lattice of open subsets of X.
It follows that the restriction map Shv(X) → Shv(B) is an equivalence of∞-topoi.
Warning 7.1.1.4. Let X be a topological space and B a basis of X regarded
as a partially ordered set with respect to inclusions. Then B inherits a Grothendieck topology, and we can define Shv(B) as above. However, the induced
map Shv(X) → Shv(B) is generally not an equivalence of ∞-categories: thisrequires the assumption that B is stable under finite intersections. In otherwords, a sheaf (of spaces) on X generally cannot be recovered by knowing
its sections on a basis for the topology of X; see Counterexample 6.5.4.8.
7.1.2 Spaces over X
Let X be a topological space with a specified basis B fixed throughout thissection. We wish to study the homotopy theory of spaces over X: that is,
spaces Y equipped with a map p : Y → X. We should emphasize that we do
not wish to assume that the map p is a fibration or that p is equivalent to afibration in any reasonable sense: we are imagining that p encodes a sheaf ofspaces on X, and we do not wish to impose any condition of local triviality
on this sheaf.
Let Top denote the category of topological spaces and Top/X the categoryof topological spaces mapping to X. For each p : Y → X and every open
subset U ⊆ X, we define a simplicial set SingX (Y, U ) by the formula
SingX (Y, U )n = HomX (U × |∆n |, Y ).
Face and degeneracy maps are defined in the obvious way. We note that thesimplicial set SingX (Y, U ) is always a Kan complex. We will simply write
SingX (Y ) to denote the simplicial presheaf on X given by
U 7→ SingX (Y, U ).
Proposition 7.1.2.1. There exists a model structure on the category Top/X
uniquely determined by the following properties:

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HIGHER TOPOS THEORY IN TOPOLOGY(W ) A morphism
/Z

Y
p
q

X

~

is a weak equivalence if and only if, for every U ⊆ X belonging to B, theinduced map SingX (Z, U )• → SingX (Y, U )• is a homotopy equivalenceof Kan complexes.
(F ) A morphism
/Z

Y
p

X

~

q

is a fibration if and only if, for every U ⊆ X belonging to B, the induced
map SingX (Z, U )• → SingX (Y, U )• is a Kan fibration.
Remark 7.1.2.2. The model structure on Top/X described in Proposition
7.1.2.1 depends on the chosen basis B for X and not only on the topologicalspace X itself.
Proof of Proposition 7.1.2.1. The proof uses the theory of cofibrantly generated model categories; we give a sketch and refer the reader to [38] for moredetails. We will say that a morphism Y → Z in Top/X is a cofibration if it
has the left lifting property with respect to every trivial fibration in Top/X .
We begin by observing that a map Y → Z in Top/X is a fibration if
and only if it has the right lifting property with respect to every inclusion
U × Λni ⊆ U × ∆n , where 0 ≤ i ≤ n and U is in B. Let I denote the weakly
saturated class of morphisms in Top/X generated by these inclusions. Using
the small object argument, one can show that every morphism Y → Z in
Top/X admits a factorizationfg

Y → Y 0 → Z,
where f belongs to I and g is a fibration. (Although the objects in Top/X
generally are not small, one can still apply the small object argument since
they are small relative to the class I of morphisms: see [38].)
Similarly, a map Y → Z is a trivial fibration if and only if it has the right
lifting property with respect to every inclusion U ×| ∂ ∆n | ⊆ U ×|∆n |, whereU ∈ B. Let J denote the weakly saturated class of morphisms generated bythese inclusions: then every morphism Y → Z admits a factorizationfg

Y → Y 0 → Z,
where f belongs to J and g is a trivial fibration.
The only nontrivial point to verify is that every morphism which belongs
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category follow formally. Since it is clear that I is contained in J and that
J consists of cofibrations, it suffices to show that every morphism in I isa weak equivalence. To prove this, let us consider the class K of all closed
immersions k : Y → Z in Top/X such that there exist functions λ : Z →[0, ∞) and h : Z × [0, ∞) → Z such that k(Y ) = λ−1 {0}, h(z, 0) = z, andh(z, λ(z)) ∈ k(Y ). Now we make the following observations:(1) Every inclusion U × |Λni | ⊆ U × |∆n | belongs to K.
(2) The class K is weakly saturated; consequently, I ⊆ K.
(3) Every morphism k : Y → Z which belongs to K is a homotopy equivalence in Top/X and is therefore a weak equivalence.

The category Top/X is naturally tensored over simplicial sets if we defineY ⊗ ∆n = Y × |∆n | for Y ∈ Top/X . This induces a simplicial structure on
Top/X which is obviously compatible with the model structure of Proposition
7.1.2.1.
We note that SingX is a (simplicial) functor from Top/X to the category ofsimplicial presheaves on B (here we regard B as a category whose morphisms
op
are given by inclusions of open subsets of X). We regard SetB
∆ as a simplicial
model category via the projective model structure described in §A.3.3. By
construction, SingX preserves fibrations and trivial fibrations. Moreover, thefunctor SingX has a left adjoint
F 7→ |F |X ;we will refer to this left adjoint as geometric realization (in the case where X
is a point, it coincides with the usual geometric realization functor from Set∆
to the category of topological spaces). The functor |F |X is determined by theproperty that |FU |X ' U if FU denotes the presheaf (of sets) represented by
U and the requirement that geometric realization commutes with colimits
and with tensor products by simplicial sets.
We may summarize the situation as follows:
Proposition 7.1.2.3. The adjoint functors (||X , SingX ) determine a simplicial Quillen adjunction between Top/X (with the model structure of Propoop
sition 7.1.2.1) and SetB
(with the projective model structure).
∆
7.1.3 The Sheaf Condition
Let X be a topological space and B a basis for the topology of X whichop
is stable under finite intersections. Let A denote the category SetBof∆simplicial presheaves on B; we regard A as a model category with respect
to the projective model structure defined in §A.3.3. According to Proposition 5.1.1.1, the ∞-category N(A◦ ) associated to A is equivalent to thehighertopoi

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695∞-category P(B) = P(N(B)) of presheaves on B. In particular, the homotopy category hP(N(B)) is equivalent to the homotopy category hA (thecategory obtained from A by formally inverting all weak equivalences ofsimplicial presheaves). The ∞-category Shv(B) is a reflective subcategoryof P(N(B)). Consequently, we may identify the homotopy category hShv(B)with a reflective subcategory of hA. We will say that a simplicial presheaf
F : Bop → Set∆ is a sheaf if it belongs to this reflective subcategory. The
purpose of this section is to obtain an explicit criterion which will allow us
to test whether or not a given simplicial presheaf F : Bop → Set∆ is a sheaf.
Warning 7.1.3.1. The condition that a simplicial presheaf F : Bop → Set∆
be a sheaf, in the sense defined above, is generally unrelated to the condition
that F be a simplicial object in the category of sheaves of sets on X (thoughthese two notions do agree in the special case where the simplicial presheaf
F takes values in constant simplicial sets).
Let j : N(B) → P(B) be the Yoneda embedding. By definition, an object
F ∈ P(B) belongs to Shv(B) if and only if, for every U ∈ B and every
monomorphism i : U 0 → j(U ) which corresponds to a covering sieve U on
U , the induced map
MapP(B) (j(U ), F ) → MapP(B) (U 0 , F )is an isomorphism in the homotopy category H. In order to make this condition explicit in terms of simplicial presheaves, we note that i : U 0 → j(U )can be identified with the inclusion χU ⊆ χU of simplicial presheaves, where(∗ if V ⊆ U
χU (V ) =∅ otherwise.
(∗ if V ∈ U
χU (V ) =∅ otherwise.
However, we encounter a technical issue: in order to extract the correct
space of maps MapP(B) (U 0 , F ), we need to select a projectively cofibrantmodel for U 0 in A. In general, the simplicial presheaf χU defined above is
not projectively cofibrant. To address this problem, we will construct a new
simplicial presheaf, equivalent to χU , which has better mapping properties.
Definition 7.1.3.2. Let U be a linearly ordered set equipped with a map
s : U → B. We define a simplicial presheaf NU : Bop → Set∆ as follows: foreach V ∈ B, let NU (V ) be the nerve of the linearly ordered set {U ∈ U :V ⊆ s(U )}. NU may be viewed as a subobject of the constant presheaf ∆Utaking the value N(U) = ∆U .
Remark 7.1.3.3. The above notation is slightly abusive in that NU depends
not only on U but also on the map s and on the linear ordering of U. If
the map s is injective (as it will be in most applications), we will frequently
simply identify U with its image in B. In practice, U will usually be a covering
sieve on some object U ∈ B.

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Remark 7.1.3.4. The linear ordering of U is unrelated to the partial ordering of B by inclusion. We will write the former as ≤ and the latter as⊆.
Example 7.1.3.5. Let U = ∅. Then NU = ∅.
Example 7.1.3.6. Let U = {U } for some U ∈ B and let s : U → B be theinclusion. Then NU ' χU .
Proposition 7.1.3.7. Let
U
s

?B

p
s0


U0
be a commutative diagram, where p is an order-preserving injection between
linearly ordered sets. Then the induced map NU → NU0 is a projective cofibration of simplicial presheaves.
Proof. Without loss of generality, we may identify U with a linearly ordered
subset of U0 via p. Choose a transfinite sequence of simplicial subsets of N U0
K0 ⊆ K1 ⊆ · · · ,where K0 = N U, Kλ = α<λ Kα if λ is a nonzero limit ordinal, and Kα+1
is obtained from Kα by adjoining a single nondegenerate simplex (if such asimplex exists). For each ordinal α, let Fα ⊆ NU0 be defined byS

Fα (V ) = NU0 (V ) ∩ Kα ⊆ N(U0 ).
Then F0 = NU and Fλ = limα<λ Fα when α is a nonzero limit ordinal, and−→Fα ' NU0 for α  0. It therefore suffices to show that each map Fα → Fα+1
is a projective cofibration. If Kα = Kα+1 , this is clear; otherwise, we may
suppose that Kα+1 is obtained from Kα by adjoining a single nondegenerate
simplex {U0 < U1 < · · · < Un } of N(U0 ). Let U = s0 (U0 ) ∩ · · · ∩ s0 (Un ) ∈ B.
Then there is a coCartesian squareχU ⊗ ∂ ∆n/ χU ⊗ ∆ nFα/ Fα+1 .

The desired result now follows since the upper horizontal arrow is clearly aprojective cofibration.
Corollary 7.1.3.8. Let U be a linearlyopordered set and s : U → B a map.
Then the simplicial presheaf NU ∈ SetB
is projectively cofibrant.
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Note that NU (V ) is contractible if V ⊆ s(U ) for some U ∈ U and empty
otherwise. Consequently, we deduce:
Corollary 7.1.3.9. Let U ⊆ B be a sieve equipped with a linear ordering.
The unique map NU → χU is a weak equivalence of simplicial presheaves.
Notation 7.1.3.10. Let U be a linearly ordered set equipped with a map
s : U → B. For any simplicial presheaf F : Bop → Set∆ , we let F (U) denotethe simplicial set MapA (NU , F ).
Remark 7.1.3.11. Let U ∈ B, let U = {U }, and let s : U → B be theinclusion. Then F (U) = F (U ). In general, we can think of F (U) as a homotopy limit of F (V ) taken over V in the sieve generated by s : U → B. To
give a vertex of F (U), we must give for each U ∈ U a point of F (sU ), forevery pair of objects U, V ∈ U a path between the corresponding points in
F (sU ∩ sV ), and so forth.
Corollary 7.1.3.12. Let F : Bop → Kan be a (projectively fibrant) simplicial presheaf on B. Then F is a sheaf if and only if, for every U ∈ B andevery sieve U that covers U , there exists a linearly ordered set U0 equippedwith a map U0 → U, which generates U as a sieve, such that the induced
map F (U ) → F (U0 ) is a weak homotopy equivalence of simplicial sets.
Lemma 7.1.3.13. Suppose that U ⊆ X is paracompact and let U ⊆ B
be a covering of U . Choose a linear ordering of U. Then the natural map
π : |NU |X → U is a homotopy equivalence in Top/X . (In other words, there
exists a section s : U → NU of π such that s ◦ π is fiberwise homotopic to
the identity.)
Proof. Any partition of unity subordinate to the open cover U gives rise toa section of π. To check that s ◦ π is fiberwise homotopic to the identity, usea “straight-line” homotopy.
Proposition 7.1.3.14. Let X be a topological space and that B a basis forthe topology of X. Assume that B is stable under finite intersections and thateach element of B is paracompact. For every continuous map of topologicalspaces p : Y → X, the simplicial presheaf SingX (Y ) of sections of p is sheaf.
Proof. Let F = SingX (Y ). We note that F is a projectively fibrant simplicial
presheaf on B. By Corollary 7.1.3.12, it suffices to show that for every U ∈ B,
every covering U of U , and every linear ordering on U, the natural map
F (U ) → F (U) is a homotopy equivalence of simplicial sets. In other words,
it suffices to show that composition with the projection π : NU → U induces
a homotopy equivalence
Map/X (U, Y ) → Map/X (|NU |X , Y )of simplicial sets. This follows immediately from Lemma 7.1.3.13.
Remark 7.1.3.15. Under the hypotheses of Proposition 7.1.3.14, the objectof Shv(X) corresponding to the simplicial presheaf SingX (Y ) is not necessarily hypercomplete.

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7.1.4 The Main Result
Suppose that X is a paracompact topological space and B is the basis forthe topology of X described in Proposition 7.1.1.1. Our main goal is to showthat the composition of the adjoint functors
F 7→ SingX |F |X
may be identified with a “sheafification” of F , at least in the case where F
is a projectively cofibrant simplicial presheaf on B.
In proving this, we have some flexibility regarding the choice of F : itwill suffice to treat the question after replacing F by a weakly equivalent
simplicial presheaf F 0 provided that F 0 is also projectively cofibrant. Our
first step is to make a particularly convenient choice for F 0 .
Lemma 7.1.4.1. Let B be a partially ordered set (via ⊆) with a least element∅ and let F : Bop → Set∆ be an arbitrary simplicial presheaf such that F (∅)
is weakly contractible.
There exists a (linearly ordered) set V and a simplicial presheaf F 0 : Bop →Set∆ with the following properties:(1) There exists a monomorphism F 0 → ∆V from F 0 to the (constant)simplicial presheaf ∆V on B taking the value ∆V . (Recall that ∆Vdenotes the nerve of the linearly ordered set V .)(2) For every finite subset V0 ⊆ V , there exists U ∈ B such that U 0 ⊆ U if
and only if ∆V0 ⊆ F 0 (U 0 ) ⊆ ∆V .
(3) As a simplicial presheaf on B, F 0 is projectively cofibrant.
0(4) In the homotopy category of SetB
∆ , F and F are equivalent to one
another.
op

Proof. Without loss of generality, we may suppose that F is (weakly) fibrant.
We now build a “cellular model” of F . More precisely, we construct thefollowing data:(A) A transfinite sequence of simplicial sets
Y0 → Y1 → · · · ,where Yα is defined for all ordinals less than α0 .
(B) For each α < α0 , a subsheaf Fα of the constant presheaf on B taking
the value Yα .
(C) A compatible family of maps Fα → F , so that we may regard {Fα } asop
a functor from the linearly ordered set {α : α < α0 } to (Set∆ )B/F .

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HIGHER TOPOS THEORY IN TOPOLOGY(D) For each α < α0 , there exists U ∈ B, n ≥ 0, and compatible pushout
diagrams/ ∆n∂ ∆n 

limβ<α Yβ −→χU × ∂ ∆n/ χU × ∆nlimβ<α Fβ −→/ Fα ,(∗
χU (W ) =∅if W ⊆ U
otherwise.

/ Yα ,

where(E) The canonical map limβ<α Fβ → F is a weak equivalence in SetB
∆ .
−→
0
op

For every simplicial set K, let K 00 denote the simplicial set described in
Variant 4.2.3.15, so that we have a canonical cofinal map K 00 → K (inparticular, a weak homotopy equivalence) and K 00 is equivalent to the nerveof a partially ordered set. Let Y = limβ<α Yβ . We note that Y 00 can be−→
0
identified with a simplicial complex: that is, a simplicial subset of ∆V forsome linearly ordered set V . For each α, let Fα00 denote the result of applying
the functor K 7→ K 00 termwise. Let F 00 = limβ<α Fβ00 . Finally, we define F 0−→0by the coCartesian square
F 00 (∅) × χ∅/ ∆V × χ ∅


F 00


/ F 0.

The simplicial presheaf F 0 satisfies (1) by construction. Properties (2)and (3) are reasonably clear (in fact, (3) is a formal consequence of (2)).
Condition (4) holds for the simplicial presheaf F 00 as a consequence of (E).
Moreover, the assumption that F (∅) is weakly contractible ensures that (4)remains valid for the pushout F 0 .
Before we can state the next lemma, let us introduce a bit of notation. Let
F : Bop → Set∆ be a simplicial presheaf. Then we let |F | denote the presheafof topological spaces on B obtained by composing F with the geometric realization functor; similarly, if G is a presheaf of topological spaces on B, thenwe let Sing G denote the presheaf of simplicial sets obtained by composing
G with the functor Sing. We note that there is a natural transformation
Sing |F | → SingX |F |X .

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Lemma 7.1.4.2. Let X be a topological space and let B be the collectionof open Fσ subsets of X (see Proposition 7.1.1.1). Let F : Bop → Set∆ bea projectively cofibrant simplicial presheaf which is a sheaf (that is, a fibrantmodel for F satisfies the criterion of Corollary 7.1.3.12). Then the unit map
F → SingX |F |X is an equivalence.
Proof. We note that the functor F 7→ Sing |F | preserves weak equivalencesin F and that the functor F 7→ SingX |F |X preserves weak equivalences between projectively cofibrant presheaves F . Consequently, by Lemma 7.1.4.1,
we may suppose without loss of generality that there is a linearly ordered
set V and that F is a subsheaf of the constant simplicial presheaf taking thevalue ∆V , such that F (∅) = ∆V .
It will be sufficient to prove thatSing |F | → SingX |F |X
is an equivalence: in other words, we wish to show that (Sing |F |)(U ) →(SingX |F |X )(U ) is a homotopy equivalence of Kan complexes for every U ∈B. Replacing X by U , we can reduce to the problem of showing thatp : (Sing |F |)(X) → (SingX |F |X )(X)
is a homotopy equivalence. It now suffices to show that for every inclusion
K 0 ⊆ K of finite simplicial sets (that is, simplicial sets with only finitely
many nondegenerate simplices), a commutative diagram/ (Sing |F |)(X)
K 0 × {0}_
K × {0}g

/ (SingX |F |X )(X)

can be expanded to a commutative diagram`(K 0 × ∆1 ) K 0 ×{1} (K × {1})_/ (Sing |F |)(X)
K × ∆1/ (SingX |F |X )(X).

(In fact, it suffices to treat the case where K 0 ⊆ K is the inclusion ∂ ∆n ⊆∆n ; however, this will result in no simplification of the following arguments.)
Now let B = {Uα }α∈A , where A is a linearly ordered set. Since F isassumed to be a sheaf on B, the equivalent presheaf Sing |F | is also a sheaf.
Consequently, for any covering U ⊆ B (and any linear ordering of U), thenatural map (Sing |F |)(X) → (Sing |F |)(U) is an equivalence. Likewise, by
Proposition 7.1.3.14, the map (SingX |F |X )(X) → (SingX |F |X )(U) is anequivalence. Consequently, it suffices to find a covering U ⊆ B of X and adiagram`/ (Sing |F |)(U)(K 0 × ∆1 )(K × {1})0
K ×{1}

K × ∆1G/ (SingX |F |X )(U)highertopoi

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701

which extends g.
Since K is finite, the map g : K → (SingX |F |X )(X) may be identified
with a continuous fiber-preserving map X × |K| → |F |X , which we will also
denote by g. By assumption, F is a subsheaf of the constant presheaf taking
the value ∆V . We will abuse notation by identifying |F |X with its image
in the product space X × |∆V | (note however that the topology on |F |X
generally does not agree with the subspace topology). We may represent a
point of X × |∆V | by an ordered pair (x, q), where x ∈ X
P and q : V → [0, 1]
has the property that {v ∈ V : q(v) 6= 0} is finite and v∈V q(v) = 1. For
each v ∈ V , we let ∆VX,v denote the open subset of X × |∆v | consisting of allpairs (x, q) such that q(v) > 0; note that the sets {∆VX,v }v∈V form an open
cover of ∆VX . Consequently, the open sets {g −1 ∆VX,v }v∈V form an open coverof X × |K|. Let x be a point of X. The compactness of |K| implies that there
is a finite subset V0 ⊆ V , an open neighborhood Ux of X containing x, andan open covering {Wx,v : v ∈ V0 } of |K|, such that g(Ux × Wx,v ) ⊆ ∆VX,v .
Choose a partition of unity subordinate to the covering {Wx,v }, thereby
determining a map fx : |K| → |∆V0 |. The open sets {Ux } cover X; since X
is paracompact, this covering has a locally finite refinement. Shrinking theUx if necessary, we may suppose that this refinement is given by {Ux }x∈X0
and that each Ux belongs to B. Let U = {Ux }x∈X0 and choose a linear
ordering of U.
We now define a new map g 0 : K → (Sing |F |)(U). To do so, we must give,for every finite U0 = {Ux0 < · · · < Uxn } ⊆ U, a map
0gU: |∆{x0 ,...,xn } | × |K| → |F (Ux0 ∩ · · · ∩ Uxn )| ⊆ |∆V |;0

moreover, these maps are required to satisfy some obvious compatibilities.
0
Define gU
by the formula
0
X
X
0
gU
(
λi xi , z) =λi fxi (z).
00
It is clear that gU
is well-defined as a map from |∆{x0 ,...,xn } | × |K| to |∆V |.
0
We claim that, in fact, this map factors through |F (Ux0 ∩ · · · ∩ Uxn )|. Let z ∈|K| and consider the set V 0 = {v ∈ V : (∃0 ≤ i ≤ n)[fxi (z)(v) 6= 0]} ⊆ V .
Condition (2) of Lemma 7.1.4.1 ensures that there exists U ∈ B such that0
∆V ⊆ F (U 0 ) if and only if U 0 ⊆ U . We note that, for each y ∈ Ux0 ∩· · ·∩Uxn ,we have g(y, z)(v) 6= 0 for v ∈ V 0 ; it follows that y ∈ U . Consequently, we
0deduce that Uα0 ∩ · · · ∩ Uαn ⊆ U , so that ∆V ⊆ F (Ux0 ∩ · · · ∩ Uxn ). It0
U0follows that gU0 |{z} × |∆ | factors through |F (Ux0 ∩ . . . ∩ Uxn )|. Since this0holds for every z ∈ |K|, it follows that gA
is well-defined; evidently these0maps are compatible with one another and give the desired map g 0 : K →(Sing |F |)(U).
We now observe that the composite mapsg0

K → (Sing |F |)(U) → (SingX |F |X )(U)g

K → (SingX |F |X )(X) → (SingX |F |X )(U)highertopoi

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are homotopic via a “straight-line” homotopy
G : K × ∆1 → (SingX |F |X )(U),
which has the desired properties.
Now the hard work is done, and we are ready to enjoy the fruits of ourlabors.
Theorem 7.1.4.3. Let X be a paracompact topological space and B thecollection of open Fσ subsets of X (see Proposition 7.1.1.1). Then, for any
projectively cofibrant F : Bop → Set∆ , the natural map
F → SingX |F |X
exhibits SingX |F |X as a sheafification of F .
Proof. Let hTop/X be the homotopy category of the model category Top/X(the category obtained by inverting all of the weak equivalences definedopin Proposition 7.1.2.1) and hSetB
the homotopy category of the category∆of simplicial presheaves on B. It follows from Proposition 7.1.2.3 that theadjoint functors SingX and ||X induce adjoint functors
op
hSetB
∆ o

||L
X

/ hTop

SingX

/X .

Here ||L
X denotes the left derived functor of the geometric realization (since
every object of Top/X is fibrant, SingX may be identified with its rightderived functor).
We first claim that for any Y ∈ Top/X , the counit map | SingX Y |L
X →Y
is a weak equivalence. To see this, choose a projectively cofibrant model
F → SingX Y for SingX Y ; we wish to show that the induced map |F |X → Y
is a weak equivalence. By definition, this is equivalent to the assertion thatSingX |F |X → SingX Y is a weak equivalence. But we have a commutative
triangle/ SingX Y

F$
x
SingX |F |X ,where the left diagonal map is a weak equivalence by Lemma 7.1.4.2 andProposition 7.1.3.14 and the top horizontal map is a weak equivalence byconstruction; the desired result now follows from the two-out-of-three property.
It follows that we may identify hTop/X with a full subcategory C ⊆op
hSetB∆ . By Proposition 7.1.3.14, the objects of this subcategory are sheaves
on B; by Lemma 7.1.4.2, every sheaf on B is equivalent to SingX Y for an
appropriately chosen Y ; thus C consists of precisely the sheaves on B.

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The composite functor F 7→ SingX |F |L
X may be identified with a localBop
ization functor from hSet∆
to the subcategory C. In particular, when F
is projectively cofibrant, the unit of the adjunction F → SingX |F |X is alocalization of F .
Corollary 7.1.4.4. Under the hypotheses of Theorem 7.1.4.3, the functor
SingX induces an equivalence of ∞-categories
N(Top◦/X ) → Shv(X).
In particular, the ∞-category N(Top◦/X ) is an ∞-topos.
Remark 7.1.4.5. In the language of model categories, we may interpret
Corollary 7.1.4.4 as asserting that SingX and ||X furnish a Quillen equivalence between Top/X (with the model structure of Proposition 7.1.2.1) andop
SetB
where the latter is equipped with the following localization of the∆
projective model structure:
is a cofibration if it is a projective cofibration(1) A map F → F 0 in SetB∆
(in the sense of Definition A.3.3.1).
opis a weak equivalence if it induces an equiv(2) A map F → F 0 in SetB∆
alence in the ∞-category Shv(X).
op

7.1.5 Base ChangeWith Corollary 7.1.4.4 in hand, we are almost ready to deduce Theorem
7.1.0.1. Suppose we are given a paracompact space X and let B denote thecollection of all open Fσ subsets of X. Let p : Shv(X) → Shv(∗) ' S be thegeometric morphism induced by the projection X → ∗.
For any simplicial set K, let FK denotethe constant simplicial presheaf on
opwith the localized model structure
B taking the value K. If we endow SetB∆
of Remark 7.1.4.5, then FK is a model for the sheaf p∗ K. Consequently, thespace p∗ p∗ K may be identified up to homotopy with the mapping space
MapShv(X) (F∗ , FK )which, by virtue of Corollary 7.1.4.4, is equivalent to
MapTop/X (X, X ⊗ K) = (SingX (X ⊗ K))(X).
However, at this point a technical wrinkle appears: X ⊗K agrees with X×|K|
as a set, but it is equipped with a finer topology (given by the direct limit of
the product topologies X × |K0 |, where K0 ⊆ K is a finite simplicial subset).
In general, we have only an inclusion of simplicial presheaves
η : SingX (X ⊗ K) ⊆ SingX (X × |K|),
which need not be an isomorphism. However, we will complete the proof of
Theorem 7.1.0.1 by showing that η is an equivalence of simplicial presheaves.
We consider a slightly more general situation. Let p : X → Y be a continuous map between paracompact spaces and let BX and BY denote thehighertopoi

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collections of open Fσ subsets in X and Y , respectively. Note that the inverse
image along p determines a map q : BY → BX . Composition with q induces aBop
Bop
pushforward functor q∗ : Set∆X → Set∆Y , which has a left adjoint which wewill denote by q ∗ . Similarly, there is a pullback functor p∗ : Top/Y → Top/X ;however, p∗ generally does not possess a right adjoint. Consider the square
Bop

Set∆Y


||Y

/ Top/Y

q∗

Bop

Set∆X

p∗
||X


/ Top/X .

This square is lax commutative in the sense that there exists a natural transformation of functors
ηF : |q ∗ F |X → p∗ |F |Y = |F |Y ×Y X.
The map ηF is always a bijection of topological spaces but is generally nota homeomorphism. Nevertheless, we have the following:
Proposition 7.1.5.1. Under the hypotheses above, if F : Bop
Y → Set∆ is a
projectively cofibrant simplicial presheaf on Y , then the map ηF : |q ∗ F |X →|F |Y ×Y X is a weak equivalence in Top/X .
The proof is based on the following lemma:
Lemma 7.1.5.2. Let Y be a paracompact topological space and let B be
the collection of open Fσ subsets of Y (see Proposition 7.1.1.1). Let V be a
linearly ordered set. Suppose that for every nonempty finite subset V0 ⊆ V ,we are given a basic open set U (V0 ) ∈ B satisfying the following conditions:(a) If V0 ⊆ V1 , then U (V1 ) ⊆ U (V0 ).
(b) The open set U (∅) concides with X.
Let F : Bop → Set∆ be the simplicial presheaf which assigns to each U ∈ B
the simplicial subset F (U ) ⊆ ∆V spanned by those nondegenerate simplices
σ corresponding to finite subsets V0 ⊆ V such that U ⊆ U (V0 ) (see Lemma
7.1.4.1).
For every object X ∈ Top/Y , an n-simplex τ of MapTop/Y (Y, |F |Y ) determines a map of topological spaces from X × |∆n | to |∆V |, which in turn
determines a collection of maps φv : X × |∆n | → [0, 1] such that for everyx ∈ X × |∆n |, the sum Σv∈V φv (x) is equal to 1. Let Map0Top/Y (X, |F |Y ) denote the simplicial subset of MapTop/Y (X, |F |Y ) spanned by those simplices
τ which satisfy the following condition, where K = ∆n :(∗) There exists a locally finite collection of open sets {Uv ⊆ X × |K|}v∈Vsuch that
T each Uv contains the closure of the support of the function
φv and v∈V0 Uv is contained in the inverse image of U (V0 ) for every
finite subset V0 ⊆ V .

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If the topological space X is paracompact, then the inclusion
i : Map0Top/Y (X, |F |Y ) ⊆ MapTop/Y (X, |F |Y )is a homotopy equivalence of Kan complexes.
Proof. Note that for any finite simplicial set K, we can identify
HomSet∆ (K, Map0Top/Y (X, |F |Y )with the set of all collections of continuous maps {φv : X × |K| → [0, 1]}satisfying the condition (∗). Composition with a retraction of |∆n | onto ahorn |Λni | determines a section of the restriction map
HomSet∆ (∆n , Map0Top/Y (X, |F |Y )) → HomSet∆ (Λni , Map0Top/Y (X, |F |Y )),from which it follows that Map0Top/Y (X, |F |Y ) is a Kan complex.
To prove that i is a homotopy equivalence, we argue as in the proof of
Lemma 7.1.4.2: it will suffice to show that for every inclusion K 0 ⊆ K offinite simplicial sets, every commutative diagram/ Map0TopK 0 × {0}_
K × {0}g/ MapTop/Y

can be expanded to a commutative diagram`(K 0 × ∆1 ) K 0 ×{1} (K × {1})_
K × ∆1(X, |F |Y )

/Y

G



(X, |F |Y )/ Map0Top/Y(X, |F |Y )

(X,
|F |Y ).
/Y

/ MapTopThe map g is classified by a collection of continuous maps {gv : X × |K| →[0, 1]}v∈V such that Σv∈V gv (x) = 1. Let {Uv }v∈V be a collection of open
subsets of X × |K 0 | satisfying condition (∗) for the functions {gv |X × |K 0 |}.
For each v ∈ V , let Wv = {x ∈ X × |K| : gv (x) 6= 0}. Choose a locally finite
open covering {Uv0 }v∈V of X × |K| which refines {Wv }. Let {gv0 }v∈V be apartition of unity such that the closure of the support of each gv0 is containedin Uv0 . We define maps {Gv : X × |K| × [0, 1] → [0, 1]}v∈V by the formula((2t)gv0 (x) + (1 − 2t)gv (x) if t ≤ 21
Gv (x, t) =gv0 (x)
if t ≥ 12 .
Then the maps {Gv } determine a continuous map G : X ×|K|×[0, 1] → |F |Y ,which we can identify with a map of simplicial sets
K × ∆1 → MapTop/Y (X, |F |Y ).
`The restriction of this map to (K 0 × ∆1 ) K 0 ×{1} (K × {1}) factors through
Map0Top/Y (X, |F |Y ) since the open subsets {(Uv × [0, 1]) ∪ (Uv0 × {1}} satisfy
condition (∗).

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Remark 7.1.5.3. In the situation of Lemma 7.1.5.2, suppose we are given
a Kan complex
Map1Top/Y (X, |F |Y ) ⊆ MapTop/Y (X, |F |Y )which contains Map0Top/Y (X, |F |Y ) and is closed under the formation of“straight-line” homotopies. More precisely, suppose that any map G : ∆n ×
∆1 → MapTop/Y (X, |F |Y ) factors through Map1Top/Y (X, |F |Y ) provided thatit satisfies the properties listed below:(i) The map G is classified by a collection of continuous functions {Gv :X × |∆n | × [0, 1] → [0, 1]}v∈V .
(ii) Each Gv can be described by the formula((2t)gv0 (x) + (1 − 2t)gv (x)Gv (x, t) =gv0 (x)

if t ≤ 21
if t ≥ 12 .

(iii) The closure of the support of each gv0 is contained in the open set{x ∈ X × |∆n | : gv (x) 6= 0}.
(iv) The restriction of G to ∆n × {0} belongs to Map1Top/Y (X, |F |Y ). By
virtue of (iii), this implies that G|∆n × {1} factors through the inclusion
Map0Top/Y (X, |F |Y ) ⊆ Map1Top/Y (X, |F |Y ).
Then the proof of Lemma 7.1.5.2 shows that the inclusions
Map0Top/Y (X, |F |Y ) ⊆ Map1Top/Y (X, |F |Y ) ⊆ MapTop/Y (X, |F |Y )are homotopy equivalences.
Proof of Proposition 7.1.5.1. Suppose we are given a weak equivalence F →F 0 between projectively cofibrant simplicial presheaves F, F 0 : Bop
Y → Set∆ .
Both q ∗ and ||X are left Quillen functors and therefore preserve weak equivalences between cofibrant objects; it follows that |q ∗ F |X → |q ∗ F 0 |X is a weakequivalence. Similarly, |F |Y → |F 0 |Y is a weak equivalence between cofibrant objects of Top/Y . Since every object of Top/Y is fibrant, we conclude
that |F |Y → |F 0 |Y is a homotopy equivalence in Top/Y ; thus |F |Y ×Y X →|F 0 |Y ×Y X is a homotopy equivalence in Top/X . Consequently, we deducethat ηF is a weak equivalence if and only if ηF 0 is a weak equivalence.
Let F be an arbitrary projectively cofibrant simplicial presheaf; we wish
to show that ηF is a weak equivalence. There exists a trivial projective
cofibration F → F 0 , where F 0 is projectively fibrant. It now suffices to showthat ηF 0 is a weak equivalence. Replacing F by F 0 , we reduce to the casewhere F is projectively fibrant.
Let F 0 be a simplicial presheaf on BY satisfying the conditions of Lemma
7.1.4.1. Then F 0 and F are equivalent in the homotopy category of simplicial

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presheaves on BY . Since F 0 is projectively cofibrant and F is projectively
fibrant, there exists a weak equivalence F 0 → F . We may therefore once
again reduce to proving that ηF 0 is a weak equivalence. Replacing F by F 0 ,we may suppose that F satisfies the conditions of Lemma 7.1.4.1, for somelinearly ordered set V .
For each U ∈ BY , we have a commutative diagram of Kan complexes
Map0Top/X (U, |q ∗ F |X )

MapTop/X (U, |q ∗ F |X )φ0

/ Map0Top(U, |F |Y )/Y

φ/ MapTop/Y

(U, |F |Y )where the vertical maps are defined as in Lemma 7.1.5.2. We wish to showthat φ is a homotopy equivalence. This follows from the observation that
φ0 is an isomorphism, and the vertical arrows are homotopy equivalences byLemma 7.1.5.2.

Theorem 7.1.0.1 now follows immediately from Proposition 7.1.5.1 appliedin the case where Y = ∗ and F is the constant simplicial presheaf BX → Set∆
taking the value K.
Remark 7.1.5.4. There is another solution to the technical difficulty presented by the fact that the bijection X ⊗ K → X × |K| is not necessarily
a homeomorphism: one can work in a suitable category of compactly generated topological spaces where the base change functor Z 7→ X ×Y Z has aright adjoint and therefore automatically commutes with all colimits. Thisis perhaps a more conceptually satisfying approach; however, it leads to a
proof of Theorem 7.1.0.1 only in the special case where the space X is itself
compactly generated.
We close this section by describing a few applications of Proposition 7.1.5.1
and its proof to the theory of sheaves (of spaces) on a paracompact topological space X.
Corollary 7.1.5.5. Let X be a paracompact topological space, Y a closed
subset of X, and i : Y → X the inclusion map. Let F be an object of Shv(X)and let η0 be a global section of i∗ F. Then there exists an open subset U of X
which contains Y and a section η ∈ F(U ) whose image under the restriction
map F(U ) → (i∗ F)(U ∩ Y ) = (i∗ F)(Y ) lies in the path component of η0 .
Proof. Let B denote the collection of open Fσ subsets of X. Without loss ofgenerality, we may assume that F is represented by a projectively cofibrant
simplicial presheaf F ⊆ ∆V satisfying the conditions of Lemma 7.1.4.1,
where V is a linearly ordered set. Using Proposition 7.1.5.1, Corollary 7.1.4.4,
and Lemma 7.1.5.2, it will suffice to prove the following assertion:

highertopoi

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CHAPTER 7(a) Every vertex η0 of Map0Top/X (Y, |F |X ) can be lifted to a vertex of
Map0Top/X (U, |F |X ) for some sufficiently small paracompact neighborhood U of Y .
To prove (a), suppose we are given a vertex of η0 ∈ Map0Top/X (Y, |F |X )corresponding to a collection of functions {φv : Y → [0, 1]}. Since η0 belongs
to Map0Top/X (Y, |F |X ), there exist open sets Uv ⊆ Y satisfying condition (∗)
of Lemma 7.1.5.2.
For each y ∈ Y , there exists an open neighborhoodSWy of y in X for which
the set V (y) = {v ∈ V : Wy ∩ Uv } is finite. Let W = y∈Y Wy . Shrinking W
if necessary, we may suppose that W is a paracompact open neighborhood of
Y in X. Since W is paracompact, there exists a locally finite open covering{Wα0 }α of W , so that for each indexSα there exists a point yα ∈ Y suchthat Wα0 ⊆ Wy . For v ∈ V , let Uv0 = v∈V (yα ) Wα0 . The open sets Uv0 form
a locally finite open covering of W , and each intersection Uv0 ∩ Y is an open
subset of Uv which contains the closure of the support of φv .
For each v ∈ V , choose a continuous function φ0v : X → [0, 1] such that0φv |Y = φv and the closure of the support of φ0v is contained in Uv0 . Thereexists another open set Uv00 whose closure is contained in Uv0 which again
contains the closure of the support of φ0v . For every finite subset V0 ⊆ V , letT00
KV0 denote the intersection v∈V0 U v and let KV00 denote the open subset
of KV0 given by the inverse image of the open set U (V0 ) ⊆ X (the largest
open subset for which ∆V0 belongs to F (U (V0 )) ⊆ ∆V ). Then {KV0 − KV00 }is a locally finite
S collection of closed subsets of X, none of which intersects
Y . Let K = V0 (KV0 − KV00 ); then K is a closed subset of X. Let W 0 be an
open Fσ -subset of W which contains Y and does not intersect K. Replacing
X by W 0 , we may assume that W = X and that K = ∅.
Since the collection of functions {φ0v }v∈V has locally finite support, thefunction φ0 = Σv∈V φ0v is well-defined and takes the value 1 on Y . The open
set {x ∈ X : φ0 (x) > 0} is a paracompact open subset of X (Proposition 7.1.1.1). Shrinking X further, we may suppose that φ0 is everywhere
nonzero on X. Set φ00v = φφv for each v ∈ V . Then the functions φ00v determine
a vertex η ∈ MapTop/X (X, |F |X ). Moreover, the open sets {Uv00 } satisfy condition (∗) appearing in the statement of Lemma 7.1.5.2, so that η belongs
to Map0Top/X (X, |F |X ), as desired.
Corollary 7.1.5.5 admits the following refinement:
Corollary 7.1.5.6. Let X be a paracompact topological space, Y a closed
subset of X, and i : Y → X the inclusion map. Let F be an object of Shv(X).
Then the canonical map
αF : lim F(U ) → lim (i∗ F)(U ∩ Y ) ' (i∗ F)(Y )
−→
−→
Y ⊆U

Y ⊆U

is a homotopy equivalence. Here the colimit is taken over the filtered partiallyordered set of all open subsets of X which contain Y .

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Proof. We will prove by induction on n ≥ 0 that the map αF is n-connective.
The case n = 0 follows from Corollary 7.1.5.5. Suppose that n > 0. We mustshow that, for every pair of points η, η 0 ∈ limY ⊆U F(U ), the induced map of−→fiber products
0αF: ∗ ×limF(U ) ∗ → ∗ ×(i∗ F)(Y ) ∗−→Y ⊆U

is (n − 1)-connective. Without loss of generality, we may assume that η andη 0 arise from sections of F over some U ⊆ X containing Y . Shrinking U if
necessary, we may assume that U is paracompact. Replacing X by U , we
may assume that η and η 0 arise from global sections f, f 0 : 1 → F, where 1
denotes the final object of Shv(X). Let G = 1 ×F 1 ∈ Shv(X). Using the left
0
exactness of i∗ and Proposition 5.3.3.3, we can identify αF
with αG . We now
invoke the inductive hypothesis to deduce that αG is (n − 1)-connective, as
desired.
Lemma 7.1.5.7. Let Y be a paracompact topological and B the collection ofopen Fσ subsets of Y (see Proposition 7.1.1.1). Let V be a linearly ordered
set and let F : Bop → Set∆ be as in the statement of Lemma 7.1.5.2. Suppose
we are given a paracompact space X ∈ Top/Y and a closed subspace X 0 ⊆ X.
Then the map
Map0Top/Y (X, |F |Y ) → Map0Top/Y (X 0 , |F |Y )is a Kan fibration.
Proof. We must show that every lifting problem of the form
Λni_

/ Map0Top∆n/ Map0Top/Y(X, |F |Y )0(X
, |F |Y )/Y

m−1
admits a solution. Since the pair (|∆m |, |Λm|×i |) is homeomorphic to (|∆m−1
m−1[0, 1], |∆| × {0}), we can replace X by X × |∆| and thereby reduceto the case m = 1. `Let Z = (X×{0}) X 0 ×{0} (X 0 ×[0, 1]) and let η0 ∈ Map0Top/Y (Z, |F |Y ); wewish to show that η0 can be lifted to a point in Map0Top/Y (X × [0, 1], |F |Y ).
The proof of Corollary 7.1.5.5 shows that we can lift η0 to a point η1 ∈Map0Top/Y (U, |F |Y ) for some open set U ⊆ X × [0, 1] containing Z.
For each x ∈ X, there exists a real number x > 0 and an open neighborhood Vx ⊆ X such that Vx × [0, x ) ⊆ U . Since X is paracompact, the open
covering {Vx }x∈X admits a locally finite refinement {Wα }, so that for eachindex α there exists a point x(α) ∈ X such that Wα ⊆ Vx(α) . Let {φα } be apartition of unity subordinate to the covering Wα and letψ = Σα x(α) φα .

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Since the interval [0, 1] is compact, there exists an open neighborhood V ⊆ X
containing X 0 such that V ×[0, 1] ⊆ U . Choose a function ψ 0 : X → [0, 1] suchthat ψ 0 |(X − V ) = ψ|(X − V ) and ψ 0 |X 0 is equal to 1. Set K = {(x, t) ∈ X ×[0, 1] : t ≤ φ(x)}, so that Z ⊆ K ⊆ U , and let η2 ∈ Map0Top/Y (K, |F |X ) be
the restriction of η1 . Since K is a retract of X × [0, 1] in the category Top/Y ,we can lift η2 to a point η ∈ Map0Top/Y (X × [0, 1], |F |X ), as desired.
Proposition 7.1.5.8. Let X be a paracompact topological space. Suppose
we are given a sequence of closed subspaces
X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ X
with the following properties:
S(1) The union Xi coincides with X.
(2) A subset U ⊆ X is open if and only if each of the intersections U ∩ Xi
is an open subset of Xi .
Then the induced diagram
Shv(X0 ) → Shv(X1 ) → · · · → Shv(X)exhibits Shv(X) as the colimit of the sequence {Shv(Xi )}i≥0 in the ∞category RTop of ∞-topoi.
Remark 7.1.5.9. Hypotheses (1) and (2) of Proposition 7.1.5.8 can besummarized by saying that X is the direct limit of the sequence {Xi } in the
category of topological spaces. It follows from this condition that for any
locally compact space Y , the product X × Y is also the direct limit of thesequence {Xi × Y }. To prove this, we observe that for any topological space
Z we have bijections
Hom(X × Y, Z) ' Hom(X, Z Y ) ' lim Hom(Xi , Z Y ) ' lim Hom(Xi × Y, Z),←−←−
Y
where Z is endowed with the compact-open topology. In particular, we
deduce that X × ∆n is the direct limit of the topological spaces Xi × ∆n foreach n ≥ 0.
Proof. For each nonnegative integer n, let i(n) denote the inclusion fromXn to Xn+1 and j(n) the inclusion of Xn into X. These functors induce
geometric morphisms
Shv(Xn+1 ) o

Shv(X) oi(n)∗/ Shv(X )
n

i(n)∗

j(n)∗
j(n)∗

/ Shv(X ) .
n

Let C denote a homotopy inverse limit of the tower of ∞-categories∗···/ Shv(X2 ) i(1) / Shv(X1 )i(0)∗/ Shv(X0 ).

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HIGHER TOPOS THEORY IN TOPOLOGY

In view of Proposition 6.3.2.3, we can also identify C with the direct limit of
the sequence {Shv(Xi )}i≥0 in RTop. The maps j(n) determine a geometricmorphism
Shv(X) o

j∗
j∗

/C.

To complete the proof, it will suffice to show that the functor j ∗ is an equivalence of ∞-categories.
We first show that the unit map u : idShv(X) → j∗ j ∗ is an equivalence offunctors. Let F ∈ Shv(X); we wish to show that the map
uF : F → j∗ j ∗ F ' lim j(n)∗ j(n)∗ F←−
is an equivalence in Shv(X). It will suffice to prove the analogous assertion
after evaluating both sides on every open Fσ subset U ⊆ X. Replacing X
by U , we are reduced to proving that the induced map
αF : F(X) → (j∗ j ∗ F)(X) ' lim(j(n)∗ F)(Xn )←−is a homotopy equivalence. Let B be the collection of all open Fσ subsetsof X. Without loss of generality, we may assume that F is represented by aprojectively cofibrant simplicial presheaf F : Bop → S satisfying the conditions of Lemma 7.1.4.1. Using Theorem 7.1.4.3, Corollary 7.1.4.4, andProposition 7.1.5.1, we can identify F(X) with the Kan complex of sece and each (j(n)∗ F)(Xn ) with the Kan complex
tions K = Map0Top/X (X, X)
e It will therefore suffice to show thatof sections K(n) = MapTop (Xn , X).
/X

the canonical map K → lim K(n) exhibits K as a homotopy inverse limit of←−
the tower {K(n)}.
It follows from Remark 7.1.5.9 that the map K → lim K(n) is an isomor←−e ⊆phism of simplicial sets. For each n ≥ 0, let K(n)0 = Map0Top/X (Xn , X)
K(n) (with notation as in Lemma 7.1.5.2) and let K 0 = lim K(n)0 ⊆ K.
←−
Lemma 7.1.5.2 implies that each inclusion K(n)0 ⊆ K(n) is a homotopy
equivalence. Lemma 7.1.5.7 implies that the restriction maps K(n + 1)0 →K(n)0 are Kan fibrations. It follows that the inverse limit K 0 of the tower{K(n)0 } is a Kan complex and that the map K 0 ' lim{K(n)0 } exhibits←−
K 0 as the homotopy inverse limit of {K(n)0 }. Invoking Remark 7.1.5.3, wededuce that the inclusion K 0 ⊆ K is a homotopy equivalence, so that theequivalent diagram K ' lim{K(n)} exhibits K as a homotopy inverse limit←−of {K(n)}, as desired.
We now argue that the counit map v : j ∗ j∗ → id is an equivalence offunctors. Unwinding the definitions, we must prove the following: given acollection of sheaves Fn ∈ Shv(Xn ) and equivalences Fn ' i(n)∗ Fn+1 , thecanonical mapj(n)∗ (lim j(n + k)∗ Fn+k ) → Fn←−
is an equivalence of sheaves on Xn for each n ≥ 0. It will suffice to show that
this map induces a homotopy equivalence after passing to the global sections

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over every open Fσ subset U ⊆ Xn . There exists a function φ0 : Xn → [0, 1]
such that U = {x ∈ Xn : φ0 (x) > 0}. Choose a map φ : X → [0, 1] such thatφ0 = φ|Xn . Replacing X by the paracompact open subset {x ∈ X : φ(x) >0}, we can reduce to the case where U = Xn .
We will prove by induction on k that, for any compatible collection ofsheaves {Fn ∈ Shv(Xn ), Fn ' i(n)∗ Fn+1 }, the mapψ : (j(n)∗ F)(Xn ) → Fn (Xn )is k-connective, where F = lim j(m)∗ Fm . If k > 0, then it will suffice to←−show that for any pair of points η, η 0 ∈ (j(n)∗ F)(Xn ), the induced map
ψ 0 : ∗ ×(j(n)∗ F)(Xn ) ∗ → ∗ ×Fn (Xn ) ∗is (k − 1)-connective. Using Corollary 7.1.5.5, we may assume that η andη 0 arise from sections η, η 0 ∈ F(U ) for some open neighborhood U of Xn .
Shrinking U if necessary, we may assume that U is paracompact. Replacing
X by U , we may assume that η and η 0 are global sections of F. Since j(n)∗is left exact, we can identify ψ 0 with the mapj(n)∗ (∗ ×F ∗)(Xn ) → (∗ ×Fn ∗)(Xn ).
The (k − 1)-connectivity of this map now follows from the inductive hypothesis.
It remains to treat the case k = 0. Fix an element ηn ∈ Fn (Xn ); we wish
to show that ηn lies in the image of π0 ψ. For every open set U ⊆ X, thecomposition
π0 F(U ) = π0 lim(j(m)∗ Fm )(U )←−→ lim(π0 j(m)∗ Fm )(U )←−' lim π0 Fm (U ∩ Xm )←−is surjective. Consequently, to prove that ηn lies in the image of π0 ψ, it will
suffice to show that there exists an open set U containing Xn such that
ηn can be lifted to lim π0 Fm (U ∩ Xm ). By virtue of assumption (2), it will←−
suffice to construct a sequence of open Fσ subsets {Um ⊆ Xm }m≥n anda sequence of compatible sections γm ∈ π0 Fm (Um ) such that Un = Xnand γm = ηm . The construction proceeds by induction on m. Assumingthat (Um , ηm ) has already been constructed, we invoke the assumption that
Um is an Fσ to choose a continuous function f : Xm → [0, 1] such that
Um = {x ∈ Xm : f (x) > 0}. Let f 0 : Xm+1 → [0, 1] be a continuous extensionof f and let V = {x ∈ Xm+1 : f 0 (x) > 0}. Then V is a paracompact
open subset of Xm+1 , and Um can be identified with a closed subset of
V . Applying Corollary 7.1.5.5 to the restriction Fn+1 |V , we deduce theexistence of an open set Um+1 ⊆ V such that ηk can be extended to a section
ηk+1 ∈ π0 Fm+1 (Um+1 ). Shrinking Um+1 if necessary, we may assume thatUm+1 is itself an Fσ , which completes the induction.
Remark 7.1.5.10. Suppose we are given a sequence of closed embeddingsof topological spaces
X0 ⊆ X1 ⊆ X2 ⊆ · · · ,and let X be the direct limit of the sequence. Suppose further that:highertopoi

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HIGHER TOPOS THEORY IN TOPOLOGY713(a) For each n ≥ 0, the space Xn is paracompact.
(b) For each n ≥ 0, there exists an open neighborhood Yn of Xn in Xn+1
and a retraction rn of Yn onto Xn .
Then X is itself paracompact, so that the hypotheses of Proposition 7.1.5.8
are satisfied and Shv(X) is the direct limit of the sequence of ∞-topoi{Shv(Xn )}n≥0 . To prove this, it will suffice to show that every open covering {Uα }α∈A of X admits a refinement {Vβ }β∈B which
S is countably locallyfinite: that is, that there exists a decomposition B = n≥0 Bn such that eachof the collections {Vβ }β∈Bn is a locally finite collection of open sets, each ofwhich is contained in some Uα (see [60]). To construct this locally finite open
covering, we choose for each n ≥ 0 a locally finite open covering {Wβ }β∈Bn
of Xn which refines the covering {Uα ∩ Xn }α∈A . For each β ∈ Bn , we haveWβ ⊆ Uα for some α ∈ A. We now define Vβ to be the union of a collectionof open subsets {Vβ (m) ⊆ Xm }m≥n , which are constructed as follows:• If m = n, we set Vβ (m) = Wβ .
• Let m > n and let Zm−1 be an open neighborhood of Xm−1 in Xm
whose closure is contained in Ym−1 . We then set Vβ (m) = {z ∈ Zm−1 :rm−1 (z) ∈ Vβ (m − 1)} ∩ Uα .
It is clear that each Vβ (m) is an open subset of Xm contained in Uα andthat Vβ (m + 1) ∩ Xm = V
Sβ (m). Since X is equipped with the direct limit
topology, the union Vβ = m Vβ (m) is open in X. The only nontrivial point
is to verify that the collection {Vβ }β∈Bn is locally finite.
Pick a point x ∈ X; we wish to prove the existence of a neighborhood
Sx of x such that {β ∈ Bn : S ∩ Vβ 6= ∅} is finite. Then there exists somem ≥ n such that x ∈ Xm ; we will construct Sx using induction on m. If
m > n and x ∈ Z m−1 , then let x0 = rm−1 (x) andS set Sx = Sx0 . If m > n
and x ∈
/ Z m−1 , or if m = n, then we define Sx = k≥m Sx (k), where Sx (k)is an open subset of Xk containing x, defined as follows. If m > n, letSx (m) = Xm − Z m−1 , and if m = n, let Sx (m) be an open subset of Xn
which intersects only finitely many of the sets {Wβ }β∈Bn . If k > m, we let
Sx (k) = {z ∈ Yk−1 : rk−1 (z) ∈ Sx (k − 1)}. It is not difficult to verify thatthe open set Sx has the desired properties.
7.1.6 Higher Topoi and Shape Theory
If X is a sufficiently nice topological space (for example, an absolute neighborhood retract), then there exists a homotopy equivalence Y → X, where
Y is a CW complex. If X is merely assumed to be paracompact, then it isgenerally not possible to approximate X well by means of a CW complex Y
equipped with a map to X. However, in view of Theorem 7.1.4.3, one canstill extract a substantial amount of information by considering maps fromX to CW complexes. Shape theory is an attempt to summarize all of thisinformation in a single invariant called the shape of X. In this section, we
will sketch a generalization of shape theory to the setting of ∞-topoi.

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Definition 7.1.6.1. We let Pro(S) denote the full subcategory of Fun(S, S)op
spanned by accessible left exact functors f : S → S. We will refer to Pro(S)as the ∞-category of Pro-spaces, or as the ∞-category of shapes.
Remark 7.1.6.2. If C is a small ∞-category which admits finite limits,then any functor f : C → S is accessible and may be viewed as an objectof P(Cop ). The left exactness of f is then equivalent to the condition thatf belongs to Ind(Cop ) = Pro(C)op . Definition 7.1.6.1 constitutes a naturalextension of this terminology to a case where C is not necessarily small;here it is convenient to add a hypothesis of accessibility for technical reasons(which will not play any role in the discussion below).
Definition 7.1.6.3. Let X be an ∞-topos. According to Proposition 6.3.4.1,
there exists a geometric morphism q∗ : X → S which is unique up to homotopy. Let q ∗ be a left adjoint to q∗ (also unique up to homotopy). The
composition q∗ q ∗ : S → S is an accessible left exact functor, which we willrefer to as the shape of X and denote by Sh(X) ∈ Pro(S).
Remark 7.1.6.4. This definition of the shape of an ∞-topos also appearsin [79].
Remark 7.1.6.5. Let p∗ : Y → X be a geometric morphism of ∞-topoi andlet p∗ be a left adjoint to p∗ . Let q∗ : X → S and q ∗ be as in Definition
7.1.6.3. The unit map idX → p∗ p∗ induces a transformationq∗ q ∗ → q∗ p∗ p∗ q ∗ ' (q ◦ p)∗ (q ◦ p)∗ ,which we may view as a map Sh(X) → Sh(Y) in Pro(S). Via this construction, we may view Sh as a functor from the homotopy category hRTop of∞-topoi to the homotopy category hPro(S). We will say that a geometricmorphism p∗ : Y → X is a shape equivalence if it induces an equivalenceSh(Y) → Sh(X) of Pro-spaces.
Remark 7.1.6.6. By construction, the shape of an ∞-topos X is welldefined up to equivalence in Pro(S). By refining the above construction,
it is possible to construct a shape functor from RTop to the ∞-category
Pro(S) rather than on the level of homotopy.
Remark 7.1.6.7. Our terminology does not quite conform to the usage
in classical topology. Recall that if X is a compact metric space, the shapeof X is defined as a pro-object in the homotopy category of spaces. Thereis a refinement of shape, known as strong shape, which takes values in thehomotopy category of Pro-spaces. Definition 7.1.6.3 is a generalization ofstrong shape rather than shape. We refer the reader to [56] for a discussionof classical shape theory.
Proposition 7.1.6.8. Let p : X → Y be a continuous map of paracompact topological spaces. Then p∗ : Shv(X) → Shv(Y ) is a shape equivalence
if and only if, for every Kan complex K, the induced map of Kan complexes MapTop (Y, |K|) → MapTop (X, |K|) is a homotopy equivalence. (Herehighertopoi

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HIGHER TOPOS THEORY IN TOPOLOGY715

MapTop (Y, |K|) denotes the simplicial set whose n-simplices are given bycontinuous maps Y × |∆n | → |K|, and MapTop (X, |K|) is defined likewise.)
Proof. Corollary 7.1.4.4 and Proposition 7.1.5.1 imply that for any paracompact topological space Z and any Kan complex K, there is a natural
isomorphism
Sh(Shv(Z))(K) ' MapTop (Z, |K|)in the homotopy category H.
Example 7.1.6.9. Let X be a scheme, let X be the topos of étale sheaves
on X, and let X be the associated 1-localic ∞-topos (see §6.4.5). The shape
Sh(X) defined above is closely related to the étale homotopy type introduced
by Artin and Mazur (see [3]). There are three important differences:(1) Artin and Mazur work with pro-objects in the homotopy category H
rather than with actual pro-objects of S. Our definition is closer inspirit to that of Friedlander, who works instead in the homotopy category of pro-objects in Set∆ (see [30]).
(2) The étale homotopy type of [3] is constructed by considering étale
hypercoverings of X; it is therefore more closely related to the shapeof the hypercompletion X∧ .
(3) Artin and Mazur generally study a certain completion of Sh(X∧ ) withrespect to the class of truncated spaces, which has the effect of erasing
the distinction between X and X∧ and discarding a bit of (generally
irrelevant) information.
Remark 7.1.6.10. Let ∗ denote a topological space consisting of a single
point. By definition, Shv(∗) is the full subcategory of Fun(∆1 , S) spanned bythose morphisms f : X → Y , where Y is a final object of S. We observe thatShv(∗) is equivalent to the full subcategory spanned by those morphisms f
as above, where Y = ∆0 ∈ S, and that this full subcategory is isomorphic to
S.
Definition 7.1.6.11. We will say that an ∞-topos X has trivial shape if
Sh(X) is equivalent to the identity functor S → S.
Remark 7.1.6.12. Let q∗ : X → S be a geometric morphism. Then the unit
map u : idS → q∗ q ∗ induces a map of Pro-spaces Sh(X) → idS . Since idS isa final object in Pro(S), we observe that X has trivial shape if and only if
u is an equivalence; in other words, if and only if the pullback functor q ∗ isfully faithful.
We now sketch another interpretation of shape theory based on the ∞topoi associated to Pro-spaces. Let X = S, let π : S × S → S be the projection
onto the first factor, let δ : S → S × S denote the diagonal map, and let
φ : (S × S)/δS → S be defined as in §4.2.2. Proposition 4.2.2.4 implies thathighertopoi

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φ is a coCartesian fibration. We may identify the fiber of φ over an object
X ∈ S with the ∞-category S/X . To each morphism f : X → Y in S, φ
associates a functor f! : S/X → S/Y given by composition with f . Since
S admits pullbacks, each f! admits a right adjoint f ∗ , so that φ is also a
Cartesian fibration associated to some functor ψ : Sop → LTop.
b : S → S be a Pro-space. Then Xb classifies a left fibration M op → S,
Let X
where M is a filtered ∞-category. Let θ denote the composition
ψ op

M op → S → (LTop)op .
Although M is generally not small, the accessibility condition on F guarantees the existence of a cofinal map M 0 → M , where M 0 is a small filtered∞-category. Theorem 6.3.3.1 implies that the diagram θ has a limit, which
we will denote by
S/Xbb
and refer to as the ∞-topos of local systems on X.
b is a Pro-space, then Proposition 6.3.6.4 implies
Remark 7.1.6.13. If X
that the associated geometric morphism S/Xb → S is algebraic. However, theconverse is false in general.
Remark 7.1.6.14. Let G be a profinite group, which we may identify witha Pro-object in the category of finite groups. We let BG denote the corresponding Pro-object of S obtained by applying the classifying space functor
objectwise. Then S/BG can be identified with the 1-localic ∞-topos associated to the ordinary topos of sets with a continuous G-action. It follows fromthe construction of filtered limits in RTop (see §6.3.3) that we can describe
objects Y ∈ S/BG informally as follows: Y associates to each open subgroupU ⊆ G a space Y U of U -fixed points which depends functorially on the finite
G-space G/U . Moreover, if U is a normal subgroup of V , then the natural
map from Y V to the (homotopy) fixed-point space (Y U )V /U should be a
homotopy equivalence.
Remark 7.1.6.15. By refining the construction above, it is possible toconstruct a functor
Pro(S) → RTopb 7→ S b .
X/XThis functor has a left adjoint given by
X 7→ Sh(X).
b is a Pro-space, then the shape of S b is not necWarning 7.1.6.16. If X
/X
b In general we have only a counit morphism
essarily equivalent to X.
b
Sh(S/Xb ) → X.

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717

7.2 DIMENSION THEORYIn this section, we will discuss the dimension theory of topological spaces
from the point of view of higher topos theory. We begin in §7.2.1 by introducing the homotopy dimension of an ∞-topos. We will show that thefiniteness of the homotopy dimension of an ∞-topos X has pleasant consequences: it implies that every object is the inverse limit of its Postnikov
tower and, in particular, that X is hypercomplete.
In §7.2.2, we define the cohomology groups of an ∞-topos X. These cohomology groups have a natural interpretation in terms of the classificationof higher gerbes on X. Using this interpretation, we will show that the cohomology dimension of an ∞-topos X almost coincides with its homotopy
dimension.
In §7.2.3, we review the classical theory of covering dimension for paracompact topological spaces. Using the results of §7.1, we will show that thecovering dimension of a paracompact space X coincides with the homotopy
dimension of the ∞-topos Shv(X).
We conclude in §7.2.4 by introducing a dimension theory for Heyting
spaces, which generalizes the classical theory of Krull dimension for Noetherian topological spaces. Using this theory, we will prove an upper bound forthe homotopy dimension of Shv(X) for suitable Heyting spaces X. This result can be regarded as a generalization of Grothendieck’s vanishing theorem
for the cohomology of Noetherian topological spaces.
7.2.1 Homotopy Dimension
Throughout this section, we will use the symbol 1X to denote the final objectof an ∞-topos X.
Definition 7.2.1.1. Let X be an ∞-topos. We shall say that X has homotopy
dimension ≤ n if every n-connective object U ∈ X admits a global section
1X → U . We say that X has finite homotopy dimension if there exists n ≥ 0such that X has homotopy dimension ≤ n.
Example 7.2.1.2. An ∞-topos X is of homotopy dimension ≤ −1 if andonly if X is equivalent to the trivial ∞-category ∗ (the ∞-topos of sheaves
on the empty space ∅). The “if” direction is obvious. Conversely, if X has
homotopy dimension ≤ −1, then the initial object ∅ of X admits a global
section 1X → ∅. For every object X ∈ X, we have a map X → 1X → ∅, sothat X is also initial (Lemma 6.1.3.6). Since the collection of initial objectsof X span a contractible Kan complex (Proposition 1.2.12.9), we deduce that
X is itself a contractible Kan complex.
Example 7.2.1.3. The ∞-topos S has homotopy dimension 0. More generally, if C is an ∞-category with a final object 1C , then P(C) has homotopy
dimension ≤ 0. To see this, we first observe that the Yoneda embeddingj : C → P(C) preserves limits, so that j(1C ) is a final object of P(C). To

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prove that P(C) has homotopy dimension ≤ 0, we need to show that thefunctor P(C) → S corepresented by j(1C ) preserves effective epimorphisms.
This functor can be identified with evaluation at 1C . It therefore preserves
all limits and colimits and so carries effective epimorphisms to effective epimorphisms by Proposition 6.2.3.7.
Example 7.2.1.4. Let X be a Kan complex and let n ≥ −1. The following
conditions are equivalent:(1) The ∞-topos S/X has homotopy dimension ≤ n.
(2) The geometric realization |X| is a retract (in the homotopy categoryH) of a CW complex K of dimension ≤ n.
To prove that (2) ⇒ (1), let us choose an n-connective object of X/X corresponding to a Kan fibration p : Y → X whose homotopy fibers are nconnective. Choose a map K → |X| which admits a right homotopy inverse.
To prove that p admits a section up to homotopy, it will suffice to show that
there exists a dotted arrow
f

K

>

|Y |
p


/ |X|

in the category of topological spaces, rendering the diagram commutative.
The construction of f proceeds simplex by simplex on K, using the nconnectivity of p to solve lifting problems of the form
S k−1
_

/ |Y |
=


Dk


/ |X|

p

for k ≤ n.
To prove that (1) ⇒ (2), we choose any n-connective map q : K → |X|,
where K is an n-dimensional CW complex. Condition (1) guarantees thatq admits a right homotopy inverse, so that |X| is a retract of K in thehomotopy category H.
If n 6= 2, then (1) and (2) are equivalent to the following apparently
stronger condition:(3) The geometric realization |X| is homotopy equivalent to a CW complex
of dimension ≤ n.
For a proof, we refer the reader to [82].
Remark 7.2.1.5. If X is a coproduct (in the ∞-category RTop) of ∞-topoi
Xα , then X is of homotopy dimension ≤ n if and only if each Xα is ofhomotopy dimension ≤ n.

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It is convenient to introduce a relative version of Definition 7.2.1.1.
Definition 7.2.1.6. Let f : X → Y be a geometric morphism of ∞-topoi.
We will say that f is of homotopy dimension ≤ n if, for every k ≥ n and every
k-connective morphism X → X 0 in X, the induced map f∗ X → f∗ X 0 is a(k − n)-connective morphism in Y (since f∗ is well-defined up to equivalence,
this condition is independent of the choice of f∗ ).
Lemma 7.2.1.7. Let X be an ∞-topos and let F∗ : X → S be a geometricmorphism (which is unique up to equivalence). The following are equivalent:(1) The ∞-topos X is of homotopy dimension ≤ n.
(2) The geometric morphism F∗ is of homotopy dimension ≤ n.
Proof. Suppose first that (2) is satisfied and let X be an n-connective objectof X. Then F∗ X is a 0-connective object of S: that is, it is a nonempty Kancomplex. It therefore has a point 1S → F∗ X. By adjointness, we see that
there exists a map 1X → X in X, where 1X = F ∗ 1S is a final object of X
because F ∗ is left exact. This proves (1).
Now assume (1) and let s : X → Y be an k-connective morphism in X;
we wish to show that F∗ s is (k − n)-connective. The proof goes by induction
on k ≥ n. If k = n, then we are reduced to proving the surjectivity of thehorizontal maps in the diagram/ π0 MapX (1X , Y )π0 MapX (1X , X)
/ π0 MapS (1S , F∗ Y )

π0 MapS (1S , F∗ X)

of sets. Let p : 1X → Y be any morphism in X and form a pullback diagram
/X
Z


s0

1X

s
p


/ Y.

0

The map s is a pullback of s and therefore n-connective by Proposition
6.5.1.16. Using (1), we deduce the existence of a map 1X → Z, and a composite map
1X → Z → X
is a lifting of p up to homotopy.
We now treat the case where k > n. Form a diagramX
s0(
X ×Y X

/X


X


/Y

s
s

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where the square at the bottom right is a pullback in X. According to Proposition 6.5.1.18, s0 is (k − 1)-connective. Using the inductive hypothesis, we
deduce that F∗ (s0 ) is (k − n − 1)-connective. We now invoke Proposition
6.5.1.18 in the ∞-topos S and deduce that F∗ (s) is (k − n)-connective, as
desired.
Definition 7.2.1.8. We will say that an ∞-topos X is locally of homotopy
dimension ≤ n if there exists a collection {Uα } of objects of X which generate
X under colimits, such that each X/Uα is of homotopy dimension ≤ n.
Example 7.2.1.9. Let C be a small ∞-category. Then P(C) is locally of
homotopy dimension ≤ 0. To prove this, we first observe that P(C) is generated under colimits by the Yoneda embedding j : C → P(C). It therefore
suffices to prove that each of the ∞-topoi P(C)/j(C) has finite homotopy
dimension. According to Corollary 5.1.6.12, the ∞-topos P(C)/j(C) is equivalent to P(C/C ), which is of homotopy dimension 0 (see Example 7.2.1.3).
Our next goal is to prove the following result:
Proposition 7.2.1.10. Let X be an ∞-topos which is locally of homotopy
dimension ≤ n for some integer n. Then Postnikov towers in X are convergent.
Proof. It will suffice to prove that for every tower in X, conditions (1) and(2) of Proposition 5.5.6.26 are equivalent. We first show that (2) ⇒ (1) byop
verifying criterion (∗) of Remark 5.5.6.27 Let X : N(Z∞
→ X be a limit≥0 )tower and assume that the underlying pretower is highly connected. We wish
to show that X is highly connected. Choose m ≥ −1; we wish to show thatthe map X(∞) → X(k) is m-connective for k  0. Reindexing the tower
if necessary, we may suppose that for every p ≥ q, the map X(p) → X(q)is (m + q)-connective. We claim that, in this case, we can take k = 0. The
proof goes by induction on m. If m > 0, we can deduce the desired result byapplying the inductive hypothesis to the tower
X(∞) → · · · → X(∞) ×X(1) X(∞) → X(∞) ×X(0) X(∞).
Let us therefore assume that m = 0; we wish to show that the map X(∞) →X(0) is an effective epimorphism. Since the objects {Uα } generate X undercolimits, there is an effective epimorphism φ : U → X(0), where U is a
coproduct of objects of the form {Uα }. Using Remark 7.2.1.5, we deducethat X/U has homotopy dimension ≤ n. Let F : X → S denote the functor
corepresented by U . Then F factors as a compositionf∗Γ

X → X/U → S,
where f ∗ is the left adjoint to the geometric morphism X/U → X and Γ isthe global sections functor. It follows that F carries n-connective morphisms
to effective epimorphisms (Lemma 7.2.1.7). The map φ determines a pointof F (X(0)). Since each of the maps F (X(k + 1)) → F (X(k)) induces asurjection on connected components, we can lift this point successively tohighertopoi

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each F (X(k)) and thereby obtain a point in F (X(∞)) ' holim{F (X(n))}.
This point determines a diagram
X(∞)
=
φ

U

ψ

#
/ X(0)

which commutes up to homotopy. Since φ is an effective epimorphism, we
deduce that the map ψ is an effective epimorphism, as desired.
We now complete the proof by verifying the implication (1) ⇒ (2) ofop
Proposition 5.5.6.26. Let X : N(Z∞
→ X be a Postnikov tower; we wish
≥0 )op
to show that X is a limit diagram. To prove this, let X 0 : N(Z∞→X
≥0 )be a limit of the diagram X| N(Z≥0 )op , so that the equality X| N(Z≥0 )op =X 0 | N(Z≥0 )op induces a map X → X 0 . Since X 0 satisfies condition (2) of
Proposition 5.5.6.26, it also satisfies condition (1). It follows that the map α :X(∞) → X 0 (∞) is ∞-connective. We wish to prove that α is an equivalence.
For this, it will suffice to prove that for every object U ∈ X, composition withα induces a homotopy equivalence MapX (U, X(∞)) → MapX (U, X 0 (∞)).
The collection of those objects U ∈ X which satisfy this condition is closed
under small colimits. Since X is locally of homotopy dimension ≤ n, we may
assume without loss of generality that X/U has homotopy dimension ≤ n.
Replacing X by X/U , we may assume that U is the final object of X and thatthe ∞-topos X has homotopy dimension ≤ n. The desired result now follows
from Lemma 7.2.1.7 (note that θ is the image of α under the global sections
functor X → S).
op
Lemma 7.2.1.11. Let X be a presentable ∞-category, let Fun(N(Z∞≥0 ) , X)∞ op
be the ∞-category of towers in X, and let Xτ ⊆ Fun(N(Z≥0 ) , X) denote thefull subcategory spanned by the Postnikov towers. Evaluation at ∞ induces atrivial fibration of simplicial sets Xτ → X. In particular, every object X(∞) ∈X can be extended to a Postnikov tower

X(∞) → · · · → X(1) → X(0).
op
Proof. Let C be the full subcategory of X × N(Z∞
spanned by the pairs≥0 )(X, n), where X is an object of X, n ∈ Z∞,
andX
is n-truncated, and let≥0
p : C → X denote the natural projection. Since every m-truncated objectof X is also n-truncated for m ≥ n, it is easy to see that p is a Cartesianfibration. Proposition 5.5.6.18 implies that each of the inclusion functors
τ≤m X ⊆ τ≤n X has a left adjoint, so that p is also a coCartesian fibration(Corollary 5.2.2.5). By definition, Xτ can be identified with the simplicial
set
op \ op]Map[N(Z∞ ) (N(Z∞≥0 ) , (C ) ) ,≥0

and X itself can be identified with
Map[N(Z∞ ) ({∞}] , (Cop )\ )op .
≥0highertopoi

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It now suffices to observe that the inclusion {∞}] ⊆ N(Z∞≥0 ) is marked
anodyne.

Corollary 7.2.1.12 (Jardine). Let X be an ∞-topos which is locally of
homotopy dimension ≤ n for some integer n. Then X is hypercomplete.
Proof. Let X(∞) be an arbitrary object of X. By Lemma 7.2.1.11, we canfind a Postnikov tower
X(∞) → · · · → X(1) → X(0).
Since X(n) is n-truncated, it belongs to X∧ by Corollary 6.5.1.14. By Proposition 7.2.1.10, the tower exhibits X(∞) as a limit of objects of X∧ , so that
X(∞) belongs to X∧ as well since the full subcategory X∧ ⊆ X is stable
under limits.
Lemma 7.2.1.13. Let X be an ∞-topos, n ≥ 0, X an (n + 1)-connective
object of X, and f ∗ : X → X/X a right adjoint to the projection X/X → X.
Then f ∗ induces a fully faithful functor τ≤n X → τ≤n X/X which restricts toan equivalence from τ≤n−1 X to τ≤n−1 X/X .
Proof. We first prove that f ∗ is fully faithful when restricted to the ∞category of n-truncated objects of X. Let Y, Z ∈ X be objects, where Y isn-truncated. We have a commutative diagram
MapX/X (f ∗ Y, f ∗ Z)
O

MapX (X × Y, Z) o

MapX (Y, Z) o

MapX (τ≤n (X × Y ), Z)O

MapX (τ≤n Y, Z)

in the homotopy category H, where the horizontal arrows are homotopy
equivalences. Consequently, to prove that the left vertical map is a homotopy
equivalence, it suffices to show that the projection τ≤n (X × Y ) → τ≤n Y
is an equivalence. This follows immediately from Lemma 6.5.1.2 and ourassumption that X is (n + 1)-connective.
Now suppose that Y is an (n − 1)-truncated object of X/X . We wish toshow that Y lies in the essential image of f ∗ |τ≤n−1 X. Let Y denote theimage of Y in X and let Y → Z exhibit Z as an (n − 1)-truncation of Y in
X. To complete the proof, it will suffice to show that the composition
u0

u00

u : Y → f ∗Y → f ∗Z
is an equivalence in X/X . Since both Y and f ∗ Z are (n − 1)-truncated, it
suffices to prove that u is n-connective. According to Proposition 6.5.1.16,
it suffices to prove that u0 and u00 are n-connective. Proposition 5.5.6.28
implies that u00 exhibits f ∗ Z as an (n − 1)-truncation of f ∗ Y and is therefore
n-connective.
We now complete the proof by showing that u0 is n-connective. Let v 0
denote the image of u0 in the ∞-topos X. According to Proposition 6.5.1.19,

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it will suffice to show that v 0 is n-connective. We observe that v 0 is a sectionof the projection q : Y × X → Y . According to Proposition 6.5.1.20, it will
suffice to prove that q is (n + 1)-connective. Since q is a pullback of the
projection X → 1X , Proposition 6.5.1.16 allows us to conclude the proof(since X is (n + 1)-connective by assumption).
Lemma 7.2.1.13 has some pleasant consequences.
Proposition 7.2.1.14. Let X be an ∞-topos and let τ≤0 : X → τ≤0 X
denote a left adjoint to the inclusion. A morphism φ : U → X in X is aneffective epimorphism if and only if τ≤0 (φ) is an effective epimorphism inthe ordinary topos h(τ≤0 X).
Proof. Suppose first that φ is an effective epimorphism. Let U• : N ∆op+ →X
be a Čech nerve of φ, so that U• is a colimit diagram. Since τ≤0 is a left
adjoint, τ≤0 U• is a colimit diagram in τ≤0 X. Using Proposition 6.2.3.10, we
deduce easily that τ≤0 φ is an effective epimorphism.
For the converse, choose a factorization of φ as a composition
φ0

φ00

U → V → X,
where φ0 is an effective epimorphism and φ00 is a monomorphism. Applying
Lemma 7.2.1.13 to the ∞-topos X/τ≤0 X , we conclude that φ00 is the pullback
of a monomorphism i : V → τ≤0 X. Since the effective epimorphism τ≤0 (φ)
factors through i, we conclude that i is an equivalence, so that φ00 is likewise
an equivalence. It follows that φ is an effective epimorphism, as desired.
Proposition 7.2.1.14 can be regarded as a generalization of the following well-known property of the ∞-category of spaces, which can itself be
regarded as the ∞-categorical analogue of the second part of Fact 6.1.1.6:
Corollary 7.2.1.15. Let f : X → Y be a map of Kan complexes. Then fis an effective epimorphism in the ∞-category S if and only if the induced
map π0 X → π0 Y is surjective.
Remark 7.2.1.16. It follows from Proposition 7.2.1.14 that the class of∞-topoi having the form Shv(C), where C is a small ∞-category, is notsubstantially larger than the class of ordinary topoi. More precisely, every
topological localization of P(C) can be obtained by inverting morphisms between discrete objects of P(C). It follows that there exists a pullback diagramof ∞-topoi
Shv(C)/ P(C)
Shv(N(hC))

/ P(N(hC)),

where the ∞-topoi on the bottom line are 1-localic and therefore determinedby the ordinary topoi of presheaves of sets on the homotopy category h C
and sheaves of sets on h C, respectively.

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Corollary 7.2.1.17. Let X be a topological space. Suppose that Shv(X) is
locally of homotopy dimension ≤ n for some integer n. Then Shv(X) has
enough points.
Proof. Note that every point x ∈ X gives rise to a point x∗ : Shv(∗) →Shv(X) of the ∞-topos Shv(X). Let f : F → F0 be a morphism in Shv(X)such that x∗ (f ) is an equivalence in S for each x ∈ X. We wish to provethat f is an equivalence. According to Corollary 7.2.1.12, it will suffice toprove that f is ∞-connective. We will prove by induction on n that f isn-connective. If n > 0, we simply apply the inductive hypothesis to the
diagonal morphism δ : F → F ×F0 F. We may therefore reduce to the casen = 0; we wish to show that f is an effective epimorphism. Since Shv(X)is generated under colimits by the sheaves χU associated to open subsets
U ⊆ X, we may assume without loss of generality that F0 = χU . We may
now invoke Proposition 7.2.1.14 to reduce to the case where F is an object ofτ≤0 Shv(X)/χU . This ∞-category is equivalent to the nerve of the categoryof sheaves of sets on U . We are therefore reduced to proving that if F is asheaf of sets on U whose stalk Fx is a singleton at each point x ∈ U , then F
has a global section, which is clear.
7.2.2 Cohomological Dimension
In classical homotopy theory, one can analyze a space X by means of its
Postnikov towerφn· · · → τ≤n X → τ≤n−1 X → · · · .
In this diagram, the homotopy fiber F of φn (n ≥ 1) is a space which has
only a single nonvanishing homotopy group which appears in dimension n.
The space F is determined up to homotopy equivalence by πn F : in fact,
F is homotopy equivalent to an Eilenberg-MacLane space K(πn F, n) which
can be functorially constructed from the group πn F . The study of theseEilenberg-MacLane spaces is of central interest because (according to theabove analysis) they constitute basic building blocks out of which any arbitrary space can be constructed. Our goal in this section is to generalize thetheory of Eilenberg-MacLane spaces to the setting of an arbitrary ∞-topos
X.
Definition 7.2.2.1. Let X be an ∞-category. A pointed object is a morphismX∗ : 1 → X in X, where 1 is a final object of X. We let X∗ denote the fullsubcategory of Fun(∆1 , X) spanned by the pointed objects of X.
A group object of X is a groupoid object U• : N ∆op → X for which U0 is
a final object of X. Let Grp(X) denote the full subcategory of X∆ spannedby the group objects of X.
We will say that a pointed object 1 → X of an ∞-topos X is an EilenbergMacLane object of degree n if X is both n-truncated and n-connective. We
let EMn (X) denote the full subcategory of X∗ spanned by the EilenbergMacLane objects of degree n.

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Example 7.2.2.2. Let C be an ordinary category which admits finite limits.
A group object of C is an object X ∈ C which is equipped with an identity
section 1C → X, an inversion map X → X, and a multiplication m : X ×
X → X, which satisfy the usual group axioms. Equivalently, a group objectof C is an object X together with a group structure on each morphism space
HomC (Y, X) which depends functorially on Y . We will denote the categoryof group objects of C by Grp(C). The ∞-category N(Grp(C)) is equivalent to
the ∞-category of group objects of N(C) in the sense of Definition 7.2.2.1.
Thus the notion of a group object of an ∞-category can be regarded as a
generalization of the notion of a group object of an ordinary category.
Remark 7.2.2.3. Let X be an ∞-topos and n ≥ 0 an integer. Then the fullsubcategory of Fun(∆1 , X) consisting of Eilenberg-MacLane objects p : 1 →X is stable under finite products. This is an immediate consequence of thefollowing observations:(1) A finite product of n-connective objects of X is n-connective (Corollary
6.5.1.13).
(2) Any limit of n-truncated objects of X is n-truncated (since τ≤n X is alocalization of X).
Proposition 7.2.2.4. Let X be an ∞-category and let U• be a simplicial
object of X. Then U• is a group object of X if and only if the following
conditions are satisfied:(1) The object U0 is final in X.
(2) For every decomposition [n] = S ∪ S 0 , where S ∩ S 0 = {s}, the maps
U (S) ← Un → U (S 0 )exhibit Un as a product of U (S) and U (S) in X.
Proof. This follows immediately from characterization (400 ) of Proposition
6.1.2.6.
Corollary 7.2.2.5. Let X and Y be ∞-categories which admit finite products
and let f : X → Y be a functor which preserves finite products. Then the
induced functor X∆ → Y∆ carries group objects of X to group objects of Y.
Corollary 7.2.2.6. Let X be an ∞-category which admits finite products
and let Y ⊆ X be a full subcategory which is stable under finite products. LetY• be a simplicial object of Y. Then Y• is a group object of Y if and only if
it is a group object of X.
Definition 7.2.2.7. Let X be an ∞-category. A zero object of X is an object
which is both initial and final.
Lemma 7.2.2.8. Let X be an ∞-category with a final object 1X . Then the
inclusion i : X1X / ⊆ X∗ is an equivalence of ∞-categories.

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Proof. Let K be the full subcategory of X spanned by the final objects
and let 1X be an object of K. Proposition 1.2.12.9 implies that K is acontractible Kan complex, so that the inclusion {1X } ⊆ K is an equivalenceof ∞-categories. Corollary 2.4.7.12 implies that the projection X∗ → K is
a coCartesian fibration. We now apply Proposition 3.3.1.3 to deduce thedesired result.
Lemma 7.2.2.9. Let X be an ∞-category with a final object. Then the ∞category X∗ has a zero object. If X already has a zero object, then the forgetful
functor X∗ → X is an equivalence of ∞-categories.
Proof. Let 1X be a final object of X and let U = id1X ∈ X∗ . We wish to showthat U is a zero object of X∗ . According to Lemma 7.2.2.8, it will suffice toshow that U is a zero object of X1X / . It is clear that U is initial, and thefinality of U follows from Proposition 1.2.13.8.
For the second assertion, let us suppose that 1X is also an initial objectof X. We wish to show that the forgetful functor X∗ → X is an equivalenceof ∞-categories. Applying Lemma 7.2.2.8, it will suffice to show that the
projection f : X1X / → X is an equivalence of ∞-categories. But f is a trivial
fibration of simplicial sets.
Lemma 7.2.2.10. Let X be an ∞-category and let f : X∗ → X be the forgetful functor (which carries a pointed object 1 → X to X). Then f induces
an equivalence of ∞-categories
Grp(X∗ ) → Grp(X).
Proof. The functor f factors as a compositionX∗ ⊆ Fun(∆1 , X) → X,
where the first map is the inclusion of a full subcategory which is stable
under limits and the second map preserves all limits (Proposition 5.1.2.2). It
follows that f preserves limits so that composition with f induces a functor
F : Grp(X∗ ) → Grp(X) by Corollary 7.2.2.5.
Observe that the 0-simplex ∆0 is an initial object of ∆op . Consequently,
there exists a functor T : ∆1 × N(∆)op → N(∆)op which is a natural transformation from the constant functor taking the value ∆0 to the identity
functor. Composition with T induces a functor
X∆ → Fun(∆1 , X)∆ .
Restricting to group objects, we get a functor s : Grp(X) → Grp(X∗ ). It isclear that F ◦ s is the identity.
We observe that if X has a zero object, then f is an equivalence of ∞categories (Lemma 7.2.2.9). It follows immediately that F is an equivalenceof ∞-categories. Since s is a right inverse to F , we conclude that s is anequivalence of ∞-categories as well.
To complete the proof in the general case, it will suffice to show that thecomposition s ◦ F is an equivalence of ∞-categories. To prove this, we sethighertopoi

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Y = X∗ and let F 0 : Grp(Y∗ ) → Grp(X∗ ) and s0 : Grp(Y) → Grp(Y∗ ) be
defined as above. We then have a commutative diagram
Grp(Y)Fs0
Grp(Y∗ )/ Grp(X)sF0/ Grp(X∗ ),so that s ◦ F = F 0 ◦ s0 . Lemma 7.2.2.9 implies that Y has a zero object, sothat F 0 and s0 are equivalences of ∞-categories. Therefore F 0 ◦ s0 = s ◦ F is
an equivalence of ∞-categories, and the proof is complete.
The following proposition guarantees a good supply of Eilenberg-MacLane
objects in an ∞-topos X.
Lemma 7.2.2.11. Let X be an ∞-topos containing a final object 1X and letn ≥ 1. Let p denote the composition
Č

Fun(∆1 , X) → X∆+ → X∆ ,which associates to each morphism U → X the underlying groupoid of its
Čech nerve. Then
(1) Let X0 denote the full subcategory of Fun(∆1 , X) consisting of connected
pointed objects of X. Then the restriction of p induces an equivalenceof ∞-categories from X0 to the ∞-category Grp(X).
(2) The essential image of p| EMn (X) coincides with the essential imageof the composition
Grp(EMn−1 (X)) ⊆ Grp(X∗ ) → Grp(X).
Proof. Let X00 be the full subcategory of Fun(∆1 , X) spanned by the effective
epimorphisms u : U → X. Since X is an ∞-topos, p induces an equivalencefrom X00 to the ∞-category of groupoid objects of X. Consequently, to prove(1), it will suffice to show that if u : 1X → X is a morphism in X and 1X is a
final object, then u is an effective epimorphism if and only if X is connected.
We note that X is connected if and only if the map τ≤0 (u) : τ≤0 1X → τ≤0 X
is an isomorphism in the ordinary topos Disc(X). According to Proposition
7.2.1.14, u is an effective epimorphism if and only if τ≤0 (u) is an effective
epimorphism. We now observe that in any ordinary category C, an effective
epimorphism u0 : 1C → X 0 whose source is a final object of C is automaticallyan isomorphism since the equivalence relation 1C ×X 0 1C ⊆ 1C × 1C consistsof the whole of 1C × 1C ' 1C .
To prove (2), we consider an augmented simplicial object X• of X which
is a Čech nerve having the property that X0 is a final object of X. We wish
to show that the pointed object X0 → X−1 belongs to EMn (X) if and only ifeach Xk is (n − 1)-truncated and (n − 1)-connective for k ≥ 0. We conclude
by making the following observations:highertopoi

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CHAPTER 7(a) Since Xk is equivalent to a k-fold product of copies of X1 , the objects
Xk are (n − 1)-truncated ((n − 1)-connective) for all k ≥ 0 if and only
if X1 is (n − 1)-truncated ((n − 1)-connective).
(b) We have a pullback diagram
X1

f

/ X0
g


X0

g



/ X−1 .

The object X1 is (n − 1)-truncated if and only if f is (n − 1)-truncated.
Since g is an effective epimorphism, f is (n − 1)-truncated if and only
if g is (n − 1)-truncated (Proposition 6.2.3.17). Using the long exact
sequence of Remark 6.5.1.5, we conclude that this is equivalent to thevanishing of g ∗ πk X−1 for k > n. Since g is an effective epimorphism,
this is equivalent to the vanishing of πk X−1 for k > n, which is in turnequivalent to the requirement that X−1 is n-truncated.
(c) The object X1 is (n−1)-connective if and only if f is (n−1)-connective.
Arguing as above, we conclude that f is (n − 1)-connective if andonly if g is (n − 1)-connective (Proposition 6.5.1.16). Using the longexact sequence of Remark 6.5.1.5, this is equivalent to the vanishingof the homotopy sheaf g ∗ πk X−1 for k < n. Since g is an effective
epimorphism, this is equivalent to the vanishing of πk X−1 for k <n, which is in turn equivalent to the condition that X−1 is (n − 1)truncated.

Proposition 7.2.2.12. Let X be an ∞-topos, let n ≥ 0 be a nonnegative
integer, and let πn : X∗ → N(Disc(X)) denote the associated homotopy groupfunctor.
Then(1) If n = 0, then πn determines an equivalence from the ∞-category
EM0 (X) to the (nerve of the) category of pointed objects of Disc(X).
(2) If n = 1, then πn determines an equivalence from the ∞-category
EM1 (X) to the (nerve of the) category of group objects of Disc(X).
(3) If n ≥ 2, then πn determines an equivalence from the ∞-category
EMn (X) to the (nerve of the) category of commutative group objects ofDisc(X).
Proof. We use induction on n. The case n = 0 follows immediately from thedefinitions. The case n = 1 follows from the case n = 0 by combining Lemmas
7.2.2.11 and 7.2.2.10. If n = 2, we apply the inductive hypothesis together
with Lemma 7.2.2.11 and the observation that if C is an ordinary category

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729

which admits finite products, then Grp(Grp(C)) is equivalent to category
Ab(C) of commutative group objects of C. The argument in the case n > 2
makes use of the inductive hypothesis, Lemma 7.2.2.11, and the observationthat Grp(Ab(C)) is equivalent to Ab(C) for any ordinary category C which
admits finite products.
Fix an ∞-topos X, a final object 1X ∈ X, and an integer n ≥ 0. According
to Proposition 7.2.2.12, there exists a homotopy inverse to the functor π. We
will denote this functor by
A 7→ (p : 1X → K(A, n)),where A is a pointed object of the topos Disc(X) if n = 0, a group object if
n = 1, and an abelian group object if n ≥ 2.
Remark 7.2.2.13. The functor A 7→ K(A, n) preserves finite products.
This is clear since the class of Eilenberg-MacLane objects is stable underfinite products (Remark 7.2.2.3) and the homotopy inverse functor π commutes with finite products (since homotopy groups are constructed using
pullback and truncation functors, each of which commutes with finite products).
Definition 7.2.2.14. Let X be an ∞-topos, n ≥ 0 an integer, and A an
abelian group object of the topos Disc(X). We defineHn (X; A) = π0 MapX (1X , K(A, n));we refer to Hn (X; A) as the nth cohomology group of X with coefficients inA.
Remark 7.2.2.15. It is clear that we can also make sense of H1 (X; G) when
G is a sheaf of nonabelian groups, or H0 (X; E) when E is only a sheaf of(pointed) sets.
Remark 7.2.2.16. It is clear from the definition that Hn (X; A) is functorial in A. Moreover, this functor commutes with finite products by Remark
7.2.2.13 (and the fact that products in X are products in the homotopy category h X). If A is an abelian group, then the multiplication map A × A → A
induces a (commutative) group structure on Hn (X; A). This justifies our
terminology in referring to Hn (X; A) as a cohomology group.
Remark 7.2.2.17. Let C be a small category equipped with a Grothendiecktopology and let X be the ∞-topos Shv(N C) of sheaves of spaces on C, so
that the underlying topos Disc(X) is equivalent to the category of sheavesof sets on C. Let A be a sheaf of abelian groups on C. Then Hn (X; A) may
be identified with the nth cohomology group of Disc(X) with coefficients in
A in the sense of ordinary sheaf theory. To see this, choose a resolution
A → I 0 → I 1 → · · · → I n−1 → J
of A by abelian group objects of Disc(X), where each I k is injective. The
complex
I0 → · · · → Jhighertopoi

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may be identified, via the Dold-Kan correspondence, with a simplicial abeliangroup object C• of Disc(X). Regard C• as a presheaf on C with values inSet∆ . Then
(1) The induced presheaf F : N(C)op → S belongs to X = Shv(N(C)) ⊆P(N(C)) (this uses the injectivity of the objects I k ) and is equippedwith a canonical base point p : 1X → F .
(2) The pointed object p : 1X → F is an Eilenberg-MacLane object of X,
and there is a canonical identification A ' p∗ (πn F ). We may therefore
identify F with K(A, n).
(3) The set of homotopy classes of maps from 1X to F in X may be identified with the cokernel of the map Γ(Disc(X); I n−1 ) → Γ(Disc(X); J),
which is also the nth cohomology group of Disc(X) with coefficients in
A in the sense of classical sheaf theory.
For further discussion of this point, we refer the reader to [41].
We are ready to define the cohomological dimension of an ∞-topos.
Definition 7.2.2.18. Let X be an ∞-topos. We will say that X has cohomological dimension ≤ n if, for any sheaf of abelian groups A on X, thecohomology group Hk (X, A) vanishes for k > n.
Remark 7.2.2.19. For small values of n, some authors prefer to requirea stronger vanishing condition which also applies when A is a nonabeliancoefficient system. The appropriate definition requires the vanishing of cohomology for coefficient systems which are defined only up to inner automorphisms, as in [31]. With the appropriate modifications, Theorem 7.2.2.29
below remains valid for n < 2.
The cohomological dimension of an ∞-topos X is closely related to thehomotopy dimension of X. If X has homotopy dimension ≤ n, thenHm (X; A) = π0 MapX (1X , K(A, m)) = ∗for m > n by Lemma 7.2.1.7, so that X is also of cohomological dimension≤ n. We will establish a partial converse to this result.
Definition 7.2.2.20. Let X be an ∞-topos. An n-gerbe on X is an object
X ∈ X which is n-connective and n-truncated.
Let X be an ∞-topos containing an n-gerbe X and let f : X/X → X
denote the associated geometric morphism. If X is equipped with a basepoint p : 1X → X, then X is canonically determined (as a pointed object) byp∗ πn X by Proposition 7.2.2.12. We now wish to consider the case in which X
is not pointed. If n ≥ 2, then πn X can be regarded as an abelian group object
in the topos Disc(X/X ). Proposition 7.2.1.13 implies that πn X ' f ∗ A, where
A is a sheaf of abelian groups on X, which is determined up to canonicalhighertopoi

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HIGHER TOPOS THEORY IN TOPOLOGY

isomorphism. (In concrete terms, this boils down to the observation that the
1-connectivity of X allows us to extract higher homotopy groups without
specifying a base point on X.) In this situation, we will say that X is banded
by A.
Remark 7.2.2.21. For n < 2, the situation is more complicated. We refer
the reader to [31] for a discussion.
Our next goal is to show that the cohomology groups of an ∞-topos X canbe interpreted as classifying equivalence classes of n-gerbes over X. Before
we can prove this, we need to establish some terminology.
Notation 7.2.2.22. Let X be an ∞-topos. We define a category Band(X)as follows:(1) The objects of Band(X) are pairs (U, A), where U is an object of X
and A is an abelian group object of the homotopy category Disc(X/U ).
(2) Morphisms from (U, A) to (U 0 , A0 ) are given by pairs (η, f ), whereη ∈ π0 MapX (U, U 0 ) and f : A → A0 is a map which induces an isomorphism A ' η ∗ A0 of abelian group objects. Composition of morphisms
is defined in the obvious way.
For n ≥ 2, let Gerbn (X) denote the subcategory of Fun(∆1 , X) spannedby those objects f : X → S which are n-gerbes in X/S and those morphisms
which correspond to pullback diagrams
X0
f


S0

/X
f


/ S.

Remark 7.2.2.23. Since the class of morphisms f : X → S which belong
1
to X∆ is stable under pullback, we can apply Corollary 2.4.7.12 (whichasserts that p : Fun(∆1 , X) → Fun({1}, X) is a Cartesian fibration), Lemma
6.1.1.1 (which characterizes the p-Cartesian morphisms of Fun(∆1 , X)), andCorollary 2.4.2.5 to deduce that the projection Gerbn (X) → X is a rightfibration.
If f : X → U belongs to Gerbn (X), then there exists an abelian groupobject A of Disc(X/U ) such that X is banded by A. The construction(f : X → U ) 7→ (U, A)
determines a functor
χ : Gerbn (X) → N(Band(X)).
Let A be an abelian group object of Disc(X). We let BandA (X) be thecategory whose objects are triples (X, AX , φ), where X ∈ hX, AX is anhighertopoi

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abelian group object of Disc(X/X ), and φ is a map AX → A which induces
an isomorphism AX ' A × X of abelian group objects of Disc(X/X ). We
have forgetful functors
φ

BandA (X) → Band(X) → hX,
both of which are Grothendieck fibrations and whose composition is an equivalence of categories. We define GerbAn (X) by the following pullback diagram:
GerbA
n (X)

/ Gerbn (X)N(BandA (X))


/ N(Band(X)).

χ

Note that since φ is a Grothendieck fibration, N φ is a Cartesian fibration(Remark 2.4.2.2), so that the diagram above is homotopy Cartesian (Proposition 3.3.1.3). We will refer to GerbAn (X) as the sheaf of gerbes over X banded
by A.
More informally, an object of GerbAn (X) is an n-gerbe f : X → U in X/U
together with an isomorphism φX : πn X ' X × A of abelian group objects
of Disc(X/X ). Morphisms in GerbAn are given by pullback squares
X0

f


U0

/X

/U

such that the associated diagram of abelian group objects of Disc(X/X 0 )πn fπn X 0f ∗ (πn X)
9φX 0

f ∗ φX%/ A × X0

is commutative.
Lemma 7.2.2.24. Let X be an ∞-topos, n ≥ 1, and A an abelian groupobject in the topos Disc(X). Let X be an n-gerbe in X equipped with a fixed
isomorphism φ : πn X ' X ×A of abelian group objects of Disc(X/X ), and let
u : 1X → K(A, n) be an Eilenberg-MacLane object of X classified by A. Let
MapφX (K(A, n), X) be the summand of MapX (K(A, n), X) corresponding tothose maps f : K(A, n) → X for which the compositionf ∗φ

A × K(A, n) ' πn K(A, n) → f ∗ (πn X) → A × K(A, n)
is the identity (in the category of abelian group objects of X/K(A,n) ). Thencomposition with u induces a homotopy equivalenceθφ : MapφX (K(A, n), X) → MapX (1X , X).

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Proof. Let θ : MapX (K(A, n), X) → MapX (1X , X) and let f : 1X → X be
any map (which we may identify with an Eilenberg-MacLane object of X).
The homotopy fiber of θ over the point represented by f can be identified
with MapX1 / (u, f ). In view of the equivalence between X1X / and X∗ , weX
can identify this mapping space with MapX∗ (u, f ). Applying Proposition
7.2.2.12, we deduce that the homotopy fiber of θ is equivalent to the (discrete)
set of all endomorphisms v : A → A (in the category of group objects ofDisc(X)). We now observe that the homotopy fiber of θφ over f is a summandof the homotopy fiber of θ over f corresponding to those components forwhich v = idA . It follows that the homotopy fibers of θφ are contractible, sothat θφ is a homotopy equivalence, as desired.
Lemma 7.2.2.25. Let X be an ∞-topos, n ≥ 1, and A an abelian groupobject of Disc(X). Let f : K(A, n) × X → X be a trivial n-gerbe over X
banded by A and g : Ye → Y be any n-gerbe over Y banded by A. Then there
is a canonical homotopy equivalence
MapGerbA(f, g) ' MapX (X, Ye ).
n
Proof. Choose a morphism α : idX → f in X/X corresponding to a diagram/ X × K(A, n)

s

X

f


X

idX


/X

which exhibits f as an Eilenberg-MacLane object of X/X . We observe that
evaluation at {0} ⊆ ∆1 induces a trivial fibration
HomLX∆1 (idX , g) → HomLX (X, Ye ).
Consequently, we may identify MapX (X, Ye ) with the Kan complex
Z = Fun(∆1 , X)idX / ×Fun(∆1 ,X) {g}.
Similarly, the trivial fibration Fun(∆1 , X)α/ → Fun(∆1 , X)f / allows us to
identify MapGerbn (f, g) with the Kan complex
Z 0 = Fun(∆1 , X)α/ ×Fun(∆1 ,X) {g}and MapGerbn (f, g) with the summand Z 00 of Z 0 corresponding to those maps
which induce the identity isomorphism of A×(K(A, n)×X) (in the categoryof group objects of Disc(X/K(A,n)×X )). We now observe that evaluation at{1} ⊆ ∆1 gives a commutative diagram
/ Z0

Z 00
ψ

/Z

00

&

ψ0


XidX / ×X {Y }ψ/ XX/ ×X {Y },highertopoi

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where the vertical maps are Kan fibrations. If we fix a pullback square
/ Ye

e
X
g0


X

h


/ Y,

then we can identify the Kan complex ψ −1 {h} with MapX/X (idX , g 0 ), the−1
Kan complex ψ 0 {s0 h} with MapX/X (X × K(A, n), g 0 ), and the Kan com−1
plex ψ 0 {s0 h} with the summand of MapX/X (X × K(A, n), g 0 ) corresponding to those maps which induce the identity on A × (K(A, n) × X) (in the
category of group objects of Disc(X/K(A,n)×X )). Invoking Lemma 7.2.2.24
in the ∞-topos X/X , we deduce that the map θ in the diagram
θ

Z 00
ψ 00

/Z
ψ


XidX / ×X {Y }/ XX/ ×X {Y }induces homotopy equivalences from the fibers of ψ 00 to the fibers of ψ. Since
the lower horizontal map is a trivial fibration of simplicial sets, we conclude
that θ is itself a homotopy equivalence, as desired.
Theorem 7.2.2.26. Let X be an ∞-topos, n ≥ 1, and A an abelian groupobject of Disc(X). Then
(1) The composite map
1
θ : GerbAn (X) → Gerbn (X) ⊆ Fun(∆ , X) → Fun({1}, X) ' X

is a right fibration.
(2) The right fibration θ is representable by an Eilenberg-MacLane object
K(A, n + 1).
Proof. For each object X ∈ X, we let AX denote the projection A × X → X
viewed as an abelian group object of Disc(X/X ). The functor
φ : BandA (X) → Band(X)is a fibration in groupoids, so that N φ is a right fibration (Proposition
2.1.1.3). The functor θ admits a factorization
θ0

θ 00

GerbA
n (X) → Gerbn (X) → X,
where θ00 is a right fibration (Remark 7.2.2.23) and θ0 is a pullback of N φand therefore also a right fibration. It follows that θ, being a composition ofright fibrations, is a right fibration; this proves (1).
To prove (2), we consider an Eilenberg-MacLane object u : 1X → K(A, n+1). Since K(A, n + 1) is (n + 1)-truncated and 1X is n-truncated (in fact,

highertopoi

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735

HIGHER TOPOS THEORY IN TOPOLOGY(−2)-truncated), Lemma 5.5.6.14 implies that u is n-truncated. The longexact sequence· · · → u∗ πi+1 K(A, n + 1) → πi u → πi (1X ) → i∗ πi (K(A, n + 1)) → · · ·of Remark 6.5.1.5 shows that u is n-connective and provides an isomorphism
φ : A ' πn (u) in the category of group objects of Disc(X), so that wemay view the pair (u, φ) as an object of GerbAn (X). We will complete theproof by showing that (u, φ) is a final object of GerbAn (X) (so that the rightfibration θ is representable by θ(u, φ) = K(A, n + 1)). For this, we muste → X, ψ) ∈ GerbA
show that for every object (f : X
n (X), the mapping space
MapGerbA
((f,
ψ),
(u,
φ))
is
contractible.
This
assertionis local on X. We
(X)
n
e →X
may therefore assume without loss of generality that the n-gerbe f : X
is trivial, in which case the desired result follows from Lemma 7.2.2.25.
Corollary 7.2.2.27. Let X be an ∞-topos, let n ≥ 2, and let A be an abeliangroup object of Disc(X). There is a canonical bijection of Hn+1 (X; A) withthe set of equivalence classes of n-gerbes on X banded by A.
Remark 7.2.2.28. Under the correspondence of Proposition 7.2.2.27, an ngerbe X on X admits a global section 1X → X if and only if the associatedcohomology class in Hn+1 (X; A) vanishes.
Theorem 7.2.2.29. Let X be an ∞-topos and n ≥ 2. Then X has cohomological dimension ≤ n if and only if it satisfies the following condition: any
n-connective truncated object of X admits a global section.
Proof. Suppose that X has the property that every n-connective truncatedobject X ∈ X admits a global section. As in the proof of Lemma 7.2.1.7,
we deduce that for any (n + 1)-connective truncated object X ∈ X, thespace of global sections MapX (1, X) is connected. Let k > n and let G bea sheaf of abelian groups on X. Then K(G, k) is (n + 1)-connective, so that
Hk (X, G) = ∗. Thus X has cohomological dimension ≤ n.
For the converse, let us assume that X has cohomological dimension ≤ nand let X denote an n-connective k-truncated object of X. We will show that
X admits a global section by descending induction on k. If k ≤ n−1, then X
is a final object of X and there is nothing to prove. In the general case, choose
a truncation X → τ≤k−1 X; we may assume by the inductive hypothesis thatτ≤k−1 X has a global section s : 1 → τ≤k−1 X. Form a pullback square
/X

X0

1

s


/ τ≤k−1 X.

It now suffices to prove that X 0 has a global section. We note that X 0 is
k-connective, where k ≥ n ≥ 2. It follows that X 0 is a k-gerbe on X; suppose
it is banded by an abelian group object A ∈ Disc(X). According to Corollary
7.2.2.27, X 0 is classified up to equivalence by an element in Hk+1 (X, A), whichhighertopoi

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vanishes by virtue of the fact that k+1 > n and the cohomological dimensionof X is ≤ n. Consequently, X 0 is equivalent to K(A, k) and therefore admits
a global section.
Corollary 7.2.2.30. Let X be an ∞-topos. If X has homotopy dimension≤ n, then X has cohomological dimension ≤ n. The converse holds provided
that X has finite homotopy dimension and n ≥ 2.
Proof. Only the last claim requires proof. Suppose that X has cohomologicaldimension ≤ n and homotopy dimension ≤ k. We must show that every nconnective object X of X has a global section. Choose a truncation X →τ≤k−1 X. Then τ≤k−1 X is truncated and n-connective, so it admits a global
section by Theorem 7.2.2.29. Form a pullback square
X0

/X


1


/ τ≤k−1 X.

It now suffices to prove that X 0 has a global section. But X 0 is k-connective
and therefore has a global section by virtue of the assumption that X has
homotopy dimension ≤ k.
Warning 7.2.2.31. [Wieland] The converse to Corollary 7.2.2.30 is false if
we do not assume that X has finite homotopy dimension. To see this, we
discuss the following example, which we learned from Ben Wieland. Let G
denote the group Zp of p-adic integers (viewed as a profinite group). Let C
denote the category whose objects are the finite quotients {Zp /pn Zp }n≥0
and whose morphisms are given by G-equivariant maps. We regard C as
endowed with a Grothendieck topology in which every nonempty sieve is acovering. The ∞-topos Shv(N C) is 1-localic, and the underlying ordinary
topos hτ≤0 Shv(N C) can be identified with the category BG of continuous
G-sets (that is, sets C equipped with an action of G such that the stabilizerof each element x ∈ C is an open subgroup of G). Since the profinite group
G has cohomology dimension 2 (see [70]), we deduce that X is of cohomological dimension 2. However, we will show that X is not hypercomplete andtherefore cannot be of finite homotopy dimension.
Let K be a simply connected finite CW complex whose homotopy groups
consist entirely of p-torsion (for example, we could take K to be a Moore
space M (Z/pZ)) and let X = Sing K ∈ S. Let F : N(C)op → S denote theconstant functor taking the value X. We claim that F belongs to Shv(C).
Unwinding the definitions, we must show that for each m ≤ n, the diagram
F exhibits F (Zp /pm Zp ) as equivalent to the homotopy invariants for thetrivial action of pm Zp /pn Zp on F (Zp /pn Zp ). In other words, we must showthat the diagonal embedding
α : X → Fun(BH, X)

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HIGHER TOPOS THEORY IN TOPOLOGY

is a homotopy equivalence, where H denotes the quotient group pm Zp /pn Zp .
Since both sides are p-adically complete, it will suffice to show that α is ap-adic homotopy equivalence, which follows from a suitable version of the
Sullivan conjecture (see, for example, [68]).
We define another functor F 0 : N(C)op → S, which is obtained as thesimplicial nerve of the functor described by the formulan

Zp /pn Zp 7→ Sing(K R /p Z ).
m

For m ≤ n, the loop space K R /p Z can be identified with the homotopy
fixed points of the (nontrivial) action of H = pm Zp /pn Zp ' pm Z/pn Z onnthe loop space K R /p Z : this follows from the observation that H acts freely
on R /pn Z with quotient R /pm Z. Consequently, F 0 belongs to Shv(N(C)).
n
The inclusion of K into each loop space K R /p Z induces a morphism
α : F → F 0 in the ∞-topos Shv(N(C)). Using the fact that the homotopy
groups of K are p-torsion, we deduce that the morphism α is ∞-connective(this follows from the observation that the map
X ' lim F (Zp /pn Zp ) → lim F 0 (Z/pn ZP )−→−→is a homotopy equivalence). However, the morphism α is not an equivalencein Shv(N(C)) unless K is essentially discrete. Consequently, Shv(N(C)) is not
hypercomplete and therefore cannot be of finite homotopy dimension.
In spite of Warning 7.2.2.31, many situations which guarantee that a topological space (or topos) X is of bounded cohomological dimension also guarantee that the associated ∞-topos is of bounded homotopy dimension. We
will see some examples in the next two sections.
7.2.3 Covering Dimension
In this section, we will review the classical theory of covering dimension
for paracompact spaces and then show that the covering dimension of aparacompact space X coincides with its homotopy dimension.
Definition 7.2.3.1. A paracompact topological space X has covering dimension ≤ n if the following condition is satisfied: for any open covering{Uα } of X, there exists an open refinement {Vα } of X such that each intersection Vα0 ∩ · · · ∩ Vαn+1 = ∅ provided the αi are pairwise distinct.
Remark 7.2.3.2. When X is paracompact, the condition of Definition
7.2.3.1 is equivalent to the (a priori weaker) requirement that such a refinement exist whenever {Uα } is a finite covering of X. This weaker condition
gives a good notion whenever X is a normal topological space. Moreover, if
X is normal, then the covering dimension of X (by this second definition) coincides with the covering dimension of the Stone-Čech compactification of X.
Thus the dimension theory of normal spaces is controlled by the dimension
theory of compact Hausdorff spaces.

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Remark 7.2.3.3. Suppose that X is a compact Hausdorff space, which is
written as a filtered inverse limit of compact Hausdorff spaces {Xα }, each ofwhich has dimension ≤ n. Then X has dimension ≤ n. Conversely, any compact Hausdorff space of dimension ≤ n can be written as a filtered inverse
limit of finite simplicial complexes having dimension ≤ n. Thus the dimension theory of compact Hausdorff spaces is controlled by the (completely
straightforward) dimension theory of finite simplicial complexes.
Remark 7.2.3.4. There are other approaches to classical dimension theory.
For example, a topological space X is said to have small ( large ) inductive
dimension ≤ n if every point of X (every closed subset of X) has arbitrarily
small open neighborhoods U such that ∂ U has small inductive dimension≤ n − 1. These notions are well-behaved for separable metric spaces, where
they coincide with the covering dimension (and with each other). In general,the covering dimension has better formal properties.
Our goal in this section is to prove that the covering dimension of aparacompact topological space X coincides with the homotopy dimensionof Shv(X). First, we need a technical lemma.
Lemma 7.2.3.5. Let X be a paracompact space, let k ≥ 0, and let {Uα }α∈Abe a covering of X. Suppose that for every A0 ⊆ A of T
size k + 1, we are given
a covering {Vβ }β∈B(A0 ) of the intersection UA0 = α∈A0 Uα . Then theree → A with the following
exists a covering {Wα }α∈Ae of X and a map π : A
properties:
e with π(e(1) For αe∈Aα) = α, we have Wαe ⊆ Uα .
e with π(e(2) Suppose that α
e0 , · · · , α
ek is a collection of elements of A,
αi ) =
αi . Suppose further that A0 = {α0 , . . . , αk } has cardinality (k + 1)
(in other words, the αi are all disjoint from one another). Then there
exists β ∈ B(A0 ) such that Wαe0 ∩ . . . ∩ Wαek ⊆ Vβ .
Proof. Since X is paracompact, we can find a locally finite covering {Uα0 }α∈Aof X such that each closure Uα0 is contained in Uα . Let S denoteTthe set of
all subsets A0 ⊆ A having size k + 1. For A0 ∈ S, let K(A0 ) = α∈A0 Uα .
Now sete = {(α, A0 , β) : α ∈ A0 ∈ S, β ∈ B(A0 )} ∪ A.
A
e we set π(eFor αe = (α, A0 , β) ∈ A,α) = α and[Wαe = (Uα0 −K(A00 )) ∪ (Vβ ∩ Uα0 ).
α∈A00 ∈Se we let π(α) = α and Wα = Uα0 − S
If α ∈ A ⊆ A,
α∈A0 ∈S K(A0 ). The local
finiteness of the cover {Uα0 } ensures that each Wαe is an open set. It is now
easy to check that the covering {Wαe }αe∈Ae has the desired properties.

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Theorem 7.2.3.6. Let X be a paracompact topological space of covering
dimension ≤ n. Then the ∞-topos Shv(X) of sheaves on X has homotopy
dimension ≤ n.
Proof. We make use of the results and notations of §7.1. Let B denote thecollection of all open Fσ subsets of X and fix a linear ordering on B. We may
identify Shv(X) with the simplicial nerve of the category of all functors
F :S
Bop → Kan which have the property that for any U ⊆ B with U = V ∈U V ,the natural map F (U ) → F (U) is a homotopy equivalence.
Suppose that F : Bop → Set∆ represents an n-connective sheaf; we wish toshow that the simplicial set F (X) is nonempty. It suffices to prove that F (U)is nonempty for some covering U of X; in other words, it suffices to produce
a map NU → F . The idea is that since X has finite covering dimension, we
can choose arbitrarily fine covers U such that NU is n-dimensional (in otherwords, equal to its n-skeleton).
For every simplicial set K, let K (i) denote the i-skeleton of K (the unionof all nondegenerate simplices of K of dimension ≤ i). If G : Bop → Set∆ isa simplicial presheaf, we let G(i) denote the simplicial presheaf given by theformulaG(i) (U ) = (G(U ))(i) .
We will prove the following statement by induction on i, −1 ≤ i ≤ n:(i)

• There exists an open cover Ui ⊆ B of X and a map ηi : NUi → F .
Assume that this statement holds for i = n. Passing to a refinement, we
may assume that the cover Un has the property that no more than n + 1
of its members intersect (this is the step where we shall use the assumption(n)on the covering dimension of X). It follows that NUn = NUn , and the proof
will be complete.
To begin the induction in the case i = −1, we let U−1 = {X}; the (−1)skeleton of NU−1 is empty, so that η−1 exists (and is unique).
Now suppose that Ui = {Uα }α∈A and ηi have been T
constructed, i < n.
Let A0 ⊆ A have cardinality i + 2 and set U (A0 ) = α∈A0 Uα ; then A0
determines an n-simplex of NUi (U (A0 )), so that ηi restricts to give a map
ηi,A0 : ∂ ∆i+1 → F (U (A0 )).
By assumption, F is n-connective; it follows that there is an open covering{Vβ }β∈B(A0 )of U (A0 ) such that for each Vβ there is a commutative diagram∂ ∆i+1
_/ F (U (A0 ))


∆i+1


/ F (Vβ ).

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We apply Lemma 7.2.3.5 to this data to obtain an new open cover Ui+1 ={Wαe }αe∈Ae which refines {Uα }α∈A . Refining the cover further if necessary,
we may assume that each of its members belongs to B. By functoriality, we
obtain a map(i)

NUi+1 → F.
To complete the proof, it will suffice to extend f to the (i + 1)-skeleton off0 ⊆ Ae have cardinality i + 2 and let W (Af0 ) =
the nerve of {Wα }α∈Ae. Let ATe ; then we must solve a lifting problem
f0 Wα
α
e∈A
∂ ∆i+1
_/ F (W ):
∆i+1 .
e → A denote the map of Lemma 7.2.3.5. If A0 = π(Ae0 ) has
Let π : A
cardinality smaller than i + 2, then there is a canonical extension given by
f0 ) ⊆
applying π and using ηi . Otherwise, Lemma 7.2.3.5 guarantees that W (AVβ for some β ∈ B(A0 ), so that the desired extension exists by construction.
Corollary 7.2.3.7. Let X be a paracompact topological space. The following
conditions are equivalent:(1) The covering dimension of X is ≤ n.
(2) The homotopy dimension of Shv(X) is ≤ n.
(3) For every closed subset A ⊆ X, every m ≥ n, and every continuous
map f0 : A → S m , there exists f : X → S m extending f0 .
Proof. The implication (1) ⇒ (2) is Theorem 7.2.3.6. The equivalence (1) ⇔(3) follows from classical dimension theory (see, for example, [27]). We willcomplete the proof by showing that (2) ⇒ (3). Let A be a closed subset of X,
let m ≥ n, and let f0 : A → S m a continuous map. Let B be the collectionof all open Fσ subsets of X. We define a simplicial presheaf F : B → Kan,so that an n-simplex of F (U ) is a map f rendering the diagram(U ∩ A) × |∆n |

/A





U × |∆n |

f0
f

/ Sm

commutative. To prove (3), it will suffice to show that F (X) is nonempty.
By virtue of the assumption that Shv(X) has homotopy dimension ≤ n, it
will suffice to show that F is an n-connective sheaf on X.
We first show that F is a sheaf. Choose a linear ordering on B. We mustshow that for every open covering U of U ∈ B, the natural map F(U ) →highertopoi

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HIGHER TOPOS THEORY IN TOPOLOGYF(U) is a homotopy equivalence. The proof is similar to that of Proposition
7.1.3.14. Let π : |NU |X → U be the projection; then we may identify F (U)with the simplicial set parametrizing continuous maps |NU |X → S m , whose
restriction to π −1 (A) is given by f0 . The desired equivalence now follows
from the fact that |NU |X is fiberwise homotopy equivalent to U (Lemma
7.1.3.13).
Now we claim that F is n-connective as an object of Shv(X). In otherwords, we must show that for any U ∈ B, any k ≤ n, and any map g : ∂ ∆k →F (U ), there is an open covering {Uα } of U and a family of commutative
diagrams∂ ∆ _ k

∆k

g

gα

/ F (U )
/ F (Uα ).

We may identify g with a continuous map
g : S k−1 × U → S msuch that g(z, a) = f0 (a) for a ∈ A. Choose a point x ∈ U . Consider the map
g|S k−1 × {x}. Since k − 1 < n ≤ m, this map is nullhomotopic; therefore
it admits an extension gx0 : Dk × {x} → S m . Moreover, if x ∈ A, then we
may choose gx0 to be the constant map with value f0 (x). Amalgamating g,
gx0 , and f0 , we obtain a continuous map
g00 : (S k−1 × U ) ∪ (Dk × (A ∪ {x})) → S m .
Since (S k−1 × U ) ∪ (Dk × (A ∪ {x})) is a closed subset of the paracompact
space U × Dk and the sphere S m is an absolute neighborhood retract, themap g00 extends continuously to a map g 00 : W → S m , where W is an open
neighborhood of (S k−1 ×U )∪(Dk ×(A∪{x})) in U ×Dk . The compactness of
Dk implies that W contains Dk ×Ux , where Ux ⊆ U is an open neighborhoodof x. Shrinking Ux if necessary, we may suppose that Ux belongs to B;
these open sets Ux form an open cover of U , with the required extension∆k → F (Ux ) supplied by the map g 00 |Dk × Ux .
7.2.4 Heyting Dimension
For the purposes of studying paracompact topological spaces, Definition
7.2.3.1 gives a perfectly adequate theory of dimension. However, there areother situations in which Definition 7.2.3.1 is not really appropriate. For example, in algebraic geometry one often considers the Zariski topology on an
algebraic variety X. This topology is generally not Hausdorff and is typicallyof infinite covering dimension. In this setting, there is a better dimension theory: the theory of Krull dimension. In this section, we will introduce a mild
generalization of the theory of Krull dimension, which we will call the Heyting dimension of a topological space X. We will then study the relationshiphighertopoi

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between the Heyting dimension of X and the homotopy dimension of theassociated ∞-topos Shv(X).
Recall that a topological space X is said to be Noetherian if the collectionof closed subsets of X satisfies the descending chain condition. A closed
subset K ⊆ X is said to be irreducible if it cannot be written as a finite unionof proper closed subsets of K (in particular, the empty set is not irreducible
since it can be written as an empty union). The collection of irreducible
closed subsets of X forms a well-founded partially ordered set, therefore it
has a unique ordinal rank function rk, which may be characterized as follows:• If K is an irreducible closed subset of X, then rk(K) is the smallest
ordinal which is larger than rk(K 0 ) for all proper irreducible closed
subsets K 0 ⊂ K.
We call rk(K) the Krull dimension of K; the Krull dimension of X is thesupremum of rk(K), as K ranges over all irreducible closed subsets of X.
We next introduce a generalization of the Krull dimension to a suitableclass of non-Noetherian spaces. We shall say that a topological space X is a
Heyting space if satisfies the following conditions:(1) The compact open subsets of X form a basis for the topology of X.
(2) A finite intersection of compact open subsets of X is compact (in particular, X is compact).
(3) If U and V are compact open subsets of X, then the interior of U ∪(X − V ) is compact.
Remark 7.2.4.1. Recall that a Heyting algebra is a distributive lattice Lwith the property that for any x, y ∈ L, there exists a maximal element z
with the property that x ∧ z ⊆ y. It follows immediately from our definition
that the lattice of compact open subsets of a Heyting space forms a Heyting
algebra. Conversely, given any Heyting algebra, one may form its spectrum,
which is a Heyting space. This sets up a duality between the category of sober
Heyting spaces (Heyting spaces in which every irreducible closed subset hasa unique generic point) and the category of Heyting algebras. This duality isa special case of a more general duality between coherent topological spaces
and distributive lattices. We refer the reader to [42] for further details.
Remark 7.2.4.2. Suppose that X is a Noetherian topological space. Then
X is a Heyting space since every open subset of X is compact.
Remark 7.2.4.3. If X is a Heyting space and U ⊆ X is a compact open
subset, then X and X − U are also Heyting spaces. In this case, we say that
X − U is a cocompact closed subset of X.
We next define the dimension of a Heyting space. The definition is recursive. Let α be an ordinal. A Heyting space X has Heyting dimension ≤ αhighertopoi

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if and only if, for any compact open subset U ⊆ X, the boundary of U has
Heyting dimension < α (we note that the boundary of U is also a Heyting
space); a Heyting space has Heyting dimension < 0 if and only if it is empty.
Remark 7.2.4.4. A Heyting space has dimension ≤ 0 if and only if it isHausdorff. The Heyting spaces of dimension ≤ 0 are precisely the compact totally disconnected Hausdorff spaces. In particular, they are also paracompact
spaces, and their Heyting dimension coincides with their covering dimension.
Proposition 7.2.4.5. (1) Let X be a Heyting space of dimension ≤ α.
Then for any compact open subset U ⊆ X, both U and X − U have
Heyting dimension ≤ α.
(2) Let X be a Heyting space which is a union of finitely many compact
open subsets Uα of dimension ≤ α. Then X has dimension ≤ α.
(3) Let X be a Heyting space which is a union of finitely many cocompact
closed subsets Kα of Heyting dimension ≤ α. Then X has Heyting
dimension ≤ α.
Proof. All three assertions are proven by induction on α. The first two areeasy, so we restrict our attention to (3). Let U be a compact open subsetof X having boundary B. Then U ∩ Kα is a compact open subset of Kα ,so that the boundary Bα of U ∩ Kα inS Kα has dimension ≤ α. We
S see
immediately that Bα ⊆ B ∩ Kα , so that Bα ⊆ B. Conversely, if b ∈/ Bα ,
then for every β such that b ∈ Kβ , there exists a neighborhood Vβ containing
b such thatS Vβ ∩ Kβ ∩ U = ∅. Let V be the intersection of the Vβ and let
W = V − b∈K
by construction, b ∈ W and W ∩ U = ∅, so that
/ γ Kγ . Then Sb ∈ B. Consequently, B = Bα . Each Bα is closed in Kα , thus in X andalso in B. The hypothesis implies that Bα has dimension < α. Thus theinductive hypothesis guarantees that B has dimension < α, as desired.
Remark 7.2.4.6. It is not necessarily true that a Heyting space which
is a union of finitely many locally closed subsets of dimension ≤ α is alsoof dimension ≤ α. For example, a topological space with 2 points and anondiscrete nontrivial topology has Heyting dimension 1 but is a union of
two locally closed subsets of Heyting dimension 0.
Proposition 7.2.4.7. If X is a Noetherian topological space, then the Krull
dimension of X coincides with the Heyting dimension of X.
Proof. We first prove, by induction on α, that if the Krull dimension of a
Noetherian space X is ≤ α, then the Heyting dimension of X is ≤ α. Since
X is Noetherian, it is a union of finitely many closed irreducible subspaces,
each of which automatically has Krull dimension ≤ α. Using Proposition
7.2.4.5, we may reduce to the case where X is irreducible. Consider any
open subset U ⊆ X and let Y be its boundary. We must show that Y has
Heyting dimension ≤ α. Using Proposition 7.2.4.5 again, it suffices to provehighertopoi

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this for each irreducible component of Y . Now we simply apply the inductive
hypothesis and the definition of the Krull dimension.
For the reverse inequality, we again use induction on α. Assume that X
has Heyting dimension ≤ α. To show that X has Krull dimension ≤ α, wemust show that every irreducible closed subset of X has Krull dimension≤ α. Without loss of generality, we may assume that X is irreducible. Now,
to show that X has Krull dimension ≤ α, it will suffice to show that any
proper closed subset K ⊆ X has Krull dimension < α. By the inductive
hypothesis, it will suffice to show that K has Heyting dimension < α. By
the definition of the Heyting dimension, it will suffice to show that K is theboundary of X − K. In other words, we must show that X − K is dense in
X. This follows immediately from the irreducibility of X.
We now prepare the way for our vanishing theorem. First, we introduce a
modified notion of connectivity:
Definition 7.2.4.8. Let X be a Heyting space and k any integer. Let F ∈Shv(V ) be a sheaf of spaces on a compact open subset V ⊆ X. We will saythat F is strongly k-connective if the following condition is satisfied: for every
compact open subset U ⊆ V and every map ζ : ∂ ∆m → F(U ), there exists
a cocompact closed subset K ⊆ U such that K ⊆ X has Heyting dimension< m − k, an open cover {Uα } of U − K, and a collection of commutative
diagrams∂ ∆ m
_

∆m

ζ

ηα

/ F(U )
/ F(Uα ).

Remark 7.2.4.9. There is a slight risk of confusion with the terminology
of Definition 7.2.4.8. The condition that a sheaf F on V ⊆ X be strongly
k-connective depends not only on V and F but also on X: this is because the
Heyting dimension of a cocompact closed subset K ⊆ U can increase when
we take its closure K in X.
Remark 7.2.4.10. Strong k-connectivity is an unstable analogue of theconnectivity conditions on complexes of sheaves associated to the dual of
the standard perversity. For a discussion of perverse sheaves in the abeliancontext, we refer the reader to [6].
Remark 7.2.4.11. It follows easily from the definition that a strongly kconnective sheaf F on V ⊆ X is k-connective. Conversely, suppose that X
has Heyting dimension ≤ n and that F is k-connective; then F is strongly(k − n)-connective (if ∂ ∆m → F(U ) is any map, then we may take K = U
for m > n and K = ∅ for m ≤ n).
The strong k-connectivity of a sheaf F is by definition a local property.
The key to our vanishing result is that this is equivalent to an apparently
stronger global property.

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Lemma 7.2.4.12. Let X be a Heyting space, let V be a compact open subsetof X, and F : U(V )op → Kan a strongly k-connective sheaf on V . Let A ⊆ B
be an inclusion of finite simplicial sets of dimension ≤ m, let U ⊆ V , andlet ζ : A → F(U ) be a map of simplicial sets.
There exists a cocompact closed subset K ⊆ U whose closure K ⊆ X has
Heyting dimension < m − 1 − k, an open covering {Uα } of U − K, and acollection of commutative diagrams
A _

B

ζ

ηα

/ F(U )
/ F(Uα ).

Proof. Induct on the number of simplices of B which do not belong to A,
and invoke Definition 7.2.4.8.
Lemma 7.2.4.13. Let X be a Heyting space, V a compact open subset ofX, let F : U(V )op → Kan be a sheaf on X, let η : ∂ ∆m → F(V ) be a map,and form a pullback square
/ F∆

F0

∗

η

m


m
/ F∂ ∆ .

If F is strongly k-connective, then F0 is strongly (k − m)-connective.
Proof. Unwinding the definitions, we must show that for every compact U ⊂V and every mapaζ : (∂ ∆m × ∆n )(∆m × ∂ ∆n ) → F(U )∂ ∆m ×∂ ∆nmnwhose restriction ζ| ∂ ∆ × ∆ is given by η, there exists a cocompact closed
subset K ⊆ U such that K ⊆ X has Heyting dimension < n + m − k, an
open covering {Uα } of U − K, and a collection of maps
ζα : ∆m × ∆n → F(Uα )which extend ζ. This follows immediately from Lemma 7.2.4.12.
Theorem 7.2.4.14. Let X be a Heyting space of dimension ≤ n, let W ⊆ X
be a compact open set, and let F ∈ Shv(W ). The following conditions areequivalent:(1) For any compact open sets U ⊆ V ⊆ W and any commutative diagram∂ ∆ m
_

∆m

ζ

η

/ F(V )
/ F(U ),

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there exists a cocompact closed subset K ⊆ V − U such that K ⊆ X
has dimension < m − k and a commutative diagram/ F(V )ζ∂ ∆ m_∆m/ F(V − K)η0η0such that the composition ∆m → F(V − K) → F(U ) is homotopic toη relative to ∂ ∆m .
(2) For any compact open sets V ⊆ W and any map ζ : ∂ ∆m → F(V ),there exists a commutative diagram/ F(V )ζ∂ ∆ m
_

∆m

η0


/ F(V − K),

where K ⊆ V is a cocompact closed subset and K ⊆ X has dimension< m − k.
(3) The sheaf F is strongly k-connective.
Proof. It is clear that (1) implies (2) (take U to be empty) and that (2)implies (3) (by definition). We must show that (3) implies (1). So let F be astrongly k-connective sheaf on W andζ∂ ∆ m_∆mη/ F(V )
/ F(U )

a commutative diagram as above. Without loss of generality, we may replace
W by V and F by F |V .
We may identify F with a functor from U(V )op into the category Kan of
Kan complexes. Form a pullback square
/ F∆

m

F0

∗
U(V )opζm/ F∂ ∆

in Set∆. The right vertical map is a projective fibration, so that the diagram is homotopy Cartesian (with respect to the projective model structure).
It follows that F0 is also a sheaf on V , which is strongly (k − m)-connective
by Lemma 7.2.4.13. Replacing F by F0 , we may reduce to the case m = 0.

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The proof now proceeds by induction on k. For our base case, we takek = −n − 1, so that there is no connectivity assumption on the stack F. We
are then free to choose K = V − U (it is clear that K has dimension ≤ n).
Now suppose that the theorem is known for strongly (k − 1)-connective
stacks on any compact open subset of X; we must show that for any strongly
k-connective F on V and any η ∈ F(U ), there exists a closed subset K ⊆ V −U such that K ⊆ X has Heyting dimension < −k and a point η 0 ∈ F(V − K)
whose restriction to U lies in the same component of F(U ) as η.
Since F is strongly k-connective, we deduce that there exists an open
cover {Vα } of some open subset V − K0 , where K0 has dimension < −k in
X, together with points ψα ∈ F(Vα ). Adjoining the open set U and the pointη if necessary, we may suppose that K0 ∩ U = ∅. Replacing V by V − K0 ,we may reduce to the case K0 = ∅.
Since V is compact, we may assume that there exist only finitely many
indices α. Proceeding by induction on the number of indices, we may reduceto the case where V = U ∪ Vα for some α. Let η 0 and ψ 0 denote the imagesof η and ψ in U ∩ Vα and form a pullback diagram/ (F |(U ∩ Vα ))∆1F0
∗(η 0 ,ψ 0 )
/ (F |(U ∩ Vα ))∂ ∆1 .

Again, this diagram is a homotopy pullback, so that F0 is a sheaf on U ∩ Vα
which is strongly (k − 1)-connective by Lemma 7.2.4.13. According to the
inductive hypothesis, there exists a closed subset K ⊂ U ∩ Vα such that K ⊆X has dimension < −k + 1, such that the images of ψα and η belong to thesame component of F((U ∩ Vα ) − K). Since K has dimension < −k + 1 in X,
the boundary ∂ K of K has codimension < −k in X. Let V 0 = Vα −(Vα ∩K).
Since F is a sheaf, we have a homotopy pullback diagram/ F(U )
F(V 0 ∪ U )
F(V 0 )
/ F(V 0 ∩ U ).

We observe that there is a path joining the images of η and ψα in F(V 0 ∩U ) =F((U ∩Vα )−K), so that there is a vertex ηe ∈ F(V 0 ∪U ) whose image in F(U )lies in the same component as η. We now observe that V 0 ∪U = V −(V ∩∂ K)
and that V ∩ ∂ K is contained in ∂ K and therefore has Heyting dimension≤ −k.
Corollary 7.2.4.15. Let π : X → Y be a continuous map between Heyting
spaces of finite dimension. Suppose that π has the property that for any
cocompact closed subset K ⊆ X of dimension ≤ n, π(K) is contained in a
cocompact closed subset of dimension ≤ n. Then the functor π∗ : Shv(X) →Shv(Y ) carries strongly k-connective sheaves on X to strongly k-connective
sheaves on Y .

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Proof. This is clear from the characterization (2) of Theorem 7.2.4.14.
Corollary 7.2.4.16. Let X be a Heyting space of finite Heyting dimension
and let F be a strongly k-connective sheaf on X. Then F(X) is k-connective.
Proof. Apply Corollary 7.2.4.15 in the case where Y is a point.
Corollary 7.2.4.17. Let X be a Heyting space of Heyting dimension ≤ nand let F be an n-connective sheaf on X. Then for any compact open U ⊆X, the map π0 F(X) → π0 F(U ) is surjective. In particular, Shv(X) has
homotopy dimension ≤ n.
Proof. Suppose first that (1) is satisfied. Let F be an n-connective sheaf
on X. Then F is strongly 0-connective; by characterization (2) of Theorem
7.2.4.14, we deduce that F(X) → F(U ) is surjective. The last claim follows
by taking U = ∅.
Remark 7.2.4.18. Let X be a Heyting space of Heyting dimension ≤ n.
Then any compact open subset of X also has Heyting dimension ≤ n. It
follows that Shv(X) is locally of homotopy dimension ≤ n and therefore
hypercomplete by Corollary 7.2.1.12.
Remark 7.2.4.19. It is not necessarily true that a Heyting space X suchthat Shv(X) has homotopy dimension ≤ n is itself of Heyting dimension≤ n. For example, if X is the Zariski spectrum of a discrete valuation ring(that is, a 2-point space with a nontrivial topology), then X has homotopy
dimension zero (see Example 7.2.1.3).
In particular, we obtain Grothendieck’s vanishing theorem (see [34] for theoriginal, quite different, proof):
Corollary 7.2.4.20. Let X be a Noetherian topological space of Krull dimension ≤ n. Then X has cohomological dimension ≤ n.
Proof. Combine Proposition 7.2.4.7 with Corollaries 7.2.4.17 and 7.2.2.30.

Example 7.2.4.21. Let V be a real algebraic variety (defined over the real
numbers, say). Then the lattice of open subsets of V that can be defined bypolynomial equations and inequalities is a Heyting algebra, and the spectrumof this Heyting algebra is a Heyting space X having dimension at most equal
to the dimension of V . The results of this section therefore apply to X.
More generally, let T be an o-minimal theory (see for example [81]) and let
Sn denote the set of complete n-types of T . We endow Sn with the topology
generated by the sets Uφ = {p : φ ∈ p}, where φ ranges over formulas withn free variables such that the openness of the set of points satisfying φ isprovable in T . Then Sn is a Heyting space of Heyting dimension ≤ n.

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Remark 7.2.4.22. The methods of this section can be adapted to slightly
more general situations, such as the Nisnevich topology on a Noetherian
scheme of finite Krull dimension. It follows that the ∞-topoi associatedto such sites have (locally) finite homotopy dimension and are therefore
hypercomplete. We will discuss this matter in more detail in [51].

7.3 THE PROPER BASE CHANGE THEOREM
Let
X0
p0

q0

/X
p



q
/Y
Y0
be a pullback diagram in the category of locally compact Hausdorff spaces.
One has a natural isomorphism of pushforward functorsq∗ p0∗ ' p∗ q∗0
from the category of sheaves of sets on Y to the category of sheaves of sets
on X 0 . This isomorphism induces a natural transformation∗
η : q ∗ p∗ → p0∗ q 0 .
If p (and therefore also p0 ) is a proper map, then η is an isomorphism: thisis a simple version of the classical proper base change theorem.
The purpose of this section is to generalize the above result, allowing
sheaves which take values in the ∞-category S of spaces rather than in theordinary category of sets. Our generalization can be viewed as a proper basechange theorem for nonabelian cohomology.
We will begin in §7.3.1 by defining the notion of a proper morphism of∞-topoi. Roughly speaking, a geometric morphism π∗ : X → Y of ∞-topoi
is proper if and only if it satisfies the conclusion of the proper base change
theorem. Using this language, our job is to prove that a proper map of
topological spaces p : X → Y induces a proper morphism p∗ : Shv(X) →Shv(Y ) of ∞-topoi. We will outline the proof of this result in §7.3.1 byreducing to two special cases: the case where p is a closed embedding andthe case where Y is a point. We will treat the first case in §7.3.2, after
introducing a general theory of closed immersions of ∞-topoi. This allows
us to reduce to the case where Y is a point and X is a compact Hausdorff
space. Our approach is now in two parts:(1) In §7.3.3, we will show that we can identify the ∞-category Shv(X 0 ) =Shv(X × Y 0 ) with an ∞-category of sheaves on X taking values in
Shv(Y 0 ).
(2) In §7.3.4, we give an analysis of the category of sheaves on a compact
Hausdorff space X taking values in a general ∞-category C. Combining
this analysis with (1), we will deduce the desired base change theorem.

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The techniques used in §7.3.4 to analyze Shv(X) can also be applied inthe (easier) setting of coherent topological spaces, as we explain in §7.3.5.
Finally, we conclude in §7.3.6 by reformulating the classical theory of cell-likemaps in the language of ∞-topoi.
7.3.1 Proper Maps of ∞-Topoi
In this section, we introduce the notion of a proper geometric morphism
between ∞-topoi. Here we follow the ideas of [59] and turn the conclusionof the proper base change theorem into a definition. First, we require a bitof terminology.
Suppose we are given a diagram of categories and functors
C0

q∗0

p0∗


C

/ D0
p∗

q∗


/D

which commutes up to a specified isomorphism η : p∗ q∗0 → q∗ p0∗ . Suppose
furthermore that the functors q∗ and q∗0 admit left adjoints, which we will∗
denote by q ∗ and q 0 . Consider the composition
∗ η

u

∗ v

∗

γ : q ∗ p∗ → q ∗ p∗ q∗0 q 0 → q ∗ q∗ p0∗ q 0 → p0∗ q 0 ,∗where u denotes a unit for the adjunction (q 0 , q∗0 ) and v a counit for the
adjunction (q ∗ , q∗ ). We will refer to γ as the push-pull transformation associated to the above diagram.
Definition 7.3.1.1. A diagram of categories
C0

q∗0

p0∗

/ D0
p∗



q∗
/D
C
which commutes up to a specified isomorphism is left adjointable if the func∗
tors q∗ and q∗0 admit left adjoints q ∗ and q 0 and the associated push-pulltransformation
γ : q ∗ p∗ → p0∗ q 0∗

is an isomorphism of functors.
Definition 7.3.1.2. A diagram of ∞-categories
C0

q∗0

p0∗


C

/ D0
p∗

q∗


/D

which commutes up to (specified) homotopy is left adjointable if the associated diagram of homotopy categories is left adjointable.

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Remark 7.3.1.3. Suppose we are given a diagram of simplicial setsfP

M0 → M → ∆1 ,where both f and f ◦ P are Cartesian fibrations. Then we may view M
as a correspondence from D = f −1 {0} to C = f −1 {1} associated to somefunctor q∗ : C → D. Similarly, we may view M0 as a correspondence from
D0 = (f ◦ P )−1 {0} to C0 = (f ◦ P )−1 {1} associated to some functor q∗0 :C0 → D0 . The map P determines functors p0∗ : C0 → C, q∗0 : D0 → D and(up to homotopy) a natural transformation α : p∗ q∗0 → q∗ p0∗ , which is anequivalence if and only if the map P carries (f ◦ P )-Cartesian edges of M0
to f -Cartesian edges of M. In this case, we obtain a diagram of homotopy
categories
q∗0

hC0
p0∗

/ hD0
p∗


hC

q∗


/ hD

which commutes up to canonical isomorphism.
Now suppose that the functors q∗ and q∗0 admit left adjoints, which we∗will denote by q ∗ and q 0 , respectively. Then the maps f and f ◦ P are coCartesian fibrations. Moreover, the associated push-pull transformation canbe described as follows. Choose an object D0 ∈ D0 and a (f ◦ P )-coCartesianmorphism φ : D0 → C 0 , where C 0 ∈ C. Let D = P (D0 ) and choose an f coCartesian morphism ψ : D → C in M, where C ∈ C. Using the fact that
ψ is f -coCartesian, we can choose a 2-simplex in M depicted as follows:
?C
ψ

θ
P (φ)

D

"
/ P (C 0 ).
∗We may then identify C with q ∗ p∗ D0 , P (C 0 ) with p0∗ q 0 D0 , and θ with the∗value of the push-pull transformation q ∗ p∗ → p0∗ q 0 D0 on the object D0 ∈ D0 .
The morphism θ is an equivalence if and only if P (φ) is f -coCartesian.
Consequently, we deduce that the original diagram
hC0

q∗0

p0∗


hC

/ hD0
p∗

q∗


/ hD

is left adjointable if and only if P carries (f ◦ P )-coCartesian edges to f coCartesian edges. We will make use of this criterion in §7.3.4.
Definition 7.3.1.4. Let p∗ : X → Y be a geometric morphism of ∞-topoi.
We will say that p∗ is proper if the following condition is satisfied:highertopoi

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CHAPTER 7(∗) For every Cartesian rectangle
X00

/ X0

/X


Y00


/ Y0


/Y

p∗

of ∞-topoi, the left square is left adjointable.
Remark 7.3.1.5. Let X be an ∞-topos and let J be a small ∞-category.
The diagonal functor δ : X → Fun(J, X) preserves all (small) limits andcolimits, by Proposition 5.1.2.2, and therefore admits both a left adjoint
δ! and a right adjoint δ∗ . If J is filtered, then δ! is left exact (Proposition
5.3.3.3). Consequently, we have a diagram of geometric morphismsδδ∗
X → Fun(J, X) →X.

Now suppose that p∗ : X → Y is a proper geometric morphism of ∞-topoi.
We obtain a rectangle
X
p∗


Y

/X

/ Fun(J, X)
pJ∗

/ Fun(J, Y)/Y

which commutes up to (specified) homotopy. One can show that this is a
Cartesian rectangle in RTop, so that the square on the left is left adjointable.
Unwinding the definitions, we conclude that p∗ commutes with filtered colimits. Conversely, if p∗ : X → Y is an arbitrary geometric morphism of∞-topoi which commutes with colimits indexed by filtered Y-stacks (overeach object of Y), then p∗ is proper. To give a proof (or even a precise formulation) of this statement would require ideas from relative category theory
which we will not develop in this book. We refer the reader to [59], where
the analogous result is established for proper maps between ordinary topoi.
The following properties of the class of proper morphisms follow immediately from Definition 7.3.1.4:
Proposition 7.3.1.6.

(1) Every equivalence of ∞-topoi is proper.

(2) If p∗ and p0∗ are equivalent geometric morphisms from an ∞-topos X
to another ∞-topos Y, then p∗ is proper if and only if p0∗ is proper.
(3) Let
X0
p0∗


Y0

/X

/Y

p∗

be a pullback diagram of ∞-topoi. If p∗ is proper, then so is p0∗ .

highertopoi

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HIGHER TOPOS THEORY IN TOPOLOGY(4) Let
q∗

p∗

X→Y→Z
be proper geometric morphisms between ∞-topoi. Then q∗ ◦ p∗ is aproper geometric morphism.
In order to relate Definition 7.3.1.4 to the classical statement of the properbase change theorem, we need to understand the relationship between products in the category of topological spaces and products in the ∞-categoryof ∞-topoi. A basic result asserts that these are compatible provided that acertain local compactness condition is met.
Definition 7.3.1.7. Let X be a topological space which is not assumed to
be Hausdorff. We say that X is locally compact if, for every open set U ⊆ X
and every point x ∈ U , there exists a (not necessarily closed) compact set
K ⊆ U , where K contains an open neighborhood of x.
Example 7.3.1.8. If X is Hausdorff space, then X is locally compact in thesense defined above if and only if X is locally compact in the usual sense.
Example 7.3.1.9. Let X be a topological space for which the compact open
subsets of X form a basis for the topology of X. Then X is locally compact.
Remark 7.3.1.10. Local compactness of X is precisely the condition needed
for function spaces Y X , endowed with the compact-open topology, to represent the functor Z 7→ Hom(Z × X, Y ).
Proposition 7.3.1.11. Let X and Y be topological spaces and assume that
X is locally compact. The diagram
Shv(X × Y )/ Shv(X)
Shv(Y )

/ Shv(∗)

is a pullback square in the ∞-category RTop of ∞-topoi.
Proof. Let C ⊆ RTop be the full subcategory spanned by the 0-localic ∞topoi. Since C is a localization of RTop, the inclusion C ⊆ RTop preserves
limits. It therefore suffices to prove that
Shv(X × Y )/ Shv(X)
Shv(Y )

/ Shv(∗)

gives a pullback diagram in C. Note that Cop is equivalent to the (nerve ofthe) ordinary category of locales. For each topological space M , let U(M )denote the locale of open subsets of M . Let
ψXψY

U(X) → P ← U(Y )highertopoi

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be a diagram which exhibits P as a coproduct of U(X) and U(Y ) in the
category of locales and let φ : P → U(X × Y ) be the induced map. We wish
to prove that φ is an isomorphism. This is a standard result in the theory of
locales; we will include a proof for completeness.
Given open subsets U ⊆ X and V ⊆ Y , let U ⊗ V = (ψX U ) ∩ (ψY V ) ∈ P,
so that φ(U ⊗ V ) = U × V ∈ U(X × Y ). We define a map θ : U(X × Y ) → P
by the formula[θ(W ) =U ⊗ V.
U ×V ⊆W

Since every open subset of X × Y can be written as a union of products
U × V , where U is an open subset of X and V is an open subset of Y , it isclear that φ ◦ θ : U(X × Y ) → U(X × Y ) is the identity. To complete theproof, it will suffice to show
S that θ ◦ φ : P → P is the identity. Every elementof P can be written as α Uα ⊗ Vα for Uα ⊆ X and Vα ⊆ Y appropriately
chosen. We therefore wish to show that[[U ×V =
Uα ⊗ Vα .
U ×V ⊆S

α Uα ⊗Vαα

It is clear that the right hand side is contained in the left hand
S side. The
reverse containmentisequivalent
totheassertion
that
if
U
×V
⊆
α Uα ×Vα ,Sthen U ⊗ V ⊆ α Uα ⊗ Vα .
S
We now invoke the local compactness of X. Write U =Kβ , where
each Kβ S
is a compact subset of U and the interiors {Kβ◦ } cover
S U . ThenU ⊗V = β Kβ◦ ⊗V ; it therefore suffices to prove that Kβ◦ ⊗V ⊆S α Uα ⊗Vα .
Let v be a point of V . Then Kβ × {v} is a compact subset of α Uα × Vα .
Consequently, there exists a finite set of indices {α1 , . . . , αn } such that v ∈Vv,β = Vα1 ∩ · · · ∩ Vαn and Kβ ⊆ Uα1 ∪ · · · ∪ Uαn . It follows that Kβ◦ ⊗ Vv,β ⊆S
α Uα ⊗ Vα . Taking a union over all v ∈ V , we deduce the desired result.
Let us now return to the subject of the proper base change theorem.
We have essentially defined a proper morphism of ∞-topoi to be one forwhich the proper base change theorem holds. The challenge, then, is to
produce examples of proper geometric morphisms. The following results will
be proven in §7.3.2 and §7.3.4, respectively:(1) If p : X → Y is a closed embedding of topological spaces, then p∗ :Shv(X) → Shv(Y ) is proper.
(2) If X is a compact Hausdorff space, then the global sections functor
Γ : Shv(X) → Shv(∗) is proper.
Granting these statements for the moment, we can deduce the main resultof this section. First, we must recall a bit of point-set topology:
Definition 7.3.1.12. A topological space X is said to be completely regular
if every point of X is closed in X and if for every closed subset Y ⊆ X andevery point x ∈ X − Y there is a continuous function f : X → [0, 1] suchthat f (x) = 0 and f |Y takes the constant value 1.

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Remark 7.3.1.13. A topological space X is completely regular if and only
if it is homeomorphic to a subspace of a compact Hausdorff space X (see[60]).
Definition 7.3.1.14. A map p : X → Y of (arbitrary) topological spaces issaid to be proper if it is universally closed. In other words, p is proper if andonly if for every pullback diagram of topological spaces
X0
p0


Y0

/X

/Y

p

the map p0 is closed.
Remark 7.3.1.15. A map p : X → Y of topological spaces is proper if
and only if it is closed and each of the fibers of p is compact (though notnecessarily Hausdorff).
Theorem 7.3.1.16. Let p : X → Y be a proper map of topological spaces,
where X is completely regular. Then p∗ : Shv(X) → Shv(Y ) is proper.
Proof. Let q : X → X be an identification of X with a subspace of a compact
Hausdorff space X. The map p admits a factorization
q×p

π

Y
X → X ×Y →Y.

Using Proposition 7.3.1.6, we can reduce to proving that (q × p)∗ and (πY )∗are proper.
Because q identifies X with a subspace of X, q × p identifies X with asubspace over X × Y . Moreover, q × p factors as a composition
X → X × X → X × Y,
where the first map is a closed immersion (since X is Hausdorff) and thesecond map is closed (since p is proper). It follows that q × p is a closed
immersion, so that (q × p)∗ is a proper geometric morphism by Proposition
7.3.2.12.
Proposition 7.3.1.11 implies that the geometric morphism (πY )∗ is a pullback of the global sections functor Γ : Shv(X) → Shv(∗) in the ∞-category
RTop. Using Proposition 7.3.1.6, we may reduce to proving that Γ is proper,
which follows from Corollary 7.3.4.11.
Remark 7.3.1.17. The converse to Theorem 7.3.1.16 holds as well (anddoes not require the assumption that X is completely regular): if p∗ :Shv(X) → Shv(Y ) is a proper geometric morphism, then p is a proper mapof topological spaces. This can be proven easily using the characterizationof properness described in Remark 7.3.1.5.

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Corollary 7.3.1.18 (Nonabelian Proper Base Change Theorem). Let
X0

q0

p0


Y0

q

/X

/Y

p

be a pullback diagram of locally compact Hausdorff spaces and suppose thatp is proper. Then the associated diagram
Shv(X 0 )

q∗0

p0∗


Shv(Y 0 )/ Shv(X)p∗q∗/ Shv(Y )is left adjointable.
Proof. In view of Theorem 7.3.1.16, it suffices to show that
Shv(X 0 )

q∗0

p0∗


Shv(Y 0 )/ Shv(X)p∗q∗/ Shv(Y )is a pullback diagram of ∞-topoi. Let X denote a compactification of X (forexample, the one-point compactification) and consider the larger diagram of∞-topoiShv(X 0 )/ Shv(X)
Shv(X × Y 0 )/ Shv(X × Y )/ Shv(X)
Shv(Y 0 )/ Shv(Y )/ Shv(∗).

The upper square is a (homotopy) pullback by Proposition 7.3.2.12 andCorollary 7.3.2.10. Both the lower right square and the lower rectangle are(homotopy) Cartesian by Proposition 7.3.1.11, so that the lower left squareis (homotopy) Cartesian as well. It follows that the vertical rectangle is also
(homotopy) Cartesian, as desired.
Remark 7.3.1.19. The classical proper base change theorem, for sheaves ofabelian groups on locally compact topological spaces, is a formal consequenceof Corollary 7.3.1.18. We give a brief sketch. The usual formulation of thehighertopoi

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HIGHER TOPOS THEORY IN TOPOLOGY

proper base change theorem (see, for example, [46]) is equivalent to thestatement that if
X0

q0

p0


Y0

q

/X

/Y

p

is a pullback diagram of locally compact Hausdorff topological spaces and p
is proper, then the associated diagram
D+ (X 0 )

q∗0

p0∗


D+ (Y 0 )/ D+ (X)p∗q∗/ D+ (Y )is left adjointable. Here D+ (Z) denotes the (bounded below) derived category of abelian sheaves on a topological space Z.
Let A denote the category whose objects are chain complexes· · · → A−1 → A0 → A1 → · · ·of abelian groups. Then A admits the structure of a combinatorial model
category in which the weak equivalences are given by quasi-isomorphisms.
Let C = N(A◦ ) be the underlying ∞-category. For any topological space
Z, one can define an ∞-category Shv(Z; C) of sheaves on Z with values in
C; see §7.3.3. The homotopy category hShv(Z ; C) is an unbounded versionof the derived category D+ (Z); in particular, it contains D+ (Z) as a fullsubcategory. Consequently, we obtain a natural generalization of the properbase change theorem where boundedness hypotheses have been removed,
which asserts that the diagram
Shv(X 0 ; C)

q∗0

p0∗


Shv(Y 0 ; C)

/ Shv(X; C)
p∗

q∗


/ Shv(Y ; C)

is left adjointable. Using the fact that C has enough compact objects, one
can deduce this statement formally from Corollary 7.3.1.18.
7.3.2 Closed Subtopoi
If X is a topological space and U ⊆ X is an open subset, then we may viewthe closed complement X − U ⊆ X as a topological space in its own right.
Moreover, the inclusion (X − U ) ,→ X is a proper map of topological spaces(that is, a closed map whose fibers are compact). The purpose of this section
is to present an analogous construction in the case where X is an ∞-topos.

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Lemma 7.3.2.1. Let X be an ∞-topos and ∅ an initial object of X. Then ∅is (−1)-truncated.
Proof. Let X be an object of X. The space MapX (X, ∅) is contractible if X
is an initial object of X and empty otherwise (by Lemma 6.1.3.6). In either
case, MapX (X, ∅) is (−1)-truncated.
Lemma 7.3.2.2. Let X be an ∞-topos and let f : ∅ → X be a morphism in
X, where ∅ is an initial object. Then f is a monomorphism.
Proof. Apply Lemma 7.3.2.1 to the ∞-topos X/X .
Proposition 7.3.2.3. Let X be an ∞-topos and let U be an object of X.
Let SU be the smallest strongly saturated class of morphisms of X whichcontains all morphisms of the form ∅ → V , where ∅ is an initial object of
X and MapX (V, U ) 6= ∅. Then SU is topological (in the sense of Definition
6.2.1.4).
Proof. It is clear that SU is generated by monomorphisms. If X0 ⊆ X is an
essentially small full subcategory which generates X under small colimits,then SU is generated by those morphisms of the form ∅ → V , where V ∈ X0
and MapX (V, U ) 6= ∅. It follows that SU is of small generation, so that SU−1 X
is an accessible localization of X. Let L : X → SU−1 X be a left adjoint to theinclusion; we wish to show that the functor L is left exact.
Since X is an ∞-topos, we may assume without loss of generality that it
is given as an accessible left exact localization of P(C), for some small ∞category C. We will identify X with its essential image in P(C). Under thisidentification, we can think of U as a functor Cop → S. Let L0 : P(C) → X bea left adjoint to the inclusion and let j : C → P(C) denote the Yoneda embedding. Let C(0) denote the full subcategory of C spanned by those objects
C ∈ C for which U (C) 6= ∅. Note that X is generated under small colimits by
the essential image of L0 ◦ j, so that SU is generated (as a strongly saturated
class of morphisms) by morphisms of the form L0 (f ), where f : ∅ → j(C) isa morphism from the initial object of P(C) to a representable presheaf j(C)where C ∈ C(0) .
(1)
Fix an object C ∈ C. We will say that a sieve C/C ⊆ C/C is covering if(1)either C ∈ C(0) or C/C = C/C . Using the fact that C(0) is a sieve on C, we seethat the collection of covering sieves determines a Grothendieck topology on
C. Note that L factors as a composition
L◦L0

X ⊆ P(C) −→ SU−1 X .
It will therefore suffice to show that the functor L ◦ L0 is left exact. Since X
and Shv(C) are both accessible left exact localizations of P(C) (Proposition
6.2.2.7), this follows from Lemma 6.3.3.4 together with the following:(∗) We have SU−1 X = Shv(C) ∩ X.

highertopoi

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759

HIGHER TOPOS THEORY IN TOPOLOGYTo prove (∗), we must show that an object X ∈ X is SU -local if and only
if it is a sheaf with respect to the topology considered above. We know that
SU is generated by morphisms of the form L0 (f ), where f is a morphismfrom the initial object of P(C) to a representable presheaf j(C) for C ∈ C(0) .
It follows that X is SU -local if and only if X(C) is contractible for C ∈ C(0) .
Since the empty sieve ∅ ⊆ C/C is covering for C ∈ C(0) , the containment
Shv(C) ∩ X ⊆ SU−1 X is clear. To prove the converse, let us suppose that
X(C) is contractible for each C ∈ C(0) ; we wish to show that X is a sheaf.
(1)
Fix an arbitrary object D ∈ C and a covering sieve C/D ⊆ C/D ; we wish
to show that the canonical map X(D) → limD0 ∈C(1) X(D0 ) is a homotopy←−/Dequivalence. If D ∈ C(0) , then both sides are contractible and there is nothing(1)
to prove. If D ∈/ C(0) , then C/D = C/D contains D as a final object, and theresult is again obvious.
In the situation of Proposition 7.3.2.3, we will say that a morphism of X
is an equivalence away from U if it belongs to SU .
Lemma 7.3.2.4. Let X be an ∞-topos containing a pair of objects U, X ∈ X
and let SU denote the class of morphism in X which are equivalences awayfrom U . The following are equivalent:(1) The object X is SU -local.
e → U in X, the space MapX (Ue , X) is contractible.
(2) For every map U(3) There exists a morphism g : U → X such that the diagram
U
g


U
X
exhibits U as a product of U and X in X.
idU

Proof. The equivalence (1) ⇔ (2) is immediate fro the defintions. Now supe = U , we deduce that there exists a
pose that (2) is satisfied. Taking U
morphism g : U → X. We will prove that g and idX exhibit U as a productof U and X. As explained in §4.4.1, this is equivalent to the assertion thatfor every Z ∈ X, the map
MapX (Z, U ) → MapX (Z, U ) × MapX (Z, X)
is an isomorphism in H. If there are no morphisms from Z to U in X, thenboth sides are empty and the result is obvious. Otherwise, we may invoke (2)to deduce that MapX (Z, X) is contractible, and the desired result follows.
This completes the proof that (2) ⇒ (3).
Suppose now that (3) is satisfied for some morphism g : U → X. For any
object Z ∈ X, we have a homotopy equivalence
MapX (Z, U ) → MapX (Z, U ) × MapX (Z, X).

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If MapX (Z, U ) is nonempty, then we may pass to the fiber over a point of
MapX (Z, U ) to obtain a homotopy equivalence ∗ → MapX (Z, X), so that
MapX (Z, X) is contractible. This proves (2).
If X is an ∞-topos and U ∈ X, then we will say that an object X ∈ X is
trivial on U if it satisfies the equivalent conditions of Lemma 7.3.2.4. We let
X /U denote the full subcategory of X spanned by the objects X which aretrivial on U . It follows from Proposition 7.3.2.3 that X /U is a topologicallocalization of X and, in particular, that X /U is an ∞-topos. We next showthat X /U depends only on the support of U .
Lemma 7.3.2.5. Let X be an ∞-topos and let g : U → V be a morphism
in X. Then X /V ⊆ X /U . Moreover, if g is an effective epimorphism, then
X /U = X /V .
Proof. Consider an arbitrary object X ∈ X, and form a pullback diagram/ X ×V

X ×U

U

g


/ V.

It is clear that the left vertical map is an equivalence whenever the rightvertical map is an equivalence. Moreover, the converse holds if g is an effective
epimorphism (Lemma 6.2.3.16). The desired result now follows from criterion(3) of Lemma 7.3.2.4.
If X is an ∞-topos, we let Sub(1X ) denote the partially ordered set of
equivalence classes of (−1)-truncated objects of X. We note that this set isindependent of the choice of a final object 1X ∈ X up to canonical isomorphism. Any U ∈ Sub(1X ) can be represented by a (−1)-truncated objecte ∈ X. We define X /U = X /Ue ⊆ X. It follows from Lemma 7.3.2.5 thatUe representing U and that for any obX /U is independent of the choice of U
ject X ∈ X, we have X /X = X /U , where U ∈ Sub(1X ) is the “support” ofX (namely, the equivalence class of the truncation τ−1 X).
Definition 7.3.2.6. If X is an ∞-topos and U ∈ Sub(1X ), then we will refer
to X /U as the closed subtopos of X complementary to U . More generally, we
will say that a geometric morphism π : Y → X is a closed immersion if there
exists U ∈ Sub(1X ) such that π∗ induces an equivalence of ∞-categories
from Y to X /U .
Proposition 7.3.2.7. Let X be an ∞-topos and let U ∈ Sub(1X ). Then theclosed immersion
π : X /U → X
induces an isomorphism of partially ordered sets from Sub(1X /U ) to {V ∈Sub(1X ) : U ⊆ V }).

highertopoi

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HIGHER TOPOS THEORY IN TOPOLOGY761

e ∈ X representing U . Since π ∗ is
Proof. Choose a (−1)-truncated object U
left exact, an object X of X /U is (−1)-truncated as an object of X /U if
and only if it is (−1)-truncated as an object of X. It therefore suffices toprove that if Ve is a (−1)-truncated object of X representing an element
V ∈ Sub(1X ), then Ve is SU -local if and only if U ⊆ V . One direction isclear: if Ve is SU -local, then we have an isomorphisme , Ve ) → MapX (∅, Ve ) = ∗MapX (Uin the homotopy category H, so that U ⊆ V . The converse follows fromcharacterization (3) given in Lemma 7.3.2.4.
Corollary 7.3.2.8. Let X be an ∞-topos and let U, V ∈ Sub(1X ). Then
SU ⊆ SV if and only if U ⊆ V .
Proof. The “if” direction follows from Lemma 7.3.2.5, and the converse from
Proposition 7.3.2.7.
Corollary 7.3.2.9. Let X be a 0-localic ∞-topos associated to the locale U
and let U ∈ U. Then X /U is a 0-localic ∞-topos associated to the locale{V ∈ U : U ⊆ V }.
Proof. The ∞-topos X /U is a topological localization of a 0-localic ∞-topos
and is therefore also 0-localic (Proposition 6.4.5.9). The identification of theunderlying locale follows from Proposition 7.3.2.7.
Corollary 7.3.2.10. Let X be a topological space, U ⊆ X an open subset,
and Y = X − U . The inclusion of Y in X induces a closed immersion of∞-topoi Shv(Y ) → Shv(X) and an equivalence Shv(Y ) → Shv(X)/U .
Lemma 7.3.2.11. Let X and Y be ∞-topoi and let U ∈ Y be an object. The
map
Fun∗ (X, Y /U ) → Fun∗ (X, Y)
identifies Fun∗ (X, Y /U ) with the full subcategory of Fun∗ (X, Y) spanned bythose geometric morphisms π∗ : X → Y such that π ∗ U is an initial object ofX (here π ∗ denotes a left adjoint to π∗ ).
Proof. Let π∗ : X → Y be a geometric morphism. Using the adjointnessof π∗ and π ∗ , it is easy to see that π∗ X is SU -local if and only if X isπ ∗ (SU )-local. In particular, π∗ factors through Y /U if and only if π ∗ (SU )consists of equivalences in X. By definition, SU is generated (as a strongly
saturated class of morphisms) by morphisms of the form ∅ → V , where
MapY (V, U ) 6= ∅. It follows that π∗ factors through Y /U if and only if π ∗ V
is an initial object of X whenever MapY (V, U ) 6= ∅. In particular, we seethat if π∗ factors through Y /U , then π ∗ U is initial. Conversely, if π ∗ U is
initial and the mapping space MapY (V, U ) is nonempty, then the existencehighertopoi

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of a commutative diagramπ∗ V#/ π∗ Uπ∗ ∅shows that π ∗ V is a retract of an initial object of X, and therefore initial.
Proposition 7.3.2.12. Let π∗ : X → Y be a geometric morphism of ∞-topoi
and let π ∗ : Sub(1X ) → Sub(1Y ) denote the induced map of partially orderedsets. Let U ∈ Sub(1X ). There is a commutative diagram
X /π ∗ UX

π∗ |(X /π ∗ U )π∗/ Y /U/Yof ∞-topoi and geometric morphisms, where the vertical maps are given by
the natural inclusions. This diagram is left adjointable and exhibits X /(π ∗ U )as a fiber product of X and Y /U over Y in the ∞-category RTop.
Proof. Let π ∗ denote a left adjoint to π∗ . Our first step is to show that theupper horizontal map π∗ |(X /π ∗ U ) is well-defined. In other words, we mustshow that if X ∈ X is trivial on π ∗ U , then π∗ X ∈ Y is trivial on U . Supposethat Y ∈ Y has support contained in U ; we must show that MapY (Y, π∗ X)
is contractible. But this space is homotopy equivalent to MapX (π ∗ Y, X) ' ∗since π ∗ Y has support contained in π ∗ U and X is trivial on π ∗ U .
We also note that π ∗ carries Y /U into X /π ∗ U . This follows immediatelyfrom characterization (3) of Lemma 7.3.2.4 because π ∗ is left exact. Therefore
π ∗ | Y /U is a left adjoint of π∗ | X /π ∗ U . From the fact that π ∗ is left exact,
we easily deduce that π ∗ | Y /U is left exact. It follows that π∗ | X /π ∗ U has
a left exact left adjoint and is therefore a geometric morphism of ∞-topoi.
Moreover, the diagram
X /π ∗ U oXoπ ∗ | Y /Y

π∗

Y /U

Y

is (strictly) commutative, which proves that the diagram of pushforward
functors is left adjointable.
We now claim that the diagram
X /π ∗ U
X

π∗ | X /π ∗ U/ Y /U/Yhighertopoi

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is a pullback diagram of ∞-topoi. For every pair of ∞-topoi A and B, let[A, B] denote the largest Kan complex contained in Fun∗ (A, B). According to
Theorem 4.2.4.1, it will suffice to show that for any ∞-topos Z, the associateddiagram of Kan complexes[Z, X /π ∗ U ]/ [Z, Y /U ]

[Z, X]

/ [Z, Y]is homotopy Cartesian. Lemma 7.3.2.11 implies that the vertical maps areinclusions of full simplicial subsets. It therefore suffices to show that if
φ∗ : Z → Y is a geometric morphism such that π∗ ◦ φ∗ factors throughY /U , then φ∗ factors through X /π ∗ U . This follows immediately from thecharacterization given in Lemma 7.3.2.11.
Corollary 7.3.2.13. Let
X0
p0∗


Y0

/X

/Y

p∗

be a pullback diagram in the ∞-category RTop of ∞-topoi. If p∗ is a closed
immersion, then p0∗ is a closed immersion.
7.3.3 Products of ∞-Topoi
In §6.3.4, we showed that the ∞-category RTop of ∞-topoi admits all (small)
limits. Unfortunately, the construction of general limits was rather inexplicit.
Our goal in this section is to give a very concrete description of the productof two ∞-topoi, at least in a special case.
Definition 7.3.3.1. Let X be a topological space and let C be an ∞category. We let U(X) denote the collection of all open subsets of X partially ordered by inclusion. A presheaf on X with values in C is a functor
U(X)op → C.
Let F : U(X)op → C be a presheaf on X with values in C. We will say that
F is a sheaf with values in C if, for every U ⊆ X and every covering sieve(0)
U(X)/U ⊆ U(X)/U , the composition(0)F

N(U(X)/U ). ⊆ N(U(X)/U ). → N(U(X)) → Cop
is a colimit diagram.
We let P(X; C) denote the ∞-category Fun(U(X)op , C) consisting of allpresheaves on X with values in C, and Shv(X; C) the full subcategory ofP(X; C) spanned by the sheaves on X with values in C.

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Remark 7.3.3.2. We can phrase the sheaf condition informally as follows:
a C-valued presheaf F on a topological space X is a sheaf if, for every open
subset U ⊆ X and every covering sieve {Uα ⊆ U }, the natural map F(U ) →limα F(Uα ) is an equivalence in C.
←−
Remark 7.3.3.3. If X is a topological space, then Shv(X) = Shv(X, S),
where S denotes the ∞-category of spaces.
Lemma 7.3.3.4. Let C, D, and E be ∞-categories which admit finite limits
and let C0 ⊆ C and D0 ⊆ D be the full subcategories of C and D consistingof final objects. Let F : C × D → E be a functor. The following conditions
are equivalent:(1) The functor F preserves finite limits.
(2) The functors F | C0 × D and F | C × D0 preserve finite limits, and forevery pair of morphisms C → 1C , D → 1D where 1C ∈ C and 1D ∈ D
are final objects, the associated diagram
F (1C , D) ← F (C, D) → F (C, 1D )exhibits F (C, D) as a product of F (1C , D) and F (C, 1D ) in E.
(3) The functors F | C0 × D and F | C × D0 preserve finite limits, and F isa right Kan extension of the restrictionaF 0 = F |(C × D0 )(C0 × D).
C0 × D0

Proof. The implication (1) ⇒ (2) is obvious. To see that (2) ⇒ (1), we choose
final objects 1C ∈ C, 1D ∈ D and natural transformations α : idC → 1C ,β : idD → 1D (where X denotes the constant functor with value X). Let
FC : C → E denote the compositionF

C ' C ×{1D } ⊆ C × D → E
and define FD similarly. Then α and β induce natural transformations
FC ◦ π C ← F → FD ◦ π D .
Assumption (2) implies that the functors FC , FD preserve finite limits andthat the above diagram exhibits F as a product of FC ◦ πC and FD ◦ πD
in the ∞-category EC × D . We now apply Lemma 5.5.2.3 to deduce that F
preserves finite limits as well.
We now show that (2) ⇔ (3). Assume that F | C0 × D and F | C × D0 preserve finite limits, so that in particular F | C0 × D0 takes values in the fullsubcategory E0 ⊆ E spanned by the final objects. Fix morphisms u : C → 1C ,v : D → 1D , where 1C ∈ C and 1D ∈ D are final obejcts. We will show thatthe diagram
F (1C , D) ← F (C, D) → F (C, 1D )highertopoi

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exhibits F (C, D) as a product of F (1C , D) and F (C, 1D ) if and only if F isa right Kan extension of F 0 at (C, D).
The morphisms u and v determine a map u × v : ∆1 × ∆1 → C × D, which
we may identify with a map
aw : Λ22 → ((C0 × D)
(C × D0 ))(C,D)/ .
C0 × D0

Using Theorem 4.1.3.1, it is easy to see that wop is cofinal. Consequently, F
is a right Kan extension of F 0 at (C, D) if and only if the diagram
F (C, D)/ F (C, 1D )

F (1C , D)
/ F (1C , 1D )is a pullback square. Since F (1C , 1D ) is a final object of E, this is equivalent
to assertion (2).
Lemma 7.3.3.5. Let C and D be small ∞-categories which admit finite
limits, let 1C ∈ C, 1D ∈ D be final objects, and let X be an ∞-topos. The
projectionsp∗q∗

P(C ×{1D }) ← P(C × D) → P({1C } × D)
induce a categorical equivalence
Fun∗ (X, P(C × D)) → Fun∗ (X, P(C)) × Fun∗ (X, P(D)).
In particular, P(C × D) is a product of P(C) and P(D) in the ∞-category
RTop of ∞-topoi.
Proof. For every ∞-category Y which admits finite limits, let [Y, X] denotethe full subcategory of Fun(Y, X) spanned by the left exact functors Y → X.
If Y is an ∞-topos, we let [Y, X]0 denote the full subcategory of [Y, X] spannedby the colimit-preserving left exact functors Y → X. In view of Proposition
5.2.6.2 and Remark 5.2.6.4, it will suffice to prove that composition with theleft adjoints to p∗ and q∗ induces an equivalence of ∞-categories[P(C × D), X]0 → [P(C), X]0 × [P(D), X]0 .
Applying Proposition 6.2.3.20, we may reduce to the problem of showingthat the map[C × D, X] → [C, X] × [D, X]
is an equivalence of ∞-categories.
Let C0 ⊆ C and D0 ⊆ D denote the full subcategories consisting of final
objects of C and D, respectively. Proposition 1.2.12.9 implies that C0 and D0
are contractible. It will therefore suffice to prove that the restriction map
φ : [C × D, X] → [C × D0 , X] ×[C0 × D0 ,X] [C0 × D, X]
is a trivial fibration of simplicial sets. This follows immediately from Lemma
7.3.3.4 and Proposition 4.3.2.15.

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Notation 7.3.3.6. Let X be an ∞-topos and let p∗ : S → X be a geometric morphism (essentially unique in view of Proposition 6.3.4.1). Let
πX : X × S → X and πS : X × S → S denote the projection functors. Let ⊗be a product of πX and p∗ ◦ πS in the ∞-category of functors from X × S
to X. Then ⊗ is uniquely defined up to equivalence, and we have naturaltransformations
X ← X ⊗ S → p∗ S
which exhibit X ⊗ S as product of X and p∗ S for all X ∈ X, S ∈ S. We
observe that ⊗ preserves colimits separately in each variable.
If C is a small ∞-category, we let ⊗C denote the composition◦⊗

P(C; X) × P(C) ' P(C; X × S) → P(C, X).
We observe that if F ∈ P(C; X) and G ∈ P(C), then F ⊗C G can be identified
with a product of F and p∗ ◦ G in P(C; X).
Lemma 7.3.3.7. Let C be a small ∞-category and X an ∞-topos. Let g :X → S a functor corepresented by an object X ∈ X and let G : P(C; X) →P(C) be the induced functor. Let X ∈ P(C; X) denote the constant functor
with the value X. Then the functor
F = X ⊗C idP(C)is a left adjoint to G.
Proof. Since adjoints and ⊗C can both be computed pointwise on C, it suffices to treat the case where C = ∆0 . In this case, we deduce the existence ofa left adjoint F 0 to G using Corollary 5.5.2.9 (the accessibility of G follows
from the fact that X is κ-compact for sufficiently large κ since X is accessible). Now F and F 0 are both colimit-preserving functors S → X. By virtueof Theorem 5.1.5.6, to prove that F and F 0 are equivalent, it will suffice toshow that the objects F (∗), F 0 (∗) ∈ X are equivalent. In other words, wemust prove that F 0 (∗) ' X. By adjointness, we have natural isomorphisms
MapX (F 0 (∗), Y ) ' MapH (∗, G(Y )) ' MapX (X, Y )in H for each Y ∈ X, so that F 0 (∗) and X corepresent the same functor on thehomotopy category hX and are therefore equivalent by Yoneda’s lemma.
Lemma 7.3.3.8. Let C be a small ∞-category which admits finite limits
and contains a final object 1C , let X and Y be ∞-topoi, and let p∗ : X → S bea geometric morphism (essentially unique by virtue of Proposition 6.3.4.1).
Then the maps
P

e1

P(C) ←∗ P(C; X) →C X
induce equivalences of ∞-categories
Fun∗ (Y, P(C; X)) → Fun∗ (Y, X) × Fun∗ (Y, P(C)).
In particular, P(C; X) is a product of P(C) and X in the ∞-category RTopof ∞-topoi. Here e1C denotes the evaluation map at the object 1C ∈ C, andP∗ : P(C; X) → P(C) is given by composition with p∗ .

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Proof. According to Proposition 6.1.5.3, we may assume without loss of generality that there exists a small ∞-category D such that X is the essential
image of an accessible left exact localization functor L : P(D) → P(D) andthat p∗ is given by evaluation at a final object 1D ∈ D. We have a commutative diagram
Fun∗ (Y, P(C; X))/ Fun∗ (Y, P(C)) × Fun∗ (Y, X)Fun∗ (Y, P(C × D))/ Fun∗ (Y, P(C)) × Fun∗ (Y, P(D)),

where the vertical arrows are inclusions of full subcategories and the bottom
arrow is an equivalence of ∞-categories by Lemma 7.3.3.5. Consequently, it
will suffice to show that if q∗ : Y → P(C × D) is a geometric morphism withthe property that the compositionr∗ : Y → P(C × D) → P(D)factors through X, then q∗ factors through P(C; X).
Let Y ∈ Y and C ∈ C; we wish to show that q∗ (Y )(C) ∈ X. It will suffice
to show that if s : D → D0 is a morphism in P(D) such that L(s) is anequivalence in X, then q∗ (Y )(C) is s-local. Let F : P(D) → P(C × D) be aleft adjoint to the functor given by evaluation at C. We have a commutative
diagram
MapP(D) (D0 , q∗ (Y )(C))/ MapY (q ∗ F (D0 ), Y )
MapP(D) (D, q∗ (Y )(C))

/ MapY (q ∗ F (D), Y ),where the horizontal arrows are homotopy equivalences. Consequently, toprove that the left vertical map is an equivalence, it will suffice to provethat q ∗ F (s) is an equivalence in Y. According to Lemma 7.3.3.7, the functor
F can be identified with a product of a left adjoint r∗ to the projectionr∗ : P(C × D) → P(D) with a constant functor. Since q ∗ preserves finite
products, it will suffice to show that (q ∗ ◦ r∗ )(s) is an equivalence in Y. This
follows immediately from our assumption that r∗ ◦ q∗ : Y → P(D) factors
through X.
The main result of this section is the following:
Theorem 7.3.3.9. Let X be a topological space, X an ∞-topos, and π∗ :X → S a geometric morphism (which is essentially unique by virtue of Proposition 6.3.4.1). Then Shv(X; X) is an ∞-topos, and the diagramΓπ

X ← Shv(X; X) →∗ Shv(X)exhibits Shv(X; X) as a product of Shv(X) and X in the ∞-category RTopof ∞-topoi. Here Γ denotes the global sections functor given by evaluation
at X ∈ U(X).

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Proof. We first show that Shv(X; X) is an ∞-topos. Since X is an ∞-topos,
Proposition 6.1.5.3 implies that we can realize X as an accessible left exact localization of P(C), where C is a small ∞-category which admits finite limits. In what follows, we will identify X with its essential imagein P(C). This determines an identification of Shv(X : X) with the intersection Shv(X; P(C)) ∩ P(X; X) in the ambient ∞-category P(X; P(C)) 'P(C × N(U(X))) (here U(X) denotes the partially ordered set of open subsets of X). Note that both Shv(X; P(C)) ' Fun(Cop , Shv(X)) and P(X; X)
are accessible left exact localizations of P(C × N(U(X))) (see Proposition
6.2.2.7), so that Shv(X : X) is also an accessible left-exact localization ofP(C × N(U(X))) (Lemma 6.3.3.4) and therefore an ∞-topos.
Without loss of generality, we may assume that the functor π∗ : X → S isgiven by evaluation on a final object of C. We have a commutative diagramof ∞-topoi and geometric morphisms/X

Shv(X; X)/ P(X; X)
Shv(X; P(C))

/ P(X; P(X)) Γ/ P(C)
Shv(X)/ P(X)

/ S,

Γand we wish to show that the outer square is a pullback diagram in RTop.
Using Lemma 7.3.3.8, we deduce that the lower right square, the right rectangle, and the bottom rectangle are pullback squares. It follows from Lemma
4.4.2.1 that the upper right and lower left squares are pullbacks as well. Since
the upper left square is evidently a pullback diagram, we conclude (using
Lemma 4.4.2.1 again) that the outer square is a pullback as desired.
7.3.4 Sheaves on Locally Compact Spaces
By definition, a sheaf of sets F on a topological space X is determinedby the sets F(U ) as U ranges over the open subsets of X. If X is a locallycompact Hausdorff space, then there is an alternative collection of data which
determines X: the values F(K), where K ranges over the compact subsetsof X. Here F(K) denotes the direct limit limK⊆U F(U ) taken over all open−→
neighborhoods of K (or equivalently, the collection of global sections of therestriction F |K). The goal of this section is to prove a generalization ofthis result where the sheaf F is allowed to take values in a more general∞-category C.
Definition 7.3.4.1. Let X be a locally compact Hausdorff space. We let
K(X) denote the collection of all compact subsets of X. If K, K 0 ⊆ X, wewrite K b K 0 if there exists an open subset U ⊆ X such that K ⊆ U ⊆ K 0 .
If K ∈ K(X), we let KKb (X) = {K 0 ∈ K(X) : K b K 0 }.

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769

HIGHER TOPOS THEORY IN TOPOLOGYLet F : N(K(X))op → C be a presheaf on N(K(X)) (here K(X) is viewed
as a partially ordered set with respect to inclusion) with values in C. We will
say that F is a K-sheaf if the following conditions are satisfied:(1) The object F(∅) ∈ C is final.
(2) For every pair K, K 0 ∈ K(X), the associated diagram/ F(K)F(K ∪ K 0 )
F(K 0 )
/ F(K ∩ K 0 )is a pullback square in C.
(3) For each K ∈ K(X), the restriction of F exhibits F(K) as a colimit of
F | N(KKb (X))op .
We let ShvK (X; C) denote the full subcategory of Fun(N(K(X))op , C)spanned by the K-sheaves. In the case where C = S, we will write ShvK (X)instead of ShvK (X; C).
Definition 7.3.4.2. Let C be a presentable ∞-category. We will say thatfiltered colimits in C are left exact if the following condition is satisfied: forevery small filtered ∞-category I, the colimit functor Fun(I, C) → C is left
exact.
Example 7.3.4.3. A Grothendieck abelian category is an abelian category
A whose nerve N(A) is a presentable ∞-category with left exact filteredcolimits in the sense of Definition 7.3.4.2. We refer the reader to [34] forfurther discussion.
Example 7.3.4.4. Filtered colimits are left exact in the ∞-category S ofspaces; this follows immediately from Proposition 5.3.3.3. It follows thatfiltered colimits in τ≤n S are left exact for each n ≥ −2 since the full subcategory τ≤n S ⊆ S is stable under filtered colimits and finite limits (in fact,
under all limits).
Example 7.3.4.5. Let C be a presentable ∞-category in which filtered colimits are left exact and let X be an arbitrary simplicial set. Then filteredcolimits are left exact in Fun(X, C). This follows immediately from Proposition 5.1.2.2, which asserts that the relevant limits and colimits can becomputed pointwise.
Example 7.3.4.6. Let C be a presentable ∞-category in which filteredcolimits are left exact and let D ⊆ C be the essential image of an (accessible)
left exact localization functor L. Then filtered colimits in D are left exact. To
prove this, we consider an arbitrary filtered ∞-category I and observe thatthe colimit functor lim : Fun(I, D) → D is equivalent to the composition−→
L
Fun(I, D) ⊆ Fun(I, C) → C → D,
where the second arrow is given by the colimit functor lim Fun(I, C) → C.
−→highertopoi

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Example 7.3.4.7. Let X be an n-topos, 0 ≤ n ≤ ∞. Then filtered colimits
in X are left exact. This follows immediately from Examples 7.3.4.4, 7.3.4.5,
and 7.3.4.6.
Our goal is to prove that if X is a locally compact Hausdorff space andC is a presentable ∞-category in which filtered colimits are left exact, then
the ∞-categories Shv(X) and ShvK (X) are equivalent. As a first step, we
prove that a K-sheaf on X is determined locally.
Lemma 7.3.4.8. Let X be a locally compact Hausdorff space and C apresentable ∞-category in which filtered colimits are left exact. Let W bea collection of open subsets of X which covers X and let KW (X) = {K ∈K(X) : (∃W ∈ W)[K ⊆ W ]}. Suppose that F ∈ ShvK (X; C). Then F is aright Kan extension of F | N(KW (X))op .
Proof. Let us say that an open covering W of a locally compact Hausdorff
space X is good if it satisfies the conclusion of the lemma. Note that W isa good covering of X if, for every compact subset K ⊆ X, the open sets
{K ∩ W : W ∈ W} form a good covering of K. We wish to prove that every
covering W of a locally compact topological space X is good. By virtue ofthe preceding remarks, we can reduce to the case where X is compact andthereby assume that W has a finite subcover.
We will prove, by induction on n ≥ 0, that if W is a collection of open
subsets of a locally compact Hausdorff space X such that there exist elements
W1 , · · · , Wn ∈ W with W1 ∪ . . . ∪ Wn = X, then W is a good covering of X.
If n = 0, then X = ∅. In this case, we must prove that F(∅) is final, which ispart of the definition of K-sheaf.
Suppose that W ⊆ W0 are coverings of X and that for every W 0 ∈ W0
the induced covering {W ∩ W 0 : W ∈ W} is a good covering of W 0 . It thenfollows from Proposition 4.3.2.8 that W0 is a good covering of X if and only
if W is a good covering of X.
Now suppose n > 0. Let V = W2 ∪ · · · ∪ Wn , and let W0 = W ∪{V }. Using
the above remark and the inductive hypothesis, it will suffice to show that
W0 is a good covering of X. Now W0 contains a pair of open sets W1 andV which cover X. We thereby reduce to the case n = 2; using the above
remark, we can furthermore suppose that W = {W1 , W2 }.
We now wish to show that for every compact K ⊆ X, F exhibits F(K) as
the limit of F | N(KW (X))op . Let P be the collection of all pairs K1 , K2 ∈K(X) such that K1 ⊆ W1 , K2 ⊆ W2 , and K1 ∪ K2 = K. We observethat P is filtered when ordered by inclusion. For α = (K1 , K2 ) ∈ P , let
Kα = {K 0 ∈ K(X) : (K 0 ⊆ K1 ) ∨ (K 0 ⊆ K2 )}. We note that a compact
subset K 0 ⊆ K belongs to KW (X) if and only if it belongs to Kα , for someα ∈ P . Moreover, Theorem 4.1.3.1 implies that for α = (K1 , K2 ) ∈ P , theinclusion N{K1 , K2 , K1 ∩ K2 } ⊆ N(Kα ) is cofinal. Since F is a K-sheaf,
we deduce that F exhibits F(K) as a limit of the diagram F | N(Kα )op foreach α ∈ P . Using Proposition 4.2.3.4, we deduce that F(K) is a limitof F | N(KW (K))op if and only if F(K) is a limit of the constant diagramhighertopoi

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N(P )op → S taking the value F(K). This is clear since P is filtered so thatthe map N(P ) → ∆0 is cofinal by Theorem 4.1.3.1.
Theorem 7.3.4.9. Let X be a locally compact Hausdorff space and let C
be a presentable ∞-category in which filtered colimits are left exact. Let F :N(K(X) ∪ U(X))op → C be a presheaf on the partially ordered set K(X) ∪U(X). The following conditions are equivalent:(1) The presheaf FK = F | N(K(X))op is a K-sheaf, and F is a right Kan
extension of FK .
(2) The presheaf FU = F | N(U(X))op is a sheaf, and F is a left Kan extension of FU .
Proof. Suppose first that (1) is satisfied. We first prove that F is a left Kan
extension of FU . Let K be a compact subset of X and let UK⊆ (X) = {U ∈U(X) : K ⊆ U }. Consider the diagram
N(UK⊆ (X))op
N(UK⊆ (X)op ).

p/ N(UK⊆ (X) ∪ KKb (X))op o/ N(UK⊆ (X)) ∪ KKb (X))op ). oψ
N(U(X) ∪ K(X))opp0

N(KKb (X))op

.
N(Kop
Kb )

ψ0

F

%  y
C.

We wish to prove that ψ is a colimit diagram. Since FK is a K-sheaf, wededuce that ψ 0 is a colimit diagram. It therefore suffices to check that p and
p0 are cofinal. According to Theorem 4.1.3.1, it suffices to show that for every
Y ∈ UK⊆ (X) ∪ KKb (X), the partially ordered sets {K 0 ∈ K(X) : K b K 0 ⊆Y } and {U ∈ U(X) : K ⊆ U ⊆ Y } have contractible nerves. We now observethat both of these partially ordered sets is filtered since they are nonempty
and stable under finite unions.
We now show that FU is a sheaf. Let U be an open subset of X and let
W be a sieve which covers U . Let K⊆U (X) = {K ∈ K(X) : K ⊆ U } and let
KW (X) = {K ∈ K(X) : (∃W ∈ W)[K ⊆ W ]}. We wish to prove that the
diagramF

N(Wop )/ → N(U(X))op →U S
is a limit. Using Theorem 4.1.3.1, we deduce that the inclusion
N(W) ⊆ N(W ∪ KW (X))
is cofinal. It therefore suffices to prove that F |(W ∪ KW (X) ∪ {U })op isa right Kan extension of F |(W ∪ KW (X))op . Since F |(W ∪ KW (X))op is ahighertopoi

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right Kan extension of F | KW (X)op by assumption, it suffices to prove that
F |(W ∪ KW (X) ∪ {U })op is a right Kan extension of F | KW (X)op . This isclear at every object distinct from U ; it will therefore suffice to prove that
F |(KW (X) ∪ {U })op is a right Kan extension of F | KW (X)op .
By assumption, the functor F | N(K⊆U (X) ∪ {U })op is a right Kan extension of F | N(K⊆U (X))op and Lemma 7.3.4.8 implies that F | N(K⊆U (X))op isa right Kan extension of F | N(KW (X))op . Using Proposition 4.3.2.8, we deduce that F | N(KW (X)∪{U })op is a right Kan extension of F | N(KW (X))op .
This shows that FU is a sheaf and completes the proof that (1) ⇒ (2).
Now suppose that F satisfies (2). We first verify that FK is a K-sheaf. The
space FK (∅) = FU (∅) is contractible because FU is a sheaf (and because theempty sieve is a covering sieve on ∅ ⊆ X). Suppose next that K and K 0 arecompact subsets of X. We wish to prove that the diagram
F(K ∪ K 0 )/ F(K)F(K 0 )
/ F(K ∩ K 0 )is a pullback in S. Let us denote this diagram by σ : ∆1 × ∆1 → S. Let P
be the set of all pairs U, U 0 ∈ U(X) such that K ⊆ U and K 0 ⊆ U 0 . The
1
1functor F induces a map σP : N(P op ). → S∆ ×∆ , which carries each pair
(U, U 0 ) to the diagram
F(U ∪ U 0 )/ F(U )F(U 0 )
/ F(U ∩ U 0 )and carries the cone point to σ. Since FU is a sheaf, each σP (U, U 0 ) is apullback diagram in C. Since filtered colimits in C are left exact, it will suffice
to show that σP is a colimit diagram. By Proposition 5.1.2.2, it suffices toshow that each of the four maps
N(P op ). → S,given by evaluating σP at the four vertices of ∆1 × ∆1 , is a colimit diagram.
We will treat the case of the final vertex; the other cases are handled in
the same way. Let Q = {U ∈ U(X) : K ∩ K 0 ⊆ U }. We are given a map
g : N(P op ). → S which admits a factorization
g 00

g0

F

N(P op ). → N(Qop ). → N(U(X) ∪ K(X))op → C .
Since F is a left Kan extension of FU , the diagram F ◦g 00 is a colimit. It
therefore suffices to show that g 00 induces a cofinal map N(P )op → N(Q)op .
Using Theorem 4.1.3.1, it suffices to prove that for every U 00 ∈ Q, the partially ordered set PU 00 = {(U, U 0 ) ∈ P : U ∩ U 0 ⊆ U 00 } has contractible nerve.
It now suffices to observe that PUop00 is filtered (since PU 00 is nonempty andstable under intersections).

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We next show that for any compact subset K ⊆ X, the map
F

N(KKb (X)op ). → N(K(X) ∪ U(X))op → C
is a colimit diagram. Let V = U(X) ∪ KKb (X) and let V0 = V ∪{K}. It
follows from Proposition 4.3.2.8 that F | N(V)op and F | N(V0 )op are left Kan
extensions of F | N(U(X))op , so that F | N(V0 )op is a left Kan extension of
F | N(V)op . Therefore the diagram
F(N(KKb (X) ∪ {U ∈ U(X) : K ⊆ U })op ). → N(K(X) ∪ U(X))op → C
is a colimit. It therefore suffices to show that the inclusion
N(KKb (X))op ⊆ N(KKb (X) ∪ {U ∈ U(X) : K ⊆ U })op
is cofinal. Using Theorem 4.1.3.1, we are reduced to showing that if
Y ∈ KKb (X) ∪ {U ∈ U(X) : K ⊆ U },then the nerve of the partially ordered set R = {K 0 ∈ K(X) : K b K 0 ⊂ Y }is weakly contractible. It now suffices to observe that Rop is filtered since R
is nonempty and stable under intersections. This completes the proof that
FK is a K-sheaf.
We now show that F is a right Kan extension of FK . Let U be an open
subset of X and for V ∈ U(X) write V b U if the closure V is compact
and contained in U . Let UbU (X) = {V ∈ U(X) : V b U } and consider the
diagram
N(UbU (X))op
N(UbU (X)op )/f/ N(UbU (X) ∪ K⊆U (X))op o
N(UbU (X) ∪ K⊆U (X))op )/φ
N(K(X) ∪ U(X))opf0

N(K⊆U (X))op
N(K⊆U (X)op )/

φ0

F

%  y
C.

We wish to prove that φ0 is a limit diagram. Since the sieve UbU (X) covers U
and FU is a sheaf, we conclude that φ is a limit diagram. It therefore suffices
to prove that f op and (f 0 )op are cofinal maps of simplicial sets. According
to Theorem 4.1.3.1, it suffices to prove that if Y ∈ K⊆U (X) ∪ UbU (X), thenthe partially ordered sets {V ∈ U(X) : Y ⊆ V b U } and {K ∈ K(X) : Y ⊆K ⊆ U } have weakly contractible nerves. We now observe that both of thesepartially ordered sets are filtered (since they are nonempty and stable underunions). This completes the proof that F is a right Kan extension of FK .
Corollary 7.3.4.10. Let X be a locally compact topological space and C apresentable ∞-category in which filtered colimits are left exact. Let
ShvKU (X; C) ⊆ Fun(N(K(X) ∪ U(X))op , C)highertopoi

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be the full subcategory spanned by those presheaves which satisfy the equivalent conditions of Theorem 7.3.4.9. Then the restriction functors
Shv(X; C) ← ShvKU (X; C) → ShvK (X; C)
are equivalences of ∞-categories.
Corollary 7.3.4.11. Let X be a compact Hausdorff space. Then the global
sections functor Γ : Shv(X) → S is a proper morphism of ∞-topoi.
Proof. The existence of fiber products Shv(X) ×S Y in RTop follows from
Theorem 7.3.3.9. It will therefore suffice to prove that for any (homotopy)
Cartesian rectangle
X00

Y00

f∗

/ X0

/ Shv(X)


/ Y0


/ S,

the square on the left is left adjointable. Using Theorem 7.3.3.9, we canidentify the square on the left with/ Shv(X; Y0 )Shv(X; Y00 )

Y00

f∗


/ Y0 ,

where the vertical morphisms are given by taking global sections.
Choose a correspondence M from Y0 to Y00 which is associated to thefunctor f∗ . Since f∗ admits a left adjoint f ∗ , the projection M → ∆1 is botha Cartesian fibration and a coCartesian fibration. For every simplicial set K,
let MK = Fun(K, M)×Fun(K,∆1 ) ∆1 . Then MK determines a correspondence
from Fun(K, Y0 ) to Fun(K, Y00 ). Using Proposition 3.1.2.1, we conclude that
MK → ∆1 is both a Cartesian and a coCartesian fibration, and that it isassociated to the functors given by composition with f∗ and f ∗ .
Before proceeding further, let us adopt the following convention for theremainder of the proof: given a simplicial set Z with a map q : Z → ∆1 , wewill say that an edge of Z is Cartesian or coCartesian if it is q-Cartesian or
q-coCartesian, respectively. The map q to which we are referring should beclear from the context.
Let MU denote the full subcategory of MN(U(X))op whose objects correspond to sheaves on X (with values in either Y0 or Y00 ). Since f∗ preserves
limits, composition with f∗ carries Shv(X; Y00 ) into Shv(X; Y0 ). We conclude
that the projection MU → ∆1 is a Cartesian fibration and that the inclusion
MU ⊆ MN(U(X))op preserves Cartesian edges.
Similarly, we define MK to be the full subcategory of MN(K(X))op whose
objects correspond to K-sheaves on X (with values in either Y0 or Y00 ). Sincef ∗ preserves finite limits and filtered colimits, composition with f ∗ carries
ShvK (X; Y0 ) into ShvK (X; Y00 ). It follows that the projection MK → ∆1 ishighertopoi

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a coCartesian fibration and that the inclusion MK ⊆ MN(U(X))op preserves
coCartesian edges.
Now let M0KU = MN(K(X)∪U(X))op and let MKU be the full subcategory of0
MKU spanned by the objects of ShvKU (X; Y0 ) and ShvKU (X; Y00 ). We havea commutative diagram
MKU
φU

MU

{

#

φK

MK

ΓU

#

M,

{

ΓK

where ΓU and ΓK denote the global sections functors (given by evaluation
at X ∈ U(X) ∩ K(X)). According to Remark 7.3.1.3, to complete the proof
it will suffice to show that MU → ∆1 is a coCartesian fibration and that ΓUpreserves both Cartesian and coCartesian edges. It is clear that ΓU preserves
Cartesian edges since it is a composition of maps
MU ⊆ MN(U(X))op → M
which preserve Cartesian edges. Similarly, we already know that MK →∆1 is a coCartesian fibration and that ΓK preserves coCartesian edges. To
complete the proof, it will therefore suffice to show that φU and φK areequivalences of ∞-categories. We will give the argument for φU ; the proof in
the case of φK is identical and is left to the reader.
According to Corollary 7.3.4.10, the map φU induces equivalences
ShvKU (X; Y0 ) → Shv(X; Y0 )ShvKU (X; Y00 ) → Shv(X; Y00 )after passing to the fibers over either vertex of ∆1 . We will complete theproof by applying Corollary 2.4.4.4. In order to do so, we must verify thatp : MKU → ∆1 is a Cartesian fibration and that φU preserves Cartesianedges.
To show that p is a Cartesian fibration, we begin with an arbitrary F ∈ShvKU (X; Y00 ). Using Proposition 3.1.2.1, we conclude the existence of a
p0 -Cartesian morphism α : F0 → F, where p0 denotes the projection M0KU
and F0 = F ◦p∗ ∈ Fun(N(K(X) ∪ U(X))op , Y0 ). Since p∗ preserves limits, weconclude that F0 | N(U(X))op is a sheaf on X with values in Y0 ; however, F0
is not necessarily a left Kan extension of F0 | N(U(X))op . Let C denote thefull subcategory of Fun(N(K(X) ∪ U(X))op , Y0 ) spanned by those functors
G : N(K(X) ∪ U(X))op which are left Kan extensions of G | N(U(X))op , andop
let s a section of the trivial fibration C → (Y0 )N(U(X)) , so that s is a left
00 N(U(X))op
adjoint to the restriction map r : MKU → (Y ). Let F00 = (s ◦ r) F0000
be a left Kan extension of F | N(U(X))op . Then F is an initial object of thehighertopoi

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fiber M0KU ×Fun(N(U(X))op ,Y0 ) {F0 | N(U(X))op }, so that there exists a map
β : F00 → F0 which induces the identity on F00 | N(U(X))op = F0 | N(U(X))op .
Let σ : ∆2 → M0KU classify a diagram
>F
β

0
α

γ

F00


/ F,

so that γ is a composition of α and β. It is easy to see that φU (γ) is a
Cartesian edge of MU (since it is a composition of a Cartesian edge with anequivalence in Shv(X; Y0 )). We claim that γ is p-Cartesian. To prove this,consider the diagram
ShvKU (X; Y0 ) ×M0KU (M0KU )/ση0/ (MKU )/γθ0

ShvKU (X; Y0 )/β ×ShvKU (X;Y0 )/ F0 (M0KU )/α

η

θ1


ShvKU (X; Y0 ) ×M0KU (M0KU )/α

θ2


/ Z,

where Z denotes the fiber product ShvKU (X; Y0 )×MKU (MKU )/ F . We wish toshow that η is a trivial fibration. Since η is a right fibration, it suffices to showthat the fibers of η are contractible. The map η 0 is a trivial fibration (since
the inclusion ∆{0,2} ⊆ ∆2 is right anodyne), so it will suffice to prove thatη◦η 0 is a trivial fibration. In view of the commutativity of the diagram, it will
suffice to show that θ0 , θ1 , and θ2 are trivial fibrations. The triviality of θ0follows from the fact that the horn inclusion Λ21 ⊆ ∆2 is right anodyne. The
triviality of θ2 follows from the fact that α is p0 -Cartesian. Finally, we observethat θ1 is a pullback of the map θ10 : ShvKU (X; Y0 )/β → ShvKU (X; Y0 )/ F0 .
op
Let C = (Y0 )N(K(X)∪U(X)) . To prove that θ10 is a trivial fibration, we mustshow that for every G ∈ ShvKU , composition with β induces a homotopy
equivalence
MapC (G, F00 ) → MapC (G, F0 ).
Without loss of generality, we may suppose that G = s(G0 ), where G0 ∈Shv(X; Y0 ); now we simply invoke the adjointness of s with the restriction
functor r and the observation that r(β) is an equivalence.
Corollary 7.3.4.12. Let X be a compact Hausdorff space. The global sections functor Γ : Shv(X) → S preserves filtered colimits.
Proof. Applying Theorem 7.3.4.9, we can replace Shv(X) by ShvK (X). Now
observe that the full subcategory ShvK (X) ⊆ P(N(K(X))op ) is stable underfiltered colimits. We thereby reduce to proving that the evaluation functor P(N(K(X))op ) → S commutes with filtered colimits, which follows fromhighertopoi

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Proposition 5.1.2.2. Alternatively, one can apply Corollary 7.3.4.10 and Remark 7.3.1.5.
Remark 7.3.4.13. One can also deduce Corollary 7.3.4.12 using the geometric model for Shv(X) introduced in §7.1. Using the characterization ofproperness in terms of filtered colimits described in Remark 7.3.1.5, one canformally deduce Corollary 7.3.4.11 from Corollary 7.3.4.12. This leads to another proof of the proper base change theorem, which does not make use of
Theorem 7.3.4.9 or the other ideas of this section. However, this alternative
proof is considerably more difficult than the one described here since it requires a rigorous justification of Remark 7.3.1.5. We also note that Theorem
7.3.4.9 and Corollary 7.3.4.10 are interesting in their own right and could
conceivably be applied in other contexts.
7.3.5 Sheaves on Coherent Spaces
Theorem 7.3.4.9 has an analogue in the setting of coherent topological spaces
which is somewhat easier to prove. First, we need the analogue of Lemma
7.3.4.8:
Lemma 7.3.5.1. Let X be a coherent topological space, let U0 (X) denotethe collection of compact open subsets of X, and let F : N(U0 (X))op → C bea presheaf taking values in an ∞-category C having the following properties:(1) The object F(∅) ∈ C is final.
(2) For every pair of compact open sets U, V ⊆ X, the diagram
F(U ∩ V )/ F(U )F(V )

/ F(U ∪ V )is a pullback.
Let W be a covering of X by compact open subsets and let U1 (X) ⊆ U0 (X)be the collection of all compact open subsets of X which are contained insome element of W. Then F is a right Kan extension of F | N(U1 (X))op .
Proof. The proof is similar to that of Lemma 7.3.4.8 but slightly easier. Let
us say that a covering W of a coherent topological space X by compact open
subsets is good if it satisfies the conclusions of the lemma. We observe that W
automatically has a finite subcover. We will prove, by induction on n ≥ 0,that if W is a collection of open subsets of a locally coherent topologicalspace X such that there exist W1 , · · · , Wn ∈ W with W1 ∪ . . . ∪ Wn = X,
then W is a good covering of X. If n = 0, then X = ∅. In this case, we must
prove that F(∅) is final, which is one of our assumptions.
Suppose that W ⊆ W0 are coverings of X by compact open sets and thatfor every W 0 ∈ W0 the induced covering {W ∩ W 0 : W ∈ W} is a goodhighertopoi

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covering of W 0 . It then follows from Proposition 4.3.2.8 that W0 is a goodcovering of X if and only if W is a good covering of X.
Now suppose n > 0. Let V = W2 ∪ · · · ∪ Wn , and let W0 = W ∪{V }. Using
the above remark and the inductive hypothesis, it will suffice to show that
W0 is a good covering of X. Now W0 contains a pair of open sets W1 andV which cover X. We thereby reduce to the case n = 2; using the above
remark, we can furthermore suppose that W = {W1 , W2 }.
We now wish to show that for every compact U ⊆ X, F exhibits F(U )as the limit of F | N(U1 (X)/U )op . Without loss of generality, we may replace X by U and thereby reduce to the case U = X. Let U2 (X) ={W1 , W2 , W1 ∩ W2 } ⊆ U1 (X). Using Theorem 4.1.3.1, we deduce that theinclusion N(U2 (X)) ⊆ N(U1 (X)) is cofinal. Consequently, it suffices to provethat F(X) is the limit of the diagram F | N(U2 (X))op . In other words, wemust show that the diagram
F(X)/ F(W1 )F(W2 )

/ F(W1 ∩ W2 )is a pullback in C, which is true by assumption.
Theorem 7.3.5.2. Let X be a coherent topological space and let U0 (X) ⊆U(X) denote the collection of compact open subsets of X. Let C be an ∞category which admits small limits. The restriction map
Shv(X; C) → Fun(N(U0 (X))op , C)is fully faithful, and its essential image consists of precisely those functors
F0 : N(U0 (X))op → C satisfying the following conditions:(1) The object F0 (∅) ∈ C is final.
(2) For every pair of compact open sets U, V ⊆ X, the diagram
F0 (U ∩ V )/ F0 (U )
F0 (V )


/ F0 (U ∪ V )is a pullback.
op

Proof. Let D ⊆ CN(U(X)) be the full subcategory spanned by those presheaves F : N(U(X))op → C which are right Kan extensions of F0 =F | N(U0 (X))op and such that F0 satisfies conditions (1) and (2). According to Proposition 4.3.2.15, it will suffice to show that D coincides withShv(X; C).
Suppose that F : N(U(X))op → C is a sheaf. We first show that F is aright Kan extension of F0 = F | N(U0 (X))op . Let U be an open subset of

highertopoi

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X, let U(X)/U denote the collection of compact open subsets of U and let
(1)

(0)

U(X)/U denote the sieve generated by U(X)/U . Consider the diagram(1)

N(U(X)/U ).
O/ N(U(X)/U ).

/ N(U(X))

f0

F

i

*&

f

(0)

N(U(X)/U ).

/ Cop .

We wish to prove that f is a colimit diagram. Using Theorem 4.1.3.1, we(0)(1)
deduce that the inclusion N(U(X))/U ⊆ N(U(X))/U is cofinal. It therefore
suffices to prove that f 0 is a colimit diagram. Since F is a sheaf, it suffices(1)
to prove that U(X)/U is a covering sieve. In other words, we need to provethat U is a union of compact open subsets of X, which follows immediatelyfrom our assumption that X is coherent.
We next prove that F0 satisfies (1) and (2). To prove (1), we simply observe that the empty sieve is a cover of ∅ and apply the sheaf condition. To
prove (2), we may assume without loss of generality that neither U nor V is(0)
contained in the other (otherwise the result is obvious). Let U(X)/U ∪V be(1)

the full subcategory spanned by U , V , and U ∩ V , and let U(X)/U ∪V be the(0)sieve on U ∪ V generated by U(X)/U ∪V . As above, we have a diagram(1)

N(U(X)/U ∪V ).
O/ N(U(X)/U ∪V ).

/ N(U(X))

f0

F

i
(0)

(N(U(X))/U ∪V ).

+&/

f



Cop ,

and we wish to show that f is a colimit diagram. Theorem 4.1.3.1 implies(0)(1)
that the inclusion N(U(X))/U ∪V ⊆ N(U(X))/U ∪V is cofinal. It therefore
suffices to prove that f 0 is a colimit diagram, which follows from the sheaf(1)condition since U(X)/U ∪V is a covering sieve. This completes the proof that
Shv(X; C) ⊆ D.
It remains to prove that D ⊆ Shv(X; C). In other words, we must showthat if F is a right Kan extension of F0 = F | N(U0 (X))op and F0 satisfies
conditions (1) and (2), then F is a sheaf. Let U be an open subset of X and(0)
let U(X)/U be a sieve which covers U . Let U0 (X)/U denote the category(0)of compact open subsets of U and U0 (X)/U the category of compact open(0)

subsets of U which belong to the sieve U(X)/U . We wish to prove that F(U )(0)

is a limit of F | N(U(X)/U )op . We will in fact prove the slightly stronger(0)

assertion that F | N(U(X)/U )op is a right Kan extension of F | N(U(X)/U )op .

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We have a commutative diagram
U0 (X)/U(0)/ U0 (X)/U(0)
U(X)/U/ U(X)/U .

By assumption, F is a right Kan extension of F0 . It follows that the restric(0)(0)
tion F | N(U(X)/U )op is a right Kan extension of F | N(U0 (X)/U )op and thatF | N(U(X)/U )op is a right Kan extension of F | N(U0 (X)/U )op . By the transitivity of Kan extensions (Proposition 4.3.2.8), it will suffice to prove that(0)
F | N(U0 (X)/U )op is a right Kan extension of F | N(U0 (X)/U )op . This follows
immediately from Lemma 7.3.5.1.
Corollary 7.3.5.3. Let X be a coherent topological space. Then the global
sections functor Γ : Shv(X) → S is a proper map of ∞-topoi.
Proof. The proof is identical to the proof of Corollary 7.3.4.11 (using Theorem 7.3.5.2 in place of Corollary 7.3.4.10).
Corollary 7.3.5.4. Let X be a coherent topological space. Then the global
sections functor
Γ : Shv(X) → S
commutes with filtered colimits.
7.3.6 Cell-Like Maps
Recall that a topological space X is an absolute neighborhood retract if X is
metrizable and if for any closed immersion X ,→ Y of X in a metric space
Y , there exists an open set U ⊆ Y containing the image of X, such that theinclusion X ,→ U has a left inverse (in other words, X is a retract of U ).
Let p : X → Y be a continuous map between locally compact absoluteneighborhood retracts. The map p is said to be cell-like if p is proper andeach fiber Xy = X ×Y {y} has trivial shape (in the sense of Borsuk; see[56] and §7.1.6). The theory of cell-like maps plays an important role ingeometric topology: we refer the reader to [18] for a discussion (and forseveral equivalent formulations of the condition that a map be cell-like).
The purpose of this section is to describe a class of geometric morphisms
between ∞-topoi, which we will call cell-like morphisms. We will then compare our theory of cell-like morphisms with the classical theory of cell-like
maps. We will also give a “nonclassical” example which arises in the theoryof rigid analytic geometry.
Definition 7.3.6.1. Let p∗ : X → Y be a geometric morphism of ∞-topoi.
We will say that p∗ is cell-like if it is proper and if the right adjoint p∗ (which
is well-defined up to equivalence) is fully faithful.

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Warning 7.3.6.2. Many authors refer to a map p : X → Y of arbitrary compact metric spaces as cell-like if each fiber Xy = X ×Y {y} hastrivial shape. This condition is generally weaker than the condition thatp∗ : Shv(X) → Shv(Y ) be cell-like in the sense of Definition 7.3.6.1. However,
the two definitions are equivalent provided that X and Y are sufficiently nice(for example, if they are locally compact absolute neighborhood retracts).
Our departure from the classical terminology is perhaps justified by the factthat the class of morphisms introduced in Definition 7.3.6.1 has good formal
properties: for example, stability under composition.
Remark 7.3.6.3. Let p∗ : X → Y be a cell-like geometric morphism between∞-topoi. Then the unit map idY → p∗ p∗ is an equivalence of functors. It
follows immediately that p∗ induces an equivalence of shapes Sh(X) → Sh(Y)(see §7.1.6).
Proposition 7.3.6.4. Let p∗ : X → Y be a proper morphism of ∞-topoi.
Suppose that Y has enough points. Then p∗ is cell-like if and only if, forevery pullback diagram
X0

/X


S


/Y

p∗

in RTop, the ∞-topos X0 has trivial shape.
Proof. Suppose first that each fiber X0 has trivial shape. Let F ∈ Y. We wish
to show that the unit map u : F → p∗ p∗ F is an equivalence. Since Y hasenough points, it suffices to show that for each point q∗ : S → Y, the map
q ∗ u is an equivalence in S, where q ∗ denotes a left adjoint to q∗ . Form apullback diagram of ∞-topoi
/X

X0

S

p∗

s∗
q∗


/ Y.

Since p∗ is proper, this diagram is left adjointable. Consequently, q ∗ u can be
identified with the unit map
K → s∗ s∗ K,
where K = q ∗ F ∈ S. If X0 has trivial shape, then this map is an equivalence.
Conversely, if p∗ is cell-like, then the above argument shows that for every
diagram
/X

X0

S

s∗
q∗


/Y

p∗

highertopoi

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as above and every F ∈ Y, the adjunction map
K → s∗ s∗ K
is an equivalence, where K = q ∗ F. To prove that X0 has trivial shape, itwill suffice to show that q ∗ is essentially surjective. For this, we observethat since S is a final object in the ∞-category of ∞-topoi, there exists ageometric morphism r∗ : Y → S such that r∗ ◦ q∗ is homotopic to idS . It
follows that q ∗ ◦ r∗ ' idS . Since idS is essentially surjective, we conclude
that q ∗ is essentially surjective.
Corollary 7.3.6.5. Let p : X → Y be a map of paracompact topologicalspaces. Assume that p∗ is proper and that Y has finite covering dimension.
Then p∗ : Shv(X) → Shv(Y ) is cell-like if and only if each fiber Xy =X ×Y {y} has trivial shape.
Proof. Combine Proposition 7.3.6.4 with Corollary 7.2.1.17.
Proposition 7.3.6.6. Let p : X → Y be a proper map of locally compact
ANRs. The following conditions are equivalent:(1) The geometric morphism p∗ : Shv(X) → Shv(Y ) is cell-like.
(2) For every open subset U ⊆ Y , the restriction map X ×Y U → U is a
homotopy equivalence.
(3) Each fiber Xy = X ×Y {y} has trivial shape.
Proof. It is easy to see that if p∗ is cell-like, then each of the restrictions p0 :X ×Y U → U induces a cell-like geometric morphism. According to Remark
7.3.6.3, p0∗ is a shape equivalence and therefore a homotopy equivalence by
Proposition 7.1.6.8. Thus (1) ⇒ (2).
We next prove that (2) ⇒ (1). Let F ∈ Shv(Y ) and let u : F → p∗ p∗ F
be a unit map; we wish to show that u is an equivalence. It will suffice toshow that the induced map F(U ) → (p∗ p∗ F)(U ) is an equivalence in S foreach paracompact open subset U ⊆ Y . Replacing Y by U , we may reduceto the problem of showing that the map F(Y ) → (p∗ F)(X) is a homotopy
equivalence. According to Corollary 7.1.4.4, we may assume that F is thesimplicial nerve of SingY Ye , where Ye is a fibrant-cofibrant object of Top/Y .
e whereAccording to Proposition 7.1.5.1, we may identify p∗ F with SingX X,
e = X ×Y Ye . It therefore suffices to prove that the induced map of simplicialXfunction spaces
e ' MapY (X, Ye )MapY (Y, Ye ) → MapX (X, X)
is a homotopy equivalence, which follows immediately from (2).
The implication (1) ⇒ (3) follows from the proof of Proposition 7.3.6.6,
and the implication (3) ⇒ (2) is classical (see [37]).

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Remark 7.3.6.7. It is possible to prove the following generalization of
Proposition 7.3.6.6: a proper geometric morphism p∗ : X → Y is cell-like
if and only if, for each object U ∈ Y, the associated geometric morphism
X/p∗ U → Y/U is a shape equivalence (and, in fact, it is necessary to checkthis only on a collection of objects U ∈ Y which generates Y under colimits).
Remark 7.3.6.8. Another useful property of the class of cell-like morphisms, which we will not prove here, is stability under base change: givena pullback diagram
X0
p0∗


Y0

/X
p∗


/ Y,

where p∗ is cell-like, p0∗ is also cell-like.
If p∗ : X → Y is a cell-like morphism of ∞-topoi, then many properties of
Y are controlled by the analogous properties of X. For example:
Proposition 7.3.6.9. Let p∗ : X → Y be a cell-like morphism of ∞-topoi. If
X has homotopy dimension ≤ n, then Y also has homotopy dimension ≤ n.
Proof. Let 1Y be a final object of Y, U an n-connective object of Y, andp∗ a left adjoint to p∗ . We wish to prove that Homh Y (1Y , U ) is nonempty.
Since p∗ is fully faithful, it will suffice to prove that HomhX (p∗ 1Y , p∗ U ). We
now observe that p∗ 1Y is a final object of X (since p is left exact), p∗ U isn-connective (Proposition 6.5.1.16), and X has homotopy dimension ≤ n, sothat HomhX (p∗ 1Y , p∗ U ) is nonempty, as desired.
We conclude with a different example of a class of cell-like maps. We willassume in the following discussion that the reader is familiar with the basic
ideas of rigid analytic geometry; for an account of this theory we refer thereader to [29]. Let K be a field which is complete with respect to a nonArchimedean absolute value ||K : K → R. Let A be an affinoid algebra over
K: that is, a quotient of an algebra of convergent power series (in several
variables) with values in K. Let X be the rigid space associated to A. One
can associate to X two different “underlying” topological spaces:(ZR1) The category C of rational open subsets of X has a Grothendieck topology given by admissible affine covers. The topos of sheaves of sets on C
is localic, and the underlying locale has enough points: it is therefore
isomorphic to the locale of open subsets of a (canonically determined)
topological space XZR , the Zariski-Riemann space of X.
(ZR2) In the case where K is a discretely valued field with ring of integers R,
one may define XZR to be the inverse limit of the underlying spaces
b → Spf R which have generic fiber X.
of all formal schemes X

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CHAPTER 7(ZR3) Concretely, XZR can be identified with the set of all isomorphismclasses of continuous multiplicative seminorms ||A : A → M ∪ {∞},
where M is an ordered abelian group containing the value group|K ∗ |K ⊆ R∗
and the restriction of ||A to K coincides with ||K .
(B1) The category of sheaves of sets on C contains a full subcategory, consisting of overconvergent sheaves. This category is also a localic topos,
and the underyling locale is isomorphic to the lattice of open subsets ofa (canonically determined) topological space XB , the Berkovich spaceof X. The category of overconvergent sheaves is a localization of thecategory of all sheaves on C, and there is an associated map of topological spaces p : XZR → XB .
(B2) Concretely, XB can be identified with the set of all continuous multiplicative seminorms ||A : A → R ∪{∞} which extend ||K . It isequipped with the topology of pointwise convergence and is a compact Hausdorff space.
The relationship between the Zariski-Riemann space XZR and the Berkovich space XB (or more conceptually, the relationship between the categoryof all sheaves on X and the category of overconvergent sheaves on X) is neatly
summarized by the following result.
Proposition 7.3.6.10. Let K be a field which is complete with respect to a
non-Archimedean absolute value ||K , let A be an affinoid algebra over K, let
X be the associated rigid space, and let p : XZR → XB be the natural map.
Then p induces a cell-like morphism of ∞-topoi p∗ : Shv(XZR ) → Shv(XB ).
Before giving the proof, we need an easy lemma. Recall that a topologicalspace X is irreducible if every finite collection of nonempty open subsets of
X has nonempty intersections.
Lemma 7.3.6.11. Let X be an irreducible topological space. Then Shv(X)
has trivial shape.
Proof. Let π : X → ∗ be the projection from X to a point and letπ∗ :Shv(X) → Shv(∗) be the induced geometric morphism. We will construct aleft adjoint π ∗ to π∗ such that the unit map id → π∗ π ∗ is an equivalence.
We begin by defining G : P(X) → P(∗) to be the functor given by composition with π −1 , so that G| Shv(X) = π∗ . Leti : N(U(X))op → N(U(∗))op
be defined so that(∅i(U ) ={∗}

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HIGHER TOPOS THEORY IN TOPOLOGY

and let F : P(∗) → P(U ) be given by composition with i. We observe that
F is a left Kan extension functor, so that the identity map
idP(∗) → G ◦ F
exhibits F as a left adjoint to G. We will show that F (Shv(∗)) ⊆ Shv(X).
Setting π ∗ = F | Shv(∗), we conclude that the identity map
idShv(∗) → π∗ π ∗is the unit of an adjunction between π∗ and π ∗ , which will complete theproof.
Let U ⊆ U(X) be a sieve which covers the open set U ⊆ X. We wish toprove that the diagramFip : N(Uop )/ → N(U(X))op → N(U(∗))op → S
is a limit. Let U0 = {V ∈ U : V 6= ∅}. Since F(∅) is a final object of S, p is a/limit if and only if p| N(Uop0 ) is a limit diagram. If U = ∅, then this follows/
from the fact that F(∅) is final in S. If U 6= ∅, then p| N(Uop0 ) is a constantdiagram, so it will suffice to prove that the simplicial set N(U0 )op is weakly
contractible. This follows from the observation that Uop
0 is a filtered partially ordered set since U0 is nonempty and stable under finite intersections(because X is irreducible).
Proof of Proposition 7.3.6.10. We first show that p∗ is a proper map of ∞topoi. We note that p factors as a composition
p0

p00

XZR → XZR × XB → XB .
0
The map p is a pullback of the diagonal map XB → XB × XB . Since XB is
Hausdorff, p0 is a closed immersion. It follows that p0∗ is a closed immersionof ∞-topoi (Corollary 7.3.2.9) and therefore a proper morphism (Proposition
7.3.2.12). It therefore suffices to prove that p00 is a proper map of ∞-topoi.
We note the existence of a commutative diagram/ Shv(XZR )Shv(XZR × XB )
p00
∗

g∗


/ Shv(∗).


Shv(XB )Using Proposition 7.3.1.11, we deduce that this is a homotopy Cartesian
diagram of ∞-topoi. It therefore suffices to show that the global sections
functor g∗ : Shv(XZR ) → Shv(∗) is proper, which follows from Corollary
7.3.5.3.
We now observe that the topological space XB is paracompact and has
finite covering dimension ([5], Corollary 3.2.8), so that Shv(XB ) has enoughpoints (Corollary 7.2.1.17). According to Proposition 7.3.6.4, it suffices toshow that for every fiber diagram/ Shv(XZR )X0
Shv(∗)q∗/ Shv(XB ),highertopoi

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CHAPTER 7

the ∞-topos X0 has trivial shape. Using Lemma 6.4.5.6, we conclude that q∗
is necessarily induced by a homomorphism of locales U(XB ) → U(∗), whichcorresponds to an irreducible closed subset of XB . Since XB is Hausdorff,
this subset consists of a single (closed) point x. Using Proposition 7.3.2.12
and Corollary 7.3.2.9, we can identify X0 with the ∞-topos Shv(Y ), where
Y = XZR ×XB {x}. We now observe that the topological space Y is coherent
and irreducible (it contains a unique “generic” point), so that Shv(Y ) has
trivial shape by Lemma 7.3.6.11.
Remark 7.3.6.12. Let p∗ : Shv(XZR ) → Shv(XB ) be as in Proposition
7.3.6.10. Then p∗ has a fully faithful left adjoint p∗ . We might say that an
object of Shv(XZR ) is overconvergent if it belongs to the essential image ofp∗ ; for sheaves of sets, this agrees with the classical terminology.
Remark 7.3.6.13. One can generalize Proposition 7.3.6.10 to rigid spaces
which are not affinoid; we leave the details to the reader.

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Appendix

This appendix is comprised of three parts. In §A.1, we will review someideas from classical category theory, such as monoidal structures, enrichedcategories, and Quillen’s small object argument. We give a brief overview of
the theory of model categories in §A.2. The main result here is Proposition
A.2.6.15, which will allow us to establish the existence of model category
structures in a variety of situations with a minimal amount of effort. In §A.3,
we will use this result to make a detailed study of the theory of simplicial
categories. Our exposition is rather dense; for a more leisurely account of
the theory of model categories, we refer the reader to one of the standard
texts (such as [40]).

A.1 CATEGORY THEORY
Familiarity with classical category theory is the main prerequisite for reading
this book. In this section, we will fix some of the notation that we use when
discussing categories and summarize (generally without proofs) some of theconcepts employed in the body of the text.
If C is a category, we let Ob(C) denote the set of objects of C. We will write
X ∈ C to mean that X is an object of C. For X, Y ∈ C, we write HomC (X, Y )for the set of morphisms from X to Y in C. We also write idX for the identity
automorphism of X ∈ C (regarded as an element of HomC (X, X)).
If Z is an object in a category C, then the overcategory C/Z of objects over
Z is defined as follows: the objects of C/Z are diagrams X → Z in C. A
morphism from f : X → Z to g : Y → Z is a commutative triangle
/Y
X
f

~

g

Z.
Dually, we have an undercategory CZ/ = ((Cop )/Z )op of objects under Z.
If f : X → Z and g : Y → Z are objects in C/Z , then we will often write
HomZ (X, Y ) rather than HomC/Z (f, g).
We let Set denote the category of sets and Cat the category of (small)
categories (where the morphisms are given by functors).
If κ is a regular cardinal, we will say that a set S is κ-small if it has
cardinality less than κ. We will also use this terminology when discussing
mathematical objects other than sets, which are built out of sets. For example, we will say that a category C is κ-small if the set of all objects of C isκ-small and the set of all morphisms in C is likewise κ-small.

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We will need to discuss categories which are not small. In order to minimize the effort spent dealing with set-theoretic complications, we will adopt
the usual device of Grothendieck universes. We fix a strongly inaccessible
cardinal κ and refer to a mathematical object (such as a set or category)as small if it is κ-small, and large otherwise. It should be emphasized that
this is primarily a linguistic device and that none of our results depend inan essential way on the existence of a strongly inaccessible cardinal κ.
Throughout this book, the word “topos” will always mean Grothendiecktopos. Strictly speaking, a knowledge of classical topos theory is not required
to read this book: all of the relevant classical concepts will be introduced(though sometimes in a hurried fashion) in the course of our search forsuitable ∞-categorical analogues.
A.1.1 Compactness and Presentability
Let κ be a regular cardinal.
Definition A.1.1.1. A partially ordered set I is κ-filtered if, for any subset
I0 ⊆ I having cardinality < κ, there exists an upper bound for I0 in I.
Let C be a category which admits (small) colimits and let X be an objectof C. Suppose we are given a κ-filtered partially ordered set I and a diagram{Yα }α∈I in C indexed by I. Let Y denote a colimit of this diagram. Then
there is an associated map of sets
ψ : lim HomC (X, Yα ) → HomC (X, Y ).
−→We say that X is κ-compact if ψ is bijective for every κ-filtered partiallyordered set I and every diagram {Yα } indexed by I. We say that X is small
if it is κ-compact for some (small) regular cardinal κ. In this case, X is
κ-compact for all sufficiently large regular cardinals κ.
Definition A.1.1.2. A category C is presentable if it satisfies the following
conditions:(1) The category C admits all (small) colimits.
(2) There exists a (small) set S of objects of C which generates C undercolimits; in other words, every object of C may be obtained as thecolimit of a (small) diagram taking values in S.
(3) Every object in C is small. (Assuming (2), this is equivalent to theassertion that every object which belongs to S is small.)(4) For any pair of objects X, Y ∈ C, the set HomC (X, Y ) is small.
Remark A.1.1.3. In §5.5, we describe an ∞-categorical generalization ofDefinition A.1.1.2.
Remark A.1.1.4. For more details of the theory of presentable categories,
we refer the reader to [1]. Note that our terminology differs slightly fromthat of [1], in which our presentable categories are called locally presentable
categories.

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A.1.2 Lifting Problems and the Small Object Argument
Let C be a category and let p : A → B and q : X → Y be morphisms in C.
Recall that p is said to have the left lifting property with respect to q, and q
the right lifting property with respect to p, if given any diagram
/X
A
>
p


/Y


B

q

there exists a dotted arrow as indicated, rendering the diagram commutative.
Remark A.1.2.1. In the case where Y is a final object of C, we will instead
say that X has the extension property with respect to p : A → B.
Let S be any collection of morphisms in C. We define ⊥ S to be the classof all morphisms which have the right lifting property with respect to allmorphisms in S, and S⊥ to be the class of all morphisms which have the left
lifting property with respect to all morphisms in S. We observe thatS ⊆ (⊥ S)⊥ .
The class of morphisms (⊥ S)⊥ enjoys several stability properties which
we axiomatize in the following definition.
Definition A.1.2.2. Let C be a category with all (small) colimits and let
S be a class of morphisms of C. We will say that S is weakly saturated if it
has the following properties:(1) (Closure under the formation of pushouts) Given a pushout diagram
C

C0

/D

f


/ D0

f0

such that f belongs to S, the morphism f 0 also belongs to S.
(2) (Closure under transfinite composition) Let C ∈ C be an object, let α
be an ordinal, and let {Dβ }β<α be a system of objects of CC/ indexedby α: in other words, for each β < α, we are supplied with a morphism
C → Dβ , and for each γ ≤ β < α a commutative diagram
>
C

Dγ

φγ,β


Dβ

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APPENDIX

satisfying φγ,δ ◦ φβ,γ = φβ,δ . For β ≤ α, we let D<β be a colimit of thesystem {Dγ }γ<β taken in the category CC/ .
Suppose that, for each β < α, the natural map D<β → Dβ belongs to
S. Then the induced map C → D<α belongs to S.
(3) (Closure under the formation of retracts) Given a commutative diagram
C
f


D

/ C0
g


/ D0

/C
f


/D

in which both horizontal compositions are the identity, if g belongs to
S, then so does f .
It is worth noting that saturation has the following consequences:
Proposition A.1.2.3. Let C be a category which admits all (small) colimits
and let S be a weakly saturated class of morphism in C. Then
(1) Every isomorphism belongs to S.
(2) The class S is stable under composition: if f : X → Y and g : Y → Z
belong to S, then so does g ◦ f .
Proof. Assertion (1) is equivalent to the closure of S under transfinite composition in the special case where α = 0; (2) is equivalent to the special casewhere α = 2.
Remark A.1.2.4. A reader who is ill at ease with the style of the preceding argument should feel free to take the asserted properties as part of thedefinition of a weakly saturated class of morphisms.
The intersection of any collection of weakly saturated classes of morphisms
is itself weakly saturated. Consequently, for any category C which admits
small colimits, and any collection A of morphisms of C, there exists a smallest weakly saturated class of morphisms containing A: we will call this theweakly saturated class of morphisms generated by A. We note that (⊥ A)⊥
is weakly saturated. Under appropriate set-theoretic assumptions, Quillen’s“small object argument” can be used to establish that (⊥ A)⊥ is the weakly
saturated class generated by A:
Proposition A.1.2.5 (Small Object Argument). Let C be a presentable
category and A0 = {φi : Ci → Di }i∈I a collection of morphisms in C indexed
by a (small) set I. For each n ≥ 0, let C[n] denote the category of functors
from the linearly ordered set [n] = {0, . . . , n} into C. There exists a functor
T : C[1] → C[2] with the following properties:

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791

APPENDIX(1) The functor T carries a morphism f : X → Z to a diagram
>Y

f0

f 00



f

X

/Z

where f 0 belongs to the weakly saturated class of morphisms generated
by A0 and f 00 has the right lifting property with respect to each morphism in A0 .
(2) If κ is a regular cardinal such that each of the objects Ci , Di is κcompact, then T commutes with κ-filtered colimits.
Proof. Fix a regular cardinal κ as in (2) and fix a morphism f : X → Z in
C. We will give a functorial construction of the desired diagram
f0

>Y
f

X

f 00

/ Z.

We define a transfinite sequence of objects
Y0 → Y1 → · · ·in C/Z indexed by ordinals smaller than κ. Let Y0 = X and let Yλ =limα<λ Yα when λ is a nonzero limit ordinal. For i ∈ I, let Fi : C/Z → Set−→
be the functor(W → Z) 7→ HomC (Di , Z) ×HomC (Ci ,Z) HomC (Ci , W ).
Supposing that Yα has been defined, we define Yα+1 by the following pushout
diagram`/ Yαi∈I,η∈Fi (Yα ) Ci`

i∈I,η∈Fi (Yα ) Di
/ Yα+1 .

We conclude by defining Y to be limα<κ Yα . It is easy to check that this−→construction has the desired properties.
Remark A.1.2.6. If C is enriched, tensored, and cotensored over anotherpresentable monoidal category S (see §A.1.4), then a similar construction
shows that we can choose T to be an S-enriched functor.
Corollary A.1.2.7. Let C be a presentable category and let A be a setof morphisms of C. Then (⊥ A)⊥ is the smallest weakly saturated class ofmorphisms containing A.

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APPENDIX

Proof. Let A be the smallest weakly saturated class of morphisms containing
A, so that A ⊆ (⊥ A)⊥ . For the reverse inclusion, let us suppose that f :X → Z belongs to (⊥ A)⊥ . Proposition A.1.2.5 implies the existence of afactorization
f0

f 00

X → Y → Z,
where f 0 ∈ A and f 00 belongs to ⊥ A. It follows that f has the left lifting
property with respect to f 00 , so that f is a retract of f 0 and therefore belongs
to A.
Remark A.1.2.8. Let C be a presentable category, let S be a (small) set
of morphisms in C, and suppose that f : X → Y belongs to the weakly
saturated class of morphisms generated by S. The proofs of Proposition
A.1.2.5 and Corollary A.1.2.7 show that there exists a transfinite sequence
Y0 → Y1 → · · ·of objects of CX/ , indexed by a set of ordinals {β|β < α}, with the following
properties:(i) For each β < α, there is a pushout diagramCglimγ<β Yγ−→/D/ Yβ ,

where the colimit is formed in CX/ and g ∈ S.
(ii) The object Y is a retract of limγ<α Yγ in the category CX/ .
−→A.1.3 Monoidal Categories
A monoidal category is a category C equipped with a (coherently) associative“product” functor ⊗ : C × C → C and a unit object 1. The associativity isexpressed by demanding isomorphisms
ηA,B,C : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C),
and the requirement that 1 be unital is expressed by demanding isomorphisms
αA : A ⊗ 1 → A
βA : 1 ⊗ A → A.
We do not merely require the existence of these isomorphisms: they are partof the structure of a monoidal category. Moreover, these isomorphisms are
required to satisfy the following conditions:highertopoi

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APPENDIX• The isomorphism ηA,B,C depends functorially on the triple (A, B, C);in other words, η may be regarded as a natural isomorphism between
the functors
C×C×C → C
(A, B, C) 7→ (A ⊗ B) ⊗ C(A, B, C) 7→ A ⊗ (B ⊗ C).
Similarly, αA and βA depend functorially on A.
• Given any quadruple (A, B, C, D) of objects of C, the MacLane pentagon((A ⊗ B) ⊗ C) ⊗ D
ηA,B,C ⊗idD

ηA⊗B,C,Dv(A ⊗ (B ⊗ C)) ⊗ D((A ⊗ B) ⊗ (C ⊗ D)

ηA,B⊗C,D

ηA,B,C⊗DA ⊗ ((B ⊗ C) ⊗ D)

idA ⊗ηB,C,D/ A ⊗ (B ⊗ (C ⊗ D))

is commutative.
• For any pair (A, B) of objects of C, the triangle(A ⊗ 1) ⊗ B

ηA,1,B/ A ⊗ (1 ⊗ B)

αA ⊗idB&x
A⊗B

idA ⊗βB

is commutative.
MacLane’s coherence theorem asserts that the commutativity of this pairof diagrams implies the commutativity of all diagrams that can be written using only the isomorphisms ηA,B,C , αA , and βA . More precisely, any
monoidal category is equivalent (as a monoidal category) to a strict monoidal
category: that is, a monoidal category in which ⊗ is literally associative, 1
is literally a unit with respect to ⊗, and the isomorphisms ηA,B,C , αA , βA
are the identity maps.
Example A.1.3.1. Let C be a category which admits finite products. Then
C admits the structure of a monoidal category where the operation ⊗ is given
by Cartesian productA⊗B 'A×Band the isomorphisms ηA,B,C are induced from the evident associativity of
the Cartesian product. The identity 1 is defined to be the final object of C,

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APPENDIX

and the isomorphisms αA and βA are determined in the obvious way. We
refer to this monoidal structure on C as the Cartesian monoidal structure.
We remark that the Cartesian product A × B is well-defined only up to(unique) isomorphism (as is the final object 1), so that strictly speaking theCartesian monoidal structure on C depends on various choices; however, allsuch choices lead to (canonically) equivalent monoidal categories.
Remark A.1.3.2. Let (C, ⊗, 1, η, α, β) be a monoidal category. We will
generally abuse notation by simply saying that C is a monoidal category,that (C, ⊗) is a monoidal category, or that ⊗ is a monoidal structure on C;
the other structure is implicitly understood to be present as well.
Remark A.1.3.3. Let C be a category equipped with a monoidal structure⊗. Then we may define a new monoidal structure on C by setting A ⊗op B =B⊗A. We refer to this monoidal structure ⊗op as the opposite of the monoidal
structure ⊗.
Definition A.1.3.4. A monoidal category (C, ⊗) is said to be left-closed if,
for each A ∈ C, the functor
N 7→ A ⊗ N
admits a right adjoint
Y 7→ AY.
We say that (C, ⊗) is right-closed if the opposite monoidal structure (C, ⊗op )is left-closed; in other words, if every functor
N 7→ N ⊗ A
has a right adjoint
Y 7→ Y A .
Finally, we say that (C, ⊗) is closed if it is both right-closed and left-closed.
In the setting of monoidal categories, it is appropriate to consider only
those functors which are compatible with the monoidal structures in thefollowing sense:
Definition A.1.3.5. Let (C, ⊗) and (D, ⊗) be monoidal categories. A rightlax monoidal functor from C to D consists of the following data:• A functor G : C → D.
• A natural transformation γA,B : G(A) ⊗ G(B) → G(A ⊗ B) renderingcommutative the diagram/ G(A) ⊗ (G(B) ⊗ G(C))(G(A) ⊗ G(B)) ⊗ G(C)
γB,C

γA,B


G(A) ⊗ G(B ⊗ C)
G(A ⊗ B) ⊗ G(C)γA⊗B,CG((A ⊗ B) ⊗ C)

γA,B⊗C
G(ηA,B,C )/ G(A ⊗ (B ⊗ C)).

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APPENDIX• A map e : 1D → G(1C ) rendering commutative the diagrams
G(A) ⊗ 1Dγ/ G(A) ⊗ G(1C ) A,1C / G(A ⊗ 1C )

id ⊗e

αG(A)

1D ⊗ G(B)e⊗id(

G(αA )w

G(A)γ/ G(1C ) ⊗ G(B) 1C ,B / G(1C ⊗ B) .

βG(B)

(

G(B)

w

G(βB )

A natural transformation between right-lax monoidal functors is monoidal
if it commutes with the maps γA,B , e.
Dually, a left-lax monoidal functor from C to D consists of a right-lax
monoidal functor from Cop to Dop ; it is determined by giving a functor F :C → D together with a map e0 : F (1C ) → 1D and a natural transformation
0γA,B: F (A ⊗ B) → F (A) ⊗ F (B)satisfying the appropriate analogues of the conditions listed above.
If F is a right-lax monoidal functor via isomorphisms
e : 1D → F (1C )γA,B : F (A) ⊗ F (B) → F (A ⊗ B),
then F may be regarded as a left-lax monoidal functor by setting e0 = e−1 ,−10= γA,B. In this case, we simply say that F is a monoidal functor.
γA,BRemark A.1.3.6. Let
Co

F
G

/D

be an adjunction between categories C and D. Suppose that C and D areequipped with monoidal structures. Then endowing G with the structure ofa right-lax monoidal functor is equivalent to endowing F with the structureof a left-lax monoidal functor.
Example A.1.3.7. Let C and D be categories which admit finite products
and let F : C → D be a functor between them. If we regard C and D
as endowed with the Cartesian monoidal structure, then F acquires the
structure of a left-lax monoidal functor in a canonical way via the maps
F (A × B) → F (A) × F (B) induced from the functoriality of F . In this case,
F is a monoidal functor if and only if it commutes with finite products.

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APPENDIX

A.1.4 Enriched Category Theory
One frequently encounters categories D in which the collections of morphisms
HomD (X, Y ) between two objects X, Y ∈ D have additional structure: forexample, a topology, a group structure, or the structure of a vector space.
These situations may all be efficiently described using the language of enriched category theory, which we now introduce.
Let (C, ⊗) be a monoidal category. A C-enriched category D consists ofthe following data:(1) A collection of objects.
(2) For every pair of objects X, Y ∈ D, a mapping object MapD (X, Y ) ofC.
(3) For every triple of objects X, Y, Z ∈ D, a composition map
MapD (Y, Z) ⊗ MapD (X, Y ) → MapD (X, Z).
Composition is required to be associative in the sense that for any
W, X, Y, Z ∈ C, the diagram
MapD (Y, Z) ⊗ MapD (X, Y ) ⊗ MapD (W, X)
w
MapD (Y, Z) ⊗ MapD (W, Y )'MapD (X, Z) ⊗ MapD (W, X)'
w
MapD (W, Z)
is commutative.
(4) For every object X ∈ D, a unit map 1 → MapD (X, X) renderingcommutative the diagrams
1 ⊗ MapD (Y, X)/ MapD (X, X) ⊗ MapD (Y, X)&
x
MapD (Y, X)

MapD (X, Y ) ⊗ 1/ MapD (X, Y ) ⊗ MapD (X, X)&
x
MapD (X, Y ).

Example A.1.4.1. Suppose that (C, ⊗) is a right-closed monoidal category.
Then C is enriched over itself in a natural way if one defines MapC (X, Y ) =Y X.

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Example A.1.4.2. Let C be the category of sets endowed with the Cartesianmonoidal structure. Then a C-enriched category is simply a category in theusual sense.
Remark A.1.4.3. Let G : C → C0 be a right-lax monoidal functor between
monoidal categories. Suppose that D is a category enriched over C. We may
define a category G(D) enriched over C0 as follows:(1) The objects of G(D) are the objects of D.
(2) Given objects X, Y ∈ D, we set
MapG(D) (X, Y ) = G(MapD (X, Y )).
(3) The composition in G(D) is given by the composite map
G(MapD (Y, Z)) ⊗ G(MapD (X, Y )) → G(MapD (Y, Z) ⊗ MapD (X, Y ))→ G(MapD (X, Z)).
Here the first map is determined by the right-lax monoidal structureon the functor G, and the second is obtained by applying G to thecomposition law in the category D.
(4) For every object X ∈ D, the associated unit in G(D) is given by thecomposition
1C0 → G(1C ) → G(MapD (X, X)).
Suppose that C is any monoidal category. Consider the functor C → Set
given by
X 7→ HomC (1, X).
This is a right-lax monoidal functor from (C, ⊗) to Set, where the latter isequipped with the Cartesian monoidal structure. By the above remarks, we
see that we may equip any C-enriched category D with the structure of an
ordinary category by setting
HomD (X, Y ) = HomC (1, MapD (X, Y )).
We will generally not distinguish notationally between D as a C-enrichedcategory and this (underlying) category having the same objects. However, toavoid confusion, we use different notations for the morphisms: MapD (X, Y )is an object of C, while HomD (X, Y ) is a set.
Remark A.1.4.4. If D and D0 are categories which are enriched over thesame monoidal category C, then one can define a category of C-enriched
functors from D to D0 in the evident way. Namely, an enriched functor
F : D → D0 consists of a map from the objects of D to the objects of D0
and a collection of morphisms
ηX,Y : MapD (X, Y ) → MapD0 (F X, F Y )with the following properties:

highertopoi

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APPENDIX(i) For each object X ∈ D, the composition
ηX,X

1C → MapD (X, X) → MapD0 (F X, F X)
coincides with the unit map for F X ∈ D0 .
(ii) For every triple of objects X, Y, Z ∈ D, the diagram
MapD (Y, Z) ⊗ MapD (X, Y )/ MapD (X, Z)
MapD0 (F Y, F Z) ⊗ MapD0 (F X, F Y )/ MapD0 (F X, F Z)

is commutative.
If F and F 0 are enriched functors, an enriched natural transformation α
from F to F 0 consists of specifying, for each object X ∈ D, a morphism
αX ∈ HomD0 (F X, F 0 X) which renders commutative the diagram/ MapD0 (F X, F Y )MapD (X, Y )
MapD0 (F 0 X, F 0 Y )

αY

αX


/ MapD0 (F X, F 0 Y ).

Let C be a right-closed monoidal category and let D be a category enrichedover C. Fix objects C ∈ C, X ∈ D, and consider the functor
D→C
Y 7→ MapD (X, Y )C .
This functor may or may not be corepresentable in the sense that there exists
an object Z ∈ D and an isomorphism of functors
η : MapD (X, •)C ' MapD (Z, •).
If such an object Z exists, we will denote it by C ⊗ X. The natural isomorphism η is determined by specifying a single map
η(X) : C → MapD (X, C ⊗ X).
By general nonsense, the map η(X) determines C ⊗ X up to (unique) isomorphism provided that C ⊗ X exists. If the object C ⊗ X exists for every
C ∈ C, X ∈ D, then we say that D is tensored over C. In this case, theconstruction(X, C) 7→ C ⊗ X
determines a functor C × D → D. Moreover, one has canonical isomorphisms(C ⊗ D) ⊗ X ' C ⊗ (D ⊗ X),

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APPENDIX

which express the idea that D may be regarded as equipped with an “action”
of C. Here we imagine C as a kind of generalized monoid (via its monoidal
structure).
Dually, if C is left-closed, then an object of D which represents the functor
Y 7→C MapD (Y, X)
will be denoted by CX; the object CX (if it exists) is determined up to(unique) isomorphism by a map C → MapD (CX, X). If this object exists for
all C ∈ C, X ∈ D, then we say that D is cotensored over C.
Example A.1.4.5. Let C be a right-closed monoidal category. Then C may
be regarded as enriched over itself in a natural way. It is automaticallytensored over itself; it is cotensored over itself if and only if it is left-closed.
A.1.5 Trees
Let C be a presentable category and S a small collection of morphisms in C.
According to Remark A.1.2.8, the smallest weakly saturated class of morphisms S containing S can be obtained from S using pushouts, retracts, andtransfinite composition. It is natural to ask if the formation of retracts isnecessary: that is, does the weakly saturated class of morphisms generated
by S coincide with the class of morphisms which generated by transfinite
compositions of pushouts of morphisms of S? Our goal for the remainder ofthis section is to give an affirmative answer, at least after S has been suitably enlarged (Proposition A.1.5.12). This result is of a somewhat technicalnature and will be needed only during our discussion of combinatorial model
categories in §A.2.6.
We begin by introducing a generalization of the notion of a transfinite
chain of morphisms.
Definition A.1.5.1. Let C be a presentable category and let S be a collection of morphisms in C. An S-tree in C consists of the following data:(1) An object X ∈ C called the root of the S-tree.
(2) A partially ordered set A which is well-founded (so that every nonempty
subset of A has a minimal element).
(3) A diagram A → CX/ , which we will denote by α 7→ Yα .
(4) For each α ∈ A, a pushout diagramC
limβ<α Yβ−→
where f ∈ S.

f/D

/ Yα ,

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APPENDIX

Let κ be a regular cardinal. We will say that an S-tree in C is κ-good if eachof the objects C and D appearing above is κ-compact, and if for each α ∈ A,
the set {β ∈ A : β < α} is κ-small.
Notation A.1.5.2. Let C be a presentable category and S a collection ofmorphisms in C. We will indicate an S-tree by writing {Yα }α∈A . Here theroot X ∈ C and the relevant pushout diagrams are understood implicitly to
be part of the data.
Suppose we are given an S-tree {Yα }α∈A and a subset B ⊆ A which is
downward-closed in the following sense: if α ∈ B and β ≤ α, then β ∈ B.
Then {Yα }α∈B is an S-tree. We let YB denote the colimit limα∈B Yα formed−→in the category CX/ . In particular, we have a canonical isomorphism Y∅ ' X.
If B = {α ∈ A|α ≤ β}, then YB ' Yβ .
Remark A.1.5.3. Let C be a presentable category, S a collection of morphisms in C, and {Yα }α∈A an S-tree in C with` root X. Given a map f : X →X 0 , we can form an associated S-tree {Yα X X 0 }α∈A having root X 0 .
Example A.1.5.4. Let C be a presentable category, S a collection of morphisms in C, and {Yα }α∈A an S-tree in C with root X. If A is linearly ordered, then we may identify {Yα }α∈A with a (possibly transfinite) sequenceof morphisms belonging to S,
X → Y0 → Y1 → · · · ,as in the statement of (2) in Definition A.1.2.2.
Remark A.1.5.5. Let C be a presentable category, S a collection of morphisms in C, and {Yα }α∈A an S-tree in C. Let B ⊆ A be downward-closed.
For α ∈ A − B, let Bα = B ∪ {β ∈ A : β ≤ α} and let Zα = YBα . Then{Zα }α∈A−B is an S-tree in C with root YB .
Lemma A.1.5.6. Let C be a presentable category and let S be a collectionof morphisms in C. Let {Yα }α∈A be an S-tree in C and let A00 ⊆ A0 ⊆ A
be subsets which are downward-closed in A. Then the induced map YA00 →YA0 belongs to the weakly saturated class of morphisms generated by S. In
particular, the canonical map Y∅ → YA belongs to the weakly saturated classof morphisms generated by S.
Proof. Using Remarks A.1.5.5 and A.1.5.3, we can assume without loss ofgenerality that A00 = ∅ and A0 = A. Using the assumption that A is wellfounded, we can write A as the union of a transfinite sequence (downwardclosed) subsets {B(γ) ⊆ A}γ<β with the following property:
S(∗) For each γ < β, the set B(γ) is obtained from B 0 (γ) = γ 0 <γ B(γ 0 ) byadjoining a minimal element αγ of A − B 0 (γ).
For γ < β, let Zγ = YB(γ) . We now observe that YA ' limγ<β Zγ and that−→highertopoi

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APPENDIX

for each γ < β there is a pushout diagram/ Yαγ

limα<α Yα−→γ
limγ 0 <γ Zγ 0−→f/ Zγ ,

so that f is the pushout of a morphism belonging to S.
Lemma A.1.5.7. Let C be a presentable category, let κ be a regular cardinal,
and let S = {fs : Cs → Ds } be a collection of morphisms in C, where each of
the objects Cs and Ds is κ-compact. Suppose that {Yα }α∈A is an S-tree in
C indexed by a partially ordered set (A, ≤). Then there exists the following:(1) A new ordering  on A which refines ≤ in the following sense: ifα  β, then α ≤ β. Let A0 denote the partially ordered set A with thisnew partial ordering.
(2) A κ-good S-tree {Yα0 }α∈A0 having the same root X as {Yα }α∈A .
(3) A collection of maps fα : Yα0 → Yα which form a commutative diagram
Yα0 0


fα0

Yα0

/ Yα0
fα


/ Yα

when α0  α.
(4) For every subset B ⊆ A which is downward-closed with respect to ≤,the induced map fB : YB0 → YB is an isomorphism.
Proof. Choose a transfinite sequence of downward-closed subsets {A(γ) ⊆A}γ≤β so that the following conditions are satisfied:(i) If γ 0 ≤ γ ≤ β, then A(γ 0 ) ⊆ A(γ).
(ii) If λ ≤ β is a limit ordinal (possibly zero), then A(λ) =Sγ<λ A(γ).

(iii) If γ + 1 ≤ β, then A(γ + 1) = A(γ) ∪ {αγ }, where αγ is a minimal
element of A − A(γ).
(iv) The subset A(β) coincides with A.
We will construct a compatible family of orderings A0 (γ) = (A(γ), ),S-trees {Yα0 }α∈A0 (γ) , and collections of morphisms {Yα0 → Yα }α∈A(γ) by induction on γ, so that the analogues of conditions (1) through (4) are satisfied.
If γ is a limit ordinal, there is nothing to do; let us assume therefore thathighertopoi

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APPENDIXγ < β and that the data (A0 (γ), {Yα0 }α∈A0 (γ) , {fα }α∈A(γ) ) has already been
constructed. Let B = {α ∈ A : α < αγ }, so that we have a pushout diagram
f

C


i

/D

/ Yα

YB

where f ∈ S. By the inductive hypothesis, we may identify YB with YB0 .
Since C is κ-compact, the map i admits a factorization
i00

i0

C → YB0 0 → YB0 ,where B 0 is κ-small. Enlarging B 0 if necessary, we may suppose that B 0 is
downward-closed under . We now extend the partial ordering  to A0 (γ +1) = A0 (γ) ∪ {αγ } by declaring that α  αγ if and only if α ∈ B 0 . We define
Yα0 γ by forming a pushout diagram
f

C


i0

/D

/ Yα0 ,

YB0 0γ

and we define fαγ : Yα0 γ → Yαγ to be the map induced by i00 . It is readily
verified that these data satisfy the desired conditions.
Lemma A.1.5.8. Let C be a presentable category, κ an uncountable regularcardinal, and S a collection of morphisms in C. Let {Yα }α∈A be a κ-good
S-tree with root X and let TA : YA → YA be an idempotent endomorphism
of YA in the category CX/ . Let B0 be an arbitrary κ-small subset of A. Then
there exists a κ-small subset B ⊆ A which is downward-closed and contains
B0 and an idempotent endomorphism TB : YB → YB such that the following
diagram commutes:
/ YB

X
=


X



/ YA

TB



/ YB

TA

/ YA .

Proof. Enlarging B0 if necessary, we may assume that B0 is downwardclosed. For every pair of downward-closed subsets A00 ⊆ A0 ⊆ A, let iA00 ,A0denote the canonical map from YA00 to YA0 . Note that because {Yα }α∈A is aκ-good S-tree, if A0 ⊆ A is downward-closed and κ-small, YA0 is κ-compact
when viewed as an object of CX/ . In particular, YB0 is a κ-compact objectof CX/ . It follows that the composition
iB ,A

T

0
A
YB0 →
YA →
YA

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APPENDIX

can also be factored as a composition
T

iB ,A

1
YA ,
YB0 →0 YB1 →where B1 ⊆ A is downward-closed and κ-small. Enlarging B1 if necessary,
we may suppose that B1 contains B0 .
We now proceed to define a sequence of κ-small downward-closed subsets
B0 ⊆ B1 ⊆ B2 ⊆ · · ·of A and maps Ti : YBi → YBi+1 . Suppose that i > 0 and that Bi and Ti−1
have already been constructed. By compactness again, we conclude that thecomposite map

iB ,A

T

A
i
YA
YA →
YBi →

can be factored asiBi+1 ,AT

YBi →i YBi+1 → YA ,where Bi+1 is κ-small. Enlarging Bi+1 if necessary, we may assume that
Bi+1 contains Bi and that the following diagrams commute:
Ti−1

YBi−1


iBi−1 ,Bi
Ti

YBi

/ YBi


Ti−1

YBi−1

iBi ,Bi+1

Ti−1

/ YBi+1

/ YBi

YBi



iBi ,Bi+1

Ti/ YBi+1 .

S

Let B = Bi ; then B is κ-small by virtue of our assumption that κ is
uncountable. The collection of maps {Ti } assemble to a map TB : YB → YB
with the desired properties.
Lemma A.1.5.9. Let C be a presentable category, κ an uncountable regularcardinal, and S a collection of morphisms in C. Let {Yα }α∈A be a κ-good
S-tree with root X, let B ⊆ A be downward-closed, and suppose we are given
a commutative diagram
/ YA
YB


TB



TA

/ YA
YB
in CX/ , where TA and TB are idempotent. Let C0 ⊆ A be a κ-small subset.
Then there exists a downward-closed κ-small subset C ⊆ A containing C0
and a pair of idempotent maps
TC : YC → YC
TB∩C : YB∩C → YB∩C
such that the following diagram commutes (in CX/ ):
/ YC
/ YA
YB o
YB∩C


TB

YB o



TB∩C

YB∩C



TC

/ YC



TA

/ YA .

highertopoi

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APPENDIX

Proof. Enlarging C0 if necessary, we may suppose that C0 is downwardclosed. We will define sequences of κ-small downward-closed subsets
C0 ⊆ C1 ⊆ · · · ⊆ A
D1 ⊆ D2 ⊆ · · · ⊆ B
and idempotent maps {TCi : YCi → YCi }i≥1 , {TDi : YDi → YDi }i≥1 . Moreover, we will guarantee that the following conditions are satisfied:(i) For each i > 0, the set Di contains the intersection B ∩ Ci−1 .
(ii) For each i > 0, the set Ci contains Di .
(iii) For each i > 0, the diagrams
YDi


/ YB

TDi

YDi



TB

/ YB

/ YA

YCi
TCi





TA

YCi

/ YA

YCi−2

/ YCi−1

are commutative.
(iv) For each i > 2, the diagrams/ YDi−1

YDi−2


TDi−2

YDi−2

YDi

=



TDi−1



TCi−2

YDi−1

YCi−2


/ YDi


YCi



TCi−1

YCi−1

=


/ YCi

commute.
(v) For each i > 1, the diagram
YDi−1


TDi−1

/ YC

i−1



TCi−1

YDi−1

YCi−1


YDi


/ YC

i

is commutative.
The construction proceeds by induction on i. Using a compactness argument,
we see that conditions (iv) and (v) are satisfied provided that we choose Ci
and Di to be sufficiently large. The existence of the desired idempotent

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APPENDIX

maps satisfying (iii) then follows from Lemma A.1.5.8
applied to the SS
trees {Yα }α∈A and {Yα }α∈B . We
now
take
C
=
C
.
Conditions(i) andiS(ii) guarantee that B ∩ C =Di . Using (iv), it follows that the maps
{TCi } and {TDi } glue to give idempotent endomorphisms TC : YC → YC ,TB∩C : YB∩C → YB∩C . Using (iii) and (v), we deduce that all of the desired
diagrams are commutative.
Lemma A.1.5.10. Let C be a presentable category, let κ be a regular cardinal, and suppose that C is κ-accessible: that is, C is generated under κ-filteredcolimits by κ-compact objects (Definition 5.4.2.1). Let f : C → D be a morphism between` κ-compact objects of C, let g : X → Y be a pushout of f (sothat Y ' X C D), and let g 0 : X 0 → Y 0 be a retract of g in the categoryof morphisms of C. Then there exists a morphism f 0 : C 0 → D0 with thefollowing properties:(1) The objects C 0 , D0 ∈ C are κ-compact.
(2) The morphism g 0 is a pushout of f 0 .
(3) The morphism f 0 belongs to the weakly saturated class of morphisms
generated by f .
Proof. Since g 0 is a retract of g, there exists a commutative diagram
X0
g0


Y0

/X

/Y

g

/ X0


g0

/ Y 0.
`
Replacing g by the induced map X 0 → X 0 X Y , we can reduce to the casewhere X = X 0 and Y 0 is a retract of Y in CX/ . Then Y 0 can be identified
with the image of some idempotent i : Y → Y .
Since C is κ-accessible, we can write X as the colimit of a κ-filtered diagram {Xλ }, where each Xλ is κ-compact. The object C is κ-compact byassumption. Refining our diagram if necessary, we may assume that it takes
values` in CC/ and that Y is given as the colimit of the κ-filtered diagram{Xλ C D}.
i

Because D is κ-compact, the composition D → Y → Y admits a factorizationaj
D → XλD → Y.
CThe κ-compactness of C implies that, after enlarging λ if necessary, we may
suppose that` the composition j ◦ f coincides with the canonical map fromC to Xλ ` C D. Consequently, j and the idXλ determine a map i0 fromYλ = Xλ C D to itself. Enlarging λ once more, we may suppose that i0 ishighertopoi

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APPENDIX

idempotent and that the diagram
Yλ
i0


Yλ

/Y

/Y

i

is commutative. Let Yλ0 be the image of the idempotent i0 and let f 0 : Xλ →Yλ0 be the canonical map. Then f 0 is a retract of the map Xλ → Yλ , which
is a pushout of f . This proves (3). The objects Xλ and Yλ0 are κ-compact byconstruction, so that (1) is satisfied. We now observe that the diagram
Xλ

/ Y0


X


/ Y0

λ

is a retract of the pushout diagram
Xλ

/ Yλ


X


/Y

and therefore itself a pushout diagram. This proves (2) and completes theproof.
Lemma A.1.5.11. Let C be a presentable category, κ a regular cardinal suchthat C is κ-accessible, and S = {fs : Cs → Ds } a collection of morphisms of
C such that each Cs is κ-compact. Let {Yα }α∈A be an S-tree in C with root
X and suppose that A is κ-small. Then there exists a map X 0 → X, where
X is κ-compact, an S-tree {Yα0 }α∈A with root X 0 , and an isomorphism of
S-treesa{Yα0X}α∈A ' {Yα }α∈AX0(see Remark A.1.5.3).
Proof. Since C is κ-accessible, we can write X as the colimit of diagram{Xi }i∈I indexed by a κ-filtered partially ordered set I, where each Xi is κcompact. Choose a transfinite sequence of downward-closed subsets {A(γ) ⊆A}γ≤β so that the following conditions are satisfied:(i) If γ 0 ≤ γ ≤ β, then A(γ 0 ) ⊆ A(γ).
(ii) If λ ≤ β is a limit ordinal (possibly zero), then A(λ) =Sγ<λ A(γ).

(iii) If γ + 1 ≤ β, then A(γ + 1) = A(γ) ∪ {αγ }, where αγ is a minimal
element of A − A(γ).
(iv) The subset A(β) coincides with A.

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APPENDIX

Note that, since A is κ-small, we have β < κ.
We will construct:(a) A transfinite sequence of elements {iγ ∈ I}γ≤β such that iγ ≤ iγ 0 forγ ≤ γ0.
(b) A sequence of S-trees {Yαγ }α∈A(γ) having roots Xiγ .
(c) A collection of isomorphisms of S-treesa0{Yαγ
Xiγ 0 }α∈A(γ) ' {Yαγ }α∈A(γ)
Xiγ

{Yαγ

a

X}α∈A(γ) ' {Yα }α∈A(γ)

Xiγwhich are compatible with one another in the obvious sense.
If γ is a limit ordinal (or zero), we simply choose iγ to be any upper boundfor {iγ 0 }γ 0 <γ in I. The rest of the data is uniquely determined. The existenceof such an upper bound is guaranteed by our assumption that I is κ-filteredsince γ ≤ β < κ. Let us therefore suppose that the above data has been
constructed for all ordinals ≤ γ, and proceed to define iγ+1 . Let i = iγ , let
α = αγ , and let B = {β ∈ A : β < α}. Then we have canonical isomorphismsaa
YB ' YBγ
X ' lim{YBγ
Xj }j≥i−→
Xi

Xi

and a pushout diagram
Cs

fs

/ Ds

g



YB


/ Yα .

The κ-compactness of Cs implies that g factors as a compositionag0
Cs → YBγ
Xj
Xi

for some j ≥ i. We now define iγ+1 = j, and Yαγ+1 by forming a pushout
diagram
Cs

γ`

YB

gs

Xi Xj

/ Ds

/ Yαγ+1 .

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APPENDIX

Proposition A.1.5.12. Let C be a presentable category, κ a regular cardinal, and S a weakly saturated class of morphisms in C. Let S ⊆ S be thesubcollection consisting of those morphisms f : X → Y in S such that X
and Y are κ-compact. Assume that(i) The regular cardinal κ is uncountable, and C is κ-accessible.
(ii) The set S generates S as a weakly saturated class of morphisms.
Then, for every morphism f : X → Y belonging to S, there exists a transfinite sequence of objects {Zγ }γ<β of CX/ with the following properties:(1) For every ordinal γ < β, the natural map limγ 0 <γ Zγ 0 → Zγ is the−→
pushout of a morphism in S.
(2) The colimit limγ<β Zγ is isomorphic to Y (as objects of CX/ ).
−→Proof. Remark A.1.2.8 implies the existence of a transfinite sequence of objects
Y0 → Y1 → · · ·in CX/ indexed by a set of ordinals A = {α|α < λ}, satisfying condition (1),such that Y is a retract of limα<λ Yα in CX/ . We may view the sequence−→{Yα }α∈A as an S-tree in C having root X. According to Lemma A.1.5.7, we
can choose a new S-tree {Yα0 }α∈A0 which is κ-good, where YA0 0 ' YA , so that
Y is a retract of YA0 0 . Choose an idempotent map TA0 : YA0 0 → YA0 0 in CX/
whose image is isomorphic to Y .
We now define a transfinite sequence
B(0) ⊆ B(1) ⊆ B(2) ⊆ · · · ,indexed by ordinals γ < β, and a compatible system of idempotent maps
00
TB(γ) : YB(γ)→ YB(γ). Fix an ordinal γ and suppose that B(γ 0 ) and TB(γ 0 )Shave been defined for γ 0 < γ. Let B 0 (γ) = γ 0 <γ B(γ 0 ) and let TB 0 (γ) be theresult of amalgamating the maps {TB(γ 0 ) }γ 0 <γ . If B 0 (γ) = A0 , we set β = γand conclude the construction; otherwise, we choose a minimal element a ∈A0 −B 0 (γ). Applying Lemma A.1.5.9, we deduce the existence of a downwardclosed subset C(γ) ⊆ A0 and a compatible collection of idempotent maps
00TC(γ) : YC(γ)→ YC(γ)00TC(γ)∩B 0 (γ) : YC(γ)∩B
0 (γ) → YC(γ)∩B 0 (γ) .

We then define B(γ) = B 0 (γ) ∪ C(γ) and define TB(γ) to be the result ofamalgamating TB 0 (γ) and TC(γ) .
For every ordinal γ, there is a κ-good S-tree {Yα00 }α∈B(γ)−B 0 (γ) with root
000
0YB(γ) such that YB(γ)−B
0 (γ) ' YB(γ) (Remark A.1.5.5). Combining Lemma
A.1.5.11 with the observation that B(γ) − B 0 (γ) is κ-small, we deduce thatthe map0
YB0 0 (γ) → YB(γ)

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APPENDIX809

is the pushout of a morphism in S.
For each ordinal γ < β, let Zγ denote the image of the idempotent map
TB(γ) . Then limγ<β Zγ ' Y , so that (2) is satisfied. Condition (1) follows−→
from Lemma A.1.5.10.
Corollary A.1.5.13. Under the hypotheses of Proposition A.1.5.12, there
exists a κ-good S-tree {Yα }α∈A such that YA ' Y in CX/ .
Proof. Combine Proposition A.1.5.12 with Lemma A.1.5.7.

A.2 MODEL CATEGORIESOne of the oldest and most successful approaches to the study of highercategorical phenomena is Quillen’s theory of model categories. In this book,
Quillen’s theory will play two (related) roles:(1) The structures that we use to describe higher categories are naturallyorganized into model categories. For example, ∞-categories are precisely those simplicial sets which are fibrant with respect to the Joyalmodel structure (Theorem 2.4.6.1). The theory of model categories
provides a convenient framework for phrasing certain results and forcomparing different models of higher category theory (see, for example,§2.2.5).
(2) The theory of model categories can itself be regarded as an approach
to higher category theory. If A is a simplicial model category, then thesubcategory A◦ ⊆ A of fibrant-cofibrant objects forms a fibrant simplicial category. Proposition 1.1.5.10 implies that the simplicial nerve
N(A◦ ) is an ∞-category. We will refer to N(A◦ ) as the underlying∞-category of A. Of course, not every ∞-category arises in this way,even up to equivalence: for example, the existence of homotopy limits
and homotopy colimits in A implies the existence of various limits andcolimits in N(A◦ ) (Corollary 4.2.4.8). Nevertheless, we can often use
the theory of model categories to prove theorems about general ∞categories by reducing to the situation of ∞-categories which arise via
the above construction (every ∞-category C admits a fully faithful embedding into N(A◦ ) for an appropriately chosen simplicial model category A). For example, our proof of the ∞-categorical Yoneda lemma(Proposition 5.1.3.1) uses this strategy.
The purpose of this section is to review the theory of model categories
with an eye toward the sort of applications described above. Our exposition
is somewhat terse, and we will omit many proofs. For a more detailed account, we refer the reader to [40] (or any other text on the theory of model
categories).

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APPENDIX

A.2.1 The Model Category Axioms
Definition A.2.1.1. A model category is a category C which is equippedwith three distinguished classes of morphisms in C, called cofibrations, fibrations, and weak equivalences, in which the following axioms are satisfied:(1) The category C admits (small) limits and colimits.
fg(2) Given a composable pair of maps X → Y → Z, if any two of g ◦ f , f ,and g are weak equivalences, then so is the third.
(3) Suppose f : X → Y is a retract of g : X 0 → Y 0 : that is, suppose there
exists a commutative diagram
X

Y

i

/ X0

0


/ Y0

r

/X

0


/ Y,

g

f
i

f

r

where r ◦ i = idX and r0 ◦ i0 = idY . Then(i) If g is a fibration, so is f .
(ii) If g is a cofibration, then so is f .
(iii) If g is a weak equivalence, then so is f .
(4) Given a diagram
/X
?

A

p

i


B


/ Y,

a dotted arrow can be found rendering the diagram commutative if
either(i) The map i is a cofibration, and the map p is both a fibration anda weak equivalence.
(ii) The map i is both a cofibration and a weak equivalence, and themap p is a fibration.
(5) Any map X → Z in C admits factorizations
f

g

f0

g0

X→Y →ZX → Y 0 → Z,
where f is a cofibration, g is a fibration and a weak equivalence, f 0 isa cofibration and a weak equivalence, and g 0 is a fibration.

highertopoi

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APPENDIX

A map f in a model category C is called a trivial cofibration if it is botha cofibration and a weak equivalence; similarly, f is called a trivial fibration
if it is both a fibration and a weak equivalence. By axiom (1), any model
category C has an initial object ∅ and a final object ∗. An object X ∈ C issaid to be fibrant if the unique map X → ∗ is a fibration and cofibrant if theunique map ∅ → X is a cofibration.
Example A.2.1.2. Let C be any category which admits small limits andcolimits. Then C can be endowed with the trivial model structure:(W ) The weak equivalences in C are the isomorphisms.
(C) Every morphism in C is a cofibration.
(F ) Every morphism in C is a fibration.
A.2.2 The Homotopy Category of a Model Category
Let C be a model category containing an object X. A cylinder object for`ji
X is an object C together with a diagram X X → C → X where i is a
cofibration,` j is a weak equivalence, and the composition j ◦ i is the “foldmap” X X → X. Dually, a path object for Y ∈ C is an object P together
with a diagramqp

Y →P →Y ×Ysuch that q is a weak equivalence, p is a fibration, and p ◦ q is the diagonal
map Y → Y ×Y . The existence of cylinder and path objects follows from` thefactorization axiom (5) of Definition A.2.1.1 (factor the fold map X X →X as a cofibration followed by a trivial fibration and the diagonal map Y →Y × Y as a trivial cofibration followed by a fibration).
Proposition A.2.2.1. Let C be a model category. Let X be a cofibrant objectof C, Y a fibrant object of C, and f, g : X → Y two maps. The following
conditions are equivalent:`j(1) For every cylinder object X X → C, there exists a commutative
diagram
`
j
/C
X X
(f,g)

#
(2) There exists a cylinder object X
`
X X

`



Y.
j

X → C and a commutative diagram
/C

j
(f,g)

#

Y.



highertopoi

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812

APPENDIXp(3) For every path object P → Y × Y , there exists a commutative diagram
/P

X
(f,g)

#
{
Y × Y.

pp(4) There exists a path object P → Y × Y and a commutative diagram
/P

X
(f,g)

#
{
Y × Y.

p

If C is a model category containing a cofibrant object X and a fibrantobject Y , we say two maps f, g : X → Y are homotopic if the hypothesesof Proposition A.2.2.1 are satisfied and write f ' g. The relation ' is anequivalence relation on HomC (X, Y ). The homotopy category hC may be
defined as follows:• The objects of hC are the fibrant-cofibrant objects of C.
• For X, Y ∈ hC, the set HomhC (X, Y ) is the set of '-equivalence classesof HomC (X, Y ).
Composition is well-defined in hC by virtue of the fact that if f ' g, thenf ◦ h ' g ◦ h (this is clear from characterization (4) of Proposition A.2.2.1)
and h0 ◦ f ' h0 ◦ g (this is clear from characterization (2) of Proposition
A.2.2.1) for any maps h, h0 such that the compositions are defined in C.
There is another way of defining hC (or, at least, a category equivalent
to hC): one begins with all of C and formally adjoins inverses to all weakequivalences. Let H(C) denote the category so obtained. If X ∈ C is cofibrantand Y ∈ C is fibrant, then homotopic maps f, g : X → Y have the sameimage in H(C); consequently, we obtain a functor hC → H (C) which can beshown to be an equivalence. We will generally ignore the distinction between
these two categories, employing whichever description is more useful for the
problem at hand.
Remark A.2.2.2. Since C is (generally) not a small category, it is not
immediately clear that H(C) has small morphism sets; however, this follows
from the equivalence between H(C) and hC.
A.2.3 A Lifting Criterion
The following basic principle will be used many times throughout this book:
Proposition A.2.3.1. Let C be a model category containing cofibrant objects
A and B and a fibrant object X. Suppose we are given a cofibration i : A → B

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APPENDIX

and any map f : A → X. Suppose moreover that there exists a commutative
diagram
A
[f ]

>X

[i]
g


B
in the homotopy category h C. Then there exists a commutative diagram
A
f

>X

i
g


B
in C, with [g] = g. (Here we let [p] denote the homotopy class in hC of amorphism p in C.)
Proof. Choose a map g 0 : B → X representing the homotopy class g. Choose
a cylinder objectaA
A → C(A) → A
and a factorization
C(A)

a
A

`

(B

a

B) → C(B) → B,

A

where the first map is a cofibration and the second is a trivial fibration. We
observe that C(B) is a cylinder object for B.
`
Since g 0 ◦ i is homotopic to f , there exists a map
h0 : C(A) A B → X`
with h0 |B = g 0 and h0 |A = f . The inclusion C(A) A B → C(B) is a trivial
cofibration, so h0 extends to a map h : C(B) → X. We may regard h as ahomotopy from g 0 to g, where g ◦ i = f .
Proposition A.2.3.1 will often be applied in the following way. Suppose weare given a diagram
A0

/A
f

i


B0


/B

>X

highertopoi

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APPENDIX

which we would like to extend as indicated by the dotted arrow. If X is
fibrant, i is a cofibration between cofibrant objects, and the horizontal arrows
are weak equivalences, then it suffices to solve the (frequently easier) problem
of constructing the dotted arrow in the diagram
A0

.
'

7X

B0.
A.2.4 Left Properness and Homotopy Pushout Squares
Definition A.2.4.1. A model category C is left proper if, for any pushoutsquare
A

i

/B

0


/ B0

j0

j


A0

i

in which i is a cofibration and j is a weak equivalence, the map j 0 is also a
weak equivalence. Dually, C is right proper if, for any pullback square
X0

p0

q0


X

p

/ Y0

/Y

q

in which p is a fibration and q is a weak equivalence, the map q 0 is also a
weak equivalence.
In this book, we will deal almost exclusively with left proper model categories. The following provides a useful criterion for establishing left properness.
Proposition A.2.4.2. Let C be a model category in which every object iscofibrant. Then C is left proper.
Proposition A.2.4.2 is an immediate consequence of the following basic
lemma:
Lemma A.2.4.3. Let C be a model category containing a pushout diagram
A

i

/B

0


/ B0.

j0

j


A0

i

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APPENDIX

Suppose that A and A0 are cofibrant, i is a cofibration, and j is a weakequivalence. Then j 0 is a weak equivalence.
Proof. We wish to show that j 0 is an isomorphism in the homotopy category
hC. In other words, we need to show that for every fibrant object Z of C,
composition with j 0 induces a bijection HomhC (B 0 , Z) → HomhC (B, Z).
We first show that composition with j 0 is surjective on homotopy classes.
Suppose we are given a map f : B → Z. Since j is a weak equivalence, thecomposition f ◦ i is homotopic to g ◦ j for some g : A0 → Z. According to
Proposition A.2.3.1, there is a map f 0 : B → Z such that f 0 ◦ i = g ◦ j andsuch that f 0 is homotopic to f . The amalgamation of f 0 and g determines amap B 0 → Z which lifts f 0 .
We now show that j 0 is injective on homotopy classes. Suppose we are
given a pair of maps s, s0 : B 0 → Z. Let P be a path object for Z. If s ◦ j 0
and s0 ◦ j 0 are homotopic, then there exists a commutative diagram
/P

h

B
j0


B0


/ Z × Z.

s×s0

We now replace C by C/Z×Z and apply the surjectivity statement above todeduce that there is a map h0 : B 0 → P such that h is homotopic to h0 ◦ j 0 .
The existence of h0 shows that s and s0 are homotopic, as desired.
Suppose we are given a diagram
A0 ← A → A1
in a model category C. In general, the pushout A0
in the sense that a map of diagrams
/ A1
A
A0 o

A A1 is poorly behaved


/ B1


B


B0 o

`

`
`
need not induce a weak equivalence A0 A A1 → B0 B B1 , even if eachof the vertical arrows in the diagram is individually a weak equivalence. To
correct this difficulty, it is convenient to introduce the left derived functor
of “pushout”. The homotopy pushout of the diagram
/ A1
A0 o
A
`
is defined to be the pushout A00 A0 A01 , where we have chosen a commutative
diagram
A00 o

A0 o

j

A0

A

i

/ A01

/ A1

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APPENDIX

where the vertical maps are weak equivalences, and the top row is cofibrantdiagram in the sense that A0 is cofibrant and the maps i and j are both
cofibrations.
One can show that such a diagram exists and that the pushout`
A00 A0 A01 depends on the choice of diagram only up to weak equivalence.
(For a more systematic approach which includes a definition of “cofibrant”
for more complicated diagrams, we refer the reader to §A.3.3.)
More generally, we will say that a diagram
A

A0

~

A1

~
M
is a homotopy pushout square if the composite mapaaA00
A01 → A0
A1 → M
A0

A

is a weak equivalence. In this case we will also say that M is a homotopy
pushout of A0 and A1 over A. One can show that this condition is independent of the choice of a “cofibrant resolution”
/ A01
A00 o
A0
of the original diagram. In particular, we note that if the diagram
/ A1
A0 o
A
`
is already cofibrant, then the ordinary pushout A0 A A1 is a homotopy
pushout. However, the condition that the diagram be cofibrant is quite
strong; in good situations we can get away with quite a bit less:
Proposition A.2.4.4. Let C be a model category and let
A
j

A0

y

i

%

A1

$` z
A0 A A1
be a pushout square in C. This diagram is also a homotopy pushout squareif either of the following conditions is satisfied:(i) The objects A and A0 are cofibrant, and j is a cofibration.
(ii) The map j is a cofibration, and C is left proper.
Remark A.2.4.5. The above discussion of homotopy pushouts can be dualized; one obtains the notion of homotopy pullbacks, and the analogue of
Proposition A.2.4.4 requires either that C be a right proper model categoryor that the objects in the diagram be fibrant.

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APPENDIX

A.2.5 Quillen Adjunctions and Quillen Equivalences
Let C and D be model categories and suppose we are given a pair of adjoint
functors
Co

F
G

/D

(here F is the left adjoint and G is the right adjoint). The following conditions
are equivalent:(1) The functor F preserves cofibrations and trivial cofibrations.
(2) The functor G preserves fibrations and trivial fibrations.
(3) The functor F preserves cofibrations, and the functor G preserves fibrations.
(4) The functor F preserves trivial cofibrations, and the functor G preserves trivial fibrations.
If any of these equivalent conditions is satisfied, then we say that the pair(F, G) is a Quillen adjunction between C and D. We also say that F is a left
Quillen functor and that G is a right Quillen functor. In this case, one canshow that F preserves weak equivalences between cofibrant objects and G
preserves weak equivalences between fibrant objects.
F /Suppose that C o
D is a Quillen adjunction. We may view the homoG

topy category hC as obtained from C by first passing to the full subcategory
consisting of cofibrant objects and then inverting all weak equivalences. Applying a similar procedure with D, we see that because F preserves weakequivalence between cofibrant objects, it induces a functor hC → hD; thisfunctor is called the left derived functor of F and denoted LF . Similarly, one
may define the right derived functor RG of G. One can show that LF and
RG determine an adjunction between the homotopy categories hC and hD.
Proposition A.2.5.1. Let C and D be model categories and let
Co

F
G

/D

be a Quillen adjunction. The following are equivalent:(1) The left derived functor LF : hC → hD is an equivalence of categories.
(2) The right derived functor RG : hD → hC is an equivalence of categories.
(3) For every cofibrant object C ∈ C and every fibrant object D ∈ D, a
map C → G(D) is a weak equivalence in C if and only if the adjoint
map F (C) → D is a weak equivalence in D.

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APPENDIX

Proof. Since the derived functors LF and RG are adjoint to one another, it
is clear that (1) is equivalent to (2). Moreover, (1) and (2) are equivalent tothe assertion that the unit and counit of the adjunctionu : idC → RG ◦ LF
v : LF ◦ RG → idD
are weak equivalences. Let us consider the unit u. Choose a cofibrant object
C of C. The composite functor (RG ◦ LF )(C) is defined to be G(D), where
F (C) → D is a weak equivalence in D and D is a fibrant object of D. Thusu is a weak equivalence when evaluated on C if and only if for any weakequivalence F (C) → D, the adjoint map C → G(D) is a weak equivalence.
Similarly, the counit v is a weak equivalence if and only if the converse holds.
Thus (1) and (2) are equivalent to (3).
If the equivalent conditions of Proposition A.2.5.1 are satisfied, then we
say that the adjunction (F, G) gives a Quillen equivalence between the model
categories C and D.
A.2.6 Combinatorial Model Categories
In this section, we give an overview of Jeff Smith’s theory of combinatorialmodel categories. Our main goal is to prove Proposition A.2.6.15, whichallows us to construct model structures on a category C by specifying theclass of weak equivalences together with a small amount of additional data.
Definition A.2.6.1 (Smith). Let A be a model category. We say that A is
combinatorial if the following conditions are satisfied:(1) The category A is presentable.
(2) There exists a set I of generating cofibrations such that the collection of
all cofibrations in A is the smallest weakly saturated class of morphisms
containing I (see Definition A.1.2.2).
(3) There exists a set J of generating trivial cofibrations such that thecollection of all trivial cofibrations in A is the smallest weakly saturated
class of morphisms containing J.
If C is a combinatorial model category, then the model structure on C is
uniquely determined by the generating cofibrations and generating trivial
cofibrations. However, in practice these generators might be difficult to find.
Our goal in this section is to reformulate Definition A.2.6.1 in a manner
which puts more emphasis on the collection of weak equivalences in A.
In practice, it is often easier to describe the class of all weak equivalences
than it is to describe a class of generating trivial cofibrations.
Definition A.2.6.2. Let C be a presentable category and κ a regular cardinal. We will say that a full subcategory C0 ⊆ C is a κ-accessible subcategoryof C if the following conditions are satisfied:highertopoi

April 9, 2017

APPENDIX

819(1) The full subcategory C0 ⊆ C is stable under κ-filtered colimits.
(2) There exists a (small) set of objects of C0 which generates C0 underκ-filtered colimits.
We will say that C0 ⊆ C is an accessible subcategory if C0 is a κ-accessible
subcategory of C for some regular cardinal κ.
In general, a κ-accessible subcategory need not be τ -accessible for an arbitrary regular cardinal τ > κ. However, this is true for many regular cardinals
τ.
Definition A.2.6.3. Let κ and τ be regular cardinals. We write κ  τ if
the following condition is satisfied: for every τ0 < τ and every κ0 < κ, wehave τ0κ0 < τ .
Note that there exist arbitrarily large regular cardinals κ0 with κ0  κ:for example, one may take κ0 to be the successor of any cardinal having theform τ κ .
Remark A.2.6.4. Every (infinite) regular cardinal κ satisfies ω  κ. An
uncountable regular cardinal κ satisfies κ  κ if and only if κ is strongly
inaccessible.
Condition (2) of Definition A.2.6.2 admits the following reformulation:
Proposition A.2.6.5. Let κ be a regular cardinal, C a presentable category,
and C0 ⊆ C a full subcategory which is stable under κ-filtered colimits. Then
C0 satisfies condition (2) of Definition A.2.6.2 if and only if the following
condition is satisfied for all sufficiently large regular cardinals τ  κ:(20τ ) Let A be a τ -filtered partially ordered set and {Xα }α∈A a diagram of τ compact objects of C indexed by A. For every κ-filtered subset B ⊆ A,
we let XB denote the (κ-filtered) colimit of the diagram {Xα }α∈B .
Suppose that XA belongs to C0 . Then for every τ -small subset C ⊆ A,
there exists a τ -small κ-filtered subset B ⊆ A which contains C, suchthat XB belongs to C0 .
First, we need the following preliminary result:Lemma A.2.6.6. Let τ  κ be regular cardinals such that τ > κ, let D be apresentable category, and let {Ca }a∈A and {Db }b∈B be families of τ -compact
objects in D indexed by τ -filtered partially ordered sets A and B, such thatlima∈A Ca ' limb∈B Db .
−→−→Then, for every pair of τ -small subsets A0 ⊆ A, B0 ⊆ B, there exist τ small κ-filtered subsets A0 ⊆ A, B 0 ⊆ B such that A0 ⊆ A0 , B0 ⊆ B 0 , andlima∈A0 Ca ' limb∈B 0 Db .
−→−→highertopoi

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April 9, 2017

APPENDIX

Proof. Let A be the partially ordered set of all τ -small κ-filtered subsets
of A which contain A0 , let B be the partially ordered set of all τ -small κfiltered subsets of B which contain B0 , let X ∈ D be the common colimit
lima∈A Ca ' limb∈B Db , and let C be the full subcategory of D/X spanned by−→−→those morphisms Y → X, where Y is a τ -compact object of D. Let f : A → C
and g : B → C be the functors described by the formulasf (A0 ) = (lima∈A0 Ca → lima∈A Ca )−→−→0g(B ) = (limb∈B 0 Db → limb∈B Db ).
−→−→The desired result now follows by applying Lemma 5.4.6.3 to the associateddiagram N(A) → N(C) ← N(B).
Proof of Proposition A.2.6.5. First suppose that (20τ ) is satisfied for all sufficiently large τ  κ. Choose τ  κ large enough that C is generated undercolimits by its full subcategory Cτ of τ -compact objects and such that (20τ )is satisfied. Let D = Cτ ∩ C0 , so that D is essentially small. We will showthat D generates C0 under τ -filtered colimits. By assumption, every object
X ∈ C can be obtained as a τ -filtered colimit of τ -compact objects {Xα }α∈A .
Let S denote the collection of all τ -small κ-filtered subsets B ⊆ A such that
XB ∈ C0 . We regard S as partially ordered via inclusions. Invoking condition(20τ ), we deduce that XA is the colimit of the τ -filtered collection of objects{XB }B∈S . We now observe that each XB belongs to D.
Now suppose that condition (2) is satisfied, so that C0 is generated underκ-filtered colimits by a small subcategory D ⊆ C0 . Choose τ  κ large
enough that every object of D is τ -compact. Enlarging τ if necessary, we may
suppose that τ > κ. We claim that (20τ ) is satisfied. To prove this, we consider
any diagram {Xα }α∈A satisfying the hypotheses of (20τ ). In particular, XA
belongs to C0 , so that XA may be obtained in some other way as a κ-filteredcolimit of a system {Yβ }β∈B , where each of the objects Yβ belongs to D
and is therefore τ -compact. Let S denote the family of all τ -small κ-filteredsubsets B0 ⊆ B. Replacing B by S and the family {Yβ }β∈B by {YB0 }B0 ∈B 0 ,we may assume that B is τ -filtered.
Let A0 ⊆ A be a τ -small subset. Applying Lemma A.2.6.6 to the diagramcategory C, we deduce that A0 ⊆ A0 , where A0 is a τ -small, κ-filtered subsetof A and there is an isomorphism XA0 ' YB 0 ; here B 0 is a κ-filtered subsetof B, so that YB 0 ∈ C0 by virtue of our assumption that C0 is stable underκ-filtered colimits.
Corollary A.2.6.7. Let f : C → D be a functor between presentable categories which preserves κ-filtered colimits and let D0 ⊆ D be a κ-accessible
subcategory. Then f −1 D0 ⊆ C is a κ-accessible subcategory.
Corollary A.2.6.8 (Smith). Let A be a combinatorial model category, letA[1] be the category of morphisms in A, let W ⊆ A[1] be the full subcategory
spanned by the weak equivalences, and let F ⊆ A[1] be the full subcategory
spanned by the fibrations. Then F , W , and F ∩W are accessible subcategoriesof A[1] .

highertopoi

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APPENDIX

Proof. For every morphism i : A → B, let Fi : A[1] → Set[1] be the functor
which carries a morphism f : X → Y to the induced map of sets
HomA (B, X) → HomA (B, Y ) ×HomA (A,Y ) HomA (A, X).
We observe that if A and B are κ-compact objects of A, then Fi preserves
κ-filtered colimits.
Let C0 be the full subcategory of Set[1] spanned by the collection of surjective maps between sets. It is easy to see that C0 is an accessible categoryof Set[1] . It follows that the full subcategories R(i) = Fi−1 C0 ⊆ A[1] areaccessible subcategories of A[1] (Corollary A.2.6.7).
Let I be a set of generating cofibrations for A and let J be a set ofgenerating trivial cofibrations. Then Proposition 5.4.7.10 implies that the
subcategories
\
F =
R(j)
j∈J

W ∩F =

\

R(i)

i∈I

are accessible subcategories of A[1] .
Applying Proposition A.1.2.5, we deduce that there exists a pair of functors T 0 , T 00 : A[1] → A[1] , which carry an arbitrary morphism f : X → Z toa factorization
T 0 (f )X → Y

T 00 (f )→ Z,

where T 0 (f ) is a trivial cofibration and T 00 (f ) is a fibration. Moreover, thefunctor T 00 can be chosen to commute with κ-filtered colimits for a sufficiently
large regular cardinal κ. We now observe that W is the inverse image of F ∩W
under the functor T 00 : A[1] → A[1] and is therefore an accessible subcategoryof A[1] by Corollary A.2.6.7.
Our next goal is to prove a converse to Corollary A.2.6.8, which will allow
us to construct examples of combinatorial model categories. First, we need
the following preliminary result.
Lemma A.2.6.9. Let A be a presentable category. Suppose W and C arecollections of morphisms of A with the following properties:(1) The collection C is a weakly saturated class of morphisms of A, andthere exists a (small) subset C0 ⊆ C which generates C as a weakly
saturated class of morphisms.
(2) The intersection C ∩ W is a weakly saturated class of morphisms of A.
(3) The full subcategory W ⊆ A[1] is an accessible subcategory of A[1] .
(4) The class W has the two-out-of-three property.

highertopoi

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822

APPENDIX

Then C ∩ W is generated, as a weakly saturated class of morphisms, by a(small) subset S ⊆ C ∩ W .
Proof. Let κ be a regular cardinal such that W is κ-accessible. Choose aregular cardinal τ  κ such that W satisfies condition (20τ ) of Proposition
A.2.6.5. Enlarging τ if necessary, we may assume that τ > κ (so that τ is
uncountable), that C is τ -accessible, and that the source and target of every
morphism in C0 is τ -compact. Enlarging C0 if necessary, we may supposethat C0 consists of all morphisms f : X → Y in C such that X and Y areτ -compact. Let S = C0 ∩ W . We will show that S generates C ∩ W as aweakly saturated class of morphisms.
Let S be the weakly saturated class of morphisms generated by S and let
f : X → Y be a morphism which belongs to C ∩ W . We wish to show thatf ∈ S. Corollary A.1.5.13 implies that there exists a τ -good C0 -tree {Yα }α∈Awith root X, such that Y is isomorphic to YA as objects of CX/ . Let us saythat a subset B ⊆ A is good if it is downward-closed and the canonical map
i : X → YB belongs to W (we note that i automatically belongs to C, byvirtue of Lemma A.1.5.6).
We now make the following observations:(i) Given S
an increasing transfinite sequence of good subsets {Aγ }γ<β , theunion Aγ is good. This follows from the assumption that C ∩ W is
weakly saturated.
(ii) Let B ⊆ A be good and let B0 ⊆ B be τ -small. Then there exists a τ small subset B 0 ⊆ B containing B0 . This follows from our assumptionthat W satisfies condition (20τ ) of Proposition A.2.6.5.
(iii) Suppose that B, B 0 ⊆ A are such that B, B 0 , and B ∩ B 0 are good.
Then B ∪ B 0 is good. To prove this, we consider the pushout diagram/ YB
YB∩B 0
YB 0
/ YB∪B 0 .

Every morphism in this diagram belongs to C (Lemma A.1.5.6), andthe upper horizontal map belongs to W by virtue of assumption (4).
Since C ∩W is stable under pushouts, we conclude that the lower vertical map belongs to W . Assumption (4) now implies that the composite
map X → YB 0 → YB∪B 0 belongs to W , as desired.
The next step is to prove the following claim:(∗) Let A0 be a good subset of A and let B0 ⊆ A be τ -small. Then there
exists a τ -small subset B ⊆ A such that B0 ⊆ B, B is good and B ∩ A0
is good.
To prove (∗), we begin by setting B00 = A0 ∩ B0 . We now define sequencesof τ -small subsets
B0 ⊆ B1 ⊆ B2 ⊆ · · ·highertopoi

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APPENDIX

B00 ⊆ B10 ⊆ B20 ⊆ · · ·as follows. Suppose that Bi and Bi0 have been defined. Applying (ii), wechoose Bi+1 to be any τ -small good subset of A which contains Bi ∪ Bi0 .
00
Applying (ii) again, we select Bi+1
to
S be any τ -small good subset of A
0which contains
A ∩ Bi+1 . Let B = Bi . It follows from (i) that B andS
A0 ∩ B = i Bi0 are both good.
We now choose a transfinite sequence of good subsets {A(γ)
S ⊆ A}γ<β .
Suppose that A(γ 0 ) has been defined for γ 0 < γ and let A0 (γ) = γ 0 <γ A(γ 0 ).
It follows from (i) that A0 (γ) is good. If A0 (γ) = A, we set β = γ and conclude
the construction. Otherwise, we choose a minimal element a ∈ A − A0 (γ).
Applying (∗), we deduce that there exists a τ -small good subset B(γ) ⊆ A
containing a, such that A0 (γ) ∩ B(γ) is good. Let A(γ) = A0 (γ) ∪ B(γ). It
follows from (iii) that A(γ) is good.
We observe that {YA(γ) }γ<β is a transfinite sequence of objects of CX/
having colimit Y . To prove that f : X → Y belongs to S, it will suffice toshow that for each γ < β, the map g : YA0 (γ) → YA(γ) belongs to S. Remark
A.1.5.5 implies the existence of a C0 -tree {Zα }α∈A(γ)−A0 (γ) with root YA0 (γ)
and colimit YA(γ) . Since A(γ) − A0 (γ) is τ -small, Lemma A.1.5.11 implies
the existence of a pushout diagram/NM
YA0 (γ)/ YA(γ)where g ∈ C0 .
Since C is τ -accessible, we can write YA0 (γ) as the colimit of a family ofτ -compact objects {Zλ }λ∈P indexed by a τ -filtered partially ordered set P .
Since M is τ -compact, we can assume (reindexing the colimit if necessary)that we have `a compatible family of maps {M → Zλ }. For each λ, letgλ : Zλ → Zλ M N be the induced map. Then g is the filtered colimit of
the family {gλ }λ∈P . Since W satisfies condition (20τ ) of Proposition A.2.6.5,
we conclude that there exists a τ -small κ-filtered subset P0 ⊆ P , such thatg 0 = limλ∈P gλ belongs to W . We now observe that g 0 ∈ S and that g is a−→0
pushout of g 0 , so that g ∈ S, as desired.
Proposition A.2.6.10. Let A be a presentable category and let W and C
be classes of morphisms in A with the following properties:(1) The collection C is a weakly saturated class of morphisms of A, andthere exists a (small) subset C0 ⊆ C which generates C as a weakly
saturated class of morphisms.
(2) The intersection C ∩ W is a weakly saturated class of morphisms of A.
(3) The full subcategory W ⊆ A[1] is an accessible subcategory of A[1] .
(4) The class W has the two-out-of-three property.

highertopoi

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APPENDIX(5) If f is a morphism in A which has the right lifting property with respect
to each element of C, then f ∈ W .
Then A admits a combinatorial model structure, which may be described asfollows:(C) The cofibrations in A are the elements of C.
(W ) The weak equivalences in A are the elements of W .
(F ) A morphism in A is a fibration if it has the right lifting property withrespect to every morphism in C ∩ W .
Proof. The category A has all (small) limits and colimits since it is presentable. The two-out-three property for W is among our assumptions, and thestability of W under retracts follows from the accessibility of W ⊆ A[1](Corollary 5.3.1.10). The class of cofibrations is stable under retracts by (1),and the class of fibrations is stable under retracts by definition. The classesof fibrations and cofibrations are stable under retracts by definition.
We next establish the factorization axioms. By the small object argument,
any morphism X → Z admits a factorization
f

g

q

r

X → Y → Z,
where f ∈ C and g has the right lifting property with respect to every
morphism in C. In particular, g has the right lifting property with respect
to every morphism in C ∩ W , so that g is a fibration; assumption (5) thenimplies that g is a trivial fibration. Similarly, using Lemma A.2.6.9, we may
choose a factorization as above, where f ∈ C ∩ W and g has the right lifting
property with respect to C ∩ W ; g is then a fibration by definition.
To complete the proof, it suffices to show that cofibrations have the left
lifting property with respect to trivial fibrations, and trivial cofibrations
have the left lifting property with respect to fibrations. The second of thesestatements is clear (it is the definition of a fibration). For the first statement,
let us consider an arbitrary trivial fibration p : X → Z. By the small object
argument, there exists a factorization of p
X → Y → Z,
where q is a cofibration and r has the right lifting property with respect toall cofibrations. Then r is a weak equivalence by (3), so that q is a weakequivalence by the two-out-of-three property. Considering the diagram
X
>X

Y

q

p

r


/ Z,

we deduce the existence of the dotted arrow from the fact that p is a fibrationand q is a trivial cofibration. It follows that p is a retract of r, and therefore
p also has the right lifting property with respect to all cofibrations. This
completes the proof that A is a model category. The assertion that A is
combinatorial follows immediately from (1) and from Lemma A.2.6.9.

highertopoi

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APPENDIX

Corollary A.2.6.11. Let A be a presentable category equipped with a model
structure. Suppose that there exists a (small) set which generates the collection of cofibrations in A (as a weakly saturated class of morphisms). Thenthe following are equivalent:(1) The model category A is combinatorial; in other words, there exists a(small) set which generates the collection of trivial cofibrations in A
(as a weakly saturated class of morphisms).
(2) The collection of weak equivalences in A determines an accessible subcategory of A[1] .
Proof. The implication (1) ⇒ (2) follows from Corollary A.2.6.8, and thereverse implication follows from Proposition A.2.6.10.
Our next goal is to prove a weaker version of Proposition A.2.6.10 which
is somewhat easier to apply in practice.
Definition A.2.6.12. Let A be a presentable category. A class W of morphisms in C is perfect if it satisfies the following conditions:(1) Every isomorphism belongs to W .
fg(2) Given a pair of composable morphisms X → Y → Z, if any two of themorphisms f , g, and g ◦ f belong to W , then so does the third.
(3) The class W is stable under filtered colimits. More precisely, suppose
we are given a family of morphisms {fα : Xα → Yα } which is indexed
by a filtered partially ordered set. Let X denote a colimit of {Xα }, Ya colimit of {Yα }, and f : X → Y the induced map. If each fα belongs
to W , then so does f .
(4) There exists a (small) subset W0 ⊆ W such that every morphism
belonging to W can be obtained as a filtered colimit of morphisms
belonging to W0 .
Example A.2.6.13. If C is a presentable category, then the class W consisting of all isomorphisms in C is perfect.
The following is an immediate consequence of Corollary A.2.6.7:
Corollary A.2.6.14. Let F : C → C0 be a functor between presentable
categories which preserves filtered colimits and let WC0 be a perfect class ofmorphisms in C0 . Then WC = F −1 WC0 is a perfect class of morphisms in C.
Proposition A.2.6.15. Let A be a presentable category. Suppose we are
given a class W of morphisms of A, which we will call weak equivalences,
and a (small) set C0 of morphisms of A, which we will call generating cofibrations. Suppose furthermore that the following assumptions are satisfied:highertopoi

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APPENDIX(1) The class W of weak equivalences is perfect (Definition A.2.6.12).
(2) For any diagram
/Y

f

X

X0


/ Y0

g


X 00



g0

/ Y 00in which both squares are coCartesian, f belongs to C0 , and g belongs
to W , the map g 0 also belongs to W .
(3) If g : X → Y is a morphism in A which has the right lifting property
with respect to every morphism in C0 , then g belongs to W .
Then there exists a left proper combinatorial model structure on A which
may be described as follows:(C) A morphism f : X → Y in A is a cofibration if it belongs to the weakly
saturated class of morphisms generated by C0 .
(W ) A morphism f : X → Y in A is a weak equivalence if it belongs to W .
(F ) A morphism f : X → Y in A is a fibration if it has the right lifting
property with respect to every map which is both a cofibration and aweak equivalence.
Proof. We first show that the class of weak equivalences is stable underpushouts by cofibrations. Let P denote the collection of all morphisms f in
A with the following property: for coCartesian diagram
X

X0
g


X 00

f

/Y

/ Y0


g0

/ Y 00 ,where g belongs to W , the map g 0 also belongs to W . By assumption, C0 ⊆ P .
It is easy to see that P is weakly saturated (using the stability of W underfiltered colimits), so that every cofibration belongs to P .
It remains only to show that A is a model category. In view of Proposition
A.2.6.10, it will suffice to show that C ∩ W is a weakly saturated class ofhighertopoi

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APPENDIX

morphisms. It is clear that C ∩ W is stable under retracts. It will therefore suffice to verify the stability of C ∩ W under pushouts and transfinite
composition. The case of transfinite composition is easy: C is stable undertransfinite composition because C is weakly saturated, and W is stable under transfinite composition because it is stable under finite composition andfiltered colimits.
It remains to show that C ∩ W is stable under pushouts. Suppose we are
given a coCartesian diagram/ X 00
X
f 00

f



/ Y 00Y
in which f belongs to C ∩W ; we wish to show that f 00 also belongs to C ∩W .
Since C is weakly saturated, it will suffice to show that f 00 belongs to W .
Using the small object argument, we can factor the top horizontal map to
produce a coCartesian rectangle
X
f

g

/ X0
f0

h

/ X 00
f 00




0
/ Y 0 h / Y 00
Y
in which g is a cofibration and h has the right lifting property with respect toall the morphisms in C0 . Since W is stable under the formation of pushouts
by cofibrations, we deduce that f 0 belongs to W . Moreover, by assumption(3), h belongs to W . Since h0 is a pushout of h by the cofibration f 0 , wededuce that h0 belongs to W as well. Applying the two-out-of-three property(twice), we deduce that f 00 belongs to W .
Remark A.2.6.16. Let A be a model category. Then A arises via theconstruction of Proposition A.2.6.15 if and only if it is combinatorial andleft proper and the collection of weak equivalences in A is stable underfiltered colimits.
A.2.7 Simplicial Sets
The formalism of simplicial sets plays a prominent role throughout this book.
In this section, we will review the definition of a simplicial set and establish
some notation.
For each n ≥ 0, we let [n] denote the linearly ordered set {0, . . . , n}. We
let ∆ denote the category of combinatorial simplices: the objects of ∆ are
the linearly ordered sets [n], and morphisms in ∆ are given by (nonstrictly)
order-preserving maps.
If C is any category, a simplicial object of C is a functor ∆op → C. Dually,a cosimplicial object of C is a functor ∆ → C. A simplicial set is a simplicial
object in the category of sets. More explicitly, a simplicial set S is determinedby the following data:highertopoi

828

April 9, 2017

APPENDIX• A set Sn for each n ≥ 0 (the value of S on the object [n] ∈ ∆).
• A map p∗ : Sn → Sm for each order-preserving map [m] → [n], theformation of which is compatible with composition (including empty
composition, so that (id[n] )∗ = idSn ).
Let us recall a bit of standard notation for working with a simplicial set
S. For each 0 ≤ j ≤ n, the face map dj : Sn → Sn−1 is defined to be thepullback p∗ , where p : [n − 1] → [n] is given by(i
if i < jp(i) =
i + 1 if i ≥ j.
Similarly, the degeneracy map sj : Sn → Sn+1 is defined to be the pullbackq ∗ , where q : [n + 1] → [n] is defined by the formula(i
if i ≤ jq(i) =
i − 1 if i > j.
Because every order-preserving map from [n] to [m] can be factored as acomposition of face and degeneracy maps, the structure of a simplicial set
S is completely determined by the sets Sn for n ≥ 0 together with the faceand degeneracy operations defined above. These operations are required tosatisfy certain identities, which we will not make explicit here.
Remark A.2.7.1. The category ∆ is equivalent to the (larger) category of
all finite nonempty linearly ordered sets. We will sometimes abuse notation
by identifying ∆ with this larger subcategory and by regarding simplicial
sets (or more general simplicial objects) as functors which are defined on all
nonempty linearly ordered sets.
Notation A.2.7.2. The category of simplicial sets will be denoted by Set∆ .
If J is a linearly ordered set, we let ∆J ∈ Set∆ denote the representablefunctor [n] 7→ Hom([n], J), where the morphisms are taken in the categoryof linearly ordered sets. For each n ≥ 0, we will write ∆n in place of ∆[n] .
We observe that, for any simplicial set S, there is a natural identification ofsets Sn ' HomSet∆ (∆n , S).
Example A.2.7.3. For 0 ≤ j ≤ n, we let Λnj ⊂ ∆n denote the “jth horn.”
It is determined by the following property: an element of (Λnj )m is given
by an order-preserving map p : [m] → [n] satisfying the condition that{j} ∪ p([m]) 6= [n]. Geometrically, Λnj corresponds to the subset of an nsimplex ∆n in which the jth face and the interior have been removed.
More generally, if J is any finite linearly ordered set containing an element
j, we let ΛJj denote the simplicial subset of ∆J obtained by removing theinterior and the face opposite the vertex j.
The category Set∆ of simplicial sets has a (combinatorial, left proper,
and right proper) model structure, which we will refer to as the Kan model
structure. It may be described as follows:highertopoi

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829

APPENDIX• A map of simplicial sets f : X → Y is a cofibration if it is a monomorphism; that is, if the induced map Xn → Yn is injective for all n ≥ 0.
• A map of simplicial sets f : X → Y is a fibration if it is a Kan fibration:that is, if for any diagram
/X
Λni
>
_

∆n

f


/ Y,

it is possible to supply the dotted arrow rendering the diagram commutative.
• A map of simplicial sets f : X → Y is a weak equivalence if the induced
map of geometric realizations |X| → |Y | is a homotopy equivalence of
topological spaces.
To prove this, we observe that the class of all cofibrations is generated by
the collection of all inclusions ∂ ∆n ⊆ ∆n ; it is then easy to see that theconditions of Proposition A.2.6.15 are satisfied. The nontrivial point is toverify that the fibrations for the resulting model structure are precisely theKan fibrations and that Set∆ is right proper; these facts ultimately rely ona delicate analysis due to Quillen (see [32]).
Remark A.2.7.4. In §2.2.5, we introduce another model structure on Set∆ ,the Joyal model structure. This model structure has the same class of cofibrations, but the fibrations and the weak equivalences differ from those definedin this section. To avoid confusion, we will refer to the fibrations and weakequivalences for the usual model structure on simplicial sets as Kan fibrations
and weak homotopy equivalences, respectively.
A.2.8 Diagram Categories and Homotopy (Co)limits
Let A be a combinatorial model category and C a small category. We let
Fun(C, A) denote the category of all functors from C to A. In this section, we
will see that Fun(C, A) again admits the structure of a combinatorial model
category: in fact, it admits two such structures. Moreover, by considering
the functoriality of this construction in the category C, we will obtain thetheory of homotopy limits and homotopy colimits.
Definition A.2.8.1. Let C be a small category and let A be a model category. We will say that a natural transformation α : F → G in Fun(C, A)
is:• an injective cofibration if the induced map F (C) → G(C) is a cofibration in A for each C ∈ C.
• a projective fibration if the induced map F (C) → G(C) is a fibrationin A for each C ∈ C.

highertopoi

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830

APPENDIX• a weak equivalence if the induced map F (C) → G(C) is a weak equivalence in A for each C ∈ C.
• an injective fibration if it has the right lifting property with respect
to every morphism β in Fun(C, A) which is simultaneously a weakequivalence and an injective cofibration.
• a projective cofibration if it has the left lifting property with respect
to every morphism β in Fun(C, A) which is simultaneously a weakequivalence and a projective fibration.
Proposition A.2.8.2. Let A be a combinatorial model category and let C
be a small category. Then there exist two combinatorial model structures on
Fun(C, A):• The projective model structure determined by the projective cofibrations, weak equivalences, and projective fibrations.
• The injective model structure determined by the injective cofibrations,weak equivalences, and injective fibrations.
The following is the key step in the proof of Proposition A.2.8.2:
Lemma A.2.8.3. Let A be a presentable category and let C be a small
category. Let S0 be a (small) set of morphisms of A and let S 0 be the weakly
saturated class of morphisms generated by S0 . Let Se be the collection of
all morphisms F → G in Fun(C, A) with the following property: for every
C ∈ C, the map F (C) → G(C) belongs to S 0 . Then there exists a (small) set
of morphisms S of Fun(C, A) which generates Se as a weakly saturated classof morphisms.
We prove a generalization of Lemma A.2.8.3 in §A.3.3 (Lemma A.3.3.3).
Proof of Proposition A.2.8.2. We first treat the case of the projective model
structure. For each object C ∈ C and each A ∈ A, we defineFC
A :C→Aby the formula
0
FC
A (C ) =aA.

α∈MapC (C,C 0 )We note that if i : A → A0 is a (trivial) cofibration in A, then the inducedC
map FCA → F A0 is a projective (trivial) cofibration in Fun(C, A).
Let I0 be a set of generating cofibrations i : A → B for A and let I
Cbe the set of all induced maps FC
A → F B (where C ranges over C). Let J0
be a set of generating trivial cofibrations for A and define J likewise. It
follows immediately from the definitions that a morphism in Fun(C, A) is a
projective fibration if and only if it has the right lifting property with respecthighertopoi

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831

APPENDIX

to every morphism in J, and a projective trivial fibration if and only if it
has the right lifting property with respect to every morphism in I. Let I
and J be the weakly saturated classes of morphisms of Fun(C, A) generated
by I and J, respectively. Using the small object argument, we deduce thefollowing:(i) Every morphism f : X → Z in Fun(C, A) admits a factorization
f0

f 00

X → Y → Z,
where f 0 ∈ I and f 00 is a projective trivial fibration.
(ii) Every morphism f : X → Z in Fun(C, A) admits a factorization
f0

f 00

X → Y → Z,
where f 0 ∈ J and f 00 is a projective fibration.
(iii) The class I coincides with the class of projective cofibrations in A.
Furthermore, since the class of trivial projective cofibrations in Fun(C, A) is
weakly saturated and contains J, it contains J. This proves that Fun(C, A)
satisfies the factorization axioms. The only other nontrivial point to checkis that Fun(C, A) satisfies the lifting axioms. Consider a diagram
/X
>

A
i


C


/Y

p

in Fun(C, A), where i is a projective cofibration and p is a projective fibration. We wish to show that there exists a dotted arrow as indicated provided
that either i or p is a weak equivalence. If p is a weak equivalence, thenthis follows immediately from the definition of a projective fibration. Suppose instead that i is a trivial projective cofibration. We wish to show thati has the left lifting property with respect to every projective fibration. It
will suffice to show that every trivial projective fibration belongs to J (thiswill also imply that J is a set of generating trivial projective cofibrations for
Fun(C, A), which shows that the projective model structure on Fun(C, A)
is combinatorial). Suppose then that i is a trivial projective cofibration andchoose a factorizationi0

i00

A → B → C,
where i0 ∈ J and i00 is a projective fibration. Then i0 is a weak equivalence,
so that i00 is a weak equivalence by the two-out-of-three property. Consider
the diagram
A

i0

i00

i


C

/B
>

=


/ C.

highertopoi

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832

APPENDIX

Since i is a cofibration, there exists a dotted arrow as indicated. This provesthat i is a retract of i0 and therefore belongs to J, as desired.
We now prove the existence of the injective model structure on Fun(C, A).
Here it is difficult to proceed directly, so we will instead apply Proposition
A.2.6.10. It will suffice to check each of the hypotheses in turn:(1) The collection of injective cofibrations in Fun(C, A) is generated (as aweakly saturated class) by some small set of morphisms. This follows
from Lemma A.3.3.3.
(2) The collection of trivial injective cofibrations in Fun(C, A) is weakly
saturated: this follows immediately from the fact that the class of injective cofibrations in A is weakly saturated.
(3) The collection of weak equivalences in Fun(C, A) is an accessible subcategory of Fun(C, A)[1] : this follows from the proof of Proposition
5.4.4.3 since the collection of weak equivalences in A form an accessible subcategory of A[1] .
(4) The collection of weak equivalences in Fun(C, A) satisfy the two-outof-three property: this follows immediately from the fact that the weakequivalences in A satisfy the two-out-of-three property.
(5) Let f : X → Y be a morphism in A which has the right lifting property
with respect to every injective cofibration. In particular, f has theright lifting property with respect to each of the morphisms in theclass I defined above, so that f is a trivial projective fibration and, inparticular, a weak equivalence.

Remark A.2.8.4. In the situation of Proposition A.2.8.2, if A is assumed
to be right or left proper, then Fun(C, A) is likewise right or left proper (withrespect to either the projective or the injective model structures).
Remark A.2.8.5. It follows from the proof of Proposition A.2.8.2 thatthe class of projective cofibrations is generated (as a weakly saturated classC0of morphisms) by the maps j : FC
A → F A0 , where C ∈ C and A → A is a
cofibration in A. We observe that j is an injective cofibration. It follows thatevery projective cofibration is an injective cofibration; dually, every injective
fibration is a projective fibration.
Remark A.2.8.6. The construction of Proposition A.2.8.2 is functorial in
the following sense: given a Quillen adjunction of combinatorial model cateF /gories A o
B and a small category C, composition with F and G deterG

mines a Quillen adjunction
Fun(C, A) oFC
GC/ Fun(C, B)

highertopoi

APPENDIX

April 9, 2017833

(with respect to either the injective or the projective model structures).
Moreover, if (F, G) is a Quillen equivalence, then so is (F C , GC ).
Because the projective and injective model structures on Fun(C, A) havethe same weak equivalences, the identity functor idFun(C,A) is a Quillen equivalence between them. However, it is important to distinguish between these
two model structures because they have different variance properties, as we
now explain.
Let f : C → C0 be a functor between small categories. Then composition
with f yields a pullback functor f ∗ : Fun(C0 , A) → Fun(C, A). Since A
admits small limits and colimits, f ∗ has a right adjoint, which we will denoteby f∗ , and a left adjoint, which we shall denote by f! .
Proposition A.2.8.7. Let A be a combinatorial model category and let
f : C → C0 be a functor between small categories. Then
(1) The pair (f! , f ∗ ) determines a Quillen adjunction between the projective model structures on Fun(C, A) and Fun(C0 , A).
(2) The pair (f ∗ , f∗ ) determines a Quillen adjunction between the injectivemodel structures on Fun(C, A) and Fun(C0 , A).
Proof. This follows immediately from the simple observation that f ∗ preserves weak equivalences, projective fibrations, and weak cofibrations.
We now review the theory of homotopy limits and colimits in a combinatorial model category A. For simplicity, we will discuss homotopy limits
and leave the analogous theory of homotopy colimits to the reader. Let A
be a combinatorial model category and let f : C → C0 be a functor betweeen(small) categories. We wish to consider the right derived functor Rf∗ of theright Kan extension f∗ : Fun(C, A) → Fun(C0 , A). This derived functor iscalled the homotopy right Kan extension functor. The usual way of defining it
involves choosing a fibrant replacement functor Q : Fun(C, A) → Fun(C, A)
and setting Rf∗ = f∗ ◦Q. The assumption that A is combinatorial guarantees
that such a fibrant replacement functor exists. However, for our purposes it
is more convenient to address the indeterminacy in the definition of Rf∗ inanother way.
Let F ∈ Fun(C, A), let G ∈ Fun(C0 , A), and let η : G → f∗ F be a map
in Fun(C0 , A). We will say that η exhibits G as the homotopy right Kan extension of F if, for some weak equivalence F → F 0 where F 0 is injectively
fibrant in Fun(C, A), the composite map G → f∗ F → f∗ F 0 is a weak equivalence in Fun(C0 , A). Since f∗ preserves weak equivalences between injectively
fibrant objects, this condition is independent of the choice of F 0 .
Remark A.2.8.8. Given an object F ∈ Fun(C, A), it is not necessarily thecase that there exists a map η : G → f∗ F which exhibits G as a homotopy
right Kan extension of F . However, such a map can always be found after
replacing F by a weakly equivalent object; for example, if F is injectively
fibrant, we may take G = f∗ F and η to be the identity map.

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Let [0] denote the final object of Cat: that is, the category with one object
and only the identity morphism. For any category C, there is a unique functor
f : C → [0]. If A is a combinatorial model category, F : C → A is afunctor, and A ∈ A ' Fun([0], A) is an object, then we will say that anatural transformation α : f ∗ A → F exhibits A as a homotopy limit of F
if it exhibits A as a homotopy right Kan extension of F . Note that we canidentify α with a map A → limC∈C F (C) in the model category A.
The theory of homotopy right Kan extensions in general can be reduced
to the theory of homotopy limits in view of the following result:
Proposition A.2.8.9. Let A be a combinatorial model category, let f :C → D be a functor between small categories, and let F : C → A andG : D → A be diagrams. A natural transformation α : f ∗ G → F exhibits G
as a homotopy right Kan extension of F if and only if for each object D ∈ D,
α exhibits G(D) as a homotopy limit of the composite diagramF

FD/ : C ×D DD/ → C → A.
To prove Proposition A.2.8.9, we can immediately reduce to the case where
F is an injectively fibrant diagram. In this case, α exhibits G as a homotopy
right Kan extension of F if and only if it induces a weak homotopy equivalence G(D) → lim FD/ for each D ∈ D. It will therefore suffice to prove thefollowing result (in the case C0 = C ×D DD/ ):Lemma A.2.8.10. Let A be a combinatorial model category and let g :C0 → C be a functor which exhibits C0 as cofibered in sets over C. Then the
pullback functor g ∗ : Fun(C, A) → Fun(C0 , A) preserves injective fibrations.
Proof. It will suffice to show that the left adjoint g! preserves injective trivial
cofibrations. Let α : F → F 0 be a map in Fun(C0 , A). We observe that foreach object C ∈ C, the map (q! α)(C) : (q! F )(C) → (q! F 0 )(C) can be identified with the coproduct of the maps {α(C 0 ) : F (C 0 ) → F 0 (C 0 )}C 0 ∈g−1 {C} .
If α is an injective trivial cofibration, then each of these maps is a trivial cofibration in A, so that q! α is again an injective trivial cofibration, as
desired.
Remark A.2.8.11. In the preceding discussion, we considered injectivemodel structures, Rf∗ , and homotopy limits. An entirely dual discussion
may be carried out with projective model structures and Lf! ; one obtains
a notion of homotopy colimit which is the dual of the notion of homotopy
limit.
Example A.2.8.12. Let A be a combinatorial model category and consider
a diagramfg

X 0 ← X → X 00 .
This diagram is projectively cofibrant if and only if the object X is cofibrantand the maps f and g are both cofibrations. Consequently, the definitionof homotopy colimits given above recovers, as a special case, the theory ofhomotopy pushouts presented in §A.2.4.

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A.2.9 Reedy Model Structures
Let A be a combinatorial model category and J a small category. In §A.2.8,
we saw that the diagram category Fun(J, A) can again be regarded as a
combinatorial model category via either the projective or the injective model
structure of Proposition A.2.8.2. In the special case where J is a Reedy category (see Definition A.2.9.1), it is often useful to consider still another model
structure on Fun(J, A): the Reedy model structure. We will sketch the definition and some of the basic properties of Reedy model categories below; werefer the reader to [38] for a more detailed treatment.
Definition A.2.9.1. A Reedy category is a small category J equipped witha factorization system JL , JR ⊆ J satisfying the following conditions:(1) Every isomorphism in J is an identity map.
(2) Given a pair of objects X, Y ∈ J, let us write X 0 Y if either there
exists a morphism f : X → Y belonging to JR or there exists a morphism g : Y → X belonging to JL . We will write X ≺0 Y if X 0 Y
and X 6= Y . Then there are no infinite descending chains· · · ≺0 X2 ≺0 X1 ≺0 X0 .
Remark A.2.9.2. Let J be a category equipped with a factorization system(JL , JR ) and let 0 be the relation described in Definition A.2.9.1. This
relation is generally not transitive. We will denote its transitive closure by
. Then condition (2) of Definition A.2.9.1 guarantees that  is a wellfounded partial ordering on the set of objects of J. In other words, every
nonempty set S of objects of J contains a -minimal element.
Remark A.2.9.3. In the situation of Definition A.2.9.1, we will often abuseterminology and simply refer to J as a Reedy category, implicitly assumingthat a factorization system on J has been specified as well.
Warning A.2.9.4. Condition (1) of Definition A.2.9.1 is not stable underequivalence of categories. Suppose that J is equivalent to a Reedy category.
Then J can itself be regarded as a Reedy category if and only if every isomorphism class of objects in J contains a unique representative. (Definition
A.2.9.1 can easily be modified so as to be invariant under equivalence, but
it is slightly more convenient not to do so.)
Example A.2.9.5. The category ∆ of combinatorial simplices is a Reedy
category with respect to the factorization system (∆L , ∆R ); here a morphism f : [m] → [n] belongs to ∆L if and only if f is surjective, and to ∆R
if and only if f is injective.
Example A.2.9.6. Let J be a Reedy category with respect to the factorization system (JL , JR ). Then Jop is a Reedy category with respect to thefactorization system ((JR )op , (JL )op ).

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Notation A.2.9.7. Let J be a Reedy category, C a category which admits
small limits and colimits, and X : J → C a functor. For every object J ∈ J,
we define the latching object LJ (X) to be the colimit
lim
−
→0
R

X(J 0 ).

J 0 ∈J/J ,J 6=J

Similarly, we define the matching object to be the limitlim←−

X(J 0 ).

J 0 ∈JL,J 0 6=JJ/

We then have canonical maps LJ (X) → X(J) → MJ (X).
Example A.2.9.8. Let X : ∆op → Set be a simplicial set and regard ∆op asa Reedy category using Examples A.2.9.5 and A.2.9.6. For every nonnegative
integer n, the latching object L[n] X can be identified with the collection of
all degenerate simplices of X. In particular, the map L[n] (X) → X([n]) isalways a monomorphism.
More generally, we observe that a map of simplicial sets f : X → Y is amonomorphism if and only if, for every n ≥ 0, the mapa
L[n] (Y )X([n]) → Y ([n])
L[n] (X)is a monomorphism of sets. The “if” direction is obvious. For the converse, let
us suppose that f is a monomorphism; we must show that if σ is an n-simplexof X such that f (σ) is degenerate, then σ is already degenerate. If f (σ) is
degenerate, then f (σ) = α∗ f (σ) = f (α∗ σ), where α : [n] → [n] is a map oflinearly ordered sets other than the identity. Since f is a monomorphism, we
deduce that σ = α∗ σ, so that σ is degenerate, as desired.
Remark A.2.9.9. Let X : J → C be as in Notation A.2.9.7. Then the Jthmatching object MJ (X) can be identified with the Jth latching object of
the induced functor X op : Jop → Cop .
Remark A.2.9.10. Let X : J → C be as in Notation A.2.9.7. Then the Jthmatching object MJ (X) can also be identified with the colimitlim−→

X(J 0 ),

(f :J 0 →J)∈S

where S is any full subcategory of J/J with the following properties:(1) Every morphism f : J 0 → J which belongs to JR and is not an isomorphism also belongs to S.
(2) If f : J 0 → J belongs to S, then J  J 0 .
This follows from a cofinality argument since every morphism f : J 0 → J in
S admits a canonical factorization
f0

f 00

J 0 → J 00 → J,highertopoi

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APPENDIX

where f 0 belongs to JL and f 00 belongs to JR . Assumption (2) guarantees
that the map f 00 is not an isomorphism.
Similarly, when convenient, we can replace the limit limf :J→J 0 X(J 0 ) defin←−ing the matching object MJ (X) by a limit over a slightly larger category.
Notation A.2.9.11. Let J be a Reedy category. A good filtration of J is atransfinite sequence{Jβ }β<αof full subcategories of J with the following properties:(a) The filtration is exhaustive in the following sense: every object of J
belongs to Jβ for sufficiently large β < α.
(b) For each ordinal
S β < α, the category Jβ is obtained from the subcategory J<β = γ<β Jγ by adjoining a single new object Jβ satisfying
the following condition: if J ∈ J satisfies J ≺ Jβ , then J ∈ J<β .
Remark A.2.9.12. Let J be a Reedy category. Then there exists a good
filtration of J. In fact, the existence of a good filtration is equivalent to thesecond assumption of Definition A.2.9.1.
Remark A.2.9.13. Let J be a Reedy category with respect to the factorization system (JL , JR ) and let {Jβ }β<α be a good filtration of J. Theneach Jβ admits a factorization system (JL ∩ Jβ , JR ∩ Jβ ). In other words, if
f : I → K is a morphism in Jβ which admits a factorization
f0

f 00

I → J → K,
where f 0 belongs to JL and f 00 belongs to JR , then the object J also belongs
to Jβ . This is clear: either f 00 is an isomorphism, in which case J = K ∈ Jβ ,or f 00 is not an isomorphism, so that J ≺ K implies that J ∈ J<β .
The following result summarizes the essential features of a good filtration:
Proposition A.2.9.14. Let J be a Reedy category with a good filtration{Jβ }β<α and let β < α be an ordinal, so that Jβ is obtained from J<β byadjoining a single new object J. Then we have a homotopy pushout square(with respect to the Joyal model structure):N(J<β )/J ? N(J<β )J/_/ N(J<β )_N(J<β )/J ? {J} ? N(J<β )J// N(Jβ ).

Corollary A.2.9.15. Let J be a Reedy category equipped with a good filtration {Jβ }β<α and let let β < α be an ordinal, so that Jβ is obtained fromJ<β by adjoining a single new object J. Let C be a category which admits
small limits and colimits, let X : J<β → C be a functor, and let the latchinghighertopoi

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APPENDIX

and matching objects LJ (X) and MJ (X) be defined as in Notation A.2.9.7(note that this does not require that the functor X be defined on the objectJ), so that we have a canonical map α : LJ (X) → MJ (X). The following
data are equivalent:(1) A functor X : Jβ → C extending X.
(2) A commutative diagram<Cα

LJ (X)#/ MJ (X)in the category C.
The equivalence carries a functor X to the evident diagram with C = X(J).
Proof. Using Proposition A.2.9.14, we see that giving an extension X : Jβ →C of X is equivalent to giving an extension Y : (J<β )/J ? {J} ? (J<β )J/ → Cof the composite functor
X

Y : (J<β )/J ? (J<β )J/ → J<β → C .
This, in turn, is equivalent to giving a commutative diagram
9C

lim Y |(J<β )/J
−→

α0

%
/ lim Y |(J<β )J/ ,←−where α0 is the map induced by the diagram Y . The equivalence of this withthe data (2) follows immediately from Remark A.2.9.10.
Remark A.2.9.16. The proof of Corollary A.2.9.15 carries over without
essential change to the case where C is an ∞-category which admits small
limits and colimits. In this case, to extend a functor X : N(J<β ) → C to afunctor X defined on the whole of Jβ , it will suffice to specify the object
X(J) ∈ CX|(J<β )/J / X|(J<β )J/ ' CLJ (X)/ /MJ (X) ,where the latching and matching objects LJ (X), MJ (X) ∈ C are defined in
the obvious way.
The proof of Proposition A.2.9.14 will require a few preliminaries.
Lemma A.2.9.17. Let J be a Reedy category equipped with a good filtration{Jβ }β<α . Fix β < α and let Jβ be obtained from J<β by adjoining the object
J. Let f : J → J be a map which is not the identity, let I denote the category(JJ/ )/f ' (J/J )f / of factorizations of the morphism f , and let I0 = I ×J J<β .
Then the nerve N I0 is weakly contractible.

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APPENDIX

Proof. Let I1 denote the full subcategory of I0 spanned by those diagrams
@I

f0

f

J

f 00



/ J,

where I ∈ J<β and f 00 is a morphism in JR . The inclusion I1 ⊆ I0 admits
a left adjoint, so that N I1 is a deformation retract of N I0 . It will therefore suffice to show that N I1 is weakly contractible. Let I2 denote the fullsubcategory of I1 spanned by those diagrams as above where, in addition,
the morphism f 0 belongs to JL . Then the inclusion I2 ⊆ I1 admits a rightadjoint, so that N I2 is a deformation retract of N I1 . It will therefore suffice
to show that N I2 is weakly contractible. This is clear since the category I2
consists of a single object (with no nontrivial endomorphisms).
Lemma A.2.9.18. Let n ≥ 1 and suppose we are given a sequence of weakly
contractible simplicial sets {Ai }1≤i≤n . Let L denote the iterated join
{J0 } ? A1 ? {J1 } ? A2 ? · · · ? An ? {Jn }and let K denote the simplicial subset of L spanned by those simplices which
do not contain all of the vertices {Ji }0≤i≤n . Then the inclusion K ⊆ L is a
categorical equivalence of simplicial sets.
Proof. If n = 1, then this follows immediately from Lemma 5.4.5.10. Supposethat n > 1. Let X denote the iterated join A1 ? {J1 } ? A2 ? · · · ? {Jn−1 } ? An .
For every subset S ⊆ {1, . . . , n − 1}, let X(S) denote the simplicial subsetof X spanned by
S those simplices which do not contain any vertex Ji for
i ∈ S. Let X 0 = S6=∅ X(S) ⊆ X(∅) = X. Then X 0 is the homotopy colimitof the diagram of simplicial sets {X(S)}S6=∅ . Each X(S) is a join of weakly
contractible simplicial sets, and is therefore weakly contractible. Since n > 1,
the partially ordered set {S ⊆ {1, . . . , n − 1} : S 6= ∅} has a largest element
and is therefore weakly contractible. It follows that the simplicial set X 0 is
weakly contractible.
The assertion that the inclusion K ⊆ L is a categorical equivalence isequivalent to the assertion that the diagram`
0/ ({J0 } ? X) ` (X ? {J0 })({J0 } ? X 0 )0 (X ? {Jn })
X

X



{J0 } ? X 0 ? {Jn } / {J0 } ? X ? {Jn }is a homotopy pushout square (with respect to the Joyal model structure). To
prove this, it suffices to observe that the vertical maps are both categoricalequivalences (Lemma 5.4.5.10).
Proof of Proposition A.2.9.14. Let S denote the collection of all composablechains of morphisms
f1

f2

fn

f : J → J → · · · → J.

highertopoi

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APPENDIX

where n ≥ 1 and each fi 6= idJ . For every subset S 0 ⊆ S, let X(S 0 ) denotethe simplicial subset of N(Jβ ) spanned by those simplices σ satisfying thefollowing condition:(∗) For every nondegenerate face τ of σ of positive dimension, if every
vertex of τ coincides with J, then τ belongs to S 0 .
Note that X(S) coincides with N(Jβ ), while X(∅) coincides with the pushoutaN(J<β ).
(N(J<β )/J ? {J} ? N(J<β )J/ )N(J<β )/J ?N(J<β )J/

It will therefore suffice to show that the inclusion X(∅) ⊆ X(S) is a categorical equivalence of simplicial sets.
Choose a well-ordering
S = {f 0 < f 1 < f 2 < · · · }with the following property: if f has length shorter than g (when regarded asa chain of morphisms), then f < g. For every ordinal α, let Sα = {f β }β<α .
We will prove that for every ordinal α, the inclusion X(∅) ⊆ X(Sα ) is a
categorical equivalence. The proof proceeds by induction on α. If α = 0,
there is nothing to prove, and if α is a limit ordinal, then the desired result
follows from the inductive hypothesis and the fact that the class of categorical equivalences is stable under filtered colimits. We may therefore assume
that α = β + 1 is a successor ordinal. The inductive hypothesis guarantees
that X(∅) ⊆ X(Sβ ) is a categorical equivalence. It will therefore suffice toshow that the inclusion j : X(Sβ ) ⊆ X(Sα ) is a categorical equivalence. We
may also suppose that β is smaller than the order type of S, so that f β is
well-defined (otherwise, the inclusion j is an isomorphism and the result isobvious).
Let f = f β be the composable chain of morphisms
f1

f2

fn

f : J → J → · · · → J.
For 1 ≤ i ≤ n, let Ai denote the nerve of the categoryJ<β ×J (JJ/ )/fi ' J<β ×J (J/J )fi / .
Let K denote the simplicial subset of{J0 } ? A1 ? {J1 } ? A2 ? · · · ? An ? {Jn }spanned by those simplices which do not contain every vertex Jn . We thenhave a homotopy pushout diagram
N(J<β )/J ? K ? N(J<β )J/_
N(J<β )/J ? {J0 } ? A1 ? · · · ? An ? {Jn } ? N(J<β )J// X(Sβ )/ X(Sα ).

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APPENDIX

It will therefore suffice to prove that the left vertical map is a categoricalequivalence. In view of Corollary 4.2.1.3, it will suffice to show that theinclusionK ⊆ {J0 } ? A1 ? {J1 } ? A2 ? · · · ? An ? {Jn }is a categorical equivalence. Since each Ai is weakly contractible (Lemma
A.2.9.17), this follows immediately from Lemma A.2.9.18.
Proposition A.2.9.19. Let J be a Reedy category and let A be a model
category. Then there exists a model structure on the category of functors
Fun(J, A) with the following properties:(C) A morphism X → Y in Fun(J, A) is a Reedy cofibration
if and only if,`for every object J ∈ J, the induced map X(J) LJ (X) LJ (Y ) → Y (J)is a cofibration in A.
(F ) A morphism X → Y in Fun(J, A) is a Reedy fibration if and only if,for every object J ∈ J, the induced map X(J) → Y (J) ×MJ (Y ) MJ (X)
is a fibration in A.
(W ) A morphism X → Y in Fun(J, A) is a weak equivalence if and only if,for every J ∈ J, the map X(J) → Y (J) is a weak equivalence.
Moreover, a morphism f : X → Y in Fun(J, A) is a trivial cofibration if andonly if the following condition is satisfied:`(W C) For every object J ∈ J, the map X(J) LJ (X) LJ (Y ) → Y (J) is atrivial cofibration in A.
Similarly, f is a fibration if and only if it satisfies the dual condition:(W F ) For every object J ∈ J, the map X(J) → Y (J) ×MJ (Y ) MJ (X) is atrivial fibration in A.
The model structure of Proposition A.2.9.19 is called the Reedy model
structure on Fun(J, A). Note that Proposition A.2.9.19 does not require themodel category A to be combinatorial.
Lemma A.2.9.20. Let J be a Reedy category containing an object J, let
A be a model category, and let f : F → G be a natural transformation in
R
Fun(J, A). Let I ⊆ JR
/J be a sieve: that is, I is a full subcategory of J/J with0
the property that if I → I 0 is a morphism in JR/J such that I ∈ I, then I ∈ I.
0
Let I ⊆ I be another sieve. Then(a) If the map f satisfies condition (C) of Proposition A.2.9.19 for every
object I ∈ I, then the induced mapaχI0 ,I : lim(F | I)lim(G| I0 ) → lim(G| I)
−→
−→
−→
0
lim(F | I )−→is a cofibration in A.

highertopoi

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842

APPENDIX(b) If the map f satisfies condition (W C) of Proposition A.2.9.19 for every
object I ∈ I, then the map χI0 ,I is a trivial cofibration in A.
Proof. We will prove (a); the proof of (b) is identical. Choose a transfinite
sequence of sieves {Iβ ⊆ I}β<α with the following properties:
S(i) The union β<α Iβ coincides with I.
S(ii) For each β < α, the sieve Iβ is obtained from I<β = I0 ∪( γ<β Iγ ) byadjoining a single new object Jβ ∈ JR/J .
For every triple δ ≤ γ ≤ β ≤ α, let χδ,γ,β denote the induced mapaalim(F | I<β )lim(G| I<δ ) → lim(F | I<β )lim(G| I<γ ).
−→
−→
−→
−→
lim(F | I<δ )lim(F | I<γ )−→−→We wish to prove that χ0,α,α is a cofibration. We will prove more generallythat χδ,γ,β is an equivalence for every δ ≤ γ ≤ β ≤ α. The proof uses induction on γ. If γ is a limit ordinal, then we can write χδ,γ,β as the transfinite
composition of the maps {χ,+1,β }δ≤<γ , which are cofibrations by the inductive hypothesis. We may therefore assume that γ = γ0 + 1 is a successor
ordinal. If δ = γ, then χδ,γ,β is an isomorphism; otherwise, we have δ ≤ γ0 .
In this case, we haveχδ,γ,β = χγ0 ,γ,β ◦ χδ,γ0 ,β .
Using the inductive hypothesis, we can reduce to the case δ = γ0 . The
map χγ0 ,γ,β is a pushout of the map χγ0 ,γ,γ . We are therefore reduced toproving`that χγ0 ,γ,γ is a cofibration. But χγ0 ,γ,γ is a pushout of the map
LI (G) LI (F ) F (I) → G(I) for I = Jγ0 . This map is a cofibration by virtueof our assumption that f satisfies (C).
Proof of Proposition A.2.9.19. Let f : X → Z be a morphism in Fun(J, A).
We will prove that f admits a factorization
f0

f 00

X → Y → Z,
where(i) The map f 00 is a fibration, and f 0 satisfies (W C).
(ii) The map f 0 is a cofibration, and f 00 satisfies (W F ).
By symmetry, it will suffice to consider case (i). Choose a good filtration{Jβ }β<α of J. For β < α, let Xβ = X| Jβ , let Zβ = Z| Jβ , and let fβ :Xβ → Zβ be the restriction of f . We will construct a compatible family offactorizations of fβ as a composition
fβ0

fβ00

Xβ → Yβ → Zβ .
Suppose that Jβ is obtained from J<β by adjoining a single new object
J. Assuming that (fγ0 , fγ00 ) has been constructed for all γ < β, we note thathighertopoi

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APPENDIX

constructing (fβ0 , fβ00 ) is equivalent (by virtue of Corollary A.2.9.15) to giving
a commutative diagram
LJ (X)/ LJ (Y<β )


X(J)


/ Yβ (J)/ Z(J)
MJ (Y<β )/ MJ (Z).

In other words, we must factor a certain map
ag : LJ (Y<β )X(J) → MJ (Y<β ) ×MJ (Z) Z(J)LJ (X)

as a composition
LJ (Y<β )

a

g0

g 00

X(J) → Yβ (J) → MJ (Y<β ) ×MJ (Z) Z(J).

LJ (X)

Using the fact that A is a model category, we can choose a factorization
where g 0 is a trivial cofibration and g 00 a fibration. It is readily verified that
this construction has the desired properties.
We now prove the following:(i0 ) A morphism f : X → Y in Fun(J, A) satisfies (W C) if and only if f isboth a fibration and a weak equivalence.
(ii0 ) A morphism f : X → Y in Fun(J, A) satisfies (W F ) if and only if f isboth a cofibration and a weak equivalence.
By symmetry, it will suffice to prove (i0 ). The “only if” direction follows
from Lemma A.2.9.20. For the “if” direction, it will suffice to show that foreach β < α, the induced transformation fβ : Xβ → Yβ satisfies (W C) whenregarded as a morphism of Fun(Jβ , A). Suppose that Jβ is obtained fromJ<β by adjoining a single new element J. We have a commutative diagram`
LJ (Y ) LJ (X) X(J)
7
p

X(J)

q

r

'

/ Y (J).

We wish to prove that q is a trivial cofibration in A. Since f is a cofibrationin Fun(J, A), the map q is a cofibration in A. It will therefore suffice toshow that q is a weak equivalence. By the two-out-of-three property, it will
suffice to show that p and r are weak equivalences. For r, this follows from ourassumption that f is a weak equivalence in Fun(J, A). The map p is a pushoutof the map of latching objects LJ (X) → LJ (Y ), which is a cofibration in A
by virtue of the inductive hypothesis and Lemma A.2.9.20.

highertopoi

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APPENDIX

Combining (i) and (i0 ) (and the analogous assertions (ii) and (ii0 )), we
deduce that Fun(J, A) satisfies the factorization axioms for a model category.
To complete the proof, it will suffice to verify the lifting axioms:(i00 ) Every fibration in Fun(J, A) has the right lifting property with respect
to morphisms in Fun(J, A) which satisfy (W C).
(ii00 ) Every cofibration in Fun(J, A) has the left lifting property with respect
to morphisms in Fun(J, A) which satisfy (W F ).
Again, by symmetry it will suffice to prove (i00 ). Consider a diagram
/X
?

A
f

h


B

g


/ Y,

where f satisfies (W C) and g satisfies (F ); we wish to prove that there
exists a dotted arrow h as indicated, rendering the diagram commutative. To
prove this, we will construct a compatible family of natural transformations{hβ : B| Jβ → X| Jβ }β<α which render the diagram
A| Jβ
hβ



B| Jβ

/ X| Jβ;g/ Y | Jβ

commutative. Suppose that Jβ is obtained from J<β by adjoining a single new
object J. Assume that the maps {hγ }γ<β have already been constructed andcan be amalgamated into a single natural transformation h<β : B| J<β →X| J<β . Using Corollary A.2.9.15, we see that extending h<β to a map hβ
with the desired properties is equivalent to solving a lifting problem of thekind depicted in the following diagram:`/ X(J)LJ (B) LJ (A) A(J)
5
f0


B(J)



g0

/ Y (J) ×M (Y ) MJ (X).
J

0Since our assumptions guarantee that f is a trivial cofibration and that g 0
is a fibration, this lifting problem has a solution, as desired.
Example A.2.9.21. Let A be the category of bisimplicial sets, which we will
identify with Fun(∆op , Set∆ ) and endow with the Reedy model structure. It
follows from Example A.2.9.8 that a morphism f : X → Y of bisimplicial
sets is a Reedy cofibration if and only if it is a monomorphism. Consequently,
the Reedy model structure on A coincides with the injective model structureon A.

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Example A.2.9.22. Let J be a Reedy category with JL = J and let A bea model category. Then the weak equivalences and cofibrations of the Reedy
model structure (Proposition A.2.9.19) are the injective cofibrations and theweak equivalences appearing in Definition A.2.8.1. It follows that the Reedy
model structure on Fun(J, A) coincides with the injective model structureof Proposition A.2.8.2 (in particular, the injective model structure is welldefined in this case even without the assumption that A is combinatorial).
Similarly, if JR = J, then we can identify the Reedy model structure on
Fun(J, A) with the projective model structure of Proposition A.2.8.2.
In the general case, we can regard the Reedy model structure on Fun(J, A)
as a mixture of the projective and injective model structures. More precisely,
we have the following:(i) A natural transformation F → G in Fun(J, A) satisfies condition (C) of
Proposition A.2.9.19 if and only if the induced transformation F | JR →G| JR is a projective cofibration in Fun(JR , A).
(ii) A natural transformation F → G in Fun(J, A) satisfies condition (F ) of
Proposition A.2.9.19 if and only if the induced transformation F | JL →G| JL is an injective fibration in Fun(JL , A).
(iii) A natural transformation F → G in Fun(J, A) satisfies condition(W C) of Proposition A.2.9.19 if and only if the induced transformation
F | JR → G| JR is a trivial projective cofibration in Fun(JR , A).
(iv) A natural transformation F → G in Fun(J, A) satisfies condition(W F ) of Proposition A.2.9.19 if and only if the induced transformation
F | JL → G| JL is a trivial injective fibration in Fun(JL , A).
Remark A.2.9.23. Let J be a Reedy category and A a combinatorial model
category, so that the injective and projective model structures on Fun(J, A)
are well-defined. The identity functor from Fun(J, A) to itself can be regarded as a left Quillen equivalence from the projective model structure
to the Reedy model structure and from the Reedy model structure to theinjective model structure.
Corollary A.2.9.24. Let C be a small category. Suppose that there exists
a well-ordering ≤ on the collection of objects of C satisfying the following
condition: for every pair of objects X, Y ∈ C, we have(∅
if X  Y
HomC (X, Y ) ={idX } if X = Y.
Let A be a model category. Then(i) A natural transformation F → G in Fun(C, A) is a (trivial) projective
cofibration if and only if, for every object C ∈ C, the induced map
a
F (C)
lim
G(D) → G(C)−→lim
F (D) D→C,D6=C−→D→C,D6=C
is a (trivial) cofibration in A.

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APPENDIX(ii) A natural transformation F → G in Fun(Cop , A) is a (trivial) injective
fibration if and only if, for every object C ∈ C, the induced map
F (D)F (C) → G(C) ×lim
lim
G(D)
←−
←
−D→C,D6=C
D→C,D6=C
is a (trivial) fibration in A.
Proof. Combine Example A.2.9.22 with Proposition A.2.9.19.
Corollary A.2.9.25. Let A be a model category, let α be an ordinal, andlet (α) denote the linearly ordered set {β < α} regarded as a category. Then
(1) Let F → F 0 be a natural transformation of diagrams (α) → A. Suppose
that, for each β < α, the maps
limγ<β F (γ) → F (β)−→
limγ<β F 0 (γ) → F 0 (β)−→
are cofibrations, while the map F (β) → F 0 (β) is a weak equivalence.
Then the induced map
limγ<α F (γ) → limγ<α F 0 (γ)−→−→
is a weak equivalence.
(2) Let G → G0 be a natural transformation of diagrams (α)op → A.
Suppose that, for each β < α, the maps
G(β) → lim G(γ)γ<β

G0 (β) → lim G0 (γ)γ<β

are fibrations, while the map G(β) → G0 (β) is a weak equivalence.
Then the induced map
lim G(γ) → lim G0 (γ)

γ<α

γ<α

is a weak equivalence.
Proof. We will prove (1); (2) follows by the same argument. Let p : (α) → ∗be the unique map, let p∗ : A → A(α) be the diagonal map, and let p! :A(α) → A be a left adjoint to p∗ . Then p! can be identified with the functor
F 7→ limγ<α F (γ). We observe that (p! , p∗ ) is a Quillen adjunction (where−→A(α) is endowed with the projective model structure) so that p! preserves
weak equivalence between projectively cofibrant objects. The desired result
now follows from Corollary A.2.9.24.

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Suppose that we are given a bifunctor⊗ : A × B → C,
where C is a category which Radmits small limits. For any small category J,
we define the coend functor J : Fun(J, A) × Fun(Jop , B) → C so that theRintegral J (F, G) is defined to be the coequalizer of the diagram`0
J→J 0 F (J) ⊗ G(J )//`J F (J) ⊗ G(J) .

We then have the following result:
Proposition A.2.9.26. Let ⊗ : A × B → C be a left Quillen bifunctor (seeProposition A.3.1.1) and let J be a Reedy category. Then the coend functor
Z: Fun(J, A) × Fun(Jop , B) → C
J

is also a left Quillen bifunctor, where we regard Fun(J, A) and Fun(Jop , B)
as endowed with the Reedy model structure.
Proof. Let f : F → F 0 be a Reedy cofibration→ G0 aR in Fun(J,`RA) andR g : G
op00
Reedy cofibration in Fun(J , B). Set C = J (F, G )(F,G) J (F , G) ∈ C
J
R
R
0
0
0
and C = J (F , G ). We wish to show that the induced map C → J (F 0 , G0 )is a cofibration, which is trivial if either f or g is trivial.
Choose a good filtration {Jβ }β<α of J. For β ≤ α, we define
Z
Z
a
Cβ =
(F | J<β , G0 | J<β )(F 0 | J<β , G| J<β )
Jβ

R

Cβ0 =

Z
Jβ

Jβ (F | J<β ,G| J<β )Jβ(F 0 | J<β , G0 | J<β ).

We wish to show that the map
Cα ' CαaCαC00 → Cα

a

Cα0

C0

is a cofibration (which is trivial if either f or g is trivial). We will prove moregenerally that for δ ≤ γ ≤ β ≤ α, the mapaaηδ,γ,β : Cβ
Cδ0 → Cβ
Cγ0
Cδ

Cγ

is a cofibration (trivial if either f or g is trivial). The proof proceeds by
induction on γ. If γ is a limit ordinal, then ηδ,γ,β can be obtained as atransfinite composition of the maps {η,+1,β }δ≤<γ , and the result follows
from the inductive hypothesis. We may therefore assume that γ = γ0 + 1 isa successor ordinal. Since ηδ,γ,β = ηγ0 ,γ,β ◦ ηδ,γ0 ,β , we can use the inductive
hypothesis to reduce to the case where δ = γ0 . Since ηδ,γ,β is a pushout ofhighertopoi

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APPENDIX

ηδ,γ,γ , we can also assume that β = γ. In other words, we are reduced toproving that the mapah : Cγ0 +1
Cγ0 0 → Cγ0 0
Cγ0

is a cofibration, which is trivial if either f or g is trivial. Let J be the objectof Jγ0 which does not belong to J<γ0 . Form a pushout diagram``(F (J) LJ (F ) LJ (F 0 )) ⊗ (G(J) LJ (G) LJ (G0 ))(F (J)w
0
L(F)) ⊗ G0 (J)J
LJ (F )

' `F 0 (J) ⊗ (G(J) LJ (G) LJ (G0 ))

`

'

X.

w

We have an evident map h0 : X → F 0 (J) ⊗ G0 (J) which is a cofibration(trivial if either f or g is trivial) by virtue of our assumptions on f and g(together with the fact that ⊗ is a left Quillen bifunctor). We conclude byobserving that h is a pushout of h0 .
Remark A.2.9.27. Proposition A.2.9.26 has an analogue for the model
structures introduced in Proposition A.2.8.2. That is, suppose that A and B
are combinatorial model categories and let J be an arbitrary small category.
Then any left Quillen bifunctor ⊗ : A × B → C induces a left Quillen
bifunctor
Z: Fun(J, A) × Fun(Jop , B) → C,

J

where we regard Fun(J, A) as endowed with the projective model structure
and Fun(Jop , B) with the injective model structure. To see this, we mustshow that for any projective cofibration f : F → F 0 in Fun(J, A) and any
injective cofibration g : G → G0 in Fun(Jop , B), the induced map
ZZa Z
h : (F, G0 )(F 0 , G) → (F 0 , G0 )
J

R

J

(F,G)

J

J

is a cofibration in C which is trivial if either f or g is trivial. Without lossof generality, we may suppose that f is a generating projective cofibrationof the form FJA → FJA0 associated to an object J ∈ J and a cofibrationi : A → A0 in A, which is trivial if f is trivial (see the proof of Proposition
A.2.8.2 for an explanation of this notation). Unwinding the definitions, we
can identify h with the mapa(A ⊗ G0 (J))(A0 ⊗ G(J)) → A0 ⊗ G0 (J).
A⊗G(J)

Since i is a cofibration in A and the map G(J) → G0 (J) is a cofibration inB, we deduce that h is a cofibration in C (since ⊗ is a left Quillen bifunctor)
which is trivial if either i or h is trivial.

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Example A.2.9.28. Let A be a simplicial model category, so that we havea left Quillen bifunctor⊗ : A × Set∆ → A.
The coend construction determines a left Quillen bifunctor
Z: Fun(∆, A) × Fun(∆op , Set∆ ) → A.
∆

where Fun(∆, A) and Fun(∆op , Set∆ ) are both endowed with the Reedy
model structure. In particular, if we fix a cosimplicial object X • ∈ Fun(∆, A)
which is Reedy cofibrant, then forming the coend against X • determines aleft Quillen functor from the category of bisimplicial sets (with the Reedy
model structure, which coincides with the injective model structure by Example A.2.9.21) to A.
Example A.2.9.29. Let A be a simplicial model category, so that we havea left Quillen bifunctor⊗ : A × Set∆ → A,
and consider the coendfunctorZ∆op

Fun(∆op , A) × Fun(∆, Set∆ ) → A.

Let ∆• ∈ Fun(∆, Set∆ ) denote the standard simplex (that is, the functor[n] 7→ ∆n ) and let 1 denote the final object of Fun(∆, Set∆ ) (that is, theconstant functor given by [n] 7→ ∆0 ). The unique map ∆• → 1 is a weakequivalence, and ∆• is Reedy cofibrant: we may therefore regard ∆• as acofibrant replacementRfor the constant functor 1.
The functor X• 7→ ∆op (X• , 1) can be identified with the colimit functor
Fun(∆op , A) → A. This is a left Quillen functor if Fun(∆op , A) is endowedwith the projective model structure but not the Reedy Rmodel structure.
However, the geometric realization functor X• 7→ |X• | = ∆op (X• , ∆• ) is aleft Quillen functor with respect to the Reedy model structure.
Corollary A.2.9.30. Let A be a combinatorial simplicial model category
and let X• be a simplicial object of A. There is a canonical map
γ : hocolim X• → |X• |in the homotopy category of A. This map is an equivalence if X• is Reedy
cofibrant.
Proof. Let ∆• and ∗ be the cosimplicial objects of Set∆ described in Example
A.2.9.29. Choose a weak equivalence of simplicial objects X•0 → X• , whereX•0 is projectively cofibrant. We then have a diagram
Z
Z
Z
β
α
hocolim X• ' lim X•0 '(X•0 , ∗) ←(X•0 , ∆• ) →(X• , ∆• ).
−→
∆op
∆op
∆op
Since X•0R is projectively cofibrant, Remark A.2.9.27 implies that the coendfunctor ∆op (X•0 , •) preserves weak equivalences between injectively cofibrant cosimplicial objects of Set∆ ; in particular, α is a weak equivalence in A.

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This gives the desired map γ. Proposition A.2.9.26 implies that ∆op (•, ∆• )preserves weak equivalences between Reedy cofibrant simplicial objects of
A, which proves that γ is an isomorphism if X• is Reedy cofibrant.
R

Example A.2.9.31. If A is the category of simplicial sets, then the map γ
of Corollary A.2.9.30 is always an isomorphism; this follows from Example
A.2.9.21. In other words, if X•,• is a bisimplicial set, then we can identify
the diagonal simplicial set [n] 7→ Xn,n with the homotopy colimit of corresponding diagram ∆op → Set∆ .

A.3 SIMPLICIAL CATEGORIES
Among the many different models for higher category theory, the theory ofsimplicial categories is perhaps the most rigid. This can be either a curse ora blessing, depending on the situation. For the most part, we have chosen to
use the less rigid theory of ∞-categories (see §1.1.2) throughout this book.
However, some arguments are substantially easier to carry out in the settingof simplicial categories. For this reason, we have devoted the final section ofthis appendix to a review of the theory of simplicial categories.
There exists a model structure on the category Cat∆ of (small) simplicial
categories, which was constructed by Bergner ([7]). In §A.3.2, we will describe
an analogous model structure on the category CatS of S-enriched categories,
where S is a suitable model category. To formulate this generalization, we
will need to employ the language of monoidal model categories, which we
review in §A.3.1. Under mild assumptions on S, one can show that an Senriched category C is fibrant if and only if each of the mapping objects
MapC (X, Y ) is a fibrant object of S.
In §A.3.3, we will study the category AC of diagrams C → A, where C
is a small category and A is a model category, both enriched over somefixed model category S. In the enriched setting we can again endow ACwith projective and injective model structures, which can be used to define
homotopy limits and colimits.
Putting aside set-theoretic technicalities, every S-enriched model category
A gives rise to a fibrant object of CatS : namely, the full subcategory A◦ ⊆ A
spanned by the fibrant-cofibrant objects. In §A.3.4, we will introduce a pathobject for A◦ , which will enable us to perform some calculations in thehomotopy category of CatS .
In §A.3.5, we will consider the problem of constructing homotopy colimits
in the category CatS of S-enriched categories. Our main result, Theorem
A.3.5.15, asserts that the formation of homotopy colimits in CatS is compatible with the formation of (tensor) products in CatS . We will apply thisresult in §A.3.6 to study the homotopy theory of internal mapping objects
in CatS .
We conclude this section with §A.3.7, where we discuss localizations of(simplicial) model categories.

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A.3.1 Enriched and Monoidal Model Categories
Many of the model categories which arise naturally are enriched over the
category of simplicial sets. Our goal in this section to study enrichments ofone model category over another.
Definition A.3.1.1. Let A, B, and C be model categories. We will saythat a functor F : A × B → C is a left Quillen bifunctor if the following
conditions are satisfied:(a) Let i : A → A0 and j : B → B 0 be cofibrations in A and B, respectively.
Then the induced mapai ∧ j : F (A0 , B)
F (A, B 0 ) → F (A0 , B 0 )
F (A,B)is a cofibration in C. Moreover, if either i or j is a trivial cofibration,then i ∧ j is also a trivial cofibration.
(b) The functor F preserves small colimits separately in each variable.
Definition A.3.1.2. A monoidal model category is a monoidal category S
equipped with a model structure, which satisfies the following conditions:(i) The tensor product functor ⊗ : S × S → S is a left Quillen bifunctor.
(ii) The unit object 1 ∈ S is cofibrant.
(iii) The monoidal structure on S is closed.
Remark A.3.1.3. Some authors demand only a weakened form of axiom(ii) in the preceding definition.
Example A.3.1.4. The category of simplicial sets Set∆ is a monoidal model
category with respect to the Cartesian product and the Kan model structure
defined in §A.2.7.
Definition A.3.1.5. Let S be a monoidal model category. A S-enriched
model category is an S-enriched category A equipped with a model structure
satisfying the following conditions:(1) The category A is tensored and cotensored over S.
(2) The tensor product ⊗ : A × S → A is a left Quillen bifunctor.
In the special case where S is the category of simplicial sets (regarded as a
monoidal model category as in Example A.3.1.4), we will simply refer to A
as a simplicial model category.
Remark A.3.1.6. An easy formal argument shows that condition (2) isequivalent to either of the following statements:

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APPENDIX(20 ) Given any cofibration i : D → D0 in A and any fibration j : X → Y
in A, the induced map
k : MapA (D0 , X) → MapA (D, X) ×MapA (D,Y ) MapA (D0 , Y )is a fibration in S, which is trivial if either i or j is a weak equivalence.
(200 ) Given any cofibration i : C → C 0 in S and any fibration j : X → Y in
A, the induced map
0

k : X C → X C ×Y C Y C

0is a fibration in A, which is trivial if either i or j is trivial.
The following provides a criterion for detecting simplicial model structures:
Proposition A.3.1.7. Let C be a simplicial category that is equipped with amodel structure (not assumed to be compatible with the simplicial structureon C). Suppose that every object of C is cofibrant and that the collection of
weak equivalences in C is stable under filtered colimits. Then C is a simplicial
model category if and only if the following conditions are satisfied:(1) As a simplicial category, C is both tensored and cotensored over Set∆ .
(2) Given a cofibration of simplicial sets i : K → L and a cofibration
j : C → D in C, the induced mapa(C ⊗ L)(D ⊗ K) → D ⊗ L
C⊗Kis a cofibration in C.
(3) For every n ≥ 0 and every object C in C, the natural map
C ⊗ ∆n → C ⊗ ∆0 ' C
is a weak equivalence in C.
Proof. Suppose first that C is a simplicial model category. It is clear that (1)and (2) are satisfied. To prove (3), we note that the projection ∆n → ∆0admits a section s : ∆0 → ∆n which is a trivial cofibration. If C is a simplicial
model category, then since C is cofibrant, it follows that C⊗∆0 → C⊗∆n is atrivial cofibration and, in particular, a weak equivalence. Thus the projectionC ⊗ ∆n → C ⊗ ∆0 is a weak equivalence by the two-out-of-three property.
Now suppose that (1), (2), and (3) are satisfied. We wish to show that C
is a simplicial model category. We first show that the bifunctor(C, K) 7→ C ⊗ K
preserves weak equivalences separately in each variable.
Fix the object C ∈ C and suppose that f : K → K 0 is a weak homotopy
equivalence of simplicial sets. Choose a cofibration K → K 00 , where K 00 is acontractible Kan complex. Then we may factor f as a composition
f0

f 00

K → K 0 × K 00 → K 0 .

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To prove that idC ⊗f is a weak equivalence, it suffices to prove that idC ⊗f 0
and idC ⊗f 00 are weak equivalences. Note that the map f 00 has a section s
which is a trivial cofibration. Thus, to prove that idC ⊗f 00 is a weak equivalence, it suffices to show that idC ⊗s is a weak equivalence. In other words,
we may reduce to the case where f is itself a trivial cofibration of simplicial
sets.
Consider the collection A of all monomorphisms f : K → K 0 of simplicial
sets having the property that idC ⊗f is a weak equivalence in C. It is easy
to see that this collection of morphisms is weakly saturated. Thus, to provethat it contains all trivial cofibrations of simplicial sets, it suffices to showthat every horn inclusion Λni → ∆n belongs to A. We prove this by induction
on n > 0. Choose a vertex v belonging to Λni . We note that the inclusion{v} → Λni is a pushout of horn inclusions in dimensions < n; by the inductive
hypothesis, this inclusion belongs to A. Thus it suffices to show that {v} →∆n belongs to A, which is equivalent to assumption (3).
Now let us show that for each simplicial set K, the functor
C 7→ C ⊗ K
preserves weak equivalences. We will prove this by induction on the (possibly
infinite) dimension of K. Choose a weak equivalence g : C → C 0 in C. Let
S denote the collection of all simplicial subsets L ⊆ K such that g ⊗ idL isa weak equivalence. We regard S as a partially ordered set with respect toinclusions of simplicial subsets. Clearly, ∅ ∈ S. Since weak equivalences in C
are stable under filtered colimits, the supremum of every chain in S belongs
to S. By Zorn’s lemma, S has a maximal element L. We wish to show that
L = K. If not, we may choose some nondegenerate simplex σ of K which
does not belong to L. Choose σ of the smallest possible dimension,` so that allof the faces of σ belong to L. Thus there is an inclusion L0 = L ∂ σ σ ⊆ K.
Since C is left proper, assumption (2) implies that the diagram
D⊗∂σ

/ D⊗σ


D⊗L


/ D ⊗ L0

is a homotopy pushout for every object D ∈ C. We observe that g ⊗ idL isa weak equivalence by assumption, g ⊗ id∂ σ is a weak equivalence by the
inductive hypothesis (since ∂ σ has dimension smaller than the dimension ofK), and g⊗idσ is a weak equivalence by virtue of assumption (3) and the factthat g is a weak equivalence. It follows that g ⊗ idL0 is a weak equivalence,
which contradicts the maximality of L. This completes the proof that thebifunctor ⊗ : C × Set∆ → C preserves weak equivalences separately in eachvariable.
Now suppose we are given a cofibration i : C → C 0 in C and anothercofibration j : S → S 0 in Set∆ . We wish to prove that the induced mapai ∧ j : (C ⊗ S 0 )(C 0 ⊗ S) → C 0 ⊗ S 0
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APPENDIX

is a cofibration in C, which is trivial if either i or j is trivial. The first point
follows immediately from (2). For the triviality, we will assume that i is aweak equivalence (the case where j is a weak equivalence follows using thesame argument). Consider the diagramC ⊗ S0/ C0 ⊗ S

i⊗idS

C ⊗S

f

/ (C 0 ⊗ S)

`0C⊗S (C ⊗ S )/ C 0 ⊗ S0.

The arguments above show that i ⊗ idS and i ⊗ idS 0 are weak equivalences.
The square in the diagram is a homotopy pushout, so Proposition A.2.4.2
implies that f is a weak equivalence as well. Thus i ∧ j is a weak equivalenceby the two-out-of-three property.
If C is a simplicial model category, then there is automatically a strong
relationship between the homotopy theory of the underlying model category
and the homotopy theory of the simplicial sets MapC (•, •). For example, wehave the following:
Remark A.3.1.8. Let C be a simplicial model category, let X be a cofibrantobject of C, and let Y be a fibrant object of C. The simplicial set K =MapC (X, Y ) is a Kan complex; moreover, there is a canonical bijection
π0 K ' HomhC (X, Y ).
We conclude this section by studying a situation which arises in Chapter
3. Let C and D be model categories enriched over another model category
S, and suppose we are given a Quillen adjunction
Co

F
G

/D

between the underlying model categories. We wish to study the situation
where G (but not F ) has the structure of an S-enriched functor. Thus, forevery triple of objects X ∈ C, Y ∈ D, S ∈ S, we have a canonical map
HomC (S ⊗ X, GY ) ' HomS (S, MapC (X, GY ))→ HomS (S, MapD (F X, F GY ))' HomD (S ⊗ F X, F GY )→ HomD (S ⊗ F X, Y ).
Taking Y = F (S ⊗ X) and applying this map to the unit of the adjunction
between F and G, we obtain a map S⊗F X → F (S⊗X), which we will denoteby βX,S . The collection of maps βX,S is simply another way of encoding thedata of G as an S-enriched functor. If the maps βX,S are isomorphisms,
then F is again an S-enriched functor and (F, G) is an adjunction between
S-enriched categories. We wish to study an analogous situation where themaps βX,S are only assumed to be weak equivalences.

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Remark A.3.1.9. Suppose that S is the category Set∆ of simplicial setswith its usual model structure. Then the map βX,S is automatically a weakequivalence for every cofibrant object X ∈ C. To prove this, we consider
the collection K of all simplicial sets S such that βS,X is an equivalence. It
is not difficult to show that K is closed under weak equivalences, homotopy
pushout squares, and coproducts. Since ∆0 ∈ K, we conclude that K = Set∆ .
Proposition A.3.1.10. Let C and D be S-enriched model categories. LetF /D be a Quillen adjunction between the underlying model categories.
Co
G

Assume that every object of C is cofibrant, and that the map βX,S : S ⊗F (X) → F (S ⊗ X) is a weak equivalence for every pair of cofibrant objects
X ∈ C, S ∈ S. The following are equivalent:(1) The adjunction (F, G) is a Quillen equivalence.
(2) The restriction of G determines a weak equivalence of S-enriched categories D◦ → C◦ (see §A.3.2).
Remark A.3.1.11. Strictly speaking, in §A.3.2, we define only weak equivalences between small S-enriched categories; however, the definition extends
to large categories in an obvious way.
Proof. Since G preserves fibrant objects and every object of C is cofibrant,
it is clear that G carries D◦ into C◦ . Condition (1) is equivalent to theassertion that for every pair of fibrant-cofibrant objects C ∈ C, D ∈ D,a map g : C → GD is a weak equivalence in C if and only if the adjoint
map f : F C → D is a weak equivalence in D. Choose a factorization of f 0
f0

f 00

as a composition F C → D0 → D, where f 0 is a trivial cofibration and f 00
is a fibration. By the two-out-of-three property, f is a weak equivalence if
and only if f 00 is a weak equivalence. We note that g admits an analogous
factorization as
g0

g 00

C → GD0 → GD.
Using (2), we deduce that f 00 is an equivalence in D◦ if and only if g 00 is anequivalence in C◦ . It will therefore suffice to show that g 0 is an equivalencein C◦ . For this, it will suffice to show that C and GD0 corepresent the samefunctor on the homotopy category hC. Invoking (2) again, it will suffice toshow that for every fibrant-cofibrant object D00 ∈ D, the induced map
HomhC (GD0 , GD00 ) → HomhC (C, GD00 ) ' HomhD (F C, D00 )is bijective. Using (2), we deduce that the map
HomhD (D0 , D00 ) → HomhD (GD0 , GD00 )is bijective. The desired result now follows from the fact that f 0 is a weakequivalence in D.
We now show that (1) ⇒ (2). The S-enriched functor G◦ : D◦ → C◦ isessentially surjective since the right derived functor RG is essentially surjective on homotopy categories. It suffices to show that G◦ is fully faithful: inhighertopoi

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APPENDIX

other words, that for every pair of fibrant-cofibrant objects X, Y ∈ D, theinduced map
i : MapD (X, Y ) → MapC (G(X), G(Y ))is a weak equivalence in S.
Since the left derived functor LF is essentially surjective, there exists anobject X 0 ∈ C and a weak equivalence F X 0 → X. We may regard X as afibrant replacement for F X 0 in D; it follows that the adjoint map X 0 → GX
may be identified with the adjunction X 0 → (RG ◦ LF )X 0 and is therefore
a weak equivalence by (1). Thus we have a diagram/ MapC (G(X), G(Y ))i

MapD (X, Y )MapD (F (X 0 ), Y )i0/ MapC (X 0 , G(Y ))in which the vertical arrows are homotopy equivalences; thus, to show thati is a weak equivalence, it suffices to show that i0 is a weak equivalence. For
this, it suffices to show that i0 induces a bijection from [S, MapD (F (X 0 ), Y )]to [S, MapC (X 0 , G(Y ))] for every cofibrant object S ∈ S; here [S, K] denotes
the set of homotopy classes of maps from S into K in the homotopy category
hS. But we may rewrite this map of sets as
i0S : MaphD (F (X 0 )⊗S, Y ) → MaphC (X 0 ⊗S, G(Y )) = MaphD (F (X 0 ⊗S), Y ),and it is given by composition with βX 0 ,S . (Here hC and hD denote thehomotopy categories of C and D as model categories; these are equivalent
to the corresponding homotopy categories of C◦ and D◦ as S-enriched categories). Since βX 0 ,S is an isomorphism in the homotopy category hD, themap i0S is bijective and (2) holds, as desired.
Corollary A.3.1.12. Let C oFG/ D be a Quillen equivalence between sim-

plicial model categories, where every object of C is cofibrant. Suppose that
G is a simplicial functor. Then G induces an equivalence of ∞-categories
N(D◦ ) → N(C◦ ).
A.3.2 The Model Structure on S-Enriched Categories
Throughout this section, we will fix a symmetric monoidal model category S
and study the category of S-enriched categories. The main case of interest to
us is that in which S is the category of simplicial sets (with its usual model
structure and the Cartesian monoidal structure). However, the treatment of
the general case requires little additional effort, and there are a number ofother examples which arise naturally in other contexts:(i) The category of simplicial sets equipped with the Cartesian monoidal
structure and the Joyal model structure defined in §2.2.5.

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APPENDIX(ii) The category of complexes· · · → Mn → Mn → Mn−1 → · · · ,of vector spaces over a field k with its usual model structure (inwhich weak equivalences are quasi-isomorphisms, fibrations are epimorphisms, and cofibrations are monomorphisms) and monoidal structure
given by the formation of tensor products of complexes.
Let S be an monoidal model category and let CatS denote the category of(small) S-enriched categories in which morphisms are given by S-enriched
functors. The goal of this section is to describe a model structure on CatS . We
first note that the monoidal structure on S induces a monoidal structure on
its homotopy category hS, which is determined up to (unique) isomorphismby the requirement that there exist a monoidal structure on the functor
S → hS
given by inverting all weak equivalences. Consequently, we note that any
S-enriched category C gives rise to an hS-enriched category hC having thesame objects as C and where mapping spaces are given by
MaphC (X, Y ) = [MapC (X, Y )].
Here we let [K] denote the image in hS of an object K ∈ S. We will refer
to hC as the homotopy category of C; the passage from C to hC is a special
case of Remark A.1.4.3.
Definition A.3.2.1. Let S be an monoidal model category. We say thata functor F : C → C0 in CatS is a weak equivalence if the induced functor
hC → hC0 is an equivalence of hS-enriched categories. In other words, F is aweak equivalence if and only if:(1) For every pair of objects X, Y ∈ C, the induced map
MapC (X, Y ) → MapC0 (F (X), F (Y ))is a weak equivalence in S.
(2) Every object Y ∈ C0 is equivalent to F (X) in the homotopy category
hC0 for some X ∈ C.
Remark A.3.2.2. If S is the category Set∆ (endowed with the Kan model
structure), then Definition A.3.2.1 reduces to the definition given in §1.1.3.
Remark A.3.2.3. Suppose that the collection of weak equivalences in S isstable under filtered colimits. Then it is easy to see that the collection of
weak equivalences in CatS is also stable under filtered colimits. If S is also a
combinatorial model category, then a bit more effort shows that the class of
weak equivalences in CatS is perfect in the sense of Definition A.2.6.12.

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We now introduce a bit of notation for working with S-enriched categories.
If A is an object of S, we will let [1]A denote the S-enriched category having
two objects X and Y , with
1S if Z = Z 0 = X



1
0
S if Z = Z = Y
Map[1]A (Z, Z 0 ) =A if Z = X, Z 0 = Y∅
if Z = Y, Z 0 = X.
Here ∅ denotes the initial object of S, and 1S denotes the unit object withrespect to the monoidal structure on S. We will denote [1]1S simply by [1]S .
We let [0]S denote the S-enriched category having only a single object X,
with Map∗ (X, X) = 1S .
We let C0 denote the collection of all morphisms in S of the following
types:(i) The inclusion ∅ ,→ [0]S .
(ii) The induced maps [1]S → [1]S 0 , where S → S 0 ranges over a set ofgenerators for the weakly saturated class of cofibrations in S.
Proposition A.3.2.4. Let S be a combinatorial monoidal model category.
Assume that every object of S is cofibrant and that the collection of weakequivalences in S is stable under filtered colimits. Then there exists a leftproper combinatorial model structure on CatS characterized by the following
conditions:(C) The class of cofibrations in CatS is the smallest weakly saturated classof morphisms containing the set of morphisms C0 appearing above.
(W ) The weak equivalences in CatS are defined as in §A.3.2.1.
Proof. It suffices to verify the hypotheses of Proposition A.2.6.15. Condition (1) follows from Remark A.3.2.3. For condition (3), we must show thatany functor F : C → C0 having the right lifting property with respect to allmorphisms in C0 is a weak equivalence. Since F has the right lifting property with respect to i : ∅ → [0]S , it is surjective on objects and therefore
essentially surjective. The assumption that F has the right lifting property
with respect to the remaining morphisms of C0 guarantees that for every
X, Y ∈ C, the induced map
MapC (X, Y ) → MapC0 (F (X), F (Y ))is a trivial fibration in S and therefore a weak equivalence.
It remains to verify condition (2): namely, that the class of weak equivalences is stable under pushout by the elements of C0 . We must show thatgiven any pair of functors F : C → D, G : C → C0 with F a weakequivalence and G a`pushout of some morphism in C0 , the induced map
F 0 : C0 → D0 = D C C0 is a weak equivalence. There are two cases toconsider.

highertopoi

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APPENDIXFirst, suppose that G is a pushout of the generating cofibration i : ∅ →∗. In other words, the category C0 is obtained from C by adjoining a new
object X, which admits no morphisms to or from the objects of C (and noendomorphisms other than the identity). The category D0 is obtained fromD by adjoining X in the same fashion. It is easy to see that if F is a weakequivalence, then F 0 is also a weak equivalence.
The other basic case to consider is one in which G is a pushout of one of
the generating cofibrations [1]S → [1]T , where S → T is a cofibration in S.
Let H : [1]S → C denote the “attaching map,” so that H is determined by apair of objects x = H(X) and y = H(Y ) and a map of h : S → MapC (x, y).
By definition, C0 is universal with respect to the property that it receives a
map from C, and the map h extends to a map eh : T → MapC0 (x, y). To carryout the proof, we will give an explicit construction of an S-enriched category
C0 which has this universal property.
For the remainder of the proof, we will assume that S is the category ofsimplicial sets. This is purely for notational convenience; the same arguments
can be employed without change in the general case.
We begin by declaring that the objects of C0 are the objects of C. The
definition of the morphisms in C0 is a bit more complicated. Let w and z be
objects of C. We define a sequence of simplicial sets MCk0 as follows:
MC0 = MapC (w, z)
MC1 = MapC (y, z) × T × MapC (w, x)
MC2 = MapC (y, z) × T × MapC (y, x) × T × MapC (w, x),and so forth. More specifically, for k ≥ 1, the m-simplices of MCk are finite
sequences(σ0 , τ1 , σ1 , τ2 , . . . , τk , σk ),where σ0 ∈ MapC (y, z)m , σk ∈ MapC (w, x)m , σi ∈ MapC (y, x)m for 0 < i <k, and τi ∈ Tm for 1 ≤ i ≤ k.
`
We define MapC0 (w, z) to be the quotient of the disjoint union k MCk by
the equivalence relation which is generated by making the identification(σ0 , τ1 , . . . , σk ) ' (σ0 , τ1 , . . . , τj−1 , σj−1 ◦ h(τj ) ◦ σj , τj+1 , . . . , σk )whenever the simplex τj belongs to Sm ⊆ Tm .
We equip C0 with an associative composition law, which is given on thelevel of simplices by(σ0 , τ1 , . . . , σk ) ◦ (σ00 , τ10 , . . . , σl0 ) = (σ0 , τ1 , . . . , τk , σk ◦ σ00 , τ10 , . . . , σl0 ).
It is easy to verify that this composition law is well-defined (that is, compatible with the equivalence relation introduced above) and associative andthat the identification MC0 = MapC (w, z) gives rise to an inclusion of categories C ⊆ C0 . Moreover, the map h : S → MapC (x, y) extends to eh:T →MapC0 (x, y) given by the composition
T ' {idy } × T × {idx } ⊆ MapC (y, y) × T × MapC (x, x) = MC1 → MapC0 (x, y).

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Moreover, it is not difficult to see that C0 has the desired universal property.
We observe that, by construction, the simplicial sets MapC0 (w, z) comeequipped with a natural filtration. Namely, define MapC0 (w, z)k to be theimage of
a
MCi
0≤i≤k

in MapC0 (w, z). Then we have
MapC (w, z) = MapC0 (w, z)0 ⊆ MapC0 (w, z)1 ⊆ · · ·
and

S

k
k MapC0 (w, z) = MapC0 (w, z). Moreover, the inclusion

MapC0 (w, z)k ⊆ MapC0 (w, z)k+1
is a pushout of the inclusion NCk+1 ⊆ MCk+1 , where N k+1 is the simplicial subset of MCk+1 whose m-simplices consist of those (2m + 1)-tuples(σ0 , τ1 , . . . , σm ) such that τi ∈ Sm for at least one value of i.
Let us now return to the problem at hand: namely, we wish to prove thatF 0 : C0 → D0 is an equivalence. We note that the construction outlined above
may also be employed to produce a model for D0 and an analogous filtration
on its morphism spaces.
Since G0 : D → D0 and F : C → D are essentially surjective, we deducethat F 0 is essentially surjective. Hence it will suffice to show that, for any
objects w, z ∈ C0 , the induced map
φ : MapC0 (w, z) → MapD0 (w, z)
is a weak homotopy equivalence. For this, it will suffice to show that foreach i ≥ 0, the induced map φi : MapC0 (w, z)i → MapD0 (w, z)i is a weakhomotopy equivalence; then φ, being a filtered colimit of weak homotopy
equivalences φi , will itself be a weak homotopy equivalence.
The proof now proceeds by induction on i. When i = 0, φi is a weakhomotopy equivalence by assumption (since F is an equivalence of simplicial
categories). For the inductive step, we note that φi+1 is obtained as a pushoutaai+1
MapC0 (w, z)i
MCi+1 → MapD0 (w, z)i
MD
.
i+1
NC

i+1
ND

Since S is left proper, both of these pushouts are homotopy pushouts. Consequently, to show that φi+1 is a weak equivalence, it suffices to show that
φi is a weak equivalence and that both of the maps
i+1
NCi+1 → ND
i+1
MCi+1 → MD

are weak equivalences. These statements follow easily from the compatibilityof the monoidal structure of S with the model structure and the assumptionthat every object of S is cofibrant.

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Remark A.3.2.5. It follows from the proof of Proposition A.3.2.4 that if
f : C → C0 is a cofibration of S-enriched categories, then the induced map
MapC (X, Y ) → MapC0 (f X, f Y ) is a cofibration for every pair of objects
X, Y ∈ C.
Remark A.3.2.6. The model structure of Proposition A.3.2.4 enjoys thefollowing functoriality: suppose that f : S → S0 is a monoidal left Quillen
functor between model categories satisfying the hypotheses of Proposition
A.3.2.4, with right adjoint g : S0 → S. Then f and g induce a Quillen
adjunction
CatS oFG/ Cat 0 ,S

where F and G are as in Remark A.1.4.3. Moreover, if (f, g) is a Quillen
equivalence, then (F, G) is likewise a Quillen equivalence.
In order for Proposition A.3.2.4 to be useful in practice, we need to understand the fibrations in CatS . For this, we first introduce a few definitions.
Definition A.3.2.7. Let F : C → D be a functor between ordinary categories. We will say that F is a quasi-fibration if, for every object X ∈ C
and every isomorphism f : F (X) → Y in D, there exists an isomorphism
f : X → Y in C such that F (f ) = f .
Remark A.3.2.8. The relevance of Definition A.3.2.7 is as follows: thecategory Cat admits a model structure in which the weak equivalences are
the equivalences of categories and the fibrations are the quasi-fibrations. Thisis a special case of Theorem A.3.2.24, which we will prove below (namely, thespecial case where we take S = Set endowed with the trivial model structureof Example A.2.1.2).
Definition A.3.2.9. Let S be a monoidal model category and let C be an
S-enriched category. We will say that a morphism f in C is an equivalence if
the homotopy class [f ] of f is an isomorphism in hC.
We will say that C is locally fibrant if, for every pair of objects X, Y ∈ C,
the mapping space MapC (X, Y ) is a fibrant object of S.
We will say that an S-enriched functor F : C → C0 is a local fibration if
the following conditions are satisfied:(i) For every pair of objects X, Y ∈ C, the induced map MapC (X, Y ) →MapC0 (F X, F Y ) is a fibration in S.
(ii) The induced map hC → hC0 is a quasi-fibration of categories.
Remark A.3.2.10. Let F : C → C0 be a functor between S-enrichedcategories which satisfies condition (i) of Definition A.3.2.9. Let X ∈ C
and Y ∈ C0 be objects. If C0 is locally fibrant, then every isomorphism[f ] : F (X) → Y in hC0 can be represented by an equivalence f : F (X) → Y

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in C0 . Let Y be an object of C such that F (Y ) = Y . Since 1S is a cofibrant object of S and the map MapC (X, Y ) → MapC0 (F (X), Y ) is a fibration, Proposition A.2.3.1 implies that [f ] can be lifted to an isomorphism[f ] : X → Y in hC if and only if f can be lifted to an equivalence f : X → Y
in C. Consequently, when C0 is locally fibrant, condition (ii) is equivalent to
the following analogous assertion:(ii0 ) For every equivalence f : F (X) → Y in C0 , there exists an equivalence
f : X → Y in C such that F (f ) = f .
Notation A.3.2.11. We let [1]∼S denote the S-enriched category containing
a pair of objects X, Y , with
(Z, Z 0 ) = 1S
Map[1]∼
S
for all Z, Z 0 ∈ {X, Y }.
Definition A.3.2.12 (Invertibility Hypothesis). Let S be a monoidal model
category satisfying the hypotheses of Proposition A.3.2.4. We will say that
S satisfies the invertibility hypothesis if the following condition is satisfied:(∗) Let i : [1]S → C be a cofibration of S-enriched categories, classifying amorphism f in C which is invertible in the homotopy category hC, andform a pushout diagram
[1]S

i

/C



[1]∼
S

j

/ Chf −1 i.

Then j is an equivalence of S-enriched categories.
In other words, the invertibility hypothesis is the assertion that invertinga morphism f in an S-enriched category C does not change the homotopy
type of C when f is already invertible up to homotopy.
Remark A.3.2.13. Let S, f , and C be as in Definition A.3.2.12 and choose atrivial cofibration F : C → C0 , where C0 is a fibrant S-enriched category. Since
CatS is left proper, the induced map Chf −1 i → C0 hF (f )−1 i is an equivalenceof S-enriched categories. Consequently, assertion (∗) holds for (C, f ) if andonly if it holds for (C0 , F (f )). In other words, to test whether S satisfies theinvertibility hypothesis, we need only check (∗) in the case where C is fibrant.
Remark A.3.2.14. In Definition A.3.2.12, the condition that i be a cofibration guarantees that the construction C 7→ Chf −1 i is homotopy invariant.
Alternatively, we can guarantee this by choosing a cofibrant replacement forthe map j : [1]S → [1]∼S . Namely, choose a factorization for j as a composition
j0

j 00

[1]S → E → [1]∼S,highertopoi

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APPENDIX

863

where j 00 is a weak equivalence and j 0 is a cofibration.`For every S-enrichedcategory containing a morphism f , define C[f −1 ] = C [1]S E. Then we havea canonical map C[f −1 ] → Chf −1 i, which is an equivalence whenever themap [1]S → C classifying f is a cofibration. Moreover, the construction C 7→
C[f −1 ] preserves weak equivalences in C. Consequently, we may reformulate
the invertibility hypothesis as follows:(∗0 ) For every S-enriched category C containing an equivalence f , the map
C → C[f −1 ] is a weak equivalence of S-enriched categories.
Remark A.3.2.15. Let C be a fibrant S-enriched category containing anequivalence f : X → Y and let C[f −1 ] be defined as in Remark A.3.2.14.
The canonical map C → C[f −1 ] is a trivial cofibration and therefore admits
a section. This section determines a map of S-enriched categories h : E → C.
We observe that E is a mapping cylinder for the object [0]CatS ∈ CatS , so
we can view h as a homotopy between the maps [0]CatS → C classifying theobjects X and Y .
More generally, the same argument shows that if F : C → D is a fibrationof S-enriched categories and f : X → Y is an equivalence in C such that
F (f ) = idD for some object D ∈ D, then the functors [0]S → C classifying
the objects X and Y are homotopic in the model category (CatS )/ D .
Definition A.3.2.16. We will say that a model category S is excellent if it
is equipped with a symmetric monoidal structure and satisfies the following
conditions:(A1) The model category S is combinatorial.
(A2) Every monomorphism in S is a cofibration, and the collection of cofibrations is stable under products.
(A3) The collection of weak equivalences in S is stable under filtered colimits.
(A4) The monoidal structure on S is compatible with the model structure.
In other words, the tensor product functor ⊗ : S × S → S is a left
Quillen bifunctor.
(A5) The monoidal model category S satisfies the invertibility hypothesis.
Remark A.3.2.17. Axiom (A2) of Definition A.3.2.16 implies that every
object of S is cofibrant. In particular, S is left proper.
Example A.3.2.18 (Dwyer, Kan). The category of simplicial sets is an
excellent model category when endowed with the Kan model structure andthe Cartesian product. The only nontrivial point is to show that Set∆ satisfies
the invertibility hypothesis. This is one of the main theorems of [21].
Example A.3.2.19. Let S be a presentable category equipped with a closed
symmetric monoidal structure. Then S is an excellent model category withrespect to the trivial model structure of Example A.2.1.2.

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APPENDIXThe following lemma guarantees a good supply of examples of excellent
model categories:
Lemma A.3.2.20. Suppose we are given a monoidal left Quillen functor
T : S → S0 between model categories S and S0 satisfying axioms (A1) through(A4) of Definition A.3.2.16. If S satisfies axiom (A5), then so does S0 .
Proof. As indicated in Remark A.3.2.6, the functor T determines a Quillen
adjunction
F

CatS oG/ Cat 0 .
S

Let C be a S0 -enriched category and i : [1]S0 → C a cofibration classifying
an equivalence f in C. We wish to prove that the map C → Chf −1 i is anequivalence of S0 -enriched categories. In view of Remark A.3.2.13, we may
assume that C is fibrant.
Choose a factorization of the map [1]S → [1]∼S as a compositionjj0[1]S → E → [1]∼S
as in Remark A.3.2.14, so that we have an analogous factorization[1]S0 → F (E) → [1]∼S0in CatS0 . Using the latter factorization, we can define C[f −1 ] as in Remark
A.3.2.14; we wish to show that the map h : C → C[f −1 ] is a trivial cofibration.
Let f0 be the morphism in G(C) classified by f , and let G(C)[f0−1 ] ∈ CatS
be defined as in Remark A.3.2.14. Using the fact that C is locally fibrant(see Theorem A.3.2.24 below), we conclude that f0 is an equivalence in G(C).
Since S satisfies the invertibility hypothesis, the map h0 : G(C) → G(C)[f0−1 ]is a trivial cofibration. We now conclude by observing that h is a pushout of
F (h0 ).
Remark A.3.2.21. Using a similar argument, we can prove a converse toLemma A.3.2.20 in the case where T is a Quillen equivalence.
Example A.3.2.22. Let S be the category Set+
∆ of marked simplicial setsendowed with the Cartesian model structure defined in §3.1. Then the functor X 7→ X ] is a monoidal left Quillen functor Set∆ → S. Combining Example A.3.2.18 with Lemma A.3.2.20, we conclude that S satisfies the invertibility hypothesis, so that S is an excellent model category (with respect to
the Cartesian product).
Example A.3.2.23. Let S denote the category of simplicial sets, endowedwith the Joyal model structure. The functor X 7→ X [ determines a monoidal
left Quillen equivalence S → Set+
∆ . Using Remark A.3.2.21, we deduce that S
satisfies the invertibility hypothesis, so that S is an excellent model category(with respect to the Cartesian product).
We are now ready to state our main result:highertopoi

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APPENDIX

Theorem A.3.2.24. Let S be an excellent model category. Then
(1) An S-enriched category C is a fibrant object of CatS if and only if it islocally fibrant: that is, if and only if the mapping object MapC (X, Y ) ∈S is fibrant for every pair of objects X, Y ∈ C.
(2) Let F : C → D be an S-enriched functor, where D is a fibrant object of
CatS . Then F is a fibration in CatS if and only if F is a local fibration.
Remark A.3.2.25. In the case where S is the category of simplicial sets(with its usual model structure), Theorem A.3.2.24 is due to Bergner; see [7].
Moreover, Bergner proves a stronger result in this case: assertion (2) holds
without the assumption that D is fibrant.
Before giving the proof of Theorem A.3.2.24, we need to establish somepreliminaries. Fix an excellent model category S. We observe that CatS is
naturally cotensored over S. That is, for every S-enriched category C andevery object K ∈ S, we can define a new S-enriched category CK as follows:(i) The objects of CK are the objects of C.
(ii) Given a pair of objects X, Y ∈ C, we have
MapCK (X, Y ) = MapC (X, Y )K ∈ S.
This construction does not endow CatS with the structure of an S-enrichedcategory because the construction D 7→ DK is not compatible with colimits
in K. However, we can remedy the situation as follows. Let C and D be
S-enriched categories and let φ be a function from the set of objects of C
to the set of objects of D. Then there exists an object MapφCatS (C, D) ∈ S
which is characterized by the following universal property: for every K ∈ S,
there is a natural bijection
HomS (K, MapφCatS (C, D)) ' HomφCatS (C, DK ),where HomφCatS (C, DK ) denotes the set of all functors from C to DK which
is given on objects by the function φ.
Lemma A.3.2.26. Let S be an excellent model category. Fix a diagram of
S-enriched categories
u
/ C0
C
F0

F


D

0

u


/ D0 .

Assume that(a) For every pair of objects X, Y ∈ C, the diagram/ MapD (F X, F Y )MapC (X, Y )
MapC0 (uX, uY )/ MapD0 (u0 F X, u0 F Y )highertopoi

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APPENDIX

is a homotopy pullback square involving fibrant objects of S and thevertical arrows are fibrations.
Let G : A → B be a functor between S-enriched categories which is atransfinite composition of pushouts of generating cofibrations of the form
[1]S → [1]S 0 , where S → S 0 is a cofibration in S and let φ be a function fromthe set of objects of B (which is isomorphic to the set of objects of A) to C.
Then the diagram
MapφCatS (B, C)/ MapF φ (B, D) ×


0
Mapuφ
CatS (B, C )/ Mapu F φ (B, D0 ) ×

CatS

MapφCatS (A, C)φ
MapF
Cat (A,D)
S

0

CatS


0F φ
0
Mapu
CatS (A,D )0
Mapuφ
CatS (A, C )is a homotopy pullback square between fibrant objects of S, and the verticalarrows are fibrations.
Proof. It is easy to see that the collection of morphisms G : A → B which
satisfy the conclusion of the lemma is weakly saturated. It will therefore
suffice to show that G contains every morphism of the form [1]S → [1]S 0 ,where S → S 0 is a cofibration in S. In this case, φ determines a pair ofobjects X, Y ∈ C, and we can rewrite the diagram of interest as
MapC (X, Y )S

0

x
&
0
0
MapC0 (uX, uY )S MapC (X, Y )S ×MapD (F X,F Y )S MapD (F X, F Y )S&x0
MapC0 (uX, uY )S ×MapD0 (u0 F X,u0 F Y )S MapD0 (u0 F X, u0 F Y )S .
The desired result now follows from (a) since the map S → S 0 is a cofibration
between cofibrant objects of S.
Proof of Theorem A.3.2.24. Assertion (1) is just a special case of (2) where
we take D to be the final object of CatS . It will therefore suffice to prove (2).
We first prove the “only if” direction. If F is a fibration, then F has
the right lifting property with respect to every trivial cofibration of theform [1]S → [1]S 0 , where S → S 0 is a trivial cofibration in S. It follows
that for every pair of objects X, Y ∈ C, the induced map MapC (X, Y ) →MapD (F X, F Y ) is a fibration in S. In particular, C is locally fibrant.
To complete the proof that F is a local fibration, we will show that F
satisfies condition (ii0 ) of Remark A.3.2.10. Suppose X ∈ C and that f :F X → Y is an equivalence in D. We wish to show that we can lift f to an
equivalence f : X → Y . Let E and D[f −1 ] be defined as in Remark A.3.2.14.

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Since S satisfies the invertibility hypothesis, the map h : D → D[f −1 ] isa trivial cofibration. Because we have assumed D to be fibrant, the map hadmits a section. This section determines a map s : E → D. We now consider
the lifting problem
/
=C

X

[0]S

F


E


/ D.

s

Since F is a fibration and the left vertical map is a trivial cofibration, there
exists a solution as indicated. This solution determines a morphism f : X →Y in C lifting f . Moreover, f is the image of a morphism in E. Since every
morphism in E is an equivalence, we deduce that f is an equivalence in C.
Let us now suppose that F is a local fibration. We wish to show that F isa fibration. Choose a factorization of F as a composition
F0u

C → C0 → D,
where u is a weak equivalence and F 0 is a fibration. We will prove the following:(∗) Suppose we are given a commutative diagram of S-enriched categories
A

v

/C

v0


/ D,

G


B

F

where G is a cofibration. If there exists a functor α : B → C0 such thatαG = uv and F 0 α = v 0 , then there exists a functor β : B → C suchthat βG = v and F β = v 0 .
Since the map F 0 has the right lifting property with respect to all trivial
cofibrations, assertion (∗) implies that F also has the right lifting property
with respect to all trivial cofibrations, so that F is a fibration as desired.
We now prove (∗). Using the small object argument, we deduce that thefunctor G is a retract of some functor G0 : A → B0 , where G0 is a transfinite
composition of morphisms obtained as pushouts of generating cofibrations.
It will therefore suffice to prove (∗) after replacing G by G0 .
Reordering the transfinite composition if necessary, we may assume that
G0 factors as a composition
G0

G0

A →0 B00 →1 B0 ,where B00 is obtained from A by adjoining a collection of new objects,{Bi }i∈I , and B0 is obtained from B00 by a transfinite sequence of pushouts
by generating cofibrations of the form [1]S → [1]S 0 , where S → S 0 is a cofibration in S. Let Ci0 = α(Bi ) for each i ∈ I. Since u is an equivalence ofhighertopoi

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APPENDIX

S-enriched categories, there exists a collection of objects {Ci }i∈I and equivalences fi : uCi → Ci0 . Let gi be the image of fi in D. Since F is a local
fibration, we can lift each gi to an equivalence fi0 : Ci → Ci00 in C. Since themaps MapC0 (uCi00 , Ci0 ) → MapD (F Ci00 , F 0 Ci0 ) are fibrations, we can choose
morphisms fi00 : uCi00 → Ci0 in C0 such that F 0 (fi0 ) is the identity for each i,and the diagrams
fi0

=

uCi00
fi00

!
/ C0

fi

uCi

i

commute up to homotopy. Replacing Ci by Ci00 , we may assume that each of
the maps fi projects to the identity in D.
Let α0 = α| B00 and let α00 : B00 → C0 be defined by the formula(
α0 (A) if A ∈ A
0α0 (A) =uCi
if A = Bi , i ∈ I.

Remark A.3.2.15 implies that the maps α0 and α00 are homotopic in themodel category (CatS )A / / D . Applying Proposition A.2.3.1, we deduce theexistence of a map α0 : B0 → C which extends α0 and satisfies α0 G = uv
and F 0 α0 = v 0 . We may therefore replace α by α0 , v by α00 , and A by
B00 and thereby reduce to the case where the functor G : A → B is atransfinite composition of generating cofibrations of the form [1]S → [1]S 0 ,where S → S 0 is a cofibration in S.
Let φ be the map from the objects of B to the objects of C determined by
α. Applying Lemma A.3.2.26, we obtain a homotopy pullback diagram
MapφCatS (B, C)/ MapF φ (B, D) ×


0
Mapuφ
CatS (B, C )/ MapF φ (B, D) ×

CatS

CatS

MapφCatS (A, C)φ
MapF
Cat (A,D)
S



φ
MapF
CatS (A,D)

0
Mapuφ
CatS (A, C ).

in which the horizontal arrows are fibrations. We therefore have a weakequivalenceφ
0
MapφCatS (B, C) → M = Mapuφ
CatS (B, C ) ×MapF φ (B,D) MapCatS (A, C)
CatS

φ
0
of fibrations over N = MapF
Mapuφφ
CatS (B, D) ×MapF
CatS (A, C ). MoreCatS (A,D)
over, the pair (α, v) determines a map 1S → M lifting the map (v 0 , uv 0 ) :1S → N . Applying Proposition A.2.3.1, we deduce that (v, uv 0 ) : 1S → N
can be lifted to a map 1S → MapφCatS (B, C), which is equivalent to theexistence of the desired map β.

We conclude this section with a few easy results concerning homotopy
limits in the model category CatS .

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Proposition A.3.2.27. Let S be an excellent model category, J a small
category, and {CJ }J∈J a diagram of S-enriched categories. Suppose we are
given a compatible family of functors {fJ : C → CJ }J∈J which exhibits C asa homotopy limit of the diagram {CJ }J∈J in CatS . Then for every pair ofobjects X, Y ∈ C, the maps {MapC (X, Y ) → MapCJ (fJ X, fJ Y )}J∈J exhibit
MapC (X, Y ) as a homotopy limit of the diagram {MapCJ (fJ X, fJ Y )}J∈J in
S.
Proof. Without loss of generality, we may assume that the diagram {CJ }J∈J
is injectively fibrant and that the maps fJ exhibit C as a limit of {CJ }J∈J . It
follows that MapC (X, Y ) is a limit of the diagram {MapCJ (fJ X, fJ Y )}J∈J . It
will therefore suffice to show that the diagram {MapCJ (fJ X, fJ Y )}J∈J is injectively fibrant. For this, it will suffice to show that {MapCJ (fJ X, fJ Y )}J∈J
has the right lifting property with respect to every injective trivial cofibration α : F → F 0 of diagrams F, F 0 : J → S. Let G : J → CatS be defined by
the formula G(J) = [1]F (J) and let G0 : J → CatS be defined likewise. The
desired result now follows from the observation that α induces an injective
trivial cofibration G → G0 in Fun(J, CatS ).
Corollary A.3.2.28. Let S be an excellent model category, J a small category, and {CJ }J∈J a diagram of S-enriched categories. Suppose we are given
S-enriched functors
βα

D → C → lim{CJ }J∈J
such that α◦β exhibits D as a homotopy limit of the diagram {CJ }J∈J . Thenthe following conditions are equivalent:(1) The functor α exhibits C as a homotopy limit of the diagram {CJ }J∈J .
(2) For every pair of objects X, Y ∈ C, the functor α exhibits MapC (X, Y )as a homotopy limit of the diagram {MapCJ (αJ X, αJ Y )}J∈J .
Proof. The implication (1) ⇒ (2) follows from Proposition A.3.2.27. To
prove the converse, we may assume that the diagram {CJ }J∈J is injectively
fibrant. In view of (2), Proposition A.3.2.27 implies that α induces a fully
faithful functor between hS-enriched homotopy categories. It will therefore
suffice to show that α is essentially surjective on homotopy categories, which
follows from our assumption that α ◦ β is a weak equivalence.
A.3.3 Model Structures on Diagram Categories
In this section, we consider enriched analogues of the constructions presented
in §A.2.8. Namely, suppose that S is an excellent model category, A a combinatorial S-enriched model category, and C a small S-enriched category. Let
AC denote the category of S-enriched functors from C to A. In this section,
we will study the associated projective and injective model structures on AC .
The ideas described here will be used in §A.3.4 to construct certain mapping
objects in CatS .
We begin with the analogue of Definition A.2.8.1.

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Definition A.3.3.1. Let C be a small S-category and A a combinatorial
S-enriched model category. A natural transformation α : F → G in AC is:• an injective cofibration if the induced map F (C) → G(C) is a cofibration in A for each C ∈ C.
• a projective fibration if the induced map F (C) → G(C) is a fibrationin A for each C ∈ C.
• a weak equivalence if the induced map F (C) → G(C) is a weak equivalence in A for each C ∈ C.
• an injective fibration if it has the right lifting property with respect toevery morphism β in AC which is simultaneously a weak equivalence
and an injective cofibration.
• a projective cofibration if it has the left lifting property with respect toevery morphism β in AC which is simultaneously a weak equivalence
and a projective fibration.
Proposition A.3.3.2. Let S be an excellent model category, let A be a
combinatorial S-enriched model category, and let C be a small S-enrichedcategory. Then there exist two combinatorial model structures on AC :• The projective model structure determined by the projective cofibrations, weak equivalences, and projective fibrations.
• The injective model structure determined by the injective cofibrations,weak equivalences, and injective fibrations.
The proof of Proposition A.3.3.2 is identical to that of Proposition A.2.8.2,
except that it requires the following more general form of Lemma A.2.8.3:
Lemma A.3.3.3. Let A be a presentable category which is enriched, tensored, and cotensored over a presentable category S. Let S0 be a (small) set
of morphisms of A and let S 0 be the weakly saturated class of morphisms
generated by S0 . Let C be a small S-enriched category. Let Se be the collection of all morphisms F → G in AC with the following property: for every
C ∈ C, the map F (C) → G(C) belongs to S 0 . Then there exists a (small)set of morphisms S of AC which generates Se (as a weakly saturated class ofmorphisms).
We will defer the proof until the end of this section.
Remark A.3.3.4. In the situation of Proposition A.3.3.2, the category AC
is again enriched, tensored, and cotensored over S. The tensor product withan object K ∈ S is computed pointwise; in other words, if F ∈ AC , then we
have the formula(K ⊗ F)(A) = K ⊗ F(A).

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Using criterion (20 ) of Remark A.3.1.6, we deduce that AC is an S-enriched
model category with respect to the injective model structure. A dual argument (using criterion (200 ) of Remark A.3.1.6) shows that AC is also an
S-enriched model category with respect to the projective model structure.
C
Remark A.3.3.5. For each object C ∈ C and each A ∈ A, let FC
A ∈ A bethe functor given by

D 7→ A ⊗ MapC (C, D).
As in the proof of Proposition A.2.8.2, we learn that the class of projectiveC
cofibrations in AC is generated by cofibrations of the form j : FC
A → F A0 ,where A → A0 is a cofibration in A. It follows that every projective cofibration is an injective cofibration; dually, every injective fibration is a projective
fibration.
As in §A.2.8, the construction (C, A) 7→ AC is functorial in both C and
A. We summarize the situation in the following propositions, whose proofs
are left to the reader:
Proposition A.3.3.6. Let S be an excellent model category, C a small SF /B an S-enriched Quillen adjunction between
enriched category, and A oGcombinatorial S-enriched model categories. The composition with F and G
determines another S-enriched Quillen adjunction
AC o

FC
G

C

/ BC

with respect to either the projective or the injective model structure. Moreover, if (F, G) is a Quillen equivalence, then (F C , GC ) is also a Quillen
equivalence.
Because the projective and injective model structures on AC have thesame weak equivalences, the identity functor idAC is a Quillen equivalence
between them. However, it is important to distinguish between these twomodel structures because they have different variance properties as we now
explain.
Let f : C → C0 be an S-enriched functor. Then composition with f yields
0a pullback functor f ∗ : AC → AC . Since A has all S-enriched limits andcolimits, f ∗ has a right adjoint, which we will denote by f∗ , and a left adjoint,
which we will denote by f! .
Proposition A.3.3.7. Let S be an excellent model category, A a combinatorial S-enriched model category, and f : C → C0 an S-enriched functor0
between small S-enriched categories. Let f ∗ : AC → AC be given by composition with f . Then f ∗ admits a right adjoint f∗ and a left adjoint f! .
Moreover:(1) The pair (f! , f ∗ ) determines a Quillen adjunction between the projec0
tive model structures on AC and AC .

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APPENDIX(2) The pair (f ∗ , f∗ ) determines a Quillen adjunction between the injective0model structures on AC and AC .
We now study some aspects of the theory which are unique to the enriched
context.
Proposition A.3.3.8. Let S be an excellent model category, A a combinatorial S-enriched model category, and f : C → C0 an equivalence of small
S-enriched categories. Then
(1) The Quillen adjunction (f! , f ∗ ) determines a Quillen equivalence be0
tween the projective model structures on AC and AC .
(2) The Quillen adjunction (f ∗ , f∗ ) determines a Quillen equivalence be0
tween the injective model structures on AC and AC .
Proof. We first note that (1) and (2) are equivalent: they are both equivalent
to the assertion that f ∗ induces an equivalence on homotopy categories.
It therefore suffices to prove (1). We first prove this under the following
additional assumption:(∗) For every pair of objects C, D ∈ C0 , the map
MapC0 (C, D) → MapC (f (C), f (D))
is a cofibration in S.
0

Let Lf! : AC → AC denote the left derived functor of f! . We must showthat the unit and counit maps
hF : F 7→ f ∗ Lf! F
kG : Lf! f ∗ G → G
are isomorphisms for all F ∈ hAC , G ∈ hAC . Since f is essentially surjective
on homotopy categories, a natural transformation K → K 0 of S-enriched
functors K, K 0 : C0 → A is a weak equivalence if and only if f ∗ K → f ∗ K 0 isa weak equivalence. Consequently, to prove kG is an isomorphism, it suffices
to show that hf ∗ G is an isomorphism.
Let us say that a map F → F 0 in AC is good if the induced mapaf ∗ f! F
F 0 → f ∗ f! F 0
F

is an injective trivial cofibration. To complete the proof, it will suffice toshow that every projective cofibration is good. Since the collection of goodtransformations is weakly saturated, it will suffice to show that each of theC00
generating cofibrations FC
A → F A0 is good, where C ∈ C and j : A → A isa cofibration in A. Unwinding the definitions, we must show that for each
D ∈ C0 the induced map`(A0 ⊗ MapC0 (C, D)) A⊗MapC0 (C,D) (A ⊗ MapC (f (C), f (D)))θ


A0 ⊗ MapC (f (C), f (D))

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APPENDIX

is a trivial cofibration. This follows from the fact that j is a cofibration andour assumption (∗).
We now treat the general case. First, choose a trivial cofibration g : C →C00 , where C00 is fibrant. Then g satisfies (∗), so g! is a Quillen equivalence. By
a two-out-of-three argument, we see that f! is a Quillen equivalence if andonly if (g ◦ f )! is a Quillen equivalence. Replacing C by C00 , we may reduceto the case where C is itself `fibrant.
Choose a cofibration j : C C0 → D, where D is fibrant and equivalent to
the final object of CatS . Then f factors as a composition
f0

f 00

C0 → C × D → C .
Since C and D are fibrant, the product C × D is equivalent to C. Moreover,the map f 00 admits a section s : C → C × D. Using another two-out-of-threeargument, it will suffice to show that f!0 and s! are Quillen equivalences. For
this, it will suffice to show that f 0 and s satisfy (∗).
We first show that f 0 satisfies (∗). Fix a pair of objects X, Y ∈ C0 . Then0f induces the composite map
u

MapC0 (X, Y ) → MapC (f X, f Y ) × MapC0 (X, Y )u0→ MapC (f X, f Y ) × MapD (jX, jY )' MapC × D (f 0 X, f 0 Y ).
The map u is a monomorphism (since it admits a left inverse) and therefore
a cofibration in view of axiom (A2) of Definition A.3.2.16. The map u0 is
a product of cofibrations and therefore also a cofibration (again by axiom(A2)).
The proof that s satisfies (∗) is similar: for every pair of objects U, V ∈ C,
the map
MapC (U, V ) → MapC × D (sU, sV ) ' MapC (U, V ) × MapD (jU, jV )is a monomorphism since it admits a left inverse and is therefore a cofibration.
In the special case where f : C → C0 is a cofibration between S-enrichedcategories, we have some additional functoriality:
Proposition A.3.3.9. Let S be an excellent model category and let f : C →C0 be a cofibration of small S-enriched categories. Then
(1) For every combinatorial S-enriched model category A, the pullback map
0f ∗ : AC → AC preserves projective cofibrations.
(2) For every projectively cofibrant object F ∈ SC , the unit map F →f ∗ f! F is a projective cofibration.

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APPENDIX

Lemma A.3.3.10. Let S be an excellent model category and suppose we are
given a pushout diagram/ [1]S 0

[1]S
i


C


/ C0

f

of S-enriched categories, where j : S → S 0 is a cofibration in S. Let C be
an object of C and let F ∈ SC be the functor given by the formula D 7→
MapC (C, D). Then the unit map F → f ∗ f! F is a projective cofibration in
SC .
Proof. The map i determines a pair of objects X, Y ∈ C and a map S →MapC (X, Y ). The proof of Proposition A.3.2.4 shows that the functor f ∗ f! F
is the colimit of a sequencehh

F = F (0) →1 F (1) →2 F (2) → · · · ,where each hk is a pushout of a map FYA → FYA0 induced by a map t : A → A0
in S. Moreover, the map t can be identified with the tensor product⊗k−1
kidMapC (C,X) ⊗ idMap(Y,X) ⊗ ∧ (j),Cwhere ∧k (j) denotes the kth pushout power of j. It follows that t is a cofibration in S, so that each hk is a projective cofibration in SC .
Proof of Proposition A.3.3.9. The collection of S-enriched functors f which
satisfy (1) and (2) is clearly closed under the formation of retracts. We may
therefore assume without loss of generality that f is a transfinite composition of pushouts of generating cofibrations (see the discussion preceding
Proposition A.3.2.4). Reordering these pushouts if necessary, we can factor
f as a composition
f0

f 00

C → C → C0 ,where C is obtained from C by freely adjoining a collection of new objects
and f 00 is bijective on objects. Since f 0 clearly satisfies (1) and (2), it will
suffice to prove that f 00 satisfies (1) and (2). Replacing f by f 00 , we may
assume that f is bijective on objects.
We now show that (2) ⇒ (1). Since the functor f ∗ preserves colimits, the0collection of morphisms α in AC such that f ∗ is a projective cofibration inC
A is weakly saturated. It will therefore suffice to show that for every object
X
X ∈ C0 and every cofibration A → A0 in A, if α : FX
A → F A0 denotes the∗corresponding generating projective cofibration, then f (α) is a projective
cofibration in S.
There is a canonical left Quillen bifunctor
 : SC × A → AC
described by the formula (F  A)(C) = F (C) ⊗ A. (Here we regard SC
as endowed with the projective model structure.) We observe that f ∗ (α)

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APPENDIX0is the induced map (f ∗ F )  A → (f ∗ F )  A0 , where F ∈ SC is given by
F (C 0 ) = MapC0 (X, C 0 ). To prove (1), it will suffice to show that f ∗ F is
projectively cofibrant.
Since F is bijective on objects, we can choose an object X0 ∈ C suchthat f X0 = X. We now observe that F ' f! F0 , where F0 ∈ SC is definedby the formula F0 (C) = MapC (X0 , C). If (2) is satisfied, then the unit map
F0 → f ∗ F is a projective cofibration in SC . Since F0 is projectively cofibrant,
we conclude that f ∗ F is projectively cofibrant as well. This completes theproof that (2) ⇒ (1).
To prove (2), let us write f as a transfinite composition of S-enriched
functors
C = C0 → C1 → · · · ,each of which is a pushout of a generating cofibration of the form [1]S → [1]S 0 ,where S → S 0 is a cofibration in S. For each α ≤ β, let fαβ : Cα → Cβ be thecorresponding cofibration. We will prove that the following statement holds
for every pair of ordinals α ≤ β:
(2α,β ) For every projectively cofibrant object F ∈ SCα , the unit map u : F →(fαβ )∗ (fαβ )! F is a projective cofibration.
The proof proceeds by induction on β. We observe that u is a transfinite
composition of maps of the formuγ : (fαγ )∗ (fαγ )! F → (fαγ )∗ (fγγ+1 )∗ (fγγ+1 )! (fαγ )! F,
where γ < β. It will therefore suffice to show that each uγ is a projective
cofibration. Our inductive hypothesis therefore guarantees that (2α,γ ) holds,
so the first part of the proof shows that (fαγ )∗ preserves trivial cofibrations.
We are therefore reduced to proving assertion (2γ,γ+1 ). In other words, toprove (2) in general, it will suffice to treat the case in which f is a pushoutof a generating cofibration of the form [1]S → [1]S 0 .
We will in fact prove the following stronger version of (2):(3) For every
cofibration φ : F 0 → F in SC , the induced map` projective0∗0
φ : F F 0 f f! F → f ∗ f! F is again a projective cofibration in SC .
Consider the collection
φ : F 0 → F in SC such that the` of ∗all morphisms
00∗induced map φ : F F 0 f f! F → f f! F is a projective cofibration. It iseasy to see that this collection is weakly saturated. Consequently, to prove(3) it suffices to treat the case where φ is a generating projective cofibrationC
0of the form FC
A → F A0 , where A → A is a cofibration in S. In this case, we
0
can identify φ with the mapa(FC  A0 )(f ∗ f! FC  A) → f ∗ f! FC  A0 ,FC A

where FC ∈ SC is the functor D 7→ MapC (C, D). Since  is a left Quillen bifunctor, it will suffice to show that the unit map fC → f ∗ f! FC is a projective
cofibration in SC . This is precisely the content of Lemma A.3.3.10.

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APPENDIX

In §A.2.8, we introduced the definitions of homotopy limits and colimits inan arbitrary combinatorial model category A. We now discuss an analogous
construction in the case where A is enriched over an excellent model category
S. To simplify the exposition, we will discuss only the case of homotopy
limits; the case of homotopy colimits is entirely dual and is left to the reader.
Fix an excellent model category S and a combinatorial S-enriched model
category A. Let f : C → C0 be a functor between small S-enriched categories,
so that we have an induced Quillen adjunction
0

AC o

f∗

/ AC .

f∗

We will refer to the right derived functor Rf∗ as the homotopy right Kan
0extension functor. Suppose we are given a pair of functors F ∈ AC , G ∈ AC0
and a morphism η : G → f∗ F in AC . We will say that η exhibits G as thehomotopy right Kan extension of F if, for some weak equivalence F → F 0
where F 0 is injectively fibrant in AC , the composite map G → f∗ F → f∗ F 0
0
is a weak equivalence in AC . Since f∗ preserves weak equivalences between
injectively fibrant objects, this condition is independent of the choice of F 0 .
Remark A.3.3.11. In §A.2.8, we defined homotopy right Kan extensions
in the setting of the diagram categories Fun(C, A), where C is an ordinary
category. In fact, this is a special case of the above construction. Namely,
there is a`unique colimit-preserving monoidal functor F : Set → S given by
F (S) = s∈S 1S . We can therefore define an S-enriched category C whose
objects are the objects of C, with MapC (X, Y ) = F MapC (X, Y ). We now
observe that we have an identification Fun(C, A) ' AC which is functorial
in both C and A. This identification is compatible with the definition of theinjective model structures on both sides, so that either point of view gives
rise to the same theory of homotopy right Kan extensions.
We now discuss a special feature of the enriched theory of homotopy Kan
extensions: they can be reduced to the theory of homotopy Kan extensions
in the model category S:
Proposition A.3.3.12. Let S be an excellent model category, let A be a
combinatorial model category enriched over S, and let f : C → C0 be afunctor between small S-enriched categories. Suppose given objects F ∈ AC ,0
G ∈ AC and a map η : G → f∗ F . Assume that F and G are projectively
fibrant. The following conditions are equivalent:(1) The map η exhibits G as a homotopy right Kan extension of F .
(2) For each cofibrant object A ∈ A, the induced map
ηA : GA → f∗ FA
exhibits GA as a homotopy right Kan extension of FA . Here FA ∈ SC
0
and GA ∈ SC are defined by FA (C) = MapA (A, F (C)), GA (C) =MapA (A, G(C)).

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APPENDIX(3) For every fibrant-cofibrant object A ∈ A, the induced map
ηA : GA → f∗ FA
exhibits GA as a homotopy right Kan extension of FA .
Proof. Choose an equivalence F → F 0 , where F 0 is injectively fibrant. We
note that the induced maps FA → FA0 are weak equivalences for any cofibrant A ∈ A since MapA (A, •) preserves weak equivalences between fibrantobjects. Consequently, we may without loss of generality replace F by F 0
and thereby assume that F is injectively fibrant.
Now suppose that A is any cofibrant object of A; we claim that FA is
injectively fibrant. To show that FA has the right lifting property with respect to a trivial weak cofibration H → H 0 of functors C → S, one need only
observe that F has the right lifting property with respect to trivial injective
cofibration A ⊗ H → A ⊗ H 0 in AC .
Now we note that (1) is equivalent to the assertion that η is a weakequivalence, (2) is equivalent to the assertion that ηA is a weak equivalence
for any cofibrant object A, and (3) is equivalent to the assertion that ηA
is a weak equivalence whenever A is fibrant-cofibrant. Because MapA (A, •)preserves weak equivalences between fibrant objects, we deduce that (1) ⇒(2). It is clear that (2) ⇒ (3). We will complete the proof by showing that(3) ⇒ (1). Assume that (3) holds; we must show that η(C 0 ) : G(C 0 ) →f∗ F (C 0 ) is an isomorphism in the homotopy category hA for each C 0 ∈ C0 .
For this, it suffices to show that G(C 0 ) and f∗ F (C 0 ) represent the same Hvalued functors on the homotopy category hA, which is precisely the contentof (3).
Remark A.3.3.13. It follows from Proposition A.3.3.12 that we can makesense of homotopy right Kan extensions for diagrams which do not takevalues in a model category. Let f : C → C0 be an S-enriched functor as in
the discussion above and let A be an arbitrary locally fibrant S-enriched0
category. Suppose we are given objects F ∈ AC , G ∈ AC and η : f ∗ G → F ;we say that η exhibits G as a homotopy right Kan extension of F if, for eachobject A ∈ A, the induced map
ηA : GA → f∗ FA
0exhibits GA ∈ SC as a homotopy right Kan extension of FA ∈ SC .
Suppose that the monoidal structure on S is given by the Cartesian prod0
uct and take C0 to be the final object of CatS , so that we can identify ACwith A. In this case, we can identify G with a single object B ∈ A andthe map η with a collection of maps {B → F (C)}C∈C . We will say that η
exhibits B as a homotopy limit of F if it identifies G with a homotopy rightKan extension of F . In other words, η exhibits B as a homotopy limit of Fif, for every object A ∈ A, the induced map
MapA (A, B) → lim{MapA (A, F (C))}C∈C

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exhibits MapA (A, B) as a homotopy limit of the diagram{MapA (A, F (C))}C∈C
in the model category S.
We also have a dual notion of homotopy colimit in an arbitrary fibrant Senriched category A: a compatible family of maps {F (C) → B}C∈C exhibits
B as a homotopy colimit of F if, for every object A ∈ A, the induced maps
{MapA (B, A) → MapA (F (C), A)}C∈C exhibit MapA (B, A) as a homotopy
limit of the diagram {MapA (F (C), A)}C∈C in S.
Remark A.3.3.14. In view of Proposition A.3.3.12, the terminology introduced in Remark A.3.3.13 for general A agrees with the terminology introduced for a combinatorial S-enriched model category A if we set A = A◦ .
We remark that, in general, the two notions do not agree if we take A = A,
so that our terminology is potentially ambiguous; however, we feel that there
is little danger of confusion.
We conclude this section by giving the proof of Lemma A.3.3.3. Let A bea presentable category which is enriched, tensored, and cotensored over apresentable category S. Let C be a small S-enriched category and let S 0 bea weakly saturated class of morphisms of A generated by a (small) set S0 .
We regard this data as fixed for the remainder of this section.
Choose a regular cardinal κ satisfying the following conditions:(i) The cardinal κ is uncountable.
(ii) The category C has fewer than κ-objects.
(iii) Let X, Y ∈ C and let K = MapC (X, Y ). Then the functor from A toitself given by the formula A 7→ AK preserves κ-filtered colimits. Thisimplies, in particular, that the collection of κ-compact objects of A isstable with respect to the functors • ⊗ K.
(iv) The category A is κ-accessible. It follows also that AC is κ-accessible,
and that an object F ∈ AC is κ-compact if and only if each F (C) ∈ A isκ-compact. We prove an ∞-category generalization of this statement
as Proposition 5.4.4.3. The same proof also works in the setting of
ordinary categories.
(v) The source and target of every morphism in S0 is a κ-compact objectof A.
Enlarging S0 if necessary, we may assume that S0 consists of all morphisms
in f ∈ S 0 such that the source and target of f are κ-compact. Let S be thecollection of all injective cofibrations between κ-compact objects of A (inview of (iv), we can equally well define S to be the set of morphisms F → G
in AC such that each of the induced morphisms F (C) → G(C) belongs toS0 ). Let S be the weakly saturated class of morphisms in AC generated by S
and choose a map f : F → G in AC such that f (C) ∈ S 0 for each C ∈ C. We

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wish to show that f ∈ S. Corollary A.1.5.13 implies that, for each C ∈ C,
there exists a κ-good S0 -tree {Y (C)α }α∈A(C) with root F (C) and colimit
G(C).
Let us define a slice to be the following data:(a) For each object C ∈ C, a downward-closed subset B(C) ⊆ A(C).
(b) For every object C ∈ C, a morphisma
Y (C 0 )B(C 0 ) ⊗ MapA (C 0 , C) → Y (C)B(C) ,ηC :C 0 ∈Crendering the following diagrams commutative:
Y (C 00 )B(C 00 ) ⊗ MapA (C 00 , C 0 ) ⊗ MapA (C 0 , C)
η

0

C
w
Y (C 0 )B(C 0 ) ⊗ MapA (C 0 , C)'
Y (C 00 )B(C 00 ) ⊗ MapA (C 00 , C)ηC'w
Y (C)B(C)ηC 00/ F (C)F (C 0 ) ⊗ MapA (C 0 , C)Y (C 0 )B(C 0 ) ⊗ MapA (C 0 , C)G(C 0 ) ⊗ MapA (C 0 , C)ηC
/ Y (C)B(C)/ G(C).

We remark that (b) is precisely the data needed to endow C 7→ Y (C)B(C)with the structure of an S-enriched functor C → A lying between F and G
in AC .
Lemma A.3.3.15. Suppose we are given a collection of κ-small subsets{B0 (C) ⊆ A(C)}C∈C . Then there exists a slice {(B(C), ηC }C∈C such thateach B(C) is a κ-small subset of A(C) containing B0 (C).
Proof. Enlarging each B0 (C) if necessary, we may assume that each B0 (C)is downward-closed. Note that because each {Y (C)α }α∈A(C) is a κ-good S0 tree, if A0 ⊆ A(C) is downward-closed and κ-small, Y (C)A0 is κ-compact
when viewed as an object of AF (C)/ . It follows from (iii) that each tensor
product Y (C)B0 (C) ⊗ MapA (C, C 0 ) is a κ-compact object of the categoryA(F (C)⊗MapA (C,C 0 ))/ . Consequently, each compositionaa
Y (C 0 )B0 (C 0 ) ⊗ MapA (C 0 , C)→
G(C 0 ) ⊗ MapA (C 0 , C) → G(C)C 0 ∈CC 0 ∈Chighertopoi

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admits another factorization
1aηC
Y (C 0 )B0 (C 0 ) ⊗ MapA (C 0 , C) →Y (C)B1 (C) → G(C),C 0 ∈C

where B1 (C) is downward-closed and κ-small, and the diagram`00/ ` 0 Y (C 0 )B (C 0 )0C ∈C
C 0 ∈C F (C ) ⊗ MapA (C , C)1
ηC
/ Y (C)B (C)
1


F (C)

commutes. Enlarging B1 (C) if necessary, we may suppose that each B1 (C)contains B0 (C).
We now continue the preceding construction by defining, for each C ∈ C,a sequence of κ-small downward-closed subsets
B0 (C) ⊆ B1 (C) ⊆ B2 (C) ⊆ · · ·`iof A(C) and maps ηC: C 0 ∈C Y (C 0 )Bi−1 (C 0 ) ⊗ MapA (C 0 , C) → Y (C)Bi (C) .
jSuppose that i > 1 and that the sets Bj (C) and maps ηChave been constructed for j < i. Using a compactness argument, we conclude that thecompositionaa
Y (C 0 )Bi−1 (C 0 ) ⊗ MapA (C 0 , C) →G(C 0 ) ⊗ MapA (C 0 , C) → G(C)C 0 ∈CC 0 ∈C

coincides withiaηC
Y (C 0 )Bi−1 (C 0 ) ⊗ MapA (C 0 , C) →Y (C)Bi (C) → G(C),C 0 ∈C

where Bi (C) is κ-small and the diagram``/
F (C 0 ) ⊗ MapA (C 0 , C)0000C ∈C Y (C )Bi−1 (C 0 ) ⊗ MapA (C , C)

C ∈C

i
ηC


/ Y (C)B (C)
i


F (C)

commutes. Enlarging Bi (C) if necessary, we may suppose that Bi (C) contains Bi−1 (C) and that the following diagrams commute as well (for all
C 0 , C 00 ∈ C):
Y (C 00 )Bi−2 (C 00 ) ⊗ MapA (C 00 , C 0 ) ⊗ MapA (C 0 , C)w
Y (C 0 )Bi−1 (C 0 ) ⊗ MapA (C 0 , C)'
Y (C 00 )Bi−1 (C 00 ) ⊗ MapA (C 00 , C)

i
ηC

'

w

Y (C)Bi (C)iηChighertopoi

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Y (C 0 )Bi−2 (C 0 ) ⊗ MapA (C 0 , C)

/ Y (C 0 )B0i−1 (C )i−1
ηC⊗ MapA (C 0 , C)
i
ηC



/ Y (C)B (C) .
Y (C)Bi−1 (C)iS
We now define B(C) = Bi (C), and ηC to be the amalgam of the compositionsiaηC
Y (C 0 )Bi−1 (C 0 ) ⊗ MapA (C 0 , C) →Y (C)Bi (C) → Y (C)B(C) .
C 0 ∈C

We now introduce a bit more terminology. Suppose we are given a pair0of slices M = {(B(C), ηC )}C∈C , M 0 = {(B 0 (C), ηC}C∈C }. We will say that
M is κ-small if each B(C) is κ-small. We will say that M 0 extends M if
B(C) ⊆ B 0 (C) for each C ∈ C and each diagram/ Y (C 0 )B 0 (C 0 ) ⊗ MapA (C 0 , C)
Y (C 0 )B(C 0 ) ⊗ MapA (C 0 , C)ηC
Y (C)B(C)0ηC/ Y (C)B 0 (C)is commutative. Equivalently, M 0 extends M if B(C) ⊆ B 0 (C) for eachC ∈ C, and the induced maps Y (C)B(C) → Y (C)B(C 0 ) constitute a naturaltransformation of simplicial functors from C to A.
Lemma A.3.3.16. Let M 0 = {(A0 (C), θC )}C∈C be a slice and let {B0 (C) ⊆A(C)}C∈C be a collection of κ-small subsets of A(C). Then there exists a0)}, where
pair of slices N = {(B(C), ηC )}C∈C , N 0 = {(B(C) ∩ A0 (C), ηC
0
0
B(C) is κ-small and N is compatible with both N and M .
Proof. Let B00 (C) = A0 (C) ∩ B0 (C). For every positive integer i, we will
construct a pair of slices Ni = {(Bi (C), η(i)C )}, Ni0 = {(Bi0 (C), η 0 (i)C )} sothat the following conditions are satisfied:(a) Each Bi (C) is κ-small and contains Bi−1 (C).
0(b) Each Bi0 (C) is κ-small, contains Bi−1
(C) and the intersection Bi (C) ∩00
A (C), and is contained in A (C).

(c) Each Ni0 is compatible with M 0 .
(d) If i > 2 and C, C 0 ∈ C, then the diagram/ Y (C 0 )B
Y (C 0 )B (C 0 ) ⊗ MapA (C 0 , C)i−2η(i−2)C0
i−1 (C )⊗ MapA (C 0 , C)η(i−1)CY (C)Bi−2 (C)Y (C)Bi−1 (C)
Y (C)Bi (C)
Y (C)Bi (C)highertopoi

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commutes.
(e) If i > 2 and C, C 0 ∈ C, then the diagram00
Y (C 0 )Bi−2(C 0 ) ⊗ MapA (C , C)

/ Y (C 0 )B 00i−1 (C )⊗ MapA (C 0 , C)

η 0 (i−2)Cη 0 (i−1)C0
Y (C)Bi−2(C)0
Y (C)Bi−1(C)Y (C)Bi0 (C)
Y (C)Bi0 (C)commutes.
(f ) If i > 1 and C, C 0 ∈ C, then the diagram00
Y (C 0 )Bi−1(C 0 ) ⊗ MapA (C , C)

η 0 (i−1)C/ Y (C 0 )B0i−1 (C )⊗ MapA (C 0 , C)η(i−1)C0
Y (C)Bi−1(C)

Y (C)Bi−1 (C)
Y (C)Bi0 (C)/ Y (C)B 0 (C)icommutes.
The construction is by induction on i. The existence of Ni satisfying (a),(d), and (f ) follows from Lemma A.3.3.15 (and a compactness argument).
Similarly, the existence of Ni0 satisfying (b), (c), and (e) follows by applying Lemma A.3.3.15 after replacing G ∈ AC by the functor G0 given by
G0 (C) = Y (C)A0 (C) , and the S0 -trees {Y (C)α }α∈A(C) by the smaller trees{Y (C)α }α∈A0 (C) .
S
We now define B(C) = i Bi (C). It follows from (d) that the η(i)C assemble to a mapaηC :Y (C 0 )B(C 0 ) ⊗ MapA (C 0 , C) → Y (C)B(C) .
C 0 ∈C

Taken together, these maps determine a slice N = {(B(C), ηC )}. Similarly, (e) implies that the maps η 0 (i)C assemble to a slice N 0 = {(B(C) ∩0
A0 (C), ηC)}. The compatibility of N and N 0 follows from (f ), while thecompatibility of M 0 and N 0 follows from (c).
We now construct a transfinite sequence of compatible slices {M (γ) ={(B(γ)(C), η(γ)C )}C∈C }γ<β . The construction is by recursion. Assume that
M (γ 0 ) has been defined for γ 0 < γ and let M 0 (γ) = {(B 0 (γ)(C), η 0 (γ)C )}C∈C
denote the slice obtained by amalgamating the family of slices {M (γ 0 )}γ 0 <γ .

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If B 0 (γ)(C) = A(C) for all C ∈ C, we set β = γ and conclude the construction. Otherwise, we choose C ∈ C and a ∈ A(C) − B 0 (γ)(C). According
to Lemma A.3.3.16, there exists a pair of slices N (γ) = {(B 00 (C), θC )}C∈C ,0
N 0 (γ) = {(B 00 (C) ∩ B 0 (γ)(C), θC}C∈C such that N 0 (γ) is compatible with0both N (γ) and M (γ). We now define M (γ) to be the slice obtained byamalgamating M 0 (γ) and N (γ).
For γ < β, let G(γ) : C → A be the simplicial functor corresponding to theslice M (γ). Then we have a transfinite sequence of composable morphisms
G(0) → G(1) → · · ·Cin (A )F/ having colimit G ' limγ<β G(γ). Since S is weakly saturated, to−→prove that the map F → G belongs to S, it will suffice to show that for eachγ < β, the map
fγ : limγ 0 <γ G(γ 0 ) → G(γ)−→
belongs to S. We observe that for each C ∈ C, the map fγ (C) can be identified with the map Y (C)B 0 (γ)(C) → Y (C)B(γ)(C) . Since B(γ)(C)−B 0 (γ)(C) isκ-small, Remark A.1.5.5, Lemma A.1.5.11, and Lemma A.1.5.6 imply that fγ
is the pushout of a morphism belonging to S0 . We now conclude by applying
the following result (replacing G by G(γ) and F by limγ 0 <γ G(γ 0 )):−→Lemma A.3.3.17. Suppose that f : F → G has the property that, for eachC ∈ C, there exists a pushout diagram
XC

F (C)

gC

f (C)

/ YC

/ G(C),

where gC ∈ S0 . Then f is the pushout of a morphism in S.
Proof. In view of (iv), we can write F as the colimit of a diagram {Fλ }λ∈Pindexed by a κ-filtered partially ordered set P , where each Fλ is a κ-compact
object of AC and is therefore a functor whose values are κ-compact objects of
A. Since each XC ∈ A is κ-compact, the map XC → F (C) factors throughFλ(C) (C) for some sufficiently large λ(C) ∈ P . Since C has fewer than κ
objects and P is κ-filtered we can choose a single λ ∈ P which works forevery object C ∈ C.
Consider, for each C ∈ C, the composite mapaa
YC 0 ⊗ MapA (C 0 , C) →G(C 0 ) ⊗ MapA (C 0 , C)C 0 ∈CC 0 ∈C→ G(C)' limλ0 ∈P Fλ0 (C)
−→

a

YC .

XC

Using another compactness argument, we deduce that this map is equivalent
to a compositionaa
YC 0 ⊗ MapA (C 0 , C) → Fλ0 (C) (C)YC
C 0 ∈CXC

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for some sufficiently large λ0 (C) ∈ P . Once again, because P is κ-filtered
we can choose a single λ0 ∈ P which works for all C. Enlarging λ and λ0 ,we can assume λ = λ0 . Using another compactness argument, we can (after
enlarging λ if necessary) assume that each of the diagrams/ Fλ (C)XC 0 ⊗ MapA (C 0 , C)
YC 0 ⊗ MapA (C 0 , C)

/ Fλ (C)YC 00 ⊗ MapA (C 00 , C 0 ) ⊗ MapA (C 0 , C)(Fλ (C 0 )


`

XC YC

/ YC 00 ⊗ MapA (C 00 , C)00 ) ⊗ Map (C , C)
Y
C
A
XC 0

`


/ Fλ (C) `XC YC

is commutative. Then the above maps allow us to `define an S-enriched functor Gλ : C → A by the formula Gλ (C) = Fλ (C) XC YC . We now observethat there is a pushout diagram
Fλ

F

fλ

f

/ Gλ

/G

and that fλ ∈ S.
A.3.4 Path Spaces in S-Enriched Categories
Let S be a excellent model category. We have seen that there exists a model
structure on the category CatS of S-enriched categories whose fibrant objects
are precisely those categories which are enriched over the full subcategoryS◦ of fibrant objects of S.
The theory of model categories provides a plethora of examples: for every S-enriched model category A, the full subcategory A◦ ⊆ A of fibrantcofibrant objects is a fibrant object of CatS . In other words, A◦ is suitable
to use for computing the homotopy set [C, A◦ ] = HomhCatS (C, A◦ ): if C iscofibrant, then every map from C to A◦ in the homotopy category of CatS
is represented by an actual S-enriched functor from C to A◦ . Moreover, two
simplicial functors F, F 0 : C → A◦ represent the same morphism in hCatS if
and only if they are homotopic to one another. The relation of homotopy canbe described in terms of either a cylinder object for C or a path object forA◦ . Unfortunately, it is rather difficult to construct a cylinder object for C
explicitly since the cofibrations in CatS are difficult to describe directly evenwhen S = Set∆ (the class of cofibrations of simplicial categories is not stable
under products, so the usual procedure of constructing mapping cylinders

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via “product with an interval” cannot be applied). On the other hand, Theorem A.3.2.24 gives a good understanding of the fibrations in CatS , which
will allow us to give a very explicit construction of a path object for A◦ .
Let A be an S-enriched model category. Our goal in this section is to givea direct construction of a path space object for A◦ in CatS . In other words,
we wish to supply a diagram of S-enriched categoriesA◦ → P (A) → A◦ × A◦ ,where the composite map is the diagonal, the left map is a weak equivalence,and the right map is a fibration. For technical reasons, we will find it convenient to work not with the entire category A but with some (usually small)
subcategory thereof. For this reason, we introduce the following definition:
Definition A.3.4.1. Let S be an excellent model category and let A be a
combinatorial S-enriched model category. A chunk of A is a full subcategory
U ⊆ A with the following properties:(a) Let A be an object of U and let {φi : A → Bi }i∈I be a finite collectionof morphisms in U. Then there exists a factorization
q Y
p
Bi
A→A→
i∈I

Q

of the product map i∈I φi , where p is a trivial cofibration, q is a
fibration, and A ∈ U. Moreover, this factorization can be chosen to
depend functorially on the collection {φi } via an S-enriched functor.
(b) Let A be an object of U and let {φi : Bi → A}i∈I be a finite collectionof morphisms in U. Then there exists a factorizationapq
Bi → A → A
i∈I`of the coproduct map i∈I φi , where p is a cofibration, q is a trivial
fibration, and A ∈ U. Moreover, this factorization can be chosen to
depend functorially on the collection {φi } via an S-enriched functor.
If U is a chunk of A, we let U◦ denote the full subcategory A◦ ∩ U ⊆ Uconsisting of fibrant-cofibrant objects of A which belong to U.
We will say that two chunks U, U0 ⊆ A are equivalent if they have thesame essential image in the homotopy category hA.
Remark A.3.4.2. In particular, if U is a chunk of A, then each object
A ∈ U admits (functorial) fibrant and cofibrant replacements which also
belong to U (take the set I to be empty in (a) and (b)).
Remark A.3.4.3. If U ⊆ U0 ⊆ A are equivalent chunks of A, then the◦
inclusion U◦ ⊆ U0 is a weak equivalence of S-enriched categories.
Example A.3.4.4. Let S be an excellent model category and let A be a
combinatorial S-enriched model category. Then A is a chunk of itself; thisfollows from the small object argument.

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Example A.3.4.5. Let U ⊆ A be a chunk and let {Xα } be a collection of
objects in A. Let V ⊆ U be the full subcategory spanned by those objects
X ∈ U such that there exists an isomorphism [X] ' [Xα ] in the homotopy
category hA. Then V is also a chunk of A.
We will prove a general existence theorem for chunks below (see Lemma
A.3.4.15).
Lemma A.3.4.6. Let S be an excellent model category and let C be a small
S-enriched category. Then there exists a weak equivalence of S-enriched categories C → U◦ , where U is a chunk of a combinatorial S-enriched model
category A.
Proof. Without loss of generality, we may suppose that C is fibrant. Let
op
A = SC endowed with the projective model structure. We can identify
C with a full subcategory of A◦ via the Yoneda embedding. Using Lemma
A.3.4.15, we can enlarge C to a chunk in A having the same image in thehomotopy category hA.
Notation A.3.4.7. Let S be an excellent model category, let A be a combinatorial S-enriched model category, and let U be a chunk of A. We define
a new category P (U) as follows:(i) The objects of P (U) are fibrations φ : A → B×C in A, where A, B, C ∈U◦ and the composite maps A → B and A → C are weak equivalences.
(ii) Morphisms in P (U) are given by maps of diagrams
Bo

A

/C


B0 o


A0


/ C 0.

We let π, π 0 : P (U) → U◦ be the functors described by the formulas
π 0 (φ : A → B × C) = C.

π(φ : A → B × C) = B

We observe that both π and π 0 have the structure of S-enriched functors.
Invoking assumption (a) of Proposition A.3.4.1, we deduce the existence ofanother S-enriched functor τ : U◦ → P (U), which carries an object A ∈ A◦
to the map q appearing in a functorial factorizationpq

A→A→A×Aof the diagonal, where p is a trivial cofibration and q is a fibration.
Theorem A.3.4.8. Let S be an excellent model category, let A be a combinatorial S-enriched model category, and let U be a chunk of A. Then themorphisms π, π 0 : P (U) → U◦ and τ : U◦ → P (U) furnish P (U) with thestructure of a path object for U◦ in CatS .

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Proof. We first show that π × π 0 is a fibration of S-enriched categories. Inview of Theorem A.3.2.24, it will suffice to show that π×π 0 is a local fibration.
Let φ : A → B × C and φ0 : A0 → B 0 × C 0 be objects of P (U). We must showthat the induced map
MapP (U) (φ, φ0 ) → MapA (B, B 0 ) × MapA (C, C 0 )is a fibration in S. This map is a base change of
MapA (A, A0 ) → MapA (A, B 0 × C 0 ),which is a fibration by virtue of the assumption that φ0 is a fibration (since
A is assumed to be cofibrant).
To complete the proof that π × π 0 is a quasi-fibration, we must show that
if φ : A → B × C is an object of P (U) and we are given weak equivalencesf : B → B 0 , g : C → C 0 , then we can lift f and g to an equivalence in P (U).
To do so, we factor the composite map A → B 0 × C 0 as a trivial cofibration
A → A0 followed by a fibration φ0 : A0 → B 0 × C 0 . Since U is a chunk of A,
we may assume that A0 ∈ U so that φ0 ∈ P (U). We have an evident naturaltransformation α : φ → φ0 . We will show below that π : P (U) → U◦ is anequivalence of S-enriched categories; since π(α) = f is an isomorphism inhU◦ , we conclude that α is an isomorphism in hP (U), as required.
To complete the proof, we must show that τ is a weak equivalence of Senriched categories. By the two-out-of-three property, it will suffice to showthat π is a weak equivalence of S-enriched categories. Since τ is a section ofπ, it is clear that π is essentially surjective. It remains only to prove that π isfully faithful. Let φ : A → B × C and φ0 : A0 → B 0 × C 0 be objects of P (U);we wish to show that the induced map p : MapP (U) (φ, φ0 ) → MapA (B, B 0 )is a weak equivalence in S. We have a commutative diagram
MapP (U) (φ, φ0 )/ MapA (A, A0 )


MapA (B, B 0 ) × MapA (C, C 0 )/ MapA (A, B 0 × C 0 )
MapA (B, B 0 ) × MapA (A, C 0 )/ MapA (A, B 0 ) × MapA (A, C 0 )
MapA (B, B 0 )/ MapA (A, B 0 ).

u

We note that because the map A → B is a weak equivalence between cofibrant objects and B 0 is fibrant, the bottom horizontal map is a weak equivalence in S. Consequently, to show that the top horizontal map is a weakequivalence in S, it will suffice to show that each square in the diagram ishomotopy Cartesian. The bottom square is Cartesian and fibrant, so there
is nothing to prove. The middle square is homotopy Cartesian because bothof the middle vertical maps are weak equivalences. The upper square is ahighertopoi

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pullback square between fibrant objects of S, and the map u is a fibration;
we now complete the proof by invoking Proposition A.2.4.4.
Fix an excellent model category S. The symmetric monoidal structure on
S induces a symmetric monoidal structure on CatS : if C and D are S-enrichedcategories, then we can define a new S-enriched category C ⊗ D as follows:(i) The objects of C ⊗ D are pairs (C, D), where C ∈ C and D ∈ D.
(ii) Given a pair of objects (C, D), (C 0 , D0 ) ∈ C ⊗ D, we have
MapC ⊗ D ((C, D), (C 0 , D0 )) = MapC (C, C 0 ) ⊗ MapD (D, D0 ) ∈ S.
In the case where the tensor product on S is the Cartesian product, thissimply reduces to the usual product of S-enriched categories.
Note that the operation ⊗ : CatS × CatS → CatS is not a left Quillen bifunctor even when S = Set∆ : for example, a product of cofibrant simplicial
categories is generally not cofibrant. Nevertheless, ⊗ behaves much like aleft Quillen bifunctor at the level of homotopy categories. For example, theoperation ⊗ respects weak equivalences in each argument and therefore induces a functor ⊗ : hCatS × hCatS → hCatS , which is characterized by theexistence of natural isomorphisms [C ⊗ D] ' [C] ⊗ [D].
Our goal for the remainder of this section is to show that the monoidal
structure ⊗ on CatS is closed: that is, there exist internal mapping objects
in hCatS . This is not completely obvious. It is easy to see that the monoidal
structure ⊗ on CatS is closed: given a pair of S-enriched categories C and D,
the category of S-enriched functors DC is itself S-enriched and possesses theappropriate universal property. However, this is not necessarily the “correct”
mapping object in the sense that the homotopy equivalence class [DC ] does
not necessarily coincide with the internal mapping object [D][C] in hCatS .
Roughly speaking, the problem is that DC consists of functors which arestrictly compatible with composition; the correct mapping object should also
incorporate functors which preserve composition only up to (coherent) weakequivalence. However, when D is the category of fibrant-cofibrant objects ofan S-enriched model category A, then we can proceed more directly.
Definition A.3.4.9. Let S be an excellent model category, A a combinatorial S-enriched model category, and C an S-enriched category. We will saythat a full subcategory U ⊆ A is a C-chunk of A if it is a chunk of A andthe subcategory UC is a chunk of AC . Here we regard AC as endowed withthe projective model structure.
Lemma A.3.4.10. Let S be an excellent model category, A a combinatorial
S-enriched model category, C a (small) cofibrant S-enriched category, andU ⊆ A a C-chunk. Let f, f 0 : C → U◦ be a pair of maps. The following
conditions are equivalent:(1) The homotopy classes [f ] and [f 0 ] coincide in HomhCatS (C, U◦ ).

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(2) The maps f and f 0 are weakly equivalent when regarded as objects ofAC .
Proof. Suppose first that (1) is satisfied. Using Theorem A.3.4.8, we deduce
the existence of a homotopy h : C → P (U) from f = π ◦ h to f 0 = π 0 ◦ h. The
map h determines another simplicial functor f 00 : C → U equipped with weakequivalences f 00 → f , f 00 → f 0 . This proves that f and f 0 are isomorphic in
the homotopy category of AC , so that (2) is satisfied.
Now suppose that (2) is satisfied. Since U is a C-chunk, we can find aprojectively cofibrant f 00 : U → C equipped with a weak equivalence α : f 00 →f . Using (2), we deduce that there is also a weak equivalence β : f 00 → f 0 .
Again using the assumption that UC is a chunk of AC , we can choose afactorization of α × β as a compositionuv
f 00 → f 000 → f × f 0
where u is a trivial projective cofibration, v is a projective fibration, and
f 000 ∈ UC . The map v can be viewed as an object of P(U), which determines
a right homotopy from f to f 0 .
Corollary A.3.4.11. Let S be an excellent model category and let f : C →C0 be an S-enriched functor. Suppose that f is fully faithful in the sensethat for every pair of objects X, Y ∈ C, the induced map MapC (X, Y ) →MapC0 (f X, f Y ) is a weak equivalence in S. Let D be an arbitrary S-enrichedcategory. Then
(1) Composition with f induces an injective map φ : HomhCatS (D, C) →HomhCatS (D, C0 ).
(2) The image of φ consists of those maps g : D → C0 in hCatS such thatthe essential image of [g] in hC0 is contained in the essential image of[f ] in hC0 .
Proof. Using Lemma A.3.4.6, we may assume without loss of generality that
C0 = U◦ , where U is a chunk of an S-enriched model category. Let V ⊆ U be
the full subcategory spanned by those objects which are weakly equivalent
to an object lying in the image of f . Since f is fully faithful, the induced
map C → V◦ is a weak equivalence. We may therefore assume that C = V◦ .
Without loss of generality, we may suppose that D is cofibrant. Enlarging
U and V if necessary (using Lemma A.3.4.15), we may assume that U and
V are D-chunks. The desired results now follow immediately from Lemma
A.3.4.10.
Let π0 AC denote the collection of weak equivalence classes of objects in
A . Every equivalence class contains a fibrant-cofibrant representative which
determines an S-enriched functor C → A◦ .
C

Proposition A.3.4.12. Let S be an excellent model category, A a combinatorial S-enriched model category, and C a (small) S-enriched category. Thenthe map
φ : π0 AC → HomhCatS (C, A◦ )highertopoi

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described above is bijective.
Proof. In view of Proposition A.3.3.8, we may assume that C is cofibrant.
Lemma A.3.4.10 shows that φ is well-defined and injective. We show thatφ is surjective. Let [f ] ∈ HomhCatS (C, U◦ ). Since C is cofibrant and A◦ isfibrant in CatS , we can find an S-enriched functor f : C → A◦ representing[f ]. The simplicial functor f takes values in fibrant-cofibrant objects of A
but is not necessarily fibrant-cofibrant as an object of AC . However, we canchoose a projective trivial fibration f 0 → f , where f 0 is projectively cofibrant.
Consequently, it will suffice to show that a weak equivalence u : f 0 → f of
S-enriched functors C → A◦ guarantees that [f ] = [f 0 ] ∈ HomhCatS (C, A◦ ),which follows from Lemma A.3.4.10.
Proposition A.3.4.13. Let S be an excellent model category, A a combinatorial S-enriched model category, and C a small S-enriched category. Thenthe evaluation map e : (AC )◦ ⊗ C → A◦ has the following property: for every
small S-enriched category D, composition with e induces a bijection
HomhCatS (D, (AC )◦ ) → HomhCatS (C ⊗ D, A◦ ).
Proof. Using Proposition A.3.4.12, we can identify both sides with π0 AC ⊗ D .

It is not clear that the conclusion of Proposition A.3.4.13 characterizes(AC )◦ up to equivalence since (AC )◦ is a large S-enriched category, and theproof of the proposition applies only in the case where D is small. To remedy
this defect, we establish a more refined version:
Corollary A.3.4.14. Let S be an excellent model category, A a combinatorial S-enriched model category, and C a small S-enriched category. Let U
be a C-chunk of A. Then the evaluation map e : (UC )◦ ⊗ C → U◦ has thefollowing property: for every small S-enriched category D, composition withe induces a bijection
HomhCatS (D, (UC )◦ ) → HomhCatS (C ⊗ D, U◦ ).
Proof. Combine Proposition A.3.4.13 with Corollary A.3.4.11.
We conclude this section with a technical result which ensures the existence of a good supply of chunks of combinatorial model categories.
Lemma A.3.4.15. Let S be an excellent model category, A a combinatorial S-enriched model category, and {Cα }α∈A a (small) collection of (small)
cofibrant S-enriched categories. Let U be a small full subcategory of A. Then
there exists a small subcategory V ⊆ A containing U, such that V is a Cα chunk for each α ∈ A. Moreover, we may arrange that U and V have thesame essential image in the homotopy category hA.

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Proof. Enlarging A if necessary, we may suppose that the collection {Cα }α∈A
includes the unit category [0]S . For each α ∈ A, we can choose S-enriched
functors
Fα : ACα ⊗[1]S → ACα ⊗[2]SGα : ACα ⊗[1]S → ACα ⊗[2]S ,such that F carries each morphism u : f → g in AC to a factorization
u0

u00

f → f 0 → g,
where u0 is a projective trivial cofibration and u00 is a projective fibration,
and G carries u to a factorization
v0

v 00

f → g 0 → g,
where v 0 is a projective cofibration and v 00 is an injective trivial cofibration.
For C ∈ Cα , let FαC be the functor u 7→ f 0 (C) and define GC
α likewise.
Choose a regular cardinal κ such that each Cα is κ-small. We define asequence of full subcategories {Uα ⊆ A}α<κ as follows:(i) If α = 0, then Uα = U.
(ii) If α is a nonzero limit ordinal, then Uα =S

β<α Uβ .

(iii) If α = β + 1, then Uα is the full subcategory of A spanned by thefollowing:(a) The objects which belong to Uβ .
(b) The objects FαC (u) ∈ A, where α ∈ A, C ∈ Cα , and u : f → g isαa morphism from an object of UC
β to a finite product of objects
Cα
in Uβ .
(c) The objects GC
α (u) ∈ A, where α ∈ A, C ∈ Cα , and u : f → g is aαmorphism from a finite coproduct of objects of UC
β to an object
Cα
in Uβ .
S
It is readily verified that the subcategory V = α<κ Uα has the desired
properties.
A.3.5 Homotopy Colimits of S-Enriched Categories
Our goal in this section is to give an explicit construction of (certain) homotopy colimits in the model category CatS , where S is an excellent model
category. We begin with some general remarks concerning localization.
Notation A.3.5.1. Consider the canonical map i : [1]S → [1]∼S . We fix once
and for all a factorization of i as a compositionii0[1]S → E → [1]∼S,highertopoi

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where i is a cofibration and i0 is a weak equivalence of S-enriched categories.
For every S-enriched category C and every map W → HomCatS ([1]S , C), we
define a new S-enriched category C[W −1 ] by a pushout diagram`/Cw∈W [1]S/ C[W −1 ].

` 

w∈W E

Remark A.3.5.2. Since the model category CatS is left proper, the construction C 7→ C[W −1 ] preserves weak equivalences in C.
We now characterize C[W −1 ] by a universal property in hCatS .
Lemma A.3.5.3. Let C be a fibrant S-enriched category and let f be amorphism in C classified by a map j0 : [1]S → C. The following conditions
are equivalent:(1) The map f is an equivalence in C.
(2) The extension problem depicted in the diagram
[1]S

j0

/C
>

j
i


E
admits a solution.

Proof. The implication (2) ⇒ (1) is clear since every morphism in E is anequivalence. For the converse, we observe that the desired`lifting problem
admits a solution if and only if the induced map i0 : C → C [1]S E admits aleft inverse. Since C is fibrant, it suffices to show that i0 is a trivial cofibration.
The map i0 is a cofibration since it is a pushout of i, and a weak equivalence
because of the invertibility hypothesis.
Lemma A.3.5.3 immediately implies the following apparently stronger
claim:
Lemma A.3.5.4. Let f0 : C → D be an S-enriched functor, where D is a
fibrant S-enriched category. Let ψ : W → HomCatS ([1]S , C) be a map of sets.
The following conditions are equivalent:(1) The map f0 extends to a map f : C[W −1 ] → D.
(2) For each w ∈ W , f0 carries the morphism ψ(w) to an equivalence inD.

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Proposition A.3.5.5. Let C and D be S-enriched categories and let ψ :W → HomCatS ([1]S , C) be a map of sets. Then the induced map
φ : HomhCatS (C[W −1 ], D) → HomhCatS (C, D)
is injective, and its image is the subset HomW
hCatS (C, D) ⊆ HomhCatS (C, D)
consisting of those homotopy classes of maps which induce functors hC → hD
carrying each element of W to an isomorphism in hD.
Proof. Without loss of generality, we may suppose that C is cofibrant andD is fibrant. The description of the image of φ follows immediately from
Lemma A.3.5.4. It will therefore suffice to show that φ is injective. Suppose we are given a pair of maps [f ], [g] ∈ HomhCatS (C[W −1 ], D) such thatφ([f ]) = φ([g]). Since C[W −1 ] is cofibrant, we may assume that [f ] and [g]are represented by actual S-enriched functors f, g : C[W −1 ] → D. Moreover,the condition that φ([f ]) = φ([g]) guarantees that the restrictions f | C andg| C are homotopic. We wish to show that f and g are homotopic.
Invoking Proposition A.2.3.1, we deduce that g is homotopic to a map
g 0 : C[W −1 ] → D such that g 0 | C = f | C. Replacing g by g 0 if necessary,
we may assume that g| C = f | C. It will now suffice to show that f and g
are homotopic in the model category (CatS )C / . We observe that f and gdetermine a mapaah : C[(WW )−1 ] ' C[W −1 ]C[W −1 ] → D .
C`Using the invertibility hypothesis, we conclude that C[(W W )−1 ] is a cylinder object for C[W −1 ] in the model category (CatS )C / , so that h is the desired
homotopy from f to g.
Lemma A.3.5.6. Let f : C → D be an S-enriched functor and let M be thecategorical mapping cylinder of f defined as follows:(1) An object of M is either an object of C or an object of D.
(2) Given a pair of objects X, Y ∈ M, we have
MapC (X, Y )if X, Y ∈ CMap (X, Y )if X, Y ∈ D
D
MapM (X, Y ) =
MapD (f X, Y ) if X ∈ C, Y ∈ D∅
if X ∈ D, Y ∈ C .
Here ∅ denotes the initial object of S.
We observe that there is a canonical retraction j of M onto D described by
the formula(f X if X ∈ C
j(X) =
X
if X ∈ D .
Let W be a collection of morphisms in M with the following properties:

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APPENDIX(i) For each w ∈ W , j(w) is an identity morphism in D.
(ii) For every object C ∈ C, the morphism C → f C in M classifying theidentity map from f C to itself belongs to W .
Assumption (i) implies that the map j canonically extends to a map j :M[W −1 ] → D. The map j is a weak equivalence of S-enriched categories.
Proof. It will suffice to show that composition with j induces a bijection
HomhCatS (D, A) → HomhCatS (M[W −1 ], A)for every S-enriched category A. Equivalently, we must show that the mapt : HomhCatS (D, A) → HomW
hCatS (M, A)
is bijective, where HomW
hCatS (M, A) is defined as in Proposition A.3.5.5. The
map t has a section t0 given by composition with the inclusion D → M. It
will therefore suffice to show that t ◦ t0 is the identity on HomW
hCatS (M, A).
Using Lemma A.3.4.6 and Corollary A.3.4.11, we can reduce to the casewhere A = A◦ , where A is a combinatorial S-enriched model category. Using
Proposition A.3.4.12, we deduce that every element [g] ∈ HomhCatS (M, A)
can be represented by a diagram g : M → A◦ . We wish to prove that g
and g ◦ i ◦ j are homotopic. We observe that there is a canonical naturaltransformation α : g → g ◦ i ◦ j. Moreover, if g carries each element of W
to an equivalence in A◦ , then assumption (ii) guarantees that α is a weakequivalence in the model category AM . We now invoke Proposition A.3.4.12
to deduce that g and g ◦ i ◦ j are homotopic, as desired.
Definition A.3.5.7. Let A be a partially ordered set. An A-filtered Senriched category is an S-enriched category C together with a map r :Ob(C) → A with the following property: if C, D ∈ C and r(C)  r(D),then MapC (C, D) ' ∅, where ∅ denotes an initial object of S.
If C is an A-filtered S-enriched category and a ∈ A, then we let C≤a denotethe full subcategory of C spanned by those objects C ∈ C such that r(C) ≤ a.
Remark A.3.5.8. Let C be an A-filtered S-enriched category and let ψ :W → HomCatS ([1]S , C) be a map of sets. For each a ∈ A, we let Wa ⊆ W be
the subset consisting of those elements w ∈ W such that the morphism ψ(w)belongs to C≤a . This data determines a diagram χW : A → CatS describedby the formula a 7→ C≤a [Wa−1 ]. Moreover, we have a canonical isomorphismof S-enriched categories C[W −1 ] ' lim(χ).
−→
Using the small object argument, we easily deduce the following result:Lemma A.3.5.9. Let A be a partially ordered set and let C be an A-filtered
S-enriched category. Then there exists an S-enriched functor f : C0 → C withthe following properties:(1) The functor f is bijective on objects, and for every pair of objects
C, D ∈ C0 , the map MapC0 (C, D) → MapC (f C, f D) is a trivial fibration in S. In particular, f is a weak equivalence of S-enriched categories.

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APPENDIX(2) The A-filtration on C induces an A-filtration on C0 . In other words, if
C, D ∈ C0 and r(f C)  r(f D), then MapC0 (C, D) is an initial objectof S.
(3) The diagram A → CatS described by the formula a 7→ C0≤a is projectively cofibrant.
Proposition A.3.5.10. Let A be a partially ordered set, let C be an Afiltered S-enriched category, and let ψ : W → HomCatS ([1]S , C) be a map ofsets. Let χ : A → CatS be defined as in Remark A.3.5.8. Then the isomorphism lim χ ' C[W −1 ] exhibits C as the homotopy colimit of the diagram−→
χ.
Proof. Choose a map C0 → C as in Lemma A.3.5.9 and a map ψ 0 : W →HomCatS ([1]S , C0 ) lifting ψ, and let χ0 : A → CatS be defined as in Remark
A.3.5.8. Then we have a canonical map χ0 → χ, which is a cofibrant replacement for χ in the model category Fun(A, CatS ). It will therefore suffice
to show that the induced map C0 [W −1 ] ' lim χ0 → lim χ ' C[W −1 ] is a−→−→weak equivalence of S-enriched categories, which follows immediately from
Remark A.3.5.2.
Definition A.3.5.11. Let A be a partially ordered set and let p : A → CatS
be an A-indexed diagram of S-enriched categories. Let us denote the imageof a ∈ A under p by Ca .
The Grothendieck construction on p is a S-enriched category Groth(p)defined as follows:(1) The objects of Groth(p) are pairs (a, C), where a ∈ A and C ∈ Ca .
(2) Given a pair of objects (a, C), (a0 , C 0 ) in Groth(p), we set(
0
MapCa0 (paa C, C 0 ) if a ≤ a000
MapGroth(p) ((a, C), (a , C )) =∅otherwise.
0

Here paa denotes the functor Ca → Ca0 determined by p, and ∅ denotes
an initial object of S.
(3) Composition in Groth(p) is defined in the obvious way.
We observe that Groth(p) is A-filtered via the map r : Ob(Groth(p)) → A
given by the formula r(a, C) = a. We let W (p) denote the collection of all0morphisms in Groth(p) of the form α : (a, C) → (a0 , paa C), where a ≤ a0 andα corresponds to the identity in Ca0 .
For each a ∈ A, there is a canonical functor πa : Groth(p)≤a → Ca given by
the formula (C, a0 ) 7→ paa0 (C). We note that π carries each element of W (p)ato an identity map in Ca , so that πa canonically extends to a map π a :Groth(p)≤a [W (p)−1a ] → Ca . The maps π a are functorial in a and therefore
determine a map of diagrams χ(p) → p, where χ(p) is defined as in Remark
A.3.5.8.

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Lemma A.3.5.12. Let p : A → CatS be as in Definition A.3.5.11. Then foreach a ∈ A, the map π a : Groth(p)≤a [W (p)−1a ] → Ca is a weak equivalenceof S-enriched categories.
Proof. This is a special case of Lemma A.3.5.6.
Lemma A.3.5.13. Let p : A → CatS be as in Definition A.3.5.11. Then
there is a canonical isomorphism Groth(p)[W (p)−1 ] ' hocolim(p) in thehomotopy category hCatS .
Proof. Combine Lemma A.3.5.12 with Proposition A.3.5.10.
Lemma A.3.5.14. Let C and D be small S-enriched categories. Let W bea collection of morphisms in C and let W 0 be the collection of all morphisms
in C ⊗ D of the form w ⊗ idD , where w ∈ W and D ∈ D. Then the canonicalmap(C ⊗ D)[W 0−1] → C[W −1 ] ⊗ D

is a weak equivalence of S-enriched categories.
Proof. It will suffice to show that for every S-enriched category A, the induced map
φ : HomhCatS (C[W −1 ] ⊗ D, A) → HomhCatS ((C ⊗ D)[W 0

−1

], A)

is bijective. Using Lemma A.3.4.6 and Corollary A.3.4.11, we can reduceto the case where A = A◦ , where A is a combinatorial S-enriched model
category. We now invoke Propositions A.3.4.13 and A.3.5.5 to get a chain of
bijections
HomhCatS (C[W −1 ] ⊗ D, A◦ ) ' HomhCatS (C[W −1 ], (AD )◦ )D ◦' HomW
hCatS (C, (A ) )0◦' HomW
hCatS (C ⊗ D, A )' HomhCatS ((C ⊗ D)[W 0 ], A◦ )whose composition is the map φ.
Theorem A.3.5.15. Let A be a partially ordered set and let D be an Senriched category. Then the functor C 7→ C ⊗ D commutes with A-indexed
homotopy colimits. In other words, if p : A → CatS is a projectively cofibrantdiagram and p0 : A → CatS is defined by p0 (a) = p(a) ⊗ D, then the canonicalisomorphism lim(p0 ) ' lim(p) ⊗ D exhibits lim(p) ⊗ D as a homotopy colimit−→−→−→of the diagram p0 .
Proof. In view of Lemma A.3.5.13, it will suffice to show that the canonicalmap h : Groth(p0 )[W (p0 )−1 ] → Groth(p)[W (p)−1 ] ⊗ D is a weak equivalenceof S-enriched categories. This is a special case of Lemma A.3.5.14.

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A.3.6 Exponentiation in Model Categories
Let C be a category which admits finite products, containing a pair of objects
X and Y . An exponential of X by Y is an object X Y ∈ C together with amap e : X Y ×Y → X, with the following universal property: for every object
W ∈ C, the compositione◦

HomC (W, X Y ) → HomC (W × Y, X Y × Y ) −→ HomC (W × Y, X)
is bijective.
Our goal in this section is to study the existence of exponentials in thehomotopy category of a model category A. Suppose we are given a pair ofobjects X, Y ∈ A such that there exists an exponential of [X] by [Y ] in thehomotopy category hA. We can then represent this exponential as [Z] forsome object Z ∈ A. Without loss of generality, we may assume that X, Y ,and Z are fibrant and cofibrant, so that we have a canonical identification[Z] × [Y ] ' [Z × Y ]. However, we encounter a technical difficulty: the evaluation map [Z] × [Y ] → [X] need not be representable by any morphism fromZ × Y to X in the category A because Z × Y need not be cofibrant. We
wish to work in certain contexts where this difficulty does arise (for example, where A is the category of simplicial categories). For this reason we areforced to work with the following somewhat cumbersome definition:
Definition A.3.6.1. Let A be a model category. We will say that a diagram
P
{
Z ×Y

p

X

exhibits Z as a weak exponential of X by Y if the following conditions are
satisfied:(1) The map p exhibits P as a homotopy product of Z and Y ; in otherwords, the induced map [p] : [P ] → [Z] × [Y ] is an isomorphism in thehomotopy category hA.
[p]−1(2) The composition [Z]×[Y ] → [P ] → [X] exhibits [Z] as an exponentialof [X] by [Y ] in the homotopy category hA.
We will say that a map Z × Y → X exhibits Z as an exponential of X by
Y if the diagram
Z ×Y
y
Z ×Y
satisfies (1) and (2).

id

#

X

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Remark A.3.6.2. Suppose we are given a diagram
P
{
Z ×Y

p

X

as in Definition A.3.6.1. We will say that this diagram is standard if X, Y, Z ∈A are fibrant, and the map p is a trivial fibration.
Suppose X and Y are fibrant objects of A such that there exists an exponential of [X] by [Y ] in the homotopy category hA. Without loss of generality, this exponential has the form [Z], where Z is a fibrant object of A. We
can then choose a trivial fibration P → Z × Y , where P is cofibrant. The
evaluation map [Z × Y ] ' [Z] × [Y ] → [X] is then representable by a map
P → X in A, so that we obtain a standard diagram which exhibits Z as a
weak exponential of X by Y .
Remark A.3.6.3. Suppose we are given a diagram
P
{
Z ×Y

p

X

in a model category A. The condition that this diagram exhibits Z as a
weak exponential of X by Y depends only on the image of this diagram in
the homotopy category hA. We may therefore replace the above diagram bya weakly equivalent diagram when testing whether or not the conditions ofDefinition A.3.6.1 are satisfied.
Remark A.3.6.4. Let A oF/ B be a Quillen equivalence of model cate-Ggories. Suppose we are given a standard diagram
P
{
Z ×Y

p

X

in B. Then this diagram exhibits Z as a weak exponential of X by Y in B
if and only if the associated diagram
GPy
GZ × GY"

GX

exhibits GZ as a weak exponential of GX by GY in A.
To work effectively with weak exponentials, we need to introduce an additional assumption.

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Definition A.3.6.5. Let A be a combinatorial model category containing
a fibrant object Y . We will say that multiplication by Y preserves homotopy
colimits if the following condition is satisfied:(∗) Let A be a (small) partially ordered set, let F : A → A be a projectively
cofibrant diagram, and let F 0 : A → A be another strongly cofibrantdiagram equipped with a natural transformation F 0 (a) → F (a) × Y
which is weak equivalence for each a ∈ A. Then the induced map
lim F 0 → (lim F ) × Y exhibits lim F 0 as a homotopy product of Y with−→−→−→
lim F in A.
−→
We will say that multiplication in A preserves homotopy colimits if condition(∗) is satisfied for every fibrant object Y ∈ A.
Remark A.3.6.6. Definition A.3.6.5 refers only to homotopy colimits indexed by partially ordered sets. However, every diagram indexed by an arbitrary category can be replaced by a diagram indexed by a partially ordered
set having the same homotopy colimit. We formulate and prove a precise
statement to this effect (in the language of ∞-categories) in §4.2. However,
we will not need (or use) any such result in this appendix.
Remark A.3.6.7. Let A oF/ B be a Quillen equivalence between combi-

Gnatorial model categories and let Y ∈ B be a fibrant object. Then multiplication by Y preserves homotopy colimits in B if and only if multiplication byG(Y ) preserves homotopy colimits in A. Since the right derived functor RG
is essentially surjective on homotopy categories, we see that multiplicationin B preserves homotopy colimits if and only if multiplication in A preserves
homotopy colimits.
Example A.3.6.8. Let S be an excellent model category with respect to thesymmetric monoidal structure given by the Cartesian product in S. Then
multiplication in CatS preserves homotopy colimits. This is precisely thecontent of Theorem A.3.5.15.
Lemma A.3.6.9. Let S be a collection of simplicial sets satisfying the following conditions:(i) The simplicial set ∆0 belongs to S.
(ii) If f : X → Y is a weak homotopy equivalence, then X ∈ S if and only
if Y ∈ S.
(iii) For every small partially ordered set A, if F : A → Set∆ is a projectively cofibrant diagram such that each F (a) ∈ S, then lim F ∈ S.
−→
Then every simplicial set belongs to S.
Proof. Using (ii) and (iii), we deduce that if F : A → Set∆ is any diagramsuch that each F (a) belongs to S, then the homotopy colimit of F belongs tohighertopoi

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S. In particular, S is closed under the formation of coproducts and homotopy
pushouts.
We now prove by induction on n that every n-dimensional simplicial set X
belongs to S. For this, we observe that there is a homotopy pushout diagram/ B × ∆nB × ∂ ∆n
skn−1 X
/ X,

where B denotes the set of n-simplices of X. The simplicial sets B × ∂ ∆nand skn−1 X belong to S by the inductive hypothesis. The simplicial set
B × ∆n is weakly equivalent to the constant simplicial set B, which belongs
to S in view of (i) and the fact that S is stable under coproducts. Since S
is stable under homotopy pushouts, we conclude that X ∈ S, as desired.
An arbitrary simplicial set X can be written as the colimit of a projectively
cofibrant diagram
sk0 X ⊆ sk1 X ⊆ sk2 X ⊆ · · ·and therefore belongs to S by assumption (iii).
Proposition A.3.6.10. Let A be a combinatorial simplicial model categorycontaining a standard diagram
P
{
Z ×Y

p

X.

Assume further that multiplication by Y preserves homotopy colimits in A.
The following conditions are equivalent:(i) The above diagram exhibits Z as a weak exponential of X by Y .
(ii) Let W and W 0 be cofibrant objects of A and let W 0 → W × Y be amap which exhibits W 0 as a homotopy product of W and Y . Then the
induced map
MapA (W, Z) ×MapA (W 0 ,Z×Y ) MapA (W 0 , P ) → MapA (W 0 , X)
is a homotopy equivalence of Kan complexes.
Remark A.3.6.11. In the situation of part (ii) of Proposition A.3.6.10,
the projection map MapA (W 0 , P ) → MapA (W 0 , Z × Y ) is a trivial Kan
fibration, so the fiber product MapA (W, Z) ×MapA (W 0 ,Z×Y ) MapA (W 0 , P ) is
automatically a Kan complex which is homotopy equivalent to MapA (W, Z).
Proof of Proposition A.3.6.10. First suppose that (ii) is satisfied. We wish
to show that for every object [W ] ∈ hA, the composition
HomhA ([W ], [Z]) → HomhA ([W ] × [Y ], [Z] × [Y ])' HomhA ([W ] × [Y ], [P ])→ HomhA ([W ] × [Y ], [P ])highertopoi

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is bijective. Without loss of generality, we may assume that [W ] is the homotopy equivalence class of a fibrant-cofibrant object W ∈ A. Choose a cofibrant replacement W 0 → W × Y . We observe that the map in question canbe identified with the composition
π0 MapA (W, Z) → π0 MapA (W 0 , Z × Y )' π0 MapA (W 0 , P )→ π0 MapA (W 0 , X),
which is bijective in view of (ii) and Remark A.3.6.11.
We now assume (i) and prove (ii). It will suffice to show that for every
simplicial set K, the induced map
HomhSet∆ (K, MapA (W, Z) ×MapA (W 0 ,Z×Y ) MapA (W 0 , P ))

HomhSet∆ (K, MapA (W 0 , X))
is a bijection. Using Remark A.3.6.11, we can identify the left side with theset HomhSet∆ (K, MapA (W, Z)) ' HomhA (W ⊗ K, Z). Similarly, the rightside can be identified with HomhA (W 0 ⊗ K, X). In view of assumption (i), it
will suffice to show that the map βK : W 0 ⊗K → Y ×(W ⊗K) exhibits W 0 ⊗K
as a homotopy product of Y and W ⊗ K. The collection of simplicial sets K
with this property clearly contains ∆0 and is stable under weak homotopy
equivalence. The assumption that multiplication by Y preserves homotopy
colimits guarantees that the hypotheses of Lemma A.3.6.9 are satisfied, sothat the desired conclusion holds for every simplicial set K.
Lemma A.3.6.12. Let A be a combinatorial model category and i : B0 → B
an inclusion of partially ordered sets. Suppose that there exists a retractionr : B → B0 such that r(b) ≤ b for each b ∈ B. Let F : B → A be a diagram.
Then a map α : X → lim(F ) in A exhibits X as a homotopy limit of F if
and only if α exhibits X as a homotopy limit of i∗ F .
Proof. Without loss of generality, we may assume that F is injectively fibrant. We have a canonical isomorphism lim(F ) ' lim(i∗ F ). It will therefore
suffice to show that the functor i∗ preserves injective fibrations. It now suffices to observe that i∗ is right adjoint to r∗ and that the functor r∗ preserves
injective trivial cofibrations.
Lemma A.3.6.13. Let A be a combinatorial model category, C a small
category, F : C → A a diagram, and α : X → lim(F ) a morphism in the
category A. Suppose that(i) For each C ∈ C, the induced map X → F (C) is a weak equivalence inA.
(ii) The category C has a final object C0 .

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Then α exhibits X as a homotopy limit of the diagram F .
Proof. Without loss of generality, we may assume that the diagram F is projectively fibrant. Let F 0 : C → A be defined by the formula F 0 (C) = F (C0 ).
We observe that, for every G ∈ Fun(C, A), we have HomFun(C,A) (G, F 0 ) =HomA (G(C0 ), F (C0 )). In particular, we have a canonical map β : F → F 0 .
Condition (i) guarantees that β is a weak equivalence. Since F (C0 ) ∈ A is
fibrant, the diagram F 0 is injectively fibrant. It therefore suffices to showthat the induced map X → lim(F 0 ) ' F (C0 ) is a weak equivalence, which
follows from (i).
Lemma A.3.6.14. Let A be a combinatorial model category, let A be apartially ordered set, and set B = {(a, b) ∈ Aop × A : a ≥ b}, regarded as apartially ordered subset of Aop × A. Let π : B → Aop denote the projection
onto the first factor.
Suppose we are given diagrams F : B → A, G : A → A, and a naturaltransformation α : π ∗ (G) → F , which induces weak equivalences G(a) →F (a, b) for each (a, b) ∈ B. Then α exhibits G as a homotopy right Kan
extension of F .
Proof. In view of Proposition A.2.8.9, it will suffice to show that for eacha0 ∈ A, the transformation α exhibits G(a0 ) as a homotopy limit of the
diagram F |{(a, b) ∈ B : a ≤ a0 }. Let B0 = {(a, b) ∈ B : a = a0 }. In view ofLemma A.3.6.12, it will suffice to show that α exhibits G(a0 ) as a limit of the
diagram F0 = F |B0 . This follows immediately from Lemma A.3.6.13.
Proposition A.3.6.15. Let A be a combinatorial model category and A =A0 ∪ {∞} a partially ordered set with a largest element ∞. Let B = {(a, b) ∈Aop × A : a ≥ b}, regarded as a partially ordered subset of Aop × A.
Suppose we are given an object X ∈ A together with functors Y : A → A,
Z : Aop → A, P : B → A, and diagrams σa,b :P (a, b)xZ(a) × Y (b)"X,

which depend functorially on (a, b) ∈ B. Suppose further that(i) Each diagram σa,b exhibits P (a, b) as a homotopy product of Z(a) andY (b) in A.
(ii) The diagrams σa,a exhibit Z(a) as a weak exponential of X by Y (a).
(iii) Multiplication in A preserves homotopy colimits.
(iv) The diagram Y exhibits Y (∞) as a homotopy colimit of Y0 = Y |A0 .
Then the diagram Z exhibits Z(∞) as the homotopy limit of the diagram
Z0 = Z|Aop
0 .

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Proof. Making fibrant replacements if necessary, we may assume that eachdiagram σa,b is standard. According to the main result of [19], there exists a
F /A, where A0 is a combinatorial simplicial model
Quillen equivalence A0 oG

category. In view of Remark A.3.6.4, we may replace A by A0 and thereby
reduce to the case where A is a simplicial model category.
In view of Proposition A.3.3.12, it will suffice to prove the following: forevery fibrant-cofibrant object C ∈ A, if we define G : Aop → Set∆ by theformula G(a) = MapA (C, Z(a)), then G exhibits G(∞) as a homotopy limitof the diagram G|Aop
0 .
Let W : A → A be a cofibrant replacement for the functor a 7→ C × Y (a).
Let G0 : Aop → Set∆ be defined by the formula G0 (a) = MapA (W (a), X).
Define G00 : B → Set∆ by the formula
G00 (a, b) = MapA (C, Z(a)) ×MapA (W (a),Z(a)×Y (b)) MapA (W (a), P (a, b)).
Let π : B → Aop denote projection onto the first factor, so that we havenatural transformations of diagramsαβπ ∗ G ← G00 → π ∗ G0 .
We observe that β induces a trivial Kan fibration G00 (a, b) → G0 (a) for all(a, b) ∈ B. In particular, for a ≤ b the induced map G00 (a, a) → G00 (a, b) is a
homotopy equivalence. Condition (ii) guarantees that α induces a homotopy
equivalence G00 (a, b) → G(a) if a = b and therefore for all (a, b) ∈ B.
Using Lemma A.3.6.14, we conclude that α and β exhibit G and G0 as
homotopy right Kan extensions of G00 along π. In particular, G and G0 are
equivalent in the homotopy category hFun(Aop , A). Assumptions (iii) and(iv) guarantee that W exhibits W (∞) as the homotopy colimit of W |A0 .
Using Proposition A.3.3.12, we deduce that G0 exhibits G0 (∞) as the homotopy limit of G0 |Aop
0 . It follows that G exhibits G(∞) as the homotopy limitof G|Aop,asdesired.
0
We conclude this section with an application of Proposition A.3.6.15.
Proposition A.3.6.16. Let S be an excellent model category in which themonoidal structure is given by the Cartesian product. Let A be a combinatorial S-enriched model category, A = A0 ∪ {∞} a partially ordered set witha largest element ∞, and {Ca }a∈A a diagram of small S-enriched categoriesaindexed by A. Let U ⊆ A be a chunk. For each a ∈ A, let UCf denote theCaCafull subcategory of U ⊆ A spanned by the projectively fibrant diagrams
and let Wa denote the collection of weak equivalences in Va . Assume that(a) For each a ∈ A, the S-enriched category Ca is cofibrant and U is a
Ca -chunk of A.
(b) The diagram {Ca }a∈A exhibits C∞ as a homotopy colimit of the diagram {Ca }a∈A0 .
(c) The chunk U is small.

highertopoi

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C∞
−1
−1
a
Then the induced diagram {UCf [Wa ]}a∈A exhibits Uf [W∞ ] as a homo−1atopy limit of the diagram {UCf [Wa ]}a∈A0 .

Before proving Proposition A.3.6.16, we need a simple lemma.
Lemma A.3.6.17. Let S be an excellent model category, A a combinatorial
S-enriched model category, and U ⊆ A a chunk. Let Uf denote the fullsubcategory of U spanned by those objects which are fibrant in A and let
W denote the collection of weak equivalences in Uf . Then the induced map
U◦ → Uf [W −1 ] is a weak equivalence of S-enriched categories.
Proof. Let W0 = W ∩ U◦ . Since every weak equivalence in U◦ is actually anequivalence, we conclude that the induced map U◦ → U◦ [W0−1 ] is a weakequivalence. It will therefore suffice to prove that the map i : U◦ [W0−1 ] →Uf [W −1 ] is a weak equivalence. Let F be an S-enriched cofibrant replacement functor which carries U to itself, so that F induces a map j : Uf [W −1 ]to U◦ [W0−1 ]. We claim that j is a homotopy inverse to i. To prove this, weobserve that there is a natural transformation α : F → id, which we canidentify with a map
h : Uf ⊗[1]S → Uf .
Let W00 be the collection of all morphisms in Uf ⊗[1]S of the form e ⊗ id,
where e is an equivalence in Uf , and let W10 be the collection of all morphismsof Uf ⊗[1]S of the form id ⊗g, where g : 0 → 1 is the tautological morphismin [1]S . Let W 0 = W00 ∪ W10 , so that h determines a maph : (Uf ⊗[1]S )[W 0−1] → Uf [W −1 ].

We will prove that h determines a homotopy from the identity to j ◦ i, sothat j is a left homotopy inverse to i. Applying the same argument to therestriction h| U◦ ⊗[1]S will show that j is a right homotopy inverse to i.
To prove that h gives the desired homotopy, it will suffice to show thatthe inclusions {0}, {1} ,→ [1]S induce weak equivalences
Uf [W −1 ] → (Uf ⊗[1]S )[W 0−1].

This follows immediately from Corollary A.3.4.11 and Lemma A.3.5.14.
Proof of Proposition A.3.6.16. Let B = {(a, b) ∈ Aop × A : a ≥ b}. For each−1a(a, b) ∈ B, we define P (a, b) = (UCf × Cb )[Va,b ], where Va,b is the collection
Caof all morphisms of Uf × Cb of the form e ⊗ idC , where e ∈ Wa and C ∈ Cb .
We have an evident family of diagrams σ(a, b):
P (a, b)x

−1
aUCf [Wa ] × Cb$
Uf [W ],

where Uf denotes the full subcategory of U spanned by the fibrant objects
and W is the collection of weak equivalences in Uf ⊆ A. To complete thehighertopoi

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proof, it will suffice to show that the hypotheses of Proposition A.3.6.15 aresatisfied. Condition (i) follows from Lemma A.3.5.14, condition (iii) from
Theorem A.3.5.15, and condition (iv) from (b). To prove (ii), we observethat for each a ∈ A, the diagram σ(a, a) is weakly equivalent to the diagram(UCa )◦ × Caw
(UCa )◦ × Ca

∼

%

U◦ .

This diagram exhibits (UCa )◦ as a weak exponential of U◦ by Ca by Corollary
A.3.4.14.
Corollary A.3.6.18. Let S be an excellent model category in which themonoidal structure is given by the Cartesian product. Let A be a combinatorial S-enriched model category, let A = A0 ∪ {∞} be a partially ordered set
with a largest element ∞, and let {Ca }a∈A be a diagram of small S-enrichedcategories indexed by A.
a
For each a ∈ A, let ACf denote the collection of projectively fibrant objects
aof ACa and let Wa denote the collection of weak equivalences in AC
f . Assume
that the diagram {Ca }a∈A exhibits C∞ as a homotopy colimit of the diagram
C∞
−1
−1
a
]
{Ca }a∈A0 . Then the induced diagram {AC[W∞
f [Wa ]}a∈A exhibits A
Ca−1
as a homotopy limit of the diagram {Af [Wa ]}a∈A0 .
Proof. Without loss of generality, we may suppose that each Ca is cofibrant.
The proof of Proposition A.2.8.2 shows that there exists a (small) regular cardinal κ such that the collection of homotopy limit diagrams in Fun(A, CatS )is stable under κ-filtered colimits. This cardinal depends only on A and S
and remains invariant if we enlarge the universe. Using Lemma A.3.4.15,
we can write A as a κ-filtered union of full subcategories U ⊆ A, where U
is a Ca -chunk for each a ∈ A. We now conclude by applying Proposition
A.3.6.16.
A.3.7 Localizations of Simplicial Model Categories
Let A and A0 be two model categories with the same underlying category.
We say that A0 is a (Bousfield) localization of A if the following conditions
are satisfied:(C) A morphism f of C is a cofibration in A if and only if f is a cofibrationin A0 .
(W ) If a morphism f of C is a weak equivalence in A, then f is a weakequivalence in A0 .
It then also follows that
(F ) If a morphism f of C is a fibration in A0 , then f is a fibration in A.

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Our goal in this section is to study the localizations of a fixed model
category A and to relate this to our study of localizations of presentable∞-categories (§5.5.4).
Let A be a simplicial model category. Let hA be the homotopy categoryof A obtained from A by inverting all weak equivalences. Alternatively, we
can obtain hA by first passing to the full subcategory A◦ ⊆ A spanned by
the fibrant-cofibrant objects and then passing to the homotopy category of
the simplicial category A◦ . From the second point of view, we see that hA
has a natural enrichment over the homotopy category H: if X, Y ∈ hA are
represented by fibrant-cofibrant objects X, Y ∈ A, then we let
MaphA (X, Y ) = [MapA (X, Y )].
Here [K] ∈ H denotes the object of H represented by a Kan complex K. In
fact, this description is accurate if we assume only that X is cofibrant andY fibrant.
Let S be a collection of morphisms in hA. Then(i) We will say that an object Z ∈ hA is S-local if, for every morphism
f : X → Y in S, the induced map
MaphA (Y, Z) → MaphA (X, Z)
is a homotopy equivalence. We say that an object Z ∈ A is S-local if
its image in hA is S-local.
(ii) We will say that a morphism f : X → Y of hA is an S-equivalence if,for every S-local object Z ∈ hA, the induced map
MaphA (Y, Z) → MaphA (X, Z)
is a homotopy equivalence. We say that a morphism f in A is an
S-equivalence if its image in hA is an S-equivalence.
If S is a collection of morphisms in A with image S in hA, we will apply
the same terminology: an object of A (or hA) is said to be S-local if it is
S-local, and a morphism of A (or hA) is said to be an S-equivalence if it isan S-equivalence.
Lemma A.3.7.1. Let A be a left proper simplicial model category, let S bea collection of morphisms in hA, and let i : A → B be a cofibration in A.
The following conditions are equivalent:(1) The map i is an S-equivalence.
(2) For every fibrant object X ∈ A which is S-local, the map i induces atrivial Kan fibration MapA (B, X) → MapA (A, X).
Proof. Choose a trivial fibration f : A0 → A, where A0 is cofibrant, andchoose a factorizationi0

f0

A0 → B 0 → B

highertopoi

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APPENDIXof i ◦ f , where i0 is a cofibration and f 0 is a trivial fibration. We have acommutative diagram
A0

i0

/ B0

f


A

i /

A

`

f0

g

A0 B

0 j

$

/ B.

Since f is a weak equivalence and i0 is a cofibration, the left properness of A
guarantees that g is a weak equivalence. It follows from the two-out-of-threeproperty that j is also a weak equivalence.
Suppose first that (1) is satisfied. Let X be an S-local fibrant object of
A. The map p : MapA (B, X) → MapA (A, X) is a Kan fibration. We wish
to show that p is a trivial Kan fibration. Our assumption that X is S-local
guarantees that the map q 0 : MapA (B 0 , X) → MapA (A0 , X) is a homotopy
equivalence and therefore a trivial fibration (since i0 is a cofibration). The
mapaq : MapA (AB 0 , X) → MapA (A, X)
A0

is a pullback of q 0 and therefore also a trivial fibration. To show that p is atrivial Kan fibration, it will suffice to show that for every t : A → X, thefiber p−1 {t} is a contractible Kan complex. Since the corresponding fiberq −1 {t} is contractible, it will suffice to show that composition with j induces
a homotopy equivalencep−1 {t} → q −1 {t}.
This is clear since j is a weak equivalence between cofibrant objects of thesimplicial model category AA/ .
Now assume that (2) holds. We wish to show that i is an S-equivalence.
For this, it suffices to show that for every fibrant S-local object X ∈ A, themap
q 0 : MapA (B 0 , X) → MapA (A0 , X)
is a trivial Kan fibration. The preceding argument shows that the fiber of q 0
over a morphism t0 : A0 → X is contractible, provided that t0 factors as acomposition
f

t

u

v

A0 → A → X.
To complete the proof, it suffices to show that the same result holds for an
arbitrary vertex t0 of MapA (A0 , X). The map t0 factors as a composition
A0 → Y → X,
where u is a cofibration and v is a trivial fibration. We have a commutative
diagram/ MapA (A0 , Y )MapA (B 0 , Y )MapA (B 0 , X)
/ MapA (A0 , X)highertopoi

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APPENDIX

in which the vertical arrows are trivial Kan fibrations. It will therefore suffice0
to show that the fiber Map` A (B , Y ) ×MapA (A0 ,Y ) {u} is contractible. Choose0a trivial cofibrationA A Y → Z, where Z is fibrant. We observe that the`map Y → A0 A Y is the pushout of a weak equivalence by a cofibrationand therefore a weak equivalence (since A is left proper). It follows that themap Y → Z is a weak equivalence between fibrant objects of A. We have acommutative diagram/ MapA (A0 , Y )MapA (B 0 , Y )MapA (B 0 , Z)

q 00

/ MapA (A0 , Z)

in which the vertical maps are homotopy equivalences and the horizontal
maps are Kan fibrations. It will therefore suffice to show that the fiber ofuq 00 is contractible when taken over the composite map t00 : A0 → Y → Z.
00
We now observe that t factors through A, so that the desired result follows
from the first part of the proof.
Corollary A.3.7.2. Let A and B be simplicial model categories and suppose
we are given a simplicial adjunction
Ao

F

/ B.

G

Assume that B is left proper. The following conditions are equivalent:(1) The adjunction between F and G is a Quillen adjunction.
(2) The functor F preserves cofibrations, and the functor G preserves fibrant objects.
Proof. The implication (1) ⇒ (2) is obvious. Conversely, suppose that (2) issatisfied. We wish to prove that F is a left Quillen functor. Since F preserves
cofibrations, it will suffice to show that for every trivial cofibration u : A →A0 in A, the image F u is a weak equivalence in B. Applying Lemma A.3.7.1
in the case S = ∅, it will suffice to prove the following: for every fibrantobject B ∈ B, the induced map
MapB (F A0 , B) → MapB (F A, B)
is a trivial Kan fibration. Since F and G are adjoint simplicial functors, this isequivalent to the requirement that the map MapA (A0 , GB) → MapA (A, GB)
be a trivial Kan fibration, which follows from our assumption that u is atrivial cofibration in A and that GB ∈ A is fibrant.
Proposition A.3.7.3. Let A be a left proper combinatorial simplicial model
category and let S be a (small) set of cofibrations in A. Let S −1 A denotethe same category, with the following distinguished classes of morphisms:

highertopoi

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APPENDIX(C) A morphism g in S −1 A is a cofibration if it is a cofibration whenregarded as a morphism in A.
(W ) A morphism g in S −1 A is a weak equivalence if it is an S-equivalence.
Then(1) The above definitions endow S −1 A with the structure of a combinatorial simplicial model category.
(2) The model category S −1 A is left proper.
(3) An object X ∈ A is fibrant in S −1 A if and only if X is S-local andfibrant in A.
Proof. Enlarging S if necessary, we may assume:(a) For every morphism f : A → B in S and every n ≥ 0, the induced mapa(A × ∆n )(B × ∂ ∆n ) → B × ∆n
A×∂ ∆nbelongs to S.
(b) The set S contains a collection of generating trivial cofibrations for A.
It follows that an object X ∈ A is fibrant and S-local if and only if it has theextension property with respect to every morphism in S. Since S ⊆ C ∩ W ,we deduce that every fibrant object of S −1 A is S-local and fibrant in A.
The converse follows from Lemma A.3.7.1; this proves (3).
To prove (1), it will suffice to show that the classes C and W satisfy the
hypotheses of Proposition A.2.6.10 (the compatibility of the simplicial structure on S −1 A with its model structure follows immediately from Proposition
A.3.1.7). We observe that Lemma A.3.7.1 implies that C ∩ W is a weakly
saturated class of morphisms in A. The only other nontrivial point is toshow that W is an accessible subcategory of A[1] .
Proposition A.1.2.5 implies the existence of a functor T : A → A and anatural transformation idA → T having the following properties:(i) For every X ∈ A, the object T X ∈ A is fibrant and S-local.
(ii) For every X ∈ A, the map X → T X belongs to the smallest weakly
saturated class of morphisms containing S; in particular, it belongs to
W ∩ C and is therefore an S-equivalence.
(iii) There exists a regular cardinal κ such that T commutes with κ-filteredcolimits.
It follows that a morphism f : X → Y in A is an S-equivalence if andonly if the induced map T f : T X → T Y is an S-equivalence. Since T X andT Y are S-local, Yoneda’s lemma (in the category hA) implies that T f is an
S-equivalence if and only if T f is a weak equivalence in A. It follows thathighertopoi

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APPENDIX

W is the inverse image under T of the collection of weak equivalences in A.
Corollaries A.2.6.7 and A.2.6.8 imply that W is an accessible subcategory ofA[1] , as desired. This completes the proof of (1).
We now prove (2). We need to show that the collection of S-equivalences in
A is stable under pushouts by cofibrations. We observe that every morphism
f : X → Z admits a factorization
f0

f 00

X → Y → Z,
where f 0 is a cofibration and f 00 is a weak equivalence in A (in fact, we canchoose f 00 to be a trivial fibration in A). If f is an S-equivalence, then f 0 is
an S-equivalence, so that f 0 ∈ C ∩ W . It will therefore suffice to show thatC ∩ W and the class of weak equivalences in A are stable under pushouts
by cofibrations. The first follows from the assertion that C ∩ W is weakly
saturated, and the second from the assumption that A is left proper.
Proposition A.3.7.4. Let A be a left proper combinatorial simplicial model
category. Then
(1) Every combinatorial localization of A has the form S −1 A, where S issome (small) set of cofibrations in A.
(2) Given two (small) sets of cofibrations S and T , the localizations S −1 A
and T −1 A coincide if and only if the class of S-local objects of hA
coincides with the class of T -local objects of hA.
Proof. The “if” direction of (2) is obvious, and the converse follows fromthe characterization of the fibrant objects of S −1 A given in Proposition
A.3.7.3. We now prove (1). Let B be a combinatorial model category which
is a localization of A and let S be a set of generating trivial cofibrations
for B. We claim that B = S −1 A. The cofibrations of S −1 A and B coincide.
Moreover, the collection of trivial cofibrations in S −1 A is a weakly saturated
class of morphisms which contains S and therefore contains every trivial
cofibration in B. To complete the proof, it will suffice to show that every
trivial cofibration f : X → Y in S −1 A is a trivial cofibration in B.
Choose a diagram
X0

X

f0

f

/ Y0

/ Y,

where X 0 is cofibrant, f 0 is a cofibration, and the vertical maps are weakequivalences in A. Then f 0 is a trivial cofibration in S −1 A, and it will suffice
to show that f 0 is a trivial cofibration in B. For this, it will suffice to showthat for every fibrant object Z ∈ B, the map
MapB (Y 0 , Z) → MapB (X 0 , Z)highertopoi

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911

APPENDIX

is a trivial fibration. In view of Lemma A.3.7.1, it will suffice to show that
Z is S-local and fibrant as an object of A. The second claim is obvious, andthe first follows from the fact that S consists of trivial cofibrations in B.
Remark A.3.7.5. In the situation of Proposition A.3.7.4, we may assume
that for every cofibration f : A → B in S, the objects A and B are themselves
cofibrant. To see this, choose for each cofibration f : A → B in S a diagram
A0

gf

/ B0

u


A

f

0

/A

`

w

v

A0 B

0f

00

#

/B

as in the proof of Lemma A.3.7.1, so that u and w are trivial cofibrations,f = f 00 ◦ f 0 , and gf is a cofibration between cofibrant objects. Then gf isa trivial cofibration in S −1 A. We claim that the localizations S −1 A andT −1 A coincide, where T = {gf }f ∈S . To prove this, it will suffice to showthat for each f ∈ S, every gf -local fibrant object X ∈ A is also f -local.
Suppose that X is gf -local. We wish to prove that the map
p : MapA (B, X) → MapA (A, X)
is a trivial Kan fibration. Since p is automatically a Kan fibration, it will
suffice to show that the fiber p−1 {t} is contractible for every morphism t :−1
A → X. Since X is gf -local, we deduce that` the0 fiber q {t} is contractible,
where q is the projection map MapA (A A0 B , X) → MapA (A, X). It will
therefore suffice to show that f 00 induces a homotopy equivalence of fibersa
MapAA/ (B, X) → MapAA/ (AB 0 , X).
A000

This is clear because f is a weak equivalence between cofibrant objects ofthe simplicial model category AA/ .
Proposition A.3.7.6. Let C be an ∞-category. The following conditions
are equivalent:(1) The ∞-category C is presentable.
(2) There exists a combinatorial simplicial model category A and an equivalence C ' N(A◦ ).
Proof. According to Theorem 5.5.1.1 and Proposition 5.5.4.15, C is presentable if and only if there exists a small simplicial set K, a small set S ofmorphisms in P(K), and an equivalence C ' S −1 P(K). Let D be the simplicial category C[K]op and let B be the category SetD
∆ of simplicial functors
D → Set∆ endowed with the injective model structure. Proposition 4.2.4.4
implies that there is an equivalence P(K) ' N(B◦ ). Moreover, Propositions
A.3.7.3 and A.3.7.4 imply that there is a bijective correspondence between
accessible localizations of P(K) (as a presentable ∞-category) and combinatorial localizations of B (as a model category). This proves the implication

highertopoi

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912

APPENDIX(1) ⇒ (2). Moreover, it also shows that (2) ⇒ (1) in the special case where
A is a localization of a category of simplicial presheaves.
We now complete the proof by invoking the following result, proven in [19]:for every combinatorial model
category A, there exists a small category D, a
op
set S of morphisms of SetD,
anda Quillen equivalence of A with S −1 SetD∆∆.
Moreover, the proof given in [19] shows that when A is a simplicial model
category, then F and G can be chosen to be simplicial functors.
Remark A.3.7.7. Let A and B be combinatorial simplicial model categories. Then the underlying ∞-categories N(A◦ ) and N(B◦ ) are equivalent
if and only if A and B can be joined by a chain of simplicial Quillen equivalences. The “only if” assertion follows from Corollary A.3.1.12, and the “if”
direction can be proven using the methods described in [19].
Proposition A.3.7.8. Let A be a left proper combinatorial simplicial model
category and let C = N(A◦ ) denote its underlying ∞-category. Suppose that
C0 ⊆ C is an accessible localization of C and let L : C → C0 denote a left
adjoint to the inclusion.
Then there exists a localization A0 of A satisfying the following conditions:(1) An object X ∈ A0 is fibrant if and only if it is fibrant in A and theassociated object of the homotopy category hA ' hC belongs to the fullsubcategory hC0 .
(2) A morphism f : X → Y in A0 is a weak equivalence if and only if thefunctor L : hC → hC0 carries f to an isomorphism in the homotopy
category hC0 .
Proof. According to Proposition A.3.7.6, the ∞-category C is presentable.
The results of §5.5.4 imply that we can write C0 = S −1 C for some smallcollection of morphisms S in C. We then take Se to be a collection of representatives for the elements of S as cofibrations between cofibrant objects of
A and let A0 denote the localization Se−1 A.
We conclude this section by establishing a universal property enjoyed by
the localization of a combinatorial simplicial model category.
Proposition A.3.7.9. Suppose we are given a simplicial Quillen adjunction
Ao

F

/B

G

between left proper combinatorial simplicial model categories and let A0 be a
Bousfield localization of A. The following conditions are equivalent:(1) The adjoint functors F and G determine a Quillen adjunction between
A0 and B.
(2) Let α be a morphism in A which is a weak equivalence in A0 . Thenthe left derived functor LF : hA → hB carries α to an isomorphismin the homotopy category hB.

highertopoi

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913

APPENDIX(3) For every fibrant object X ∈ B, the image GX is a fibrant object ofA0 .
Proof. The implication (1) ⇒ (2) is obvious, and the implication (3) ⇒ (1)follows from Corollary A.3.7.2. We will complete the proof by showing that(2) ⇒ (3). According to Proposition A.3.7.4 and Remark A.3.7.5, we may
suppose that A0 = S −1 A, where S is a small collection of cofibrations between cofibrant objects of A. Let X be a fibrant object of B; we wish
to show that GX is a fibrant object of A0 . Since GX is fibrant in A,
it will suffice to show that GX is S-local (Proposition A.3.7.3). In otherwords, we must show that if α : A → B belongs to S, then the induced
map p : MapA (B, GX) → MapA (A, GX) is a weak homotopy equivalence. Since F and G are simplicial functors, we can identify p with themap MapB (F B, X) → MapB (F A, X). To prove that p is a weak homotopy
equivalence, it will suffice to show that F (α) is a weak equivalence between
cofibrant objects of B. This follows immediately from assumption (2) (because α is a cofibration between cofibrant objects of A, we can identify F (α)
with the left derived functor LF (α)).
Corollary A.3.7.10. Let A and B be left proper combinatorial simplicial
model categories and suppose we are given a simplicial Quillen adjunction
Ao

F

/ B.

G

Then
(1) There exists a new left proper combinatorial simplicial model structure
A0 on the category A with the following properties:(C) A morphism α in A0 is a cofibration if and only if it is a cofibration in A.
(W ) A morphism α in A0 is a weak equivalence if and only if the leftderived functor LF carries α to an isomorphism in the homotopy
category hB.
(F ) A morphism α in A0 is a fibration if and only if it has the right
lifting property with respect to every morphism in A0 satisfying(C) and (W ).
(2) The functors F and G determine a new simplicial Quillen adjunction
A0 o

F0
G0

/ B.

(3) Suppose that the right derived functor RG is fully faithful. Then theadjoint pair (F 0 , G0 ) is a Quillen equivalence.

highertopoi

April 9, 2017

914

APPENDIX

Proof. The functors F and G determine a pair of adjoint functors
N A◦ ofg/ N B◦

between the underlying ∞-categories (see Proposition 5.2.4.6), which arethemselves presentable (Proposition A.3.7.6). Let S be the collection of allmorphisms u in N A◦ such that f (u) is an equivalence in N B◦ . Proposition 5.5.4.16 implies that S is generated (as a strongly saturated class ofmorphisms) by a small subset S ⊆ S. Without loss of generality, we may
suppose that the morphisms of S are represented by some (small) collection
T of cofibrations between cofibrant objects of A. Let A0 = T −1 A. We claimthat A0 satisfies the description given in (1). In other words, we claim thata morphism α in A is a T -equivalence if and only if the left derived functor
LF carries α to an isomorphism in hB. Without loss of generality, we may
suppose that α is a morphism between fibrant-cofibrant objects of A, sothat we can view α as a morphism in the ∞-category N A◦ . In this case,both conditions on α are equivalent to the requirement that α belong to
S. This completes the proof of (1). Assertion (2) follows immediately from
Proposition A.3.7.9.
We now prove (3). Note that the homotopy category hA0 can be identified
with a full subcategory of the homotopy category hA and that under thisidentification the left derived functor LF restricts to the left derived functor
LF 0 . It follows that for every fibrant object X ∈ B, the counit map(LF 0 )(RG0 )X ' (LF 0 )(GX) ' (LF )(GX) ' (LF )(RG)X ' X
is an isomorphism in hB (where the last equivalence follows from our assumption that RG is fully faithful). It follows that the functor RG0 is fully
faithful. To complete the proof, it will suffice to show that the left derived
functor LF 0 is conservative. In other words, we must show that if α : X → Y
is a morphism in A0 , then α is a weak equivalence if and only if LF (α) is an
isomorphism in B; this follows immediately from (1).

highertopoi

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Bibliography[1] Adámek, J., and J. Rosicky. Locally Presentable and Accessible Categories. Cambridge University Press, Cambridge, 1994.
[2] Artin, M. Théorie des Topos et Cohomologie Étale des Schémas. SGA
4. Lecture Notes in Mathematics 269. Springer-Verlag, Berlin and New
York, 1972.
[3] Artin, M., and B. Mazur. Étale Homotopy. Lecture Notes in Mathematics 100. Springer-Verlag, Berlin and New York, 1969.
[4] Baez, J., and M. Shulman. Lectures on n-categories and cohomology.
Available for download at math.CT/0608420.
[5] Berkovich, V. Spectral Theory and Analytic Geometry over NonArchimedean Fields. Mathematical Surveys and Monographs 33. American Mathematical Society, Providence, R.I., 1990.
[6] Beilinson, A., Bernstein, J., and P. Deligne. Faisceaux pervers.
Astérisque 100, 1982.
[7] Bergner, J.E. A model category structure on the category of simplicial
categories. Trans. Amer. Math. Soc. 359, 2007, 2043–2058.
[8] Bergner, J.E. A
math.AT/0610239.

surveyof(∞, 1)-categories.

Available

at[9] Bergner, J.E. Rigidification of algebras over multi-sorted theories. Algebr. Geom. Topol. 6, 2006, 1925–1955.
[10] Boardman, J.M., and R.M. Vogt. Homotopy Invariant Structures on
Topological Spaces. Lecture Notes in Mathematics 347. Springer-Verlag,
Berlin and New York, 1973.
[11] Bourn, D. Sur les ditopos. C.R. Acad. Sci. Paris Ser. A 279, 1974, 911–913.
[12] Bousfield, A.K. The localization of spaces with respect to homology.
Topology 14, 1975, 133–150.
[13] Breen, L. On the classification of 2-gerbes and 2-stacks. Asterisque 225,
1994, 1–160.

highertopoi

916

April 9, 2017

BIBLIOGRAPHY[14] Brown, K. Abstract homotopy theory and generalized sheaf cohomology.
Trans. Amer. Math. Soc. 186, 1973, 419–458.
[15] Chapman, T. Lectures on Hilbert Cube Manifolds. American Mathematical Society, Providence, R.I., 1976.
[16] Cordier, J.M. Sur la notion de diagramme homotopiquement cohérent.
Cah. Topol. Géom. Différ. 23 (1), 1982, 93–112.
[17] Cordier, J.M., and T. Porter. Homotopy coherent category theory. Trans.
Amer. Math. Soc. 349 (1), 1997, 1–54.
[18] Dydak, J., and J. Segal. Shape Theory. Lecture Notes in Mathematics
688. Springer-Verlag, Berlin and New York, 1978.
[19] Dugger, D. Combinatorial model categories have presentations. Adv.
Math. 164, 2001, 177–201.
[20] Dugger, D., S. Hollander, and D. Isaksen. Hypercovers and simplicial
presheaves. Math. Proc. Camb. Phil. Soc. 136 (1), 2004, 9–51.
[21] Dwyer, W.G. and D.M. Kan. Simplicial localizations of categories. J.
Pure Appl. Algebra 17, (1980), 267–284.
[22] Dwyer, W.G., and D.M. Kan. Homotopy theory of simplicial groupoids.
Nederl. Akad. Wetensch. Indag. Math. 46 (4), 1984, 379–385.
[23] Dwyer, W.G., and D. M. Kan. Realizing diagrams in the homotopy category by means of diagrams of simplicial sets. Proc. Amer. Math. Soc.
91 (3), 1984, 456–460.
[24] Dwyer, W.G., P.S. Hirschhorn, D. Kan, and J. Smith. Homotopy Limit
Functors on Model Categories and Homotopical Categories. Mathematical Surveys and Monographs 113. American Mathematical Society,
Providence, R.I., 2004.
[25] Ehlers, P.J., and T. Porter. Ordinal subdivision and special pasting inquasicategories. Adv. Math. 217, 2007, 489–518.
[26] Eilenberg, S., and N.E. Steenrod. Axiomatic approach to homology theory. Proc. Nat. Acad. Sci. USA 31, 1945, 117–120.
[27] Engelking, R. Dimension Theory. North-Holland, Amsterdam, Oxford,
and New York, 1978.
[28] Freitag, E., and R. Kiehl. Étale Cohomology and the Weil Conjectures.
Springer-Verlag, Berlin and New York, 1988.
[29] Fresnel, J., and M. van der Put. Rigid Analytic Geometry and Its Applications. Boston, Birkhauser, 2004.

highertopoi

April 9, 2017

BIBLIOGRAPHY

917[30] Friedlander, E. Étale Homotopy of Simplicial Schemes. Annals of Mathematics Studies 104. Princeton University Press, Princeton, N.J.,1982.
[31] Giraud, J. Cohomologie non abelienne. Springer Verlag, Berlin and New
York, 1971.
[32] Goerss, P., and J.F. Jardine. Simplicial Homotopy Theory. Progress inMathematics, Birkhauser, Boston, 1999.
[33] Gordon, R., A.J. Power, and R. Street. Coherence for Tricategories.
Mem. Amer. Math. Soc. 117(558), American Mathematical Society,
Providence, R.I., 1995.
[34] Grothendieck, A. Sur quelques points d’algebra homologique. Tohoku
Math. J. 9, 1957, 119–221.
[35] Grothendieck, A. A la poursuite des champs. Unpublished letter to D.
Quillen.
[36] Günther, B. The use of semisimplicial complexes in strong shape theory.
Glas. Mat. 27(47), 1992, 101–144.
[37] Haver, W. Mappings between ANRs that are fine homotopy equivalences.
Pacific J. Math. 58, 1975, 457–461.
[38] Hirschhorn, P. Model Categories and Their Localizations. Mathematical
Surveys and Monographs 99. American Mathematical Society, Providence, R.I., 2003.
[39] Hirschowitz, A., and C. Simpson. Descente pour les n-champs. Available
for download at math.AG/9807049.
[40] Hovey, M. Model Categories. American Mathematical Society, Providence, R.I., 1998.
[41] Jardine, J.F. Simplicial Presheaves. J. Pure Appl. Algebra 47(1), 1987,
35–87.
[42] Johnstone, P. Stone Spaces. Cambridge University Press, Cambridge,
1982.
[43] Joyal, A. Quasi-categories and Kan complexes. J. Pure Appl. Algebra
175, 2005, 207–222.
[44] Joyal, A. Theory of quasi-categories I. In preparation.
[45] Joyal, A., and M. Tierney. Strong Stacks and Classifying Spaces. Lecture Notes in Mathematics 1488. Springer-Verlag, Berlin and New York,
1991, 213–236.

highertopoi

April 9, 2017

918

BIBLIOGRAPHY[46] Kashiwara, M., and P. Schapira. Sheaves on Manifolds. Fundamental
Principles of Mathematical Sciences 292. Springer-Verlag, Berlin and
New York, 1990.
[47] Lazard, D. Sur les modules plats. C.R. Acad. Sci. Paris 258, 1964, 63136316.
[48] Leinster, T. A survey of definitions of n-category. Theor. Appl. Categ.
10(1), 2002, 1–70.
[49] Leinster, T. Higher Operads, Higher Categories. London MathematicalSociety Lecture Note Series 298. Cambridge University Press, Cambridge, 2004.
[50] Lurie, J. Higher Algebra.
[51] Lurie, J. Spectral Algebraic Geometry. In preparation.
[52] Lurie, J. Elliptic Cohomology. In preparation.
[53] MacLane, S. Categories for the Working Mathematician. 2nd ed. Graduate Texts in Mathematics 5. Springer-Verlag, Berlin and New York,
1998.
[54] MacLane, S., and I. Moerdijk. Sheaves in Geometry and Logic. SpringerVerlag, Berlin and New York, 1992.
[55] Makkai, M., and R. Pare. Accessible Categories. Contemporary Mathematics 104. American Mathematical Society, Providence, R.I., 1989.
[56] Mardesic, S., and J. Segal. Shape Theory. North-Holland, Amsterdam,
1982.
[57] May, J.P. Simplicial Objects in Algebraic Topology. Chicago Lectures inMathematics. University of Chicago Press, Chicago, 1993.
[58] May, J.P., and J. Sigurdsson. Parametrized Homotopy Theory. Mathematical Surveys and Monographs 132, American Mathematical Society,
Providence, R.I., 2006.
[59] Moerdijk, I., and J. Vermeulen. Proper Maps of Toposes. Memoirs of
the American Mathematical Society 148(705). American MathematicalSociety, Providence, R.I., 2000.
[60] Munkres, J. Topology. Prentice Hall, Englewood Cliffs, N.J., 1975.
[61] Nichols-Barrer, J. On Quasi-Categories as a Foundation for Higher Algebraic Stacks. Doctoral dissertation, Massachusetts Institute of Technology, Cambridge, Mass., 2007.
[62] Polesello, P., and I. Waschkies. Higher monodromy. Homology Homotopy Appl. 7(1), 2005, 109–150.

highertopoi

BIBLIOGRAPHY

April 9, 2017919

[63] Quillen, D. Higher Algebraic K-theory I. In ”Algebraic K-theory I:
Higher K-Theories,” Lecture Notes in Mathematics 341. Springer,
Berlin, 1973.
[64] Quillen, D. Homotopical Algebra. Lectures Notes in Mathematics 43.
SpringerVerlag, Berlin and New York, 1967.
[65] Rezk, C. A model for the homotopy theory of homotopy theory. Trans.
Amer. Math. Soc. 35(3), 2001, 973–1007.
[66] Rezk, C., Shipley, B., and S. Schwede. Simplicial structures on model
categories and functors. Amer. J. Math. 123, 2001, 551–575.
[67] Rosicky, J. On Homotopy Varieties. Adv. Math. 214(2), 2007, 525–550.
[68] Schwartz, L. Unstable Modules over the Steenrod Algebra and Sullivan’s
Fixed Point Set Conjecture. Chicago Lectures in Mathematics. Chicago
University Press, Chicago, 1994.
[69] Segal, G. Classifying spaces and spectral sequences. Inst. Hautes Etudes
Sci. Publ. Math. 34, 1968, 105–112.
[70] Serre, J.P. Cohomologie Galoisienne. Lecture Notes in Mathematics 5.
Springer-Verlag, Berlin and New York, 1964.
[71] Simpson, C. A Giraud-type characterization of the simplicial categories
associated to closed model categories as ∞-pretopoi. Available for download at math.AT/9903167.
[72] Simpson, C. A closed model structure for n-categories, internal Hom,n-stacks and generalized Seifert-Van Kampen. Available for download
at math.AG/9704006.
[73] Spaltenstein, N. Resolutions of Unbounded Complexes. Compos. Math.
65(2), 1988, 121–154.
[74] Stasheff, J. Homotopy associativity of H-spaces I, II. Trans. Amer.
Math. Soc. 108, 1963, 275–312.
[75] Street, R. Two-dimensional sheaf theory. J. Pure Appl. Algebra 23(3),1982, 251–270.
[76] Tamsamani, Z. On non-strict notions of n-category and n-groupoid via
multisimplicial sets. Available for download at math.AG/9512006.
[77] Toën, B. Vers une axiomatisation de la théorie des catégories supériures.
K-theory 34(3), 2005, 233–263.
[78] Toën, B. Vers une interpretation Galoisienne de la theorie del’homotopie. Cah. topol. geom. différ. categ. 43, 2002, 257–312.

highertopoi

920

April 9, 2017

BIBLIOGRAPHY[79] Toën, B., and G. Vezzosi. Segal topoi and stacks over Segal categories.
Available for download at math.AG/0212330.
[80] Toën, B. and G. Vezzosi. Homotopical algebraic geometry I: Topos theory. Advances in Mathematics 193 (2005), no. 2, 257-372.
[81] van den Dries, L. Tame Topology and O-minimal Structures. Cambridge
University Press, Cambridge, 1998.
[82] Wall, C.T.C. Finiteness conditions for CW-complexes. Ann. Math. 81,
1965, 56–69.
[83] Čech and Steenrod Homotopy Theories with Applications to Geometric
Topology. Lecture Notes in Mathematics 542. Springer-Verlag, Berlinand New York, 1976.

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April 9, 2017

General Indexabelian category
Grothendieck, 765
absolute neighborhood retract, 776
accessible
adjoint functors, 447
coproducts, 445
functor, 421
functor categories, 430
homotopy fiber products, 443∞-category, 419∞-category of sections, 452
localization, 454
overcategories, 444
products, 445
subcategory, 447, 814
undercategories, 439
adjoint functor
between categories, 330
and (co)limits, 345
and composition, 338
existence of, 341
between Ind-categories, 405
between ∞-categories, 336
Quillen, 349
and unit transformations, 339
adjoint functor theorem, 462
adjunction, 336
Quillen, 813
algebraic morphism, 631
anodyne, 53
inner, 53
left, 53, 63
marked, 147
right, 53
Berkovich space, 780
bicategory, 3bifibration, 139
associated to a correspondence, 142
and smoothness, 234
canonical covering, 591
canonical topology, 592
Cartesianequivalence, 155
locally, 120

Cartesian edge, 115, 118
and composition, 117
and simplicial categories, 119
Cartesian fibration, 121
and categorical fibrations, 209classified by f : S → Catop∞ , 211and functor categories, 151
and overcategories, 127
and pullbacks, 205
and right fibrations, 122
and trivial fibrations, 132
universal, 210
Cartesian model structure, 157
Cartesian transformation, 540
categorical equivalence, 25
and products, 92
weak, 94
categorical fibration, 90
of ∞-categories, 137
category, 1, 783
cofibered in groupoids, 56
enriched, 792
homotopy, 808
model, 805
monoidal, 788
Reedy, 831
of simplices, 253, 534
simplicial, 18
topological, 7Čech nerve, 539
cell-like, 682
map of ∞-topoi, 776
map of topological spaces, 776
chunk, 881
classifying map
for a (co)Cartesian fibration, 211
for a collection of morphisms, 562
for a left fibration, 212
for objects, 562
for relatively κ-compact morphisms,566
for a right fibration, 212
for subobjects, 561
closedimmersion, 756
subtopos, 756

highertopoi

922
coCartesian edge, 118
coCartesian equivalence, 160
coCartesian fibration, 121classified by f : S → Cat∞ , 211
and overcategories, 128
and smoothness, 234
coCartesian model structure, 160
coend, 75, 98
coequalizer, 299
reflexive, 502
cofibrant, 806
cofibration, 806
covariant, 68
injective, 825
projective, 825
Reedy, 837
of simplicial sets, 53, 825
strong, 866
weak, 866
cofinal, 223
weakly, 435
coherent topological space, 679, 773
cohomological dimension, 726
cohomology groupof an ∞-topos, 725
colimit, 47
diagram, 47
in families, 247
finite, 296
functor, 390
in a functor category, 314
homotopy, 257, 874
and homotopy fiber products, 432
of ∞-categories, 218
of ∞-topoi, 600
of presentable ∞-categories, 468
preservation of, 48
relative, 261
of spaces, 219
universal, 526, 528
combinatorial model category, 814
compact
object of a category, 784
compact objectof a category, 391
completely, 324
of a functor ∞-category, 395
of an ∞-category, 392
limits of, 446
compactly generated, 497, 679
complete lattice, 638
completely compact, 324
left fibration, 326
and presheaf ∞-categories, 328
completely regular, 750
cone
left, 68

April 9, 2017

GENERAL INDEX
right, 68cone point, 41
connected
object of an ∞-topos, 657
connective
n-connective morphism, 657
n-connective object, 657
strongly, 740
continuous functor, 392
contravariant equivalence, 71
contravariant fibration, 71
contravariant model structure, 71
coproduct, 294
disjoint, 529
homotopy, 293
of ∞-topoi, 599
corepresentablefunctor, 301, 320, 324, 461, 794
left fibration, 301
correspondence
adjunction, 336
associated functor, 331
associated to a functor, 98
between categories, 96
between ∞-categories, 97
coskeletal, 672
coskeleton, 671
cotensored, 795
counit transformation, 338
covariantcofibration, 68
equivalence, 68, 86
fibration, 68
model structure, 69
covering dimension, 733
cylinder object, 807
degeneracy map, 824
derived functor, 350, 508
left, 813
right, 813
diagonal functor, 260
diagram(co)limit, 47
homotopy coherent, 37
homotopy commutative, 37
dimension
cohomological, 726
covering, 733
Heyting, 739
homotopy, 713
Krull, 738
discrete, 489
effective epimorphism, 530, 583
effective equivalence relation, 530
Eilenberg-MacLane object, 720

highertopoi

GENERAL INDEX
enough points, 681
enriched category, 792
equalizer, 299
equivalenceaway from U , 755
Cartesian, 155
coCartesian, 160
contravariant, 71
covariant, 68
in an ∞-category, 33, 34
pointwise, 80of presheaf ∞-categories, 329
of S-enriched categories, 853
in an S-enriched category, 857
of simplicial categories, 19of topological categories, 17
in a topological category, 17, 33
essentially κ-small∞-category, 417
space, 417
essentially small, 51
essentially surjective, 43étale morphism, 613
excellentmodel category, 859
exponential, 893
weak, 893
extension property, 785
face map, 824
factorization system, 368
fibrant, 806
locally, 857
simplicial category, 18
fibration, 806
Cartesian, 121categorical, 90
coCartesian, 121
contravariant, 71
covariant, 68
injective, 825
inner, 53
Kan, 53, 825
left, 53
local, 857
locally Cartesian, 123
presentable, 463
projective, 825
Reedy, 837
right, 53
strong, 866
topos, 597
trivial, 53
weak, 866
filteredcategory, 377∞-category, 378

April 9, 2017

923
partially ordered set, 377, 784
simplicial set, 379
topological category, 377
filtered colimitof colimit-preserving functors, 500
left exactness of, 390, 765
filtered limitof ∞-topoi, 600
final objectof a category, 44of an ∞-category, 44of the ∞-category of ∞-topoi, 606
uniqueness, 46
fully faithful, 43
functorassociated to a correspondence, 331
colimit, 390
continuous, 392
corepresentable, 301, 461
derived, 350, 508
enriched, 793
fully faithful, 43
between ∞-categories, 39
κ-continuous, 392
κ-right exact, 385
lax monoidal, 790
left exact, 385
localization, 361
monoidal, 791
representable, 458
representable by an object, 300
right exact, 385
fundamental n-groupoid, 3
generation under colimits, 324
generator
projective, 510
geometric morphism, 596
of n-topoi, 650
geometric realization, 75, 539
of simplicial categories, 19of simplicial sets, 8gerbe, ix, 726
banded, 727
Giraud’s axioms
for ∞-topoi, 526
for n-topoi, 636
for ordinary topoi, 524
Giraud’s theorem
for ∞-topoi, 525
for n-topoi, 636
for 0-topoi, 638
Grothendieck abelian category, 765
Grothendieck construction, 891
Grothendieck topology, 573
Grothendieck universe, 51
Grothendieck’s vanishing theorem, 744

highertopoi

April 9, 2017924
nonabelian version, 744
group object, 720
groupoid object, 531
of a category, 531
effective, 531, 539
of an ∞-category, 534, 537
n-efficient, 641
Heyting
algebra, 738
dimension, 739
space, 738
Hilbert cube, 678
homotopy
coproduct, 293
between morphisms in a model category, 808
between morphisms of C, 29
relative to K 0 ⊆ K, 106
homotopy category
enriched over H, 16
of an ∞-category, 29of a model category, 808
of an S-enriched category, 853
of a simplicial category, 19of a simplicial set, 25of spaces, 16of a topological category, 16
homotopy coherence, 37
homotopy colimit, 257, 874
homotopy dimension, 713
finite, 713
of a geometric morphism, 715
locally finite, 716
homotopy groups
in an ∞-topos, 654
homotopy limit, 873
homotopy product, 46
homotopy pullback, 46, 812
homotopy pushout, 811
homotopy right Kan extension, 872
homotopy varieties, 503
horn, 824
inner, 12
hypercomplete∞-topos, 665
object, 665
hypercovering, 670, 671
effective, 671
idempotent
completeness, 303
effective, 309
in an ∞-category, 304
in an ordinary category, 303
idempotent complete, 309
and accessibility, 427

GENERAL INDEX
idempotent completion, 317, 420
universal property of, 319
image of a geometric morphism, 631
Ind-object, 375
characterization of, 403
of an ∞-category, 399∞-bicategory, 5∞-category, x, xiv, 5, 8, 14
accessible, 419
essentially κ-small, 417
as a fibrant object of Set∆ , 136of functors, 94
of Ind-objects, 399
of ∞-categories, 144
locally small, 419
presentable, 311, 452, 453
of presentable ∞-categories, 462
of spaces, 51∞-groupoid, 5, 35
underlying an ∞-category, 35(∞, n)-category, 5∞-topos, 525
elementary, 545
of local systems, 712
of sheaves on a topological space,
677
initial object
homotopy fiber productin a homotopy fiber product, 431
in an ∞-category of sheaves, 581
injective
cofibration, 825
fibration, 825
inner anodyne, 53, 99
inner fibration, 53
and functor categories, 101
inner horn, 12
invertibility hypothesis, 858
irreducible
closed set, 738
topological space, 780
join
of categories, 40of simplicial sets, 40
κ-accessible ∞-category, 419
κ-accessible subcategory, 814
Kanmodel structure, 824
Kan complex, 8
weak, xiv, 14
Kan extension, 260, 272, 285
homotopy, 829, 872
Kan fibration, 53, 825
κ-closed, 389
κ-cofinal, 435

highertopoi

April 9, 2017925

GENERAL INDEX
κ-compact
left fibration, 392
object, 392
κ-compactly generated, 497
κ-continuous functor, 392
κ-filtered, 378
κ-right exact, 385
Krull dimension, 738
K-sheaf, 765
κ-small, 783
k-truncatedmap between Kan complexes, 490
morphism in an ∞-category, 490
object of an ∞-category, 489

and colimits, 363
cotopological, 668
left exact, 567
of a model category, 901
of an object, 363
topological, 572
locally Cartesianedge, 120
fibration, 123
locally compact, 749
locally fibrant, 857
locally small ∞-category, 419
long exact sequence of homotopy groups,
656

latching object, 832
left adjoint, 336
left adjointable, 746
left anodyne, 53, 63
left cone, 68
left derived functor, 350, 508, 813
left exact
functor, 385
at an object Z, 552
left extension, 285
left fibration, 53
classified by S → S, 212
and functor categories, 65
and Kan fibrations, 66
and undercategories, 61
left lifting property, 785
left orthogonal, 365
left proper, 810
left Quillen bifunctor, 847
limit, xi, 47of accessible ∞-categories, 445
of compactly generated ∞-categories,499
diagram, 47
in a functor category, 314
homotopy, 873
of ∞-categories, 214of presentable ∞-categories, 466
in a presentable ∞-category, 460
of spaces, 216
limitspreservation of, 48
local
class of morphisms, 544
object, 469
locale, 639
localic
n-topos, 652
localic topos, 640
localization, 361
accessible, 454
Bousfield, 901

MacLane pentagon, 789
MacLane’s coherence theorem, 789
mapping cone, 68
mapping simplex, 179
marked, 179
markedanodyne, 147edge, 146
simplicial set, 146
marked relative nerve, 199
matching object, 832
minimal∞-category, 102
inner fibration, 101
model category, 805
Cartesian, 157
coCartesian, 160
combinatorial, 814
contravariant, 71
covariant, 69
excellent, 859
injective, 826, 866
Joyal, 49, 89
left proper, 810
local, 661
monoidal, 847
perfect, 822
projective, 826, 866
Reedy, 837
right proper, 810
S-enriched, 847
simplicial, 847
of simplicial categories, 854
of spaces over X, 688
monoidal category, 788
Cartesian, 790
closed, 790
left closed, 790
right closed, 790
strict, 789
monoidal model category, 847
monomorphism, 491, 571

highertopoi

926
morphism
Cartesian, 115
coCartesian, 118
in an ∞-category, 33
multiplicationand homotopy colimits, 895
natural equivalence, 39
natural transformation, 39
enriched, 794
n-categories
and functor categories, 108
and overcategories, 110
n-category, 106, 112
underlying an ∞-category, 110
nerveof a category, 9marked relative, 199
relative, 195
of a simplicial category, 22of a topological category, 22n-gerbe, 726
n-topos, x, 635
objectcylinder, 807
Eilenberg-MacLane, 720
final, 44
group, 720
of an ∞-category, 33
latching, 832
matching, 832
path, 807
pointed, 720
projective, 508
oppositeof a category, 26of a simplicial set, 26
orthogonal, 365
left, 365
right, 365
overcategory, 41, 783
accessible, 444
of an ∞-category, 42of an ∞-topos, 612
of presentable ∞-categories, 465
overconvergent sheaf, 782
path object, 807
in simplicial categories, 881
pentagon axiom, 789
perfectclass of morphisms, 821
model category, 822
pointof an ∞-topos, 681
pointed object, 720

April 9, 2017

GENERAL INDEX
pointwise equivalence, 80
Postnikov pretower, 494
Postnikov tower, 488, 494
presentable
category, xiv, 784
fibration, 463
functor categories, 463∞-category, 311, 452, 453
overcategories, 465
undercategories, 465
presheaf, 311
and overcategories, 329
universal property of P(S), 323
with values in C, 759
pretower, 494
highly connected, 496
Pro-space, 710
product, 294
homotopy, 46
of ∞-topoi, 759
projective
cofibration, 825
fibration, 825
projective object, 508
projectively generated, 510
properbase change theorem, 678
map of topological spaces, 751
morphism of ∞-topoi, 747
proper base change theorem
nonabelian, 751
for sheaves of sets, 745
pullback, 294
homotopy, 46, 812
pullback functor, 528, 596
push-pull transformation, 746
pushforward functor, 596
pushout, 294
homotopy, 294, 811
of ∞-topoi, 599
quasi-category, xiv, 8
quasi-equivalence, 180
Quillen adjunction, 349, 813
Quillen bifunctor, 847
Quillen equivalence, 814
Quillen’s theorem A, 238
for ∞-categories, 236
Reedy
category, 831
cofibration, 837
fibration, 837
model structure, 837
reflective subcategory, 364
strongly, 479
reflexive coequalizer, 502

highertopoi

April 9, 2017927

GENERAL INDEX
relative nerve, 195
marked, 199
relatively κ-compact morphism, 563
representablefunctor, 300, 458
right fibration, 301
resolutiongroupoid, 553
simplicial, 553
retract, 303
retraction diagram
small, 304
weak, 304
right adjoint, 336
right anodyne, 53, 228
right cone, 68
right derived functor, 350, 813
right exact
and colimits, 388
right exact functor, 385
right fibration, 53
classifed by S → Sop , 212
universal, 212
right Kan extension
homotopy, 872
right lifting property, 785
right orthogonal, 365
right proper, 810
root of an S-tree, 795
saturated, 481
strongly, 472
weakly, 785
semitopos, 582
S-equivalence, 469, 902
shape, 710
equivalence, 710
of an ∞-topos, 710
trivial, 711
sheaf, 575
sieve, 573
covering, 573
sifted, 502
simplicial category, 18
simplicial model category, 847
simplicial nerve, 22
simplicial object
augmented, 532
of a category, 823
of an ∞-category, 532
simplicial set, 823
Joyal model structure, 89
Kan model structure, 824
marked, 146
sifted, 502
skeleton, 108, 671
S-local, 469

object, 902
small, 51, 784
object of a category, 784
small generation, 473, 482
small object argument, 786
smooth, 232
square, 294
Cartesian, 294
coCartesian, 294
pullback, 294
pushout, 294
stable under pullbacks, 542
stack, ix
standard diagram, 894
standard simplex, 75
straightening functor, 73, 171
straightening of diagrams, 258
S-tree, 795
associated, 796
κ-good, 796
strongcofibration, 866
fibration, 866
strong equivalence, 15
strongly k-connective, 740
strongly final, 45
strongly reflective, 479
strongly saturated, 472
subcategory
full, 44of an ∞-category, 44
reflective, 364
subobject, 561
support, 756
tensor product with spaces, 302
tensored, 794, 795
topologicalclass of morphisms, 572
localization, 572
topological category, 7
topological nerve, 22
topological space
absolute neighborhood retract, 776
coherent, 773
completely regular, 750
Heyting, 738
irreducible, 780
locally compact, 749
Noetherian, 738
topos, xiv, 524
localic, 640
tower, 494
highly connected, 496
Postnikov, 488
transformation
counit, 338

highertopoi

928
unit, 338
tree, 795
κ-good, 796
trivialcofibration, 806
fibration, 806
trivial fibration, 53
trivial on U , 756
truncatedand homotopy groups, 656
map between Kan complexes, 490
morphism in an ∞-category, 490
object of an ∞-category, 489
space, 112, 113
undercategory, 134, 783
accessibility, 439
and compact objects, 438
and homotopy fiber products, 432
of an ∞-category, 43
of a presentable ∞-category, 465
unit transformation, 338
universal
Cartesian fibration, 210
right fibration, 212
unstraightening functor, 73, 172
Wall finiteness obstruction, 419
weak
cofibration, 866
fibration, 866
weak equivalence, 806
weak exponential, 893
weak homotopy equivalenceof simplicial sets, 825
of topological spaces, 16
weak Kan complex, xiv
weakly cofinal, 435
weakly saturated, 785
Whitehead’s theorem, 16
Yoneda embedding, 316
classical, 311
and left Kan extensions, 322
and limits, 316
simplicial, 315
Yoneda’s lemma, 316

April 9, 2017

GENERAL INDEX

highertopoi

April 9, 2017

Index of Notation[1]∼
S , 858[n], 823[0]S , 854[1]S , 854[1]A , 854
Acc, 423
Accκ , 423
A◦ , 805
BandA (X), 727
Band(X), 727
C/p , 42
C/X , 43, 783
C(0) , 573
C/p , 242
Cp/ , 241Cκ , 392
Cp/ , 43
C[W −1 ], 887
CX/ , 43, 783
Cat, 783
Cat∞ , 144
Cat∆∞ , 144
Catlex∞ , 607Cat∨∞ , 423
Rex(κ)Cat∞, 499
dRex(κ) , 499
Cat∞Cat∆ , 18
CatS , 846
Cattop , 7
C∆ , 532C∆+ , 532
CG, 7
d∞ , 145
CatC / (f ), 68
C[S], 20, 23
coskn , 671
Cov(C), 575
C / (f ), 68
di , 824∆, 823∆+ , 532∆J , 824∆/K , 534∆≤n
+ , 537∆≤n , 671
∆n , 824
, 239S , 242
Disc(C), 489
EMn (X), 720
Env(C), 513
Env+ (C), 513
F† , 670
FC
A , 826
F† , 576
FX (C), 196
F+ (C), 199Xf ⊥ g, 366
Fσ , 687
Fun(C, C0 ), 39
FunA (C, D), 422
FunK (C, D), 408
FunL (C, D), 356
Fun∗ (X, Y), 650
Fun∗ (X, Y), 598
FunR (C, D), 356
FunR (C, D), 408
FunΣ (C, D), 506
Fun∗ (X, Y), 598
GerbAn (X), 728
Gerbn (X), 727
Groth(p), 891
Grp(X), 720
Gpd(C), 537
H, 16, 19hC, 16, 19, 808, 853
hn C, 110Hn (X; A), 725
HomC (X, Y ), 783
HomL
S (X, Y ), 28
HomR
S (X, Y ), 27
HomS (X, Y ), 28
HomZ (X, Y ), 783

highertopoi

April 9, 2017930
hS , 25
Idem, 304
Idem+ , 304
Ind(C), 375, 399
Indκ , 424
Indκ (C), 399
Inv, 887
K(A, n), 725
Kan, 51κ  τ , 253, 421
K(X), 764
KKb (X), 764
KF , 248
K / , 41
K . , 41
K b K 0 , 764
ΛJ
j , 824Λn
k , 824
limK , 390−→
LJ (X), 832
LTop, 597
LTopét , 614
Map[ (X, Y ), 154
MapS (X, Y ), 26
Map[S (X, Y ), 154
Map] (X, Y ), 154
Map]S (X, Y ), 154
M \ (φ), 179
Mod(k), 681
M (φ), 179
MJ (X), 832
M(K), 518
NU , 691
N(C), 9, 22
NR
F (C), 195
N+
F (C), 199
Ob(C), 783
OC , 527(n)
OX , 646
On
X , 646(S)

OX , 542
OS
X , 542
P (A), 882
P(C), 311
P(B), 691
P(f ), 357
P≤n (C), 635⊥ S, 366π≤n X, 3

INDEX OF NOTATION
πn (X), 654
0PK
K (C), 411
Post+ (C), 494
Post(C), 494
PrL , 462
PrLκ , 499
PrR , 462
PrR
κ , 499
Pro(S), 710
PΣ (C), 503
Q• , 76Q• , 76
Res(X), 553
Ret, 304
RFib(S), 83
RTopét , 613
RTop, 597
S −1 C, 479
si , 824
Set, 783
Sh(X), 710
Shv(C), 575
ShvKU (X; C), 770
ShvK (X), 765
ShvK (X; C), 765
ShvC (X), 622
Shv(X), 677
SingC • (X), 75
SingX , 688
Sing C, 19
Sing X, 8
S −1 C, 361
skn , 671
skn X, 108
S ⊥ , 366
⊥ S, 785
S, 51
Set∆ , 824
b
S, 52
Set+
∆ , 146(Set+∆ )/S , 146
S ? S 0 , 40
S⊥ , 785
Stφ , 73St+
φ , 171
StS , 73
St+
S , 172
Sub(X), 561, 572
C , 494τ≤kτ≤k , 493
τ≤k C, 490⊗, 762⊗C , 762

highertopoi

INDEX OF NOTATION
Top, 688
TopRn , 650
U(X), 677
U0 (X), 773
U◦ , 881
Unφ , 73Un+
φ , 172
UnS , 73
Un+
S , 172
X /U , 756
X pS / , 243
X∗ , 720
X [ , 146
X∧ , 665
X \ , 150
X ⊗ K, 302
X ] , 146

April 9, 2017931