COMPLETE SEGAL SPACES ARISING FROM SIMPLICIAL
CATEGORIES
JULIA E. BERGNER
Abstract. In this paper, we compare several functors which take simplicial
categories or model categories to complete Segal spaces, which are particularly
nice simplicial spaces which, like simplicial categories, can be considered to be
models for homotopy theories. We then give a characterization, up to weak
equivalence, of complete Segal spaces arising from these functors.
1. Introduction and Overview
The idea that simplicial categories, or categories enriched over simplicial sets,
model homotopy theories goes back to a series of several papers by Dwyer and
Kan [8], [10], [11], [12]. Taking the viewpoint that a model category, or more
generally a category with weak equivalences, can be considered to be a model for
a homotopy theory, they develop two methods to obtain from a model category a
simplicial category. This “simplicial localization” encodes higher-order structure
which is lost when we pass to the homotopy category associated to the model
category. Furthermore, they prove that, up to a natural notion of weak equivalence
of simplicial categories (see Definition 3.1), every simplicial category arises as the
simplicial localization of some category with weak equivalences [10, 2.1]. Thus, the
category of all (small) simplicial categories with these weak equivalences can be
regarded as the “homotopy theory of homotopy theories.”
This notion was mentioned briefly at the end of Dwyer and Spalinski’s introduc-
tion to model categories [13, 11.6], and was made precise by the author in [2, 1.1],
in which the category SC of small simplicial categories with these weak equivalences
is shown to have the structure of a model category.
However, this category is not practically useful for many purposes. Simplicial
categories are not particularly easy objects to work with, and the weak equivalences,
while natural generalizations of equivalences of categories, are difficult to identify.
Motivated by this problem, Rezk defines a model structure, which we denote CSS,
on the category of simplicial spaces, in which the fibrant-cofibrant objects are called
complete Segal spaces (see Definition 4.3). This model structure is especially nice,
in that the objects are just diagrams of simplicial sets and the weak equivalences,
at least between complete Segal spaces, are just levelwise weak equivalences of
simplicial sets. Furthermore, this model structure has the additional structures of
a simplicial model category and a monoidal model category. Because it is given by a
Date: October 25, 2018.
2000 Mathematics Subject Classification. Primary: 55U40; Secondary: 55U35, 18G55, 18G30,
18D20.
Key words and phrases. simplicial categories, model categories, complete Segal spaces, homo-
topy theories.
http://arxiv.org/abs/0704.1624v2
2 J.E. BERGNER
localization of a model structure on the category of simplicial spaces with levelwise
weak equivalences, it is an example of a presentation for a homotopy theory as
described by Dugger [7].
In his paper, Rezk defines two different functors, one from the category of sim-
plicial categories, and one from the category of model categories, to the category
of complete Segal spaces. Thus, he describes a relationship between simplicial cat-
egories and complete Segal spaces, but he does not give an inverse construction.
However, the author was able to show in [5] that the model categories SC and
CSS are Quillen equivalent to one another, and therefore are models for the same
homotopy theory. Thus, the hope is that we can answer questions about simplicial
categories by working with complete Segal spaces. This paper is the beginning of
that project.
However, the functor that we use to show that the two model categories are
Quillen equivalent is not the same as Rezk’s functor. Furthermore, the question
arises whether Rezk’s functor on model categories agrees with the composite of the
simplicial localization functor with his functor on simplicial categories. Rezk gives
the beginning of a proof that the two functors agree when the model category in
question has the additional structure of a simplicial model category. Our goal in
this paper is prove that, up to weak equivalence in CSS, the two functors from SC
to CSS are the same, that Rezk’s result holds for model categories which are not
necessarily simplicial, and that this result does imply that the functor on model
categories agrees with the one on their corresponding simplicial categories up to
weak equivalence.
We then go on to characterize, up to weak equivalence, the complete Segal space
arising from any simplicial category. It can be described at each level as the nerve
of the monoid of self weak equivalences of representing objects of the category.
We should mention that the model categories SC and CSS are only two of several
known models for the homotopy theory of homotopy theories. Our proof that the
two are Quillen equivalent actually uses two intermediate model structures SeCatc
and SeCatf on the category of Segal precategories, or simplicial spaces with a
discrete Space at level zero [5]. The fibrant-cofibrant objects in these structures are
known as Segal categories. Furthermore, Joyal and Tierney have shown that there is
a Quillen equivalence between each of these model structures and a model structure
QCat on the category of simplicial sets [17], [19]. The fibrant-cofibrant objects of
this model structure are known as quasi-categories and are generalizations of Kan
complexes [18]. One can compare our composite functor SC → CSS to one they
define which factors through QCat rather than through SeCatc and SeCatf . It is
a consequence of Joyal and Tierney’s work that these two different functors give
rise to weakly equivalent complete Segal spaces, as we will describe further in the
section on complete Segal spaces. An introduction to each of these model structures
and the Quillen equivalences can be found in the survey paper [4].
In [1], we use the results of this paper to consider the complete Segal spaces
arising from the homotopy fiber product construction for model categories, as de-
scribed by Toën in his work on derived Hall algebras [24]. It seems that results
relating diagrams of model categories to diagrams of complete Segal spaces will
prove to be useful.
COMPLETE SEGAL SPACES 3
Acknowledgments. I would like to thank Bill Dwyer, Charles Rezk, André Joyal,
and Myles Tierney for conversations about the work in this paper, as well as the
referee for helpful comments on the exposition.
2. Background on Model Categories and Simplicial Objects
In this section, we summarize some facts about model categories, simplicial sets,
and other simplicial objects that we need in the course of this paper.
A model category M is a category with three distinguished classes of mor-
phisms, fibrations, cofibrations, and weak equivalences. A morphism which is both
a (co)fibration and a weak equivalence is called an acyclic (co)fibration. The cate-
goryM with these choices of classes is required to satisfy five axioms [13, 3.3].
Axiom MC1 guarantees that M has small limits and colimits, so in particular
M has an initial object and a terminal object. An object X in a model category
M is fibrant if the unique map X → ∗ to the terminal object is a fibration. Dually,
X is cofibrant if the unique map from the initial object φ→ X is a cofibration.
The factorization axiom (MC5) can be applied in such a way that, given any
object X ofM, we can factor the map X → ∗ as a composite
∼ //X ′ //∗
of an acyclic cofibration followed by a fibration. In this case, X ′ is called a fibrant
replacement of X , since it is weakly equivalent to X and fibrant. This replacement
is not necessarily unique, but in all the model categories we consider here it can be
assumed to be functorial [16, 1.1.1]. Cofibrant replacements can be defined dually.
The structure of a model category enables us to invert the weak equivalences
formally in such a way that we still have a set, rather than a proper class, of mor-
phisms between any two objects. If we were merely to take the localizationW−1M,
there would be no guarantee that we would not have a proper class. However, with
the structure of a model category, we can define the homotopy category Ho(M) to
have the same objects asM, and as morphisms
HomHo(M)(X,Y ) = [X
cf , Y cf ]M,
where the right-hand side denotes homotopy classes of maps between fibrant-cofibrant
replacements of X and Y , respectively.
The standard notion of equivalence of model categories is given by the following
definitions. First, recall that an adjoint pair of functors F : C ⇆ D : G satisfies the
property that, for any objects X of C and Y of D, there is a natural isomorphism
ϕ : HomD(FX, Y )→ HomC(X,GY ).
The functor F is called the left adjoint and G the right adjoint [20, IV.1].
Definition 2.1. [16, 1.3.1] An adjoint pair of functors F : M ⇆ N : G between
model categories is a Quillen pair if F preserves cofibrations and G preserves fibra-
tions.
Definition 2.2. [16, 1.3.12] A Quillen pair of model categories is a Quillen equiv-
alence if if for all cofibrant X in M and fibrant Y in N , a map f : FX → Y is a
weak equivalence in D if and only if the map ϕf : X → GY is a weak equivalence
4 J.E. BERGNER
An important example of a model category is that of the standard model struc-
ture on the category of simplicial sets SSets. Recall that a simplicial set is a functor
X : ∆op → Sets, where ∆op is the opposite of the category ∆ of finite ordered sets
[n] = {0→ 1→ · · · → n} and order-preserving maps between them. We denote the
set X([n]) by Xn. In particular in X we have face maps di : Xn → Xn−1 and de-
generacy maps si : Xn → Xn+1 for each 0 ≤ i ≤ n, satisfying several compatibility
conditions. Three particularly useful examples of simplicial sets are the n-simplex
∆[n] and its boundary ∆̇[n] for each n ≥ 0, and the boundary with the kth face
removed, V [n, k], for each n ≥ 1 and 0 ≤ k ≤ n. Given a simplicial set X , we can
take its geometric realization |X |, which is a topological space [14, I.1].
In the standard model category structure on SSets, the weak equivalences are
the maps f : X → Y for which the geometric realization |f | : |X | → |Y | is a weak
homotopy equivalence of topological spaces [14, I.11.3]. In fact, this model structure
on simplicial sets is Quillen equivalent to the standard model structure on the
category of topological spaces [16, 3.6.7].
More generally, a simplicial object in a category C is a functor ∆op → C. The
two main examples which we consider in this paper are those of simplicial spaces
(also called bisimplicial sets), or functors ∆op → SSets, and simplicial groups. We
denote the category of simplicial spaces by SSets∆
A simplicial set X can be regarded as a simplicial space in two ways. It can be
considered a constant simplicial space with the simplicial set X at each level, and in
this case we will denote the constant simplicial set by cX or just X if no confusion
will arise. Alternatively, we can take the simplicial space, which we denote Xt, for
which (Xt)n is the discrete simplicial set Xn. The superscript t is meant to suggest
that this simplicial space is the “transpose” of the constant simplicial space.
A natural choice for the weak equivalences in the category SSets∆
is the class
of levelwise weak equivalences of simplicial sets. If we define the cofibrations to be
levelwise also, we obtain a model structure which is usually referred to as the Reedy
model structure on SSets∆
[22].
The Reedy model structure has the additional structure of a simplicial model
category. In particular, given any two objects X and Y of SSets∆
, there is a
mapping space, or simplicial set Map(X,Y ) satisfying compatibility conditions [15,
9.1.6]. In the case where X is cofibrant (as is true of all objects in the Reedy model
structure) and Y is fibrant, this choice of mapping space is homotopy invariant.
One way to obtain other model structures on the category of simplicial spaces is
to localize the Reedy structure with respect to a set of maps. While this process
works for much more general model categories [15, 3.3.1], we will focus here on this
particular case. Let S = {f : A → B} be a set of maps of simplicial spaces. A
Reedy fibrant simplicial space W is S-local if for each map f ∈ S, the induced map
Map(f,W ) : Map(B,W )→ Map(A,W )
is a weak equivalence of simplicial sets. A map g : X → Y is then an S-local
equivalence if for any S-local object W , the induced map
Map(g,W ) : Map(Y,W )→ Map(X,W )
is a weak equivalence of simplicial sets.
Theorem 2.3. [15, 4.1.1] There is a model structure LSSSets
on the category
of simplicial spaces in which
COMPLETE SEGAL SPACES 5
• the weak equivalences are the S-local equivalences,
• the cofibrations are levelwise cofibrations of simplicial sets, and
• the fibrant objects are the S-local objects.
Furthermore, this model category has the additional structure of a simplicial model
category.
We now turn to a few facts about simplicial groups, or functors from ∆op to the
category of groups. Given a simplicial group G, we can take its nerve, a simplicial
space with
nerve(G)n,m = Hom([m], Gn).
Taking the diagonal of this simplicial space, we obtain a simplicial set, also often
called the nerve of G.
From another perspective, for G a simplicial group (or, more generally, a sim-
plicial monoid), we can find a classifying complex of G, a simplicial set whose
geometric realization is the classifying space BG. A precise construction can be
made for this classifying space by the WG construction [14, V.4.4], [21]. However,
we are not so concerned here with the precise construction as with the fact that
such a classifying space exists, so for simplicity we will simply write BG for the
classifying complex of G.
3. Simplicial Categories and Simplicial Localizations
In this section, we consider simplicial categories and show how they arise from
Dwyer and Kan’s simplicial localization techniques. We then discuss model category
structures, first on the category of simplicial categories with a fixed object set, and
then on the category of all small simplicial categories.
First of all, we clarify some terminology. In this paper, by (small) “simplicial
category” we will mean a category with a set of objects and a simplicial set of
morphisms Map(x, y) between any two objects x and y, also known as a category
enriched over simplicial sets. This notion does not coincide with the more general
one of a simplicial object in the category of small categories, in which we would also
have a simplicial set of objects. Using this more general definition, if we impose
the additional condition that all face and degeneracy maps are the identity on the
objects, then we get our more restricted notion.
A simplicial category can be seen as a generalization of a category, since any
ordinary category can be regarded as a simplicial category with a discrete mapping
space. Given any simplicial category C, we can consider its category of components
π0C, which has the same objects as C and whose morphisms are given by
Homπ0C(x, y) = π0MapC(x, y).
The following definition of weak equivalence of simplicial categories is a natural
generalization of the notion of equivalence of categories.
Definition 3.1. A simplicial functor f : C → D is a Dwyer-Kan equivalence or
DK-equivalence if the following two conditions hold:
(1) For any objects x and y of C, the induced map Map(x, y) → Map(fx, fy)
is a weak equivalence of simplicial sets.
(2) The induced map on the categories of components π0f : π0C → π0D is an
equivalence of categories.
6 J.E. BERGNER
The idea of obtaining a simplicial category from a model categoryM goes back
to several papers of Dwyer and Kan [8], [11], [12]. In fact, they define two different
methods of doing so, the simplicial localization LM [12] and the hammock local-
ization LHM [11]. The first has the advantage of being easier to describe, while
the second is more convenient for making calculations.
It should be noted that these constructions can be made for more general cate-
gories with weak equivalences, and do not depend on the model structure if we are
willing to ignore the potential set-theoretic difficulties. However, as with the homo-
topy category construction, the hammock localization in particular can be defined
much more nicely when we have the additional structure of a model category.
We begin with the construction of the simplicial localization LM. Recall that,
given a category M with some choice of weak equivalences W , we denote the
localization ofM with respect toW byW−1M. This localization is obtained from
M by formally inverting the maps of W . Further, recall that, given a categoryM,
we denote by FM the free category on M, or category with the same objects as
M and morphisms freely generated by the non-identity morphisms of M. Note
in particular that there are natural functors FM → M and FM → F 2M [12,
2.4]. These functors can be used to define a simplicial resolution F∗M, which is a
simplicial category with the category F k+1M at level k [12, 2.5].
We can apply this same construction to the subcategoryW to obtain a simplicial
resolution F∗W . Using these two resolutions, we have the following definition.
Definition 3.2. [12, 4.1] The simplicial localization ofM with respect toW is the
localization (F∗W)
−1(F∗M). This simplicial localization is denoted L(M,W) or
simply LM.
The following result gives interesting information about the mapping spaces in
LM in the case where W is all ofM.
Proposition 3.3. [12, 5.5] Suppose that W =M and nerve(M) is connected.
(i) The simplicial localization LM is a simplicial groupoid, so for all objects x
and y, the simplicial sets MapLM(x, y) are all isomorphic. In particular, the
simplicial sets MapLM(x, x) are all isomorphic simplicial groups.
(ii) The classifying complex BMapLM(x, x) has the homotopy type of nerve(M),
and thus each simplicial set MapLM(x, y) has the homotopy type of the loop
space Ω(nerve(M)).
We now turn to the other construction, that of the hammock localization. Again,
letM be a category with a specified subcategory W of weak equivalences.
Definition 3.4. [11, 3.1] The hammock localization of M with respect to W ,
denoted LH(M,W), or simply LHM, is the simplicial category defined as follows:
(1) The simplicial category LHM has the same objects asM.
(2) Given objects X and Y of M, the simplicial set MapLHM(X,Y ) has as
k-simplices the reduced hammocks of width k and any length between X
COMPLETE SEGAL SPACES 7
and Y , or commutative diagrams of the form
· · · C0,n−1
· · · C1,n−1
~~~~~~~~~
CCCCCCCCC
Ck,1 Ck,2 · · · Ck,n−1
in which
(i) the length of the hammock is any integer n ≥ 0,
(ii) the vertical maps are all in W ,
(iii) in each column all the horizontal maps go the same direction, and if
they go to the left, then they are in W ,
(iv) the maps in adjacent columns go in opposite directions, and
(v) no column contains only identity maps.
Proposition 3.5. [8, 2.2] For a given model categoryM, the simplicial categories
LM and LHM are DK-equivalent.
We should add that the description of the hammock localization can be greatly
simplified if we make use of the model category structure on M. In this case,
Dwyer and Kan prove that it suffices to consider hammocks of length 3 such as the
following [8, §8]:
X C0,1
≃oo //C0,2 Y
≃oo .
Restricting to the category of simplicial categories with a fixed set O of ob-
jects, Dwyer and Kan prove the existence of a model structure on this category,
which we denote SCO [12, 7.2]. In this situation, the weak equivalences are the
DK-equivalences, but with the objects fixed the second condition follows imme-
diately from the first. The fibrations in this model structure are given by the
functors f : C → D inducing, for any objects x and y, fibrations of simplicial sets
MapC(x, y)→ MapD(x, y).
The cofibrations are then defined to be the maps with the left lifting property
with respect to the acyclic fibrations. However, they can be more precisely charac-
terized. To do so, we recall the definition of a free map of simplicial categories.
Definition 3.6. [12, 7.4] A map f : C → D in SCO is free if
(1) f is a monomorphism,
(2) if ∗ denotes the free product, then in each simplicial dimension k, the
category Dk admits a unique free factorization Dk = f(Ck) ∗ Fk, where Fk
is a free category, and
(3) for each k ≥ 0, all degeneracies of generators of Fk are generators of Fk+1.
8 J.E. BERGNER
Definition 3.7. [12, 7.5] A map f : C → D of simplicial categories is a strong
retract of a map f ′ : C → D′ if there exists a commutative diagram
>>}}}}}}}
id // D
Using these definitions, Dwyer and Kan prove the following result.
Proposition 3.8. [12, 7.6] The cofibrations of SCO are precisely the strong retracts
of free maps. In particular, a cofibrant simplicial category is a retract of a free
category.
This result can then be generalized to the category of all simplicial categories, in
which the DK-equivalences are the weak equivalences. If C is a simplicial category, a
morphism e ∈ HomC(a, b)0 is a homotopy equivalence if it becomes an isomorphism
in π0C.
Theorem 3.9. [2, 1.1] There is a model category structure SC on the category of
small simplicial categories in which
• the weak equivalences are the Dwyer-Kan equivalences, and
• the fibrations are the maps f : C → D satisfying the following two condi-
tions:
(i) For any objects a1 and a2 in C, the map
HomC(a1, a2)→ HomD(fa1, fa2)
is a fibration of simplicial sets.
(ii) For any object a1 in C, b in D, and homotopy equivalence e : fa1 → b
in D, there is an object a2 in C and homotopy equivalence d : a1 → a2
in C such that fd = e.
4. Complete Segal Spaces
Here we define complete Segal spaces and describe Rezk’s model structure on the
category of simplicial spaces, in which the complete Segal spaces are the fibrant-
cofibrant objects.
Recall that by a simplicial space we mean a simplicial object in the category
of simplicial sets, or functor ∆op → SSets. In section 2, we described the Reedy
model category structure on this category, in which both the weak equivalences
and cofibrations are defined levelwise. The model structure CSS is given by a
localization of this structure with respect to a set of maps.
We begin with the definition of a Segal space. In [23, 4.1], Rezk defines for each
0 ≤ i ≤ n − 1 a map αi : [1] → [n] in ∆ such that 0 7→ i and 1 7→ i + 1. There is
a corresponding map αi : ∆[1] → ∆[n]. Then for each n he defines the simplicial
space
G(n)t =
αi∆[1]
t ⊂ ∆[n]t.
COMPLETE SEGAL SPACES 9
Let X be a Reedy fibrant simplicial space. There is a weak equivalence of
simplicial sets
MapSSets∆opc (G(n)
t, X)→ X1 ×X0 · · · ×X0 X1︸ ︷︷ ︸
where the right hand side is the limit of the diagram
d0 // X0 X1
d1oo d0 // . . .
d0 // X0 X1
with n copies of X1.
Now, given any n, define the map ϕn : G(n)t → ∆[n]t to be the inclusion map.
Then for any Reedy fibrant simplicial space W there is a map
ϕn = MapSSets∆opc (ϕ
n,W ) : MapSSets∆opc (∆[n]
t,W )→ MapSSets∆opc (G(n)
t,W ).
More simply written, this map is
ϕn : Wn → W1 ×W0 · · · ×W0 W1︸ ︷︷ ︸
and is often called a Segal map. The Segal map is actually defined for any simplicial
space W , but here we assume Reedy fibrancy so that the mapping spaces involved
are homotopy invariant.
Definition 4.1. [23, 4.1] A Reedy fibrant simplicial space W is a Segal space if for
each n ≥ 2 the Segal map
ϕn : Wn →W1 ×W0 · · · ×W0 W1
is a weak equivalence of simplicial sets.
In fact, there is a model category structure SeSp on the category of simplicial
spaces in which the fibrant objects are precisely the Segal spaces [23, 7.1]. This
model structure is obtained from the Reedy structure via localization.
The idea is that in a Segal space there is a notion of “composition,” at least up
to homotopy. In fact, given a Segal space, we can sensibly use many categorical
notions. We summarize some of these ideas here; a detailed description is given by
Rezk [23]. The objects of a Segal space W are given by the set W0,0. Given the
(d1, d0) : W1 →W0 ×W0,
themapping space mapW (x, y) is given by the fiber of this map over (x, y). (The fact
that W is Reedy fibrant guarantees that this mapping space is homotopy invariant.)
Two maps f, g ∈ mapW (x, y)0 are homotopic if they lie in the same component of
the simplicial set mapW (x, y). Thus, we define the space of homotopy equivalences
Whoequiv ⊆W1 to consist of all the components containing homotopy equivalences.
Given any (x0, . . . , xn) ∈ W
0,0 , let mapW (x0, . . . , xn) denote the fiber of the
(α0, . . . , αn) : Wn →W
over (x0, . . . , xn). Consider the commutative diagram
Wn = Map(∆[n]
t,W )
ϕk //
Map(G(n)t,W )
wwppp
Wn+10
10 J.E. BERGNER
and notice that, since W is a Segal space, the horizontal arrow is a weak equivalence
and a fibration. In particular, this map induces an acyclic fibration on the fibers of
the two vertical arrows,
mapW (x0, . . . , xn)→ mapW (xn−1, xn)× · · · ×mapW (x0, x1).
Given f ∈ map(x, y)0 and g ∈ map(y, z)0, their composite is a lift of (g, f) ∈
map(y, z) × map(x, y) along ϕ2 to some k ∈ map(x, y, z)0. The result of this
composition is defined to be d1(k) ∈ map(x, z)0. It can be shown that any two
results are homotopic, so we can use g ◦ f unambiguously.
Then, the homotopy category of W , denoted Ho(W ), has as objects the elements
of the set W0,0, and
HomHo(W )(x, y) = π0mapW (x, y).
A homotopy equivalence in W is a 0-simplex of W1 whose image in Ho(W ) is an
isomorphism.
Definition 4.2. A map f : W → Z of Segal spaces is a Dwyer-Kan equivalence if
(1) for any objects x and y ofW , the induced map mapW (x, y)→ mapZ(fx, fy)
is a weak equivalence of simplicial sets, and
(2) the induced map Ho(W )→ Ho(Z) is an equivalence of categories.
Notice that the definition of these maps bears a striking resemblance to that
of the Dwyer-Kan equivalences between simplicial categories, hence the use of the
same name.
For a Segal space W , note that the degeneracy map s0 : W0 → W1 factors
through the space of homotopy equivalences Whoequiv, since the image of s0 consists
of “identity maps.” Given this fact, we are now able to give a definition of complete
Segal space.
Definition 4.3. [23, §6] A Segal space W is a complete Segal space if the map
W0 →Whoequiv given above is a weak equivalence of simplicial sets.
The idea behind this notion is that, although W0 is not required to be discrete,
as the objects are for a simplicial category, it is not heuristically too different from a
simplicial space with discrete 0-space. (This viewpoint is further confirmed by the
comparison of complete Segal spaces with Segal categories, which are essentially
the analogues of Segal spaces with discrete 0-space [5, 6.3].)
Now, we give a description of the model structure CSS. We do not give all the
details here, such as a description of an arbitrary weak equivalence, but refer the
interested reader to Rezk’s paper [23, §7].
Theorem 4.4. [23, 7.2] There is a model structure CSS on the category of simpli-
cial spaces, obtained as localization of the Reedy model structure, such that
(1) the fibrant objects are precisely the complete Segal spaces,
(2) the cofibrations are the monomorphisms; in particular, every object is cofi-
brant,
(3) the weak equivalences between Segal spaces are Dwyer-Kan equivalences,
(4) the weak equivalences between complete Segal spaces are levelwise weak
equivalences of simplicial sets.
COMPLETE SEGAL SPACES 11
Furthermore, CSS has the additional structure of a simplicial model category and
is cartesian closed.
The fact that CSS is cartesian closed allows us to consider, for any complete Segal
spaceW and simplicial space X , the complete Segal space WX . In particular, using
the simplicial structure, the simplicial set at level n is given by
(WX)n = Map(X ×∆[n]
t,W ).
If W is a (not necessarily complete) Segal space, then WX is again a Segal space;
in other words, the model category SeSp is also cartesian closed.
We denote the functorial fibrant replacement functor in CSS by LCSS . Thus,
given any simplicial space X , there is a weakly equivalent complete Segal space
LCSSX .
This model structure is connected to the model structure SC by a chain of Quillen
equivalences as follows. Each of these model categories is Quillen equivalent to a
model structure SeCatf on the category of Segal precategories. A Segal precategory
is a simplicial space X with X0 a discrete space. A Segal category is then a Segal
precategory with the Segal maps weak equivalences. In the model structure SeCatf ,
the fibrant objects are Segal categories, and so it is considered a Segal category
model structure on the category of Segal precategories. We have the following
chain of Quillen equivalences, with the left adjoint functors topmost:
SC ⇆ SeCatf ⇄ CSS.
The right adjoint SC → SeCatf is given by the nerve functor, and the left adjoint
SeCatf → CSS is given by the inclusion functor.
There is actually another chain of Quillen equivalences connecting the two model
structures; in this case, both SC and CSS are Quillen equivalent to Joyal’s model
structure QCat on the category of simplicial sets [18]. The fibrant objects in QCat
are quasi-categories, or simplicial sets K such that a dotted arrow lift exists making
the diagram
V [m, k] //
commute for any 0 < k < m. The chain of Quillen equivalences in this case is given
SC ⇄ QCat ⇆ CSS.
The right adjoint SC → QCat is given by Cordier and Porter’s coherent nerve
functor [6], [17, 2.10], and the right adjoint CSS → QCat is given given by sending
a simplicial space W to the simplicial set W∗,0 [19, 4.11]. It is a consequence of
work of Joyal [17, §1-2] and of Joyal and Tierney [19, §4-5] that the simplicial space
obtained from a simplicial category via these functors is weakly equivalent to the
one obtained from the composite functor described in the previous paragraph.
5. Obtaining Complete Segal Spaces from Simplicial Categories and
Model Categories
In this section, we describe several different ways of obtaining a complete Segal
space. First, we look at a particularly nice functor which Rezk uses to modify
12 J.E. BERGNER
the notion of a nerve of a category. Then we look at how this functor can be
generalized to one on any simplicial category, and how a similar idea can be used to
get a complete Segal space from any model category. We then consider the functors
used in the Quillen equivalences connecting SC and CSS.
Let us begin with Rezk’s classifying diagram construction, which associates to
any small category C a complete Segal space NC. First, we denote by nerve(C)
the ordinary nerve, which is the simplicial set given by (nerve(C))n = Hom([n], C).
Further, we denote by iso(C) the maximal subgroupoid of C, or subcategory of C
with all objects of C and whose only morphisms are the isomorphisms of C. By
C[n] we denote the category of functors [n]→ C, or the category whose objects are
n-chains of composable morphisms in C.
Definition 5.1. [23, 3.5] The classifying diagram NC is the simplicial space given
by (NC)n = nerve(iso(C
[n])).
Thus, (NC)0 is simply the nerve of iso(C), and (NC)1 is the nerve of the maxi-
mal subgroupoid of the morphism category of C. In particular, information about
invertible morphisms of C is encoded at level 0, while information about the other
morphisms of C does not appear until level 1.
Thus, the classifying diagram of a category can be regarded as a more refined
version of the nerve, since, unlike the ordinary nerve construction, it enables one
to recover information about whether morphisms are invertible or not. This con-
struction is also particularly useful for our purposes due to the following result.
Proposition 5.2. [23, 6.1] If C is a small category, then its classifying diagram
NC is a complete Segal space.
However, this construction, as defined above, cannot be used to assign a complete
Segal space to any simplicial category, since, beginning with level 1, we would have
homotopy invariance problems with a simplicial set of objects in C[1]. Rezk defines
an analogous functor, though, from the category of small simplicial categories which
is similar in spirit to the classifying diagram but avoids these difficulties.
Let I[m] denote the category with m + 1 objects and a single isomorphism
between any two objects, and let E(m) = nerve(I[m])t. If W is a Segal space and
X is any simplicial space, recall that WX denotes the internal hom object, which
is a Segal space. With these notations in place, we can give the definition of Rezk’s
completion functor. Let W be a Segal space. Then its completion Ŵ is defined as
a fibrant replacement in CSS of the simplicial space W̃ defined by
W̃n = diag([m] 7→ MapSSets∆op (E(m),W
∆[n]t)) = diag([m] 7→ (WE(m))n).
From a simplicial category C, then, we can takes its nerve to obtain a simplicial
space, followed by a fibrant replacement functor in the Segal space model structure,
to obtain a Segal space W . From W we can then pass to a complete Segal space
via this completion functor. We will denote this complete Segal space LC(W ), or
LC(C) where W comes from a simplicial category as just described.
The first important fact about this completion functor is that the completion
map iW : W → Ŵ = LC(W ) is not only a weak equivalence in the model category
CSS, but is also a Dwyer-Kan equivalence of Segal spaces [23, §14]. Furthermore,
this completion functor restricts nicely to the classifying diagram in the case where
C is a discrete category.
COMPLETE SEGAL SPACES 13
Proposition 5.3. [23, 14.2] If C is a discrete category, then LC(C) is isomorphic
to NC.
If we begin with a model category M with subcategory of weak equivalences
W , a functor analogous to the classifying diagram functor can be used to obtain a
complete Segal space. In this case, rather than taking the subcategory iso(M) of
isomorphisms ofM, we take the subcategory of weak equivalences, denoted we(M).
Thus, Rezk defines the classification diagram of (M,W), denoted N(M,W), by
N(M,W)n = nerve(we(M
[n])).
Unlike the classifying diagram, the classification diagram of a model category is not
necessarily a complete Segal space as stated, but taking a Reedy fibrant replacement
of it results in a complete Segal space, as we show in the next section.
Lastly, we have the two functors given by the two different chains of Quillen
equivalences between the model categories SC and CSS. As mentioned in the
previous section, these two functors are equivalent. In each case, the resulting
simplicial space is not Reedy fibrant in general, and so not a complete Segal space,
but applying the fibrant replacement functor LCSS results in a complete Segal space.
The first of these composite functors, in particular, is simple to describe ab-
stractly, as in the previous section, but it has a disadvantage over Rezk’s functor in
that it gives very little insight into what the resulting complete Segal space looks
like. In the next section, we prove that the two functors from SC to CSS result
in weakly equivalent complete Segal spaces, and that if we use Rezk’s classification
diagram construction we get a weakly equivalent complete Segal space to the one
we would obtain by taking the simplicial localization followed by his completion
functor. We then use Rezk’s functor to describe what the complete Segal space
corresponding to a simplicial category looks like.
6. Comparison of Functors from SC to CSS
Here we prove that each of the functors we have described all give rise to com-
plete Segal spaces weakly equivalent to those given in the previous section. We
begin by stating the result that establishes the equivalence between Rezk’s com-
pletion functor LC : SC → CSS and the functor arising from the chain of Quillen
equivalences factoring through SeCatf . Let LCSS denote the functorial fibrant re-
placement functor in CSS.
Theorem 6.1. If C is a simplicial category, then the complete Segal spaces LC(C)
and LCSS(nerve(C)) are weakly equivalent in CSS.
Proof. Let LS denote a fibrant replacement functor in the Segal space model struc-
ture SeSp on the category of simplicial spaces. The fact that the two functors
in question result in weakly equivalent complete Segal spaces can be shown by
considering the following chain of weak equivalences:
LCSS(nerve(C))← nerve(C)→ LSnerve(C)→ LC(C).
The map on the left is the localization functor in CSS and so is a weak equivalence
in CSS. The middle map is a weak equivalence in SeSp and therefore also a weak
equivalence in CSS, since the latter model category is a localization of the former.
The map on the right is Rezk’s completion, and it is a weak equivalence in CSS, as
given in the previous section. Therefore, the objects at the far left and right of this
14 J.E. BERGNER
zigzag, both of which are complete Segal spaces, are weakly equivalent as objects
of CSS. �
Now, we would like to compare either of these functors to the classifying diagram
construction for a model category M. In other words, we want to show that
N(M,W) is equivalent to LC(L
HM), where we first take the hammock localization
ofM to obtain a simplicial category, and then apply Rezk’s functor LC .
An initial problem here is that N(M,W) is not necessarily Reedy fibrant, and
so it is not necessarily a complete Segal space. We prove that a Reedy fibrant
replacement of it, denoted N(M,W)f , is in fact a complete Segal space in the
process of comparing the “mapping spaces” in this Reedy fibrant replacement to
the mapping spaces of the hammock localization LHM.
Theorem 6.2. Let M be a model category, and let W denote its subcategory of
weak equivalences. Then N(M,W)f , is a complete Segal space. Furthermore, for
any objects x, y of M, there is a weak equivalence of spaces mapN(M,W)f (x, y) ≃
MapLHM(x, y), and there is an equivalence of categories Ho(N(M,W)
f ) ≈ Ho(M).
This result was proved by Rezk in the case where M is a simplicial model
category [23, 8.3], namely, in the case where we do not need to pass to the simplicial
localization ofM to consider its function complexes. However, here we prove that,
as he conjectured [23, 8.4], the result continues to hold in this more general case.
We prove this theorem very similarly to the way Rezk proves it in the more
restricted case, using a proposition of Dwyer and Kan. To begin, we introduce
some terminology. LetM be a model category. A classification complex ofM, as
defined in [9, 1.2], is the nerve of any subcategory C ofM such that
(1) every map in C is a weak equivalence,
(2) if f : X → Y inM is a weak equivalence and either X or Y is in C, then f
is in C, and
(3) nerve(C) is homotopically small; i.e., each homotopy group of |nerve(C)| is
small [11, 2.2].
The special classification complex sc(X) of an object X in M is a connected
classification complex containing X .
LetM be a model category and X a fibrant-cofibrant object ofM. Denote by
Auth(X) the simplicial monoid of weak equivalences given by Auth
(X) in the
hammock localization LHM, and by BAuth(X) its classifying complex.
The following proposition was proved by Dwyer and Kan in [9, 2.3] in the case
thatM is a simplicial model category. However, the proof does not actually require
the simplicial structure; in fact, their proof is essentially the one given below, with
the extra step showing that the mapping spaces in the hammock localization are
equivalent to those given by the simplicial structure ofM [11, 4.8].
Proposition 6.3. Let X be an object of a model category M. The classifying
complex BAuth(X) is weakly equivalent to the special classification complex of X,
sc(X), and the two can be connected by a finite zig-zag of weak equivalences.
Proof. Let W be the subcategory of weak equivalences of M. Consider the con-
nected component of nerve(W) containing X . For the rest of this proof, we assume
that W is such that its nerve is connected. We further assume that nerve(W) is
homotopically small, taking an appropriate subcategory, as described in [11, 2.3],
if necessary.
COMPLETE SEGAL SPACES 15
In this case, by Proposition 3.3, the function complexes MapLW(X,X) are all iso-
morphic. Furthermore, by the same result, the classifying complex BMapLW(X,X)
has the homotopy type of nerve(W). Thus, we can take nerve(W) as sc(X).
Now, as in the statement of the proposition, we take Auth(X) to consist of the
components of MapLHM(X,X) which are invertible in π0mapL
HM(X,X). But,
by [11, 4.6(ii)], the map BMapLHW(X,X) → BAut
h(X) is a weak equivalence of
simplicial sets. Since LHW can be connected to LW by a finite string of weak
equivalences, it follows that so can MapLHW(X,X) and MapLW(X,X). Thus,
BMapLW(X,X) and BAut
h(X) can also be connected by such a string. It follows
that sc(X) has the same homotopy type as BAuth(X). �
Proof of Theorem 6.2. Consider the category M[n] of functors [n] → M. If M
is a model category, then M[n] can be given the structure of a model category
with the weak equivalences and fibrations given by levelwise weak equivalences and
fibrations inM. Given any map [m]→ [n], we obtain a functorM[n] →M[m].
Let Y = (y0 → y1 · · · → yn) be a fibrant-cofibrant object ofM
[n]. It restricts to
an object Y ′ = (y0 → y1 · · · → yn−1) inM
[n−1]. From this map, we obtain a map
of simplicial sets
BAuth
LHM[n](Y )→ BAut
LHM(yn)×BAut
LHM[n−1](Y
The homotopy fiber of this map is weakly equivalent to the union of those compo-
nents of MapLHM(yn−1, yn) containing the conjugates of the map fn−1 : yn−1 → yn,
or maps j ◦ fn−1 ◦ i, where i and j are self-homotopy equivalences.
Iterating this process, we can take the homotopy fiber of the map
BAuth
LHM[n](Y )→ BAut
LHM(yn)× · · · ×BAut
LHM(y0),
which is weakly equivalent to the union of the components of
MapLHM(yn−1, yn)× · · · ×MapLHM(y0, y1)
containing conjugates of the sequence of maps fi : yi → yi+1, 0 ≤ i ≤ n − 1.
However, applying Proposition 6.3 to the map in question shows that this simplicial
set is also the homotopy fiber of the map
sc(Y )→ sc(yn)× · · · × sc(y0).
Let U denote the simplicial space N(M,W) so that Un = nerve(we(M
[n])).
Then, let V be a Reedy fibrant replacement of U , from which we get weak equiva-
lences Un → Vn for all n ≥ 0.
For each n ≥ 0, there exists a map pn : Un → U
0 given by iterated face
maps to the “objects.” Then, for every (n+1)-tuple of objects (x0, x1, . . . , xn), the
homotopy fiber of pn over (x0, . . . , xn), given by
mapV (xn−1, xn)× · · · ×mapV (x0, x1),
is weakly equivalent to
MapLHM(x
n−1, x
n )× · · · ×MapLHM(x
0 , x
where xcf denotes a fibrant-cofibrant replacement of X inM. It follows that once
we take the Reedy fibrant replacement V of U , it is a Segal space.
Now, consider the set π0U0, which consists of the weak equivalence classes of
objects inM; it follows that π0V0 is an isomorphic set. Further, note that
HomHo(M)(x, y) = π0MapLHM(x
cf , ycf).
16 J.E. BERGNER
Thus, we have shown that Ho(M) is equivalent to Ho(V ).
It remains to show that V is a complete Segal space. Consider the space
Vhoequiv ⊆ V1, and define Uhoequiv to be the preimage of Vhoequiv under the natural
map U → V . Since V is a Reedy fibrant replacement for U , it suffices to show
that the complete Segal space condition holds, i.e., that U0 → Uhoequiv is a weak
equivalence of simplicial sets. Notice that Uhoequiv must consist precisely of the
components of U1 whose 0-simplices come from weak equivalences inM. In other
words, Uhoequiv = nerve(we(we(M))
[1]).
There is an adjoint pair of functors
F :M[1] ⇄M : G
for which F (x→ y) = x and G(x) = idx. This adjoint pair can be restricted to an
adjoint pair
F : nerve(we(we(M))[1]) ⇄ we(M) : G
which in turn induces a weak equivalence of simplicial sets on the nerves, Uhoequiv ≃
U0, which completes the proof. �
Now that we have proved that the mapping spaces and homotopy categories agree
for V and for LHM, it remains to show that they agree for LHM and LC(L
Theorem 6.4. Let M be a model category. For any x and y objects of LHM,
there is a weak equivalence of simplicial sets
MapLHM(x, y) ≃ mapLC(LHM)(x, y),
and there is an equivalence of categories π0L
HM≈ Ho(LC(L
HM)).
Note in particular that x and y are just objects of M, and that π0L
HM is
equivalent to the homotopy category Ho(M).
Proof. Given the hammock localization LHM of the model categoryM, we have
the following composite map of simplicial spaces:
X = nerve(LHM)→ Xf → LC(L
Here, Xf denotes a Reedy fibrant replacement of X . This composite is just Rezk’s
method for assigning the complete Segal space LC(L
HM) to the simplicial category
On the left-hand side, the mapping spaces of X = nerve(LHM) are precisely
those of LHM, by the definition of the nerve functor. In the nerve, one of these
mapping spaces, say mapX(x, y) for some objects x and y ofM, is given by the fiber
over (x, y) of the map (d1, d0) : X1 → X0×X0. Although these mapping spaces can
be defined for X , there is no reason that they are homotopy invariant. When we
take a Reedy fibrant replacement Xf of X , however, this map becomes a fibration,
and hence this fiber is actually a homotopy fiber and so homotopy invariant. For a
general simplicial space, we cannot assume that the mapping spaces of the Reedy
fibrant replacement are equivalent to the original ones. However, if the 0-space of
the simplicial space in question is discrete in degree zero, then the map above is
a fibration. Using an argument similar to the one in [5, §5], we can find a Reedy
fibrant replacement functor which leaves the 0-space discrete. While the space in
degree one might be changed in this process of passing to X
1 , it will still be weakly
equivalent X1. In particular, the mapping spaces in X
f will be weakly equivalent
to those in X .
COMPLETE SEGAL SPACES 17
Since the objects of Xf are just the objects of LHM, or the objects ofM, this
equivalence of mapping spaces gives us also an equivalence of homotopy categories.
The right-most map is the one defined by Rezk, iXf : X
f → X̂f , which takes a
Segal space to a complete Segal space. But, he defines this map in such a way that it
is in fact a Dwyer-Kan equivalence. In other words, it induces weak equivalences on
mapping spaces and an equivalence of homotopy categories. Thus, the composite
map induces equivalences on mapping spaces and an equivalence on homotopy
categories. �
7. A Characterization of Complete Segal Spaces Arising from
Simplicial Categories
In this section, we give a thorough description of the weak equivalence type of
complete Segal spaces which occur as images of Rezk’s functor from the category
of simplicial categories. We consider several different cases, beginning with ones for
which we can use the classifying diagram construction, i.e., discrete categories, and
then proceed to more general simplicial categories.
It should be noted that we are characterizing these complete Segal spaces up to
weak equivalence, and so the resulting descriptions are of the homotopy type of the
spaces in each simplicial degree. For example, in the case of a discrete category,
we describe the corresponding complete Segal space in terms of the isomorphism
classes of objects, rather than in terms of individual objects, in order to simplify
the description. One could just take all objects, and generally get much larger
spaces, if the more precise description were needed for the complete Segal space
corresponding to a given category.
Furthermore, notice that determining the homotopy type of the spaces in degrees
zero and one are sufficient to determine the homotopy type of all the spaces, since we
are considering Segal spaces. Thus, we focus our attention on these spaces, adding
in a few comments about how to continue the process with the higher-degree spaces.
7.1. Case 1: C is a discrete groupoid. If C = G is a group, then applying
Rezk’s classifying diagram construction results in a complete Segal space equivalent
to BG, i.e., the constant simplicial space which is the simplicial set BG at each
level. In particular, since all morphisms are invertible, we obtain essentially no new
information at level 1 that we didn’t have already at level 0.
Example 7.1. Let G = Z/2. Then (NG)0 is just the nerve, or BZ/2. Then (NG)1
has two 0-simplices, given by the two morphisms (elements) of G. However, these
two objects of G[1] are isomorphic, and the automorphism group of either one of
them is Z/2. Thus, (NG)1 is also equivalent to BZ/2.
If C has more than one object but only one isomorphism class of objects, we get
instead a simplicial space weakly equivalent to the constant simplicial space which
is BAut(x) at each level, for a representative object x. If C has more than one
isomorphism class 〈x〉, then the result will be weakly equivalent to the constant
simplicial space
〈x〉 BAut(x).
7.2. Case 2: C is a discrete category. Since in the classifying diagram NC,
(NC)0 picks out the isomorphisms of C only, we still essentially have
〈x〉 BAut(x)
18 J.E. BERGNER
at level 0. However, if C is not a groupoid, then there is new information at level
1. It instead looks like ∐
〈x〉,〈y〉
BAut(
Hom(x, y)α)
where the α index the isomorphism classes of elements of Hom(x, y). The subspace
of (NC)1 corresponding to 〈x〉, 〈y〉, denoted (NC)1(x, y), fits into a fibration
Hom(x, y)→ (NC)1(x, y)→ BAut(x)×BAut(y).
The space in dimension 2 is determined, then, by the spaces at levels 0 and 1.
The subspace corresponding to isomorphism classes of objects 〈x〉, 〈y〉, 〈z〉, denoted
(NC)2(x, y, z), fits into a fibration
Hom(x, y)×Hom(y, z)→ (NC)2(x, y, z)→ BAut(x) ×BAut(y)×BAut(z).
The whole space (NC)2, up to homotopy, looks like
〈x〉,〈y〉,〈z〉
〈α〉,〈β〉
Hom(x, y)α ×Hom(y, z)β
We could describe each (NC)n analogously.
Example 7.2. Let C denote the category with two objects and one nontrivial
morphism between them (· → ·). If {e} denotes the trivial group, then (NC)0 ≃
B{e} ∐ B{e} and (NC)1 ≃ B{e} ∐ B{e} ∐ B{e}. In particular, NC is not equiv-
alent to the classifying diagram of the trivial category with one object and one
morphism, which would be the constant simplicial space B{e}. However, note that
the nerves of these two categories are homotopy equivalent. Thus, we can see that
the classifying diagram is more refined than the nerve in distinguishing between
these two categories.
7.3. Case 3: C is a simplicial groupoid. First, consider the case where we have
a simplicial group G. Let Gn denote the group of n-simplices of G. Then
hocolim∆op(nerve(Gn)
t) = nerve(G).
Let LC denote Rezk’s completion functor which makes the nerve into a complete
Segal space. We claim that
LC(hocolim∆op(nerve(Gn)
t)) ≃ LC(hocolim∆opLC(nerve(Gn)
We actually prove the more general statement that, for any X = hocolim∆opXn,
LC(hocolim∆opXn) ≃ LC(hocolim∆opLCXn).
To prove this claim, first note that we have Rezk’s completion map
i : hocolim∆opXn → LC(hocolim∆opXn)
which is a weak equivalence. Furthermore, since in CSS any complete Segal space
Y is a local object and every object is cofibrant, we have a weak equivalence of
spaces
Map(LC(hocolim∆opXn), Y ) ≃Map(hocolim∆opXn, Y ).
COMPLETE SEGAL SPACES 19
So, for any complete Segal space Y , we have that
Map(LChocolim∆op(LCXn), Y ) ≃Map(hocolim∆op(LCXn), Y )
≃ holim∆Map(LCXn, Y )
≃ holim∆Map(Xn, Y )
≃Map(hocolim∆opXn, Y )
≃Map(LChocolim∆opXn, Y ).
Note that the above calculation depends on the fact that,
Map(hocolim∆opXn, Y ) ≃ holim∆Map(Xn, Y ),
which follows from working levelwise on simplicial sets.
Then, since Gn is a discrete group, completing its nerve is the same as taking
the classifying diagram NGn which, by case 1, is weakly equivalent to the constant
simplicial space BGn, denoted here cBGn. Thus we have:
LC(nerve(Gn)) ≃ LC [hocolim∆op(nerve(Gn))]
≃ LC [hocolim∆op(LC(nerve(Gn)))]
≃ LC [hocolim∆op(cBGn)]
≃ LC(BG)
≃ BG.
So, we obtain a simplicial space weakly equivalent to the constant simplicial
space with BG at each level. (Recall, however, that BG here is obtained by taking
the diagonal of the simplicial nerve, so it is not quite the identical case.) If we
have a simplicial groupoid, rather than a simplicial group, we obtain the analogous
result, replacing BG with ∐
BAut(x).
7.4. Case 4: C is a simplicial category with every morphism invertible up
to homotopy. Alternatively stated, this case covers the situation in which π0(C)
is a groupoid.
Recall that we have a model structure SCO on the category of categories with a
fixed object set O, in which the cofibrant objects are retracts of free objects. So,
taking a cofibrant replacement of C in this model category structure SCO essen-
tially gives a free replacement of C, denoted F (C), which is weakly equivalent to C.
(This cofibrant category can be obtained by taking a simplicial resolution F∗C and
then taking a diagonal [12, 6.1].) Now, taking the localization with respect to all
morphisms results in a simplicial groupoid. So, we have Dwyer-Kan equivalences
F (C)−1F (C) F (C)
≃oo ≃ //C
But, now F (C)−1F (C) is a simplicial groupoid weakly equivalent to C, so we have
now reduced this situation to case 3.
Note that, to write down a description of this complete Segal space in terms of the
original category C, we need to take isomorphism classes of objects in π0(C), or weak
equivalence classes, as well as self-maps which are invertible up to homotopy rather
than strict automorphisms. While we will still use 〈x〉 to denote the equivalence
class of a given object, we will use Auth(x) to signify homotopy automorphisms of
20 J.E. BERGNER
x. Thus, the complete Segal space corresponding to C in this case essentially looks
like ∐
BAuth(x)
at each level.
7.5. Case 5: C is any simplicial category. First consider the subcategory of C
containing all the objects of C and only the morphisms of C which are invertible up
to homotopy. Apply case 5 to get a complete Segal space, but take only the 0-space
of it, to be the 0-space of the desired complete Segal space.
To find the 1-space, first recall the definition of the completion functor as applied
to a Segal space W :
LC(W ) = LCSS(diag([m] 7→ (W
E(m))n)).
Recall further that (WE(m))n = Map(E(m)×∆[n]
t,W ). Thus, the Segal space we
obtain (before applying the functor LCSS) looks like
Map(E(0)×∆[0]t,W )⇐ Map(E(1)×∆[1]t,W ) ⇚ Map(E(2)×∆[2]t,W ) · · · .
If the Segal space W is a fibrant replacement of nerve(C), then the space at level 1
consists of diagrams
x′ // y
with the maps in the appropriate simplicial level.
For simplicity, we restrict to a given pair of objects x and y, representing given
equivalence classes. Consider the homotopy automorphisms of x and y. If they are
not all invertible, we take a cofibrant replacement and group completion as in case
4. So, without loss of generality, assume that Aut(x) and Aut(y) are simplicial
groups. Note that we have
Aut(x) = hocolim∆opAut(x)n
Aut(y) = hocolim∆opAut(y)n.
Now look at
Map(x, y) = hocolim∆opMap(x, y)n.
Consider for each n ≥ 0 the discrete category C(x, y)n which has as objects Map(x, y)n
and as morphisms pairs (α, β) of automorphisms in Aut(x)n × Aut(y)n making a
commutative square
with f, f ′ ∈Map(x, y)n.
Thus, the 1-space that we are interested in is also the 1-space of the complete
Segal space given by
LCSS(hocolim∆op(nerve(C(x, y)n))).
COMPLETE SEGAL SPACES 21
Using a straightforward argument about localization functors similar to the one in
case 3 (which can be found in [3, 4.1]), we can also apply the functor LCSS on the
inside to get an equivalent simplicial space
LCSS(hocolim∆op(LCSSnerve(C(x, y)n))).
But, since C(x, y)n is a discrete category, this space is just
LCSShocolim∆op(NC(x, y)n) ≃ hocolim∆op(NC(x, y)n).
Now, we restrict to the 1-space here, which is
hocolim∆op
〈x〉,〈y〉
Mapn(x, y)α
〈x〉,〈y〉
Map(x, y)α
As with the previous case, we can then go back and weaken to homotopy auto-
morphisms and equivalence classes of objects to consider categories before taking
a group completion, so our space looks like
〈x〉,〈y〉
BAuth
Map(x, y)α
We could then obtain the 2-space of our complete Segal space by considering
categories C(x, y, z)n defined similarly, and the description of the 2-space of the
classifying diagram of a discrete category as given in case 2.
We can summarize these results in the following theorem. For an object x of
a simplicial category C, let 〈x〉 denote the weak equivalence class of x in C, and
for a morphism α : x → y, let 〈α〉 denote the weak equivalence class of α in the
morphism category C[1]. Let Auth(x) denote the space of self-maps of x which are
invertible in π0C.
Theorem 7.3. Let C be a simplicial category. The complete Segal space corre-
sponding to C has the form
BAuth(x)⇐
〈x〉,〈y〉
BAuth
Map(x, y)α
 ⇚ · · · .
References
[1] J.E. Bergner, Homotopy fiber products of homotopy theories, in preparation.
[2] J.E. Bergner, A model category structure on the category of simplicial categories, Trans.
Amer. Math. Soc. 359 (2007), 2043-2058.
[3] J.E. Bergner, Simplicial monoids and Segal categories, Contemp. Math. 431 (2007) 59-83.
[4] J.E. Bergner, A survey of (∞, 1)-categories, preprint available at math.AT/0610239.
[5] J.E. Bergner, Three models for the homotopy theory of homotopy theories, Topology 46
(2007), 397-436.
[6] J.M. Cordier and T. Porter, Vogt’s theorem on categories of homotopy coherent diagrams,
Math. Proc. Camb. Phil. Soc. (1986), 100, 65-90.
[7] Daniel Dugger, Universal homotopy theories, Adv. Math. 164 (2001), no. 1, 144–176.
[8] W.G. Dwyer and D.M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18
(1980), 17-35.
[9] W.G. Dwyer and D.M. Kan, A classification theorem for diagrams of simplicial sets, Topology
23(1984), 139-155.
[10] W.G. Dwyer and D.M. Kan, Equivalences between homotopy theories of diagrams, Algebraic
topology and algebraic K-theory (Princeton, N.J., 1983), 180–205, Ann. of Math. Stud., 113,
Princeton Univ. Press, Princeton, NJ, 1987.
http://arxiv.org/abs/math/0610239
22 J.E. BERGNER
[11] W.G. Dwyer and D.M. Kan, Function complexes in homotopical algebra, Topology 19 (1980),
427-440.
[12] W.G. Dwyer and D.M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17
(1980), no. 3, 267–284.
[13] W.G. Dwyer and J. Spalinski, Homotopy theories and model categories, in Handbook of
Algebraic Topology, Elsevier, 1995.
[14] P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Math, vol. 174,
Birkhauser, 1999.
[15] Philip S. Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys and
Monographs 99, AMS, 2003.
[16] Mark Hovey, Model Categories, Mathematical Surveys and Monographs, 63. American Math-
ematical Society 1999.
[17] A. Joyal, Simplicial categories vs quasi-categories, in preparation.
[18] A. Joyal, The theory of quasi-categories I, in preparation.
[19] André Joyal and Myles Tierney, Quasi-categories vs Segal spaces, Contemp. Math. 431 (2007)
277-326.
[20] Saunders Mac Lane, Categories for the Working Mathematician, Second Edition, Graduate
Texts in Mathematics 5, Springer-Verlag, 1997.
[21] J.P. May, Simplicial Objects in Algebraic Topology, University of Chicago Press, 1967.
[22] C.L. Reedy, Homotopy theory of model categories, unpublished manuscript, available at
http://www-math.mit.edu/∼psh.
[23] Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math.
Soc. 353(3) (2001), 973-1007.
[24] Bertrand Toën, Derived Hall algebras, Duke Math. J. 135, no. 3 (2006), 587-615.
Kansas State University, 138 Cardwell Hall Manhattan, KS 66506
E-mail address: bergnerj@member.ams.org
http://www-math.mit.edu/~psh
	1. Introduction and Overview
	2. Background on Model Categories and Simplicial Objects
	3. Simplicial Categories and Simplicial Localizations
	4. Complete Segal Spaces
	5. Obtaining Complete Segal Spaces from Simplicial Categories and Model Categories
	6. Comparison of Functors from SC to CSS
	7. A Characterization of Complete Segal Spaces Arising from Simplicial Categories
	7.1. Case 1: C is a discrete groupoid
	7.2. Case 2: C is a discrete category
	7.3. Case 3: C is a simplicial groupoid
	7.4. Case 4: C is a simplicial category with every morphism invertible up to homotopy
	7.5. Case 5: C is any simplicial category
	References

  In this paper, we compare several functors which take simplicial categories
or model categories to complete Segal spaces, which are particularly nice
simplicial spaces which, like simplicial categories, can be considered to be
models for homotopy theories. We then give a characterization, up to weak
equivalence, of complete Segal spaces arising from these functors.

Introduction and Overview
The idea that simplicial categories, or categories enriched over simplicial sets,
model homotopy theories goes back to a series of several papers by Dwyer and
Kan [8], [10], [11], [12]. Taking the viewpoint that a model category, or more
generally a category with weak equivalences, can be considered to be a model for
a homotopy theory, they develop two methods to obtain from a model category a
simplicial category. This “simplicial localization” encodes higher-order structure
which is lost when we pass to the homotopy category associated to the model
category. Furthermore, they prove that, up to a natural notion of weak equivalence
of simplicial categories (see Definition 3.1), every simplicial category arises as the
simplicial localization of some category with weak equivalences [10, 2.1]. Thus, the
category of all (small) simplicial categories with these weak equivalences can be
regarded as the “homotopy theory of homotopy theories.”
This notion was mentioned briefly at the end of Dwyer and Spalinski’s introduc-
tion to model categories [13, 11.6], and was made precise by the author in [2, 1.1],
in which the category SC of small simplicial categories with these weak equivalences
is shown to have the structure of a model category.
However, this category is not practically useful for many purposes. Simplicial
categories are not particularly easy objects to work with, and the weak equivalences,
while natural generalizations of equivalences of categories, are difficult to identify.
Motivated by this problem, Rezk defines a model structure, which we denote CSS,
on the category of simplicial spaces, in which the fibrant-cofibrant objects are called
complete Segal spaces (see Definition 4.3). This model structure is especially nice,
in that the objects are just diagrams of simplicial sets and the weak equivalences,
at least between complete Segal spaces, are just levelwise weak equivalences of
simplicial sets. Furthermore, this model structure has the additional structures of
a simplicial model category and a monoidal model category. Because it is given by a
Date: October 25, 2018.
2000 Mathematics Subject Classification. Primary: 55U40; Secondary: 55U35, 18G55, 18G30,
18D20.
Key words and phrases. simplicial categories, model categories, complete Segal spaces, homo-
topy theories.
http://arxiv.org/abs/0704.1624v2
2 J.E. BERGNER
localization of a model structure on the category of simplicial spaces with levelwise
weak equivalences, it is an example of a presentation for a homotopy theory as
described by Dugger [7].
In his paper, Rezk defines two different functors, one from the category of sim-
plicial categories, and one from the category of model categories, to the category
of complete Segal spaces. Thus, he describes a relationship between simplicial cat-
egories and complete Segal spaces, but he does not give an inverse construction.
However, the author was able to show in [5] that the model categories SC and
CSS are Quillen equivalent to one another, and therefore are models for the same
homotopy theory. Thus, the hope is that we can answer questions about simplicial
categories by working with complete Segal spaces. This paper is the beginning of
that project.
However, the functor that we use to show that the two model categories are
Quillen equivalent is not the same as Rezk’s functor. Furthermore, the question
arises whether Rezk’s functor on model categories agrees with the composite of the
simplicial localization functor with his functor on simplicial categories. Rezk gives
the beginning of a proof that the two functors agree when the model category in
question has the additional structure of a simplicial model category. Our goal in
this paper is prove that, up to weak equivalence in CSS, the two functors from SC
to CSS are the same, that Rezk’s result holds for model categories which are not
necessarily simplicial, and that this result does imply that the functor on model
categories agrees with the one on their corresponding simplicial categories up to
weak equivalence.
We then go on to characterize, up to weak equivalence, the complete Segal space
arising from any simplicial category. It can be described at each level as the nerve
of the monoid of self weak equivalences of representing objects of the category.
We should mention that the model categories SC and CSS are only two of several
known models for the homotopy theory of homotopy theories. Our proof that the
two are Quillen equivalent actually uses two intermediate model structures SeCatc
and SeCatf on the category of Segal precategories, or simplicial spaces with a
discrete Space at level zero [5]. The fibrant-cofibrant objects in these structures are
known as Segal categories. Furthermore, Joyal and Tierney have shown that there is
a Quillen equivalence between each of these model structures and a model structure
QCat on the category of simplicial sets [17], [19]. The fibrant-cofibrant objects of
this model structure are known as quasi-categories and are generalizations of Kan
complexes [18]. One can compare our composite functor SC → CSS to one they
define which factors through QCat rather than through SeCatc and SeCatf . It is
a consequence of Joyal and Tierney’s work that these two different functors give
rise to weakly equivalent complete Segal spaces, as we will describe further in the
section on complete Segal spaces. An introduction to each of these model structures
and the Quillen equivalences can be found in the survey paper [4].
In [1], we use the results of this paper to consider the complete Segal spaces
arising from the homotopy fiber product construction for model categories, as de-
scribed by Toën in his work on derived Hall algebras [24]. It seems that results
relating diagrams of model categories to diagrams of complete Segal spaces will
prove to be useful.
COMPLETE SEGAL SPACES 3
Acknowledgments. I would like to thank Bill Dwyer, Charles Rezk, André Joyal,
and Myles Tierney for conversations about the work in this paper, as well as the
referee for helpful comments on the exposition.
2. Background on Model Categories and Simplicial Objects
In this section, we summarize some facts about model categories, simplicial sets,
and other simplicial objects that we need in the course of this paper.
A model category M is a category with three distinguished classes of mor-
phisms, fibrations, cofibrations, and weak equivalences. A morphism which is both
a (co)fibration and a weak equivalence is called an acyclic (co)fibration. The cate-
goryM with these choices of classes is required to satisfy five axioms [13, 3.3].
Axiom MC1 guarantees that M has small limits and colimits, so in particular
M has an initial object and a terminal object. An object X in a model category
M is fibrant if the unique map X → ∗ to the terminal object is a fibration. Dually,
X is cofibrant if the unique map from the initial object φ→ X is a cofibration.
The factorization axiom (MC5) can be applied in such a way that, given any
object X ofM, we can factor the map X → ∗ as a composite
∼ //X ′ //∗
of an acyclic cofibration followed by a fibration. In this case, X ′ is called a fibrant
replacement of X , since it is weakly equivalent to X and fibrant. This replacement
is not necessarily unique, but in all the model categories we consider here it can be
assumed to be functorial [16, 1.1.1]. Cofibrant replacements can be defined dually.
The structure of a model category enables us to invert the weak equivalences
formally in such a way that we still have a set, rather than a proper class, of mor-
phisms between any two objects. If we were merely to take the localizationW−1M,
there would be no guarantee that we would not have a proper class. However, with
the structure of a model category, we can define the homotopy category Ho(M) to
have the same objects asM, and as morphisms
HomHo(M)(X,Y ) = [X
cf , Y cf ]M,
where the right-hand side denotes homotopy classes of maps between fibrant-cofibrant
replacements of X and Y , respectively.
The standard notion of equivalence of model categories is given by the following
definitions. First, recall that an adjoint pair of functors F : C ⇆ D : G satisfies the
property that, for any objects X of C and Y of D, there is a natural isomorphism
ϕ : HomD(FX, Y )→ HomC(X,GY ).
The functor F is called the left adjoint and G the right adjoint [20, IV.1].
Definition 2.1. [16, 1.3.1] An adjoint pair of functors F : M ⇆ N : G between
model categories is a Quillen pair if F preserves cofibrations and G preserves fibra-
tions.
Definition 2.2. [16, 1.3.12] A Quillen pair of model categories is a Quillen equiv-
alence if if for all cofibrant X in M and fibrant Y in N , a map f : FX → Y is a
weak equivalence in D if and only if the map ϕf : X → GY is a weak equivalence
4 J.E. BERGNER
An important example of a model category is that of the standard model struc-
ture on the category of simplicial sets SSets. Recall that a simplicial set is a functor
X : ∆op → Sets, where ∆op is the opposite of the category ∆ of finite ordered sets
[n] = {0→ 1→ · · · → n} and order-preserving maps between them. We denote the
set X([n]) by Xn. In particular in X we have face maps di : Xn → Xn−1 and de-
generacy maps si : Xn → Xn+1 for each 0 ≤ i ≤ n, satisfying several compatibility
conditions. Three particularly useful examples of simplicial sets are the n-simplex
∆[n] and its boundary ∆̇[n] for each n ≥ 0, and the boundary with the kth face
removed, V [n, k], for each n ≥ 1 and 0 ≤ k ≤ n. Given a simplicial set X , we can
take its geometric realization |X |, which is a topological space [14, I.1].
In the standard model category structure on SSets, the weak equivalences are
the maps f : X → Y for which the geometric realization |f | : |X | → |Y | is a weak
homotopy equivalence of topological spaces [14, I.11.3]. In fact, this model structure
on simplicial sets is Quillen equivalent to the standard model structure on the
category of topological spaces [16, 3.6.7].
More generally, a simplicial object in a category C is a functor ∆op → C. The
two main examples which we consider in this paper are those of simplicial spaces
(also called bisimplicial sets), or functors ∆op → SSets, and simplicial groups. We
denote the category of simplicial spaces by SSets∆
A simplicial set X can be regarded as a simplicial space in two ways. It can be
considered a constant simplicial space with the simplicial set X at each level, and in
this case we will denote the constant simplicial set by cX or just X if no confusion
will arise. Alternatively, we can take the simplicial space, which we denote Xt, for
which (Xt)n is the discrete simplicial set Xn. The superscript t is meant to suggest
that this simplicial space is the “transpose” of the constant simplicial space.
A natural choice for the weak equivalences in the category SSets∆
is the class
of levelwise weak equivalences of simplicial sets. If we define the cofibrations to be
levelwise also, we obtain a model structure which is usually referred to as the Reedy
model structure on SSets∆
[22].
The Reedy model structure has the additional structure of a simplicial model
category. In particular, given any two objects X and Y of SSets∆
, there is a
mapping space, or simplicial set Map(X,Y ) satisfying compatibility conditions [15,
9.1.6]. In the case where X is cofibrant (as is true of all objects in the Reedy model
structure) and Y is fibrant, this choice of mapping space is homotopy invariant.
One way to obtain other model structures on the category of simplicial spaces is
to localize the Reedy structure with respect to a set of maps. While this process
works for much more general model categories [15, 3.3.1], we will focus here on this
particular case. Let S = {f : A → B} be a set of maps of simplicial spaces. A
Reedy fibrant simplicial space W is S-local if for each map f ∈ S, the induced map
Map(f,W ) : Map(B,W )→ Map(A,W )
is a weak equivalence of simplicial sets. A map g : X → Y is then an S-local
equivalence if for any S-local object W , the induced map
Map(g,W ) : Map(Y,W )→ Map(X,W )
is a weak equivalence of simplicial sets.
Theorem 2.3. [15, 4.1.1] There is a model structure LSSSets
on the category
of simplicial spaces in which
COMPLETE SEGAL SPACES 5
• the weak equivalences are the S-local equivalences,
• the cofibrations are levelwise cofibrations of simplicial sets, and
• the fibrant objects are the S-local objects.
Furthermore, this model category has the additional structure of a simplicial model
category.
We now turn to a few facts about simplicial groups, or functors from ∆op to the
category of groups. Given a simplicial group G, we can take its nerve, a simplicial
space with
nerve(G)n,m = Hom([m], Gn).
Taking the diagonal of this simplicial space, we obtain a simplicial set, also often
called the nerve of G.
From another perspective, for G a simplicial group (or, more generally, a sim-
plicial monoid), we can find a classifying complex of G, a simplicial set whose
geometric realization is the classifying space BG. A precise construction can be
made for this classifying space by the WG construction [14, V.4.4], [21]. However,
we are not so concerned here with the precise construction as with the fact that
such a classifying space exists, so for simplicity we will simply write BG for the
classifying complex of G.
3. Simplicial Categories and Simplicial Localizations
In this section, we consider simplicial categories and show how they arise from
Dwyer and Kan’s simplicial localization techniques. We then discuss model category
structures, first on the category of simplicial categories with a fixed object set, and
then on the category of all small simplicial categories.
First of all, we clarify some terminology. In this paper, by (small) “simplicial
category” we will mean a category with a set of objects and a simplicial set of
morphisms Map(x, y) between any two objects x and y, also known as a category
enriched over simplicial sets. This notion does not coincide with the more general
one of a simplicial object in the category of small categories, in which we would also
have a simplicial set of objects. Using this more general definition, if we impose
the additional condition that all face and degeneracy maps are the identity on the
objects, then we get our more restricted notion.
A simplicial category can be seen as a generalization of a category, since any
ordinary category can be regarded as a simplicial category with a discrete mapping
space. Given any simplicial category C, we can consider its category of components
π0C, which has the same objects as C and whose morphisms are given by
Homπ0C(x, y) = π0MapC(x, y).
The following definition of weak equivalence of simplicial categories is a natural
generalization of the notion of equivalence of categories.
Definition 3.1. A simplicial functor f : C → D is a Dwyer-Kan equivalence or
DK-equivalence if the following two conditions hold:
(1) For any objects x and y of C, the induced map Map(x, y) → Map(fx, fy)
is a weak equivalence of simplicial sets.
(2) The induced map on the categories of components π0f : π0C → π0D is an
equivalence of categories.
6 J.E. BERGNER
The idea of obtaining a simplicial category from a model categoryM goes back
to several papers of Dwyer and Kan [8], [11], [12]. In fact, they define two different
methods of doing so, the simplicial localization LM [12] and the hammock local-
ization LHM [11]. The first has the advantage of being easier to describe, while
the second is more convenient for making calculations.
It should be noted that these constructions can be made for more general cate-
gories with weak equivalences, and do not depend on the model structure if we are
willing to ignore the potential set-theoretic difficulties. However, as with the homo-
topy category construction, the hammock localization in particular can be defined
much more nicely when we have the additional structure of a model category.
We begin with the construction of the simplicial localization LM. Recall that,
given a category M with some choice of weak equivalences W , we denote the
localization ofM with respect toW byW−1M. This localization is obtained from
M by formally inverting the maps of W . Further, recall that, given a categoryM,
we denote by FM the free category on M, or category with the same objects as
M and morphisms freely generated by the non-identity morphisms of M. Note
in particular that there are natural functors FM → M and FM → F 2M [12,
2.4]. These functors can be used to define a simplicial resolution F∗M, which is a
simplicial category with the category F k+1M at level k [12, 2.5].
We can apply this same construction to the subcategoryW to obtain a simplicial
resolution F∗W . Using these two resolutions, we have the following definition.
Definition 3.2. [12, 4.1] The simplicial localization ofM with respect toW is the
localization (F∗W)
−1(F∗M). This simplicial localization is denoted L(M,W) or
simply LM.
The following result gives interesting information about the mapping spaces in
LM in the case where W is all ofM.
Proposition 3.3. [12, 5.5] Suppose that W =M and nerve(M) is connected.
(i) The simplicial localization LM is a simplicial groupoid, so for all objects x
and y, the simplicial sets MapLM(x, y) are all isomorphic. In particular, the
simplicial sets MapLM(x, x) are all isomorphic simplicial groups.
(ii) The classifying complex BMapLM(x, x) has the homotopy type of nerve(M),
and thus each simplicial set MapLM(x, y) has the homotopy type of the loop
space Ω(nerve(M)).
We now turn to the other construction, that of the hammock localization. Again,
letM be a category with a specified subcategory W of weak equivalences.
Definition 3.4. [11, 3.1] The hammock localization of M with respect to W ,
denoted LH(M,W), or simply LHM, is the simplicial category defined as follows:
(1) The simplicial category LHM has the same objects asM.
(2) Given objects X and Y of M, the simplicial set MapLHM(X,Y ) has as
k-simplices the reduced hammocks of width k and any length between X
COMPLETE SEGAL SPACES 7
and Y , or commutative diagrams of the form
· · · C0,n−1
· · · C1,n−1
~~~~~~~~~
CCCCCCCCC
Ck,1 Ck,2 · · · Ck,n−1
in which
(i) the length of the hammock is any integer n ≥ 0,
(ii) the vertical maps are all in W ,
(iii) in each column all the horizontal maps go the same direction, and if
they go to the left, then they are in W ,
(iv) the maps in adjacent columns go in opposite directions, and
(v) no column contains only identity maps.
Proposition 3.5. [8, 2.2] For a given model categoryM, the simplicial categories
LM and LHM are DK-equivalent.
We should add that the description of the hammock localization can be greatly
simplified if we make use of the model category structure on M. In this case,
Dwyer and Kan prove that it suffices to consider hammocks of length 3 such as the
following [8, §8]:
X C0,1
≃oo //C0,2 Y
≃oo .
Restricting to the category of simplicial categories with a fixed set O of ob-
jects, Dwyer and Kan prove the existence of a model structure on this category,
which we denote SCO [12, 7.2]. In this situation, the weak equivalences are the
DK-equivalences, but with the objects fixed the second condition follows imme-
diately from the first. The fibrations in this model structure are given by the
functors f : C → D inducing, for any objects x and y, fibrations of simplicial sets
MapC(x, y)→ MapD(x, y).
The cofibrations are then defined to be the maps with the left lifting property
with respect to the acyclic fibrations. However, they can be more precisely charac-
terized. To do so, we recall the definition of a free map of simplicial categories.
Definition 3.6. [12, 7.4] A map f : C → D in SCO is free if
(1) f is a monomorphism,
(2) if ∗ denotes the free product, then in each simplicial dimension k, the
category Dk admits a unique free factorization Dk = f(Ck) ∗ Fk, where Fk
is a free category, and
(3) for each k ≥ 0, all degeneracies of generators of Fk are generators of Fk+1.
8 J.E. BERGNER
Definition 3.7. [12, 7.5] A map f : C → D of simplicial categories is a strong
retract of a map f ′ : C → D′ if there exists a commutative diagram
>>}}}}}}}
id // D
Using these definitions, Dwyer and Kan prove the following result.
Proposition 3.8. [12, 7.6] The cofibrations of SCO are precisely the strong retracts
of free maps. In particular, a cofibrant simplicial category is a retract of a free
category.
This result can then be generalized to the category of all simplicial categories, in
which the DK-equivalences are the weak equivalences. If C is a simplicial category, a
morphism e ∈ HomC(a, b)0 is a homotopy equivalence if it becomes an isomorphism
in π0C.
Theorem 3.9. [2, 1.1] There is a model category structure SC on the category of
small simplicial categories in which
• the weak equivalences are the Dwyer-Kan equivalences, and
• the fibrations are the maps f : C → D satisfying the following two condi-
tions:
(i) For any objects a1 and a2 in C, the map
HomC(a1, a2)→ HomD(fa1, fa2)
is a fibration of simplicial sets.
(ii) For any object a1 in C, b in D, and homotopy equivalence e : fa1 → b
in D, there is an object a2 in C and homotopy equivalence d : a1 → a2
in C such that fd = e.
4. Complete Segal Spaces
Here we define complete Segal spaces and describe Rezk’s model structure on the
category of simplicial spaces, in which the complete Segal spaces are the fibrant-
cofibrant objects.
Recall that by a simplicial space we mean a simplicial object in the category
of simplicial sets, or functor ∆op → SSets. In section 2, we described the Reedy
model category structure on this category, in which both the weak equivalences
and cofibrations are defined levelwise. The model structure CSS is given by a
localization of this structure with respect to a set of maps.
We begin with the definition of a Segal space. In [23, 4.1], Rezk defines for each
0 ≤ i ≤ n − 1 a map αi : [1] → [n] in ∆ such that 0 7→ i and 1 7→ i + 1. There is
a corresponding map αi : ∆[1] → ∆[n]. Then for each n he defines the simplicial
space
G(n)t =
αi∆[1]
t ⊂ ∆[n]t.
COMPLETE SEGAL SPACES 9
Let X be a Reedy fibrant simplicial space. There is a weak equivalence of
simplicial sets
MapSSets∆opc (G(n)
t, X)→ X1 ×X0 · · · ×X0 X1︸ ︷︷ ︸
where the right hand side is the limit of the diagram
d0 // X0 X1
d1oo d0 // . . .
d0 // X0 X1
with n copies of X1.
Now, given any n, define the map ϕn : G(n)t → ∆[n]t to be the inclusion map.
Then for any Reedy fibrant simplicial space W there is a map
ϕn = MapSSets∆opc (ϕ
n,W ) : MapSSets∆opc (∆[n]
t,W )→ MapSSets∆opc (G(n)
t,W ).
More simply written, this map is
ϕn : Wn → W1 ×W0 · · · ×W0 W1︸ ︷︷ ︸
and is often called a Segal map. The Segal map is actually defined for any simplicial
space W , but here we assume Reedy fibrancy so that the mapping spaces involved
are homotopy invariant.
Definition 4.1. [23, 4.1] A Reedy fibrant simplicial space W is a Segal space if for
each n ≥ 2 the Segal map
ϕn : Wn →W1 ×W0 · · · ×W0 W1
is a weak equivalence of simplicial sets.
In fact, there is a model category structure SeSp on the category of simplicial
spaces in which the fibrant objects are precisely the Segal spaces [23, 7.1]. This
model structure is obtained from the Reedy structure via localization.
The idea is that in a Segal space there is a notion of “composition,” at least up
to homotopy. In fact, given a Segal space, we can sensibly use many categorical
notions. We summarize some of these ideas here; a detailed description is given by
Rezk [23]. The objects of a Segal space W are given by the set W0,0. Given the
(d1, d0) : W1 →W0 ×W0,
themapping space mapW (x, y) is given by the fiber of this map over (x, y). (The fact
that W is Reedy fibrant guarantees that this mapping space is homotopy invariant.)
Two maps f, g ∈ mapW (x, y)0 are homotopic if they lie in the same component of
the simplicial set mapW (x, y). Thus, we define the space of homotopy equivalences
Whoequiv ⊆W1 to consist of all the components containing homotopy equivalences.
Given any (x0, . . . , xn) ∈ W
0,0 , let mapW (x0, . . . , xn) denote the fiber of the
(α0, . . . , αn) : Wn →W
over (x0, . . . , xn). Consider the commutative diagram
Wn = Map(∆[n]
t,W )
ϕk //
Map(G(n)t,W )
wwppp
Wn+10
10 J.E. BERGNER
and notice that, since W is a Segal space, the horizontal arrow is a weak equivalence
and a fibration. In particular, this map induces an acyclic fibration on the fibers of
the two vertical arrows,
mapW (x0, . . . , xn)→ mapW (xn−1, xn)× · · · ×mapW (x0, x1).
Given f ∈ map(x, y)0 and g ∈ map(y, z)0, their composite is a lift of (g, f) ∈
map(y, z) × map(x, y) along ϕ2 to some k ∈ map(x, y, z)0. The result of this
composition is defined to be d1(k) ∈ map(x, z)0. It can be shown that any two
results are homotopic, so we can use g ◦ f unambiguously.
Then, the homotopy category of W , denoted Ho(W ), has as objects the elements
of the set W0,0, and
HomHo(W )(x, y) = π0mapW (x, y).
A homotopy equivalence in W is a 0-simplex of W1 whose image in Ho(W ) is an
isomorphism.
Definition 4.2. A map f : W → Z of Segal spaces is a Dwyer-Kan equivalence if
(1) for any objects x and y ofW , the induced map mapW (x, y)→ mapZ(fx, fy)
is a weak equivalence of simplicial sets, and
(2) the induced map Ho(W )→ Ho(Z) is an equivalence of categories.
Notice that the definition of these maps bears a striking resemblance to that
of the Dwyer-Kan equivalences between simplicial categories, hence the use of the
same name.
For a Segal space W , note that the degeneracy map s0 : W0 → W1 factors
through the space of homotopy equivalences Whoequiv, since the image of s0 consists
of “identity maps.” Given this fact, we are now able to give a definition of complete
Segal space.
Definition 4.3. [23, §6] A Segal space W is a complete Segal space if the map
W0 →Whoequiv given above is a weak equivalence of simplicial sets.
The idea behind this notion is that, although W0 is not required to be discrete,
as the objects are for a simplicial category, it is not heuristically too different from a
simplicial space with discrete 0-space. (This viewpoint is further confirmed by the
comparison of complete Segal spaces with Segal categories, which are essentially
the analogues of Segal spaces with discrete 0-space [5, 6.3].)
Now, we give a description of the model structure CSS. We do not give all the
details here, such as a description of an arbitrary weak equivalence, but refer the
interested reader to Rezk’s paper [23, §7].
Theorem 4.4. [23, 7.2] There is a model structure CSS on the category of simpli-
cial spaces, obtained as localization of the Reedy model structure, such that
(1) the fibrant objects are precisely the complete Segal spaces,
(2) the cofibrations are the monomorphisms; in particular, every object is cofi-
brant,
(3) the weak equivalences between Segal spaces are Dwyer-Kan equivalences,
(4) the weak equivalences between complete Segal spaces are levelwise weak
equivalences of simplicial sets.
COMPLETE SEGAL SPACES 11
Furthermore, CSS has the additional structure of a simplicial model category and
is cartesian closed.
The fact that CSS is cartesian closed allows us to consider, for any complete Segal
spaceW and simplicial space X , the complete Segal space WX . In particular, using
the simplicial structure, the simplicial set at level n is given by
(WX)n = Map(X ×∆[n]
t,W ).
If W is a (not necessarily complete) Segal space, then WX is again a Segal space;
in other words, the model category SeSp is also cartesian closed.
We denote the functorial fibrant replacement functor in CSS by LCSS . Thus,
given any simplicial space X , there is a weakly equivalent complete Segal space
LCSSX .
This model structure is connected to the model structure SC by a chain of Quillen
equivalences as follows. Each of these model categories is Quillen equivalent to a
model structure SeCatf on the category of Segal precategories. A Segal precategory
is a simplicial space X with X0 a discrete space. A Segal category is then a Segal
precategory with the Segal maps weak equivalences. In the model structure SeCatf ,
the fibrant objects are Segal categories, and so it is considered a Segal category
model structure on the category of Segal precategories. We have the following
chain of Quillen equivalences, with the left adjoint functors topmost:
SC ⇆ SeCatf ⇄ CSS.
The right adjoint SC → SeCatf is given by the nerve functor, and the left adjoint
SeCatf → CSS is given by the inclusion functor.
There is actually another chain of Quillen equivalences connecting the two model
structures; in this case, both SC and CSS are Quillen equivalent to Joyal’s model
structure QCat on the category of simplicial sets [18]. The fibrant objects in QCat
are quasi-categories, or simplicial sets K such that a dotted arrow lift exists making
the diagram
V [m, k] //
commute for any 0 < k < m. The chain of Quillen equivalences in this case is given
SC ⇄ QCat ⇆ CSS.
The right adjoint SC → QCat is given by Cordier and Porter’s coherent nerve
functor [6], [17, 2.10], and the right adjoint CSS → QCat is given given by sending
a simplicial space W to the simplicial set W∗,0 [19, 4.11]. It is a consequence of
work of Joyal [17, §1-2] and of Joyal and Tierney [19, §4-5] that the simplicial space
obtained from a simplicial category via these functors is weakly equivalent to the
one obtained from the composite functor described in the previous paragraph.
5. Obtaining Complete Segal Spaces from Simplicial Categories and
Model Categories
In this section, we describe several different ways of obtaining a complete Segal
space. First, we look at a particularly nice functor which Rezk uses to modify
12 J.E. BERGNER
the notion of a nerve of a category. Then we look at how this functor can be
generalized to one on any simplicial category, and how a similar idea can be used to
get a complete Segal space from any model category. We then consider the functors
used in the Quillen equivalences connecting SC and CSS.
Let us begin with Rezk’s classifying diagram construction, which associates to
any small category C a complete Segal space NC. First, we denote by nerve(C)
the ordinary nerve, which is the simplicial set given by (nerve(C))n = Hom([n], C).
Further, we denote by iso(C) the maximal subgroupoid of C, or subcategory of C
with all objects of C and whose only morphisms are the isomorphisms of C. By
C[n] we denote the category of functors [n]→ C, or the category whose objects are
n-chains of composable morphisms in C.
Definition 5.1. [23, 3.5] The classifying diagram NC is the simplicial space given
by (NC)n = nerve(iso(C
[n])).
Thus, (NC)0 is simply the nerve of iso(C), and (NC)1 is the nerve of the maxi-
mal subgroupoid of the morphism category of C. In particular, information about
invertible morphisms of C is encoded at level 0, while information about the other
morphisms of C does not appear until level 1.
Thus, the classifying diagram of a category can be regarded as a more refined
version of the nerve, since, unlike the ordinary nerve construction, it enables one
to recover information about whether morphisms are invertible or not. This con-
struction is also particularly useful for our purposes due to the following result.
Proposition 5.2. [23, 6.1] If C is a small category, then its classifying diagram
NC is a complete Segal space.
However, this construction, as defined above, cannot be used to assign a complete
Segal space to any simplicial category, since, beginning with level 1, we would have
homotopy invariance problems with a simplicial set of objects in C[1]. Rezk defines
an analogous functor, though, from the category of small simplicial categories which
is similar in spirit to the classifying diagram but avoids these difficulties.
Let I[m] denote the category with m + 1 objects and a single isomorphism
between any two objects, and let E(m) = nerve(I[m])t. If W is a Segal space and
X is any simplicial space, recall that WX denotes the internal hom object, which
is a Segal space. With these notations in place, we can give the definition of Rezk’s
completion functor. Let W be a Segal space. Then its completion Ŵ is defined as
a fibrant replacement in CSS of the simplicial space W̃ defined by
W̃n = diag([m] 7→ MapSSets∆op (E(m),W
∆[n]t)) = diag([m] 7→ (WE(m))n).
From a simplicial category C, then, we can takes its nerve to obtain a simplicial
space, followed by a fibrant replacement functor in the Segal space model structure,
to obtain a Segal space W . From W we can then pass to a complete Segal space
via this completion functor. We will denote this complete Segal space LC(W ), or
LC(C) where W comes from a simplicial category as just described.
The first important fact about this completion functor is that the completion
map iW : W → Ŵ = LC(W ) is not only a weak equivalence in the model category
CSS, but is also a Dwyer-Kan equivalence of Segal spaces [23, §14]. Furthermore,
this completion functor restricts nicely to the classifying diagram in the case where
C is a discrete category.
COMPLETE SEGAL SPACES 13
Proposition 5.3. [23, 14.2] If C is a discrete category, then LC(C) is isomorphic
to NC.
If we begin with a model category M with subcategory of weak equivalences
W , a functor analogous to the classifying diagram functor can be used to obtain a
complete Segal space. In this case, rather than taking the subcategory iso(M) of
isomorphisms ofM, we take the subcategory of weak equivalences, denoted we(M).
Thus, Rezk defines the classification diagram of (M,W), denoted N(M,W), by
N(M,W)n = nerve(we(M
[n])).
Unlike the classifying diagram, the classification diagram of a model category is not
necessarily a complete Segal space as stated, but taking a Reedy fibrant replacement
of it results in a complete Segal space, as we show in the next section.
Lastly, we have the two functors given by the two different chains of Quillen
equivalences between the model categories SC and CSS. As mentioned in the
previous section, these two functors are equivalent. In each case, the resulting
simplicial space is not Reedy fibrant in general, and so not a complete Segal space,
but applying the fibrant replacement functor LCSS results in a complete Segal space.
The first of these composite functors, in particular, is simple to describe ab-
stractly, as in the previous section, but it has a disadvantage over Rezk’s functor in
that it gives very little insight into what the resulting complete Segal space looks
like. In the next section, we prove that the two functors from SC to CSS result
in weakly equivalent complete Segal spaces, and that if we use Rezk’s classification
diagram construction we get a weakly equivalent complete Segal space to the one
we would obtain by taking the simplicial localization followed by his completion
functor. We then use Rezk’s functor to describe what the complete Segal space
corresponding to a simplicial category looks like.
6. Comparison of Functors from SC to CSS
Here we prove that each of the functors we have described all give rise to com-
plete Segal spaces weakly equivalent to those given in the previous section. We
begin by stating the result that establishes the equivalence between Rezk’s com-
pletion functor LC : SC → CSS and the functor arising from the chain of Quillen
equivalences factoring through SeCatf . Let LCSS denote the functorial fibrant re-
placement functor in CSS.
Theorem 6.1. If C is a simplicial category, then the complete Segal spaces LC(C)
and LCSS(nerve(C)) are weakly equivalent in CSS.
Proof. Let LS denote a fibrant replacement functor in the Segal space model struc-
ture SeSp on the category of simplicial spaces. The fact that the two functors
in question result in weakly equivalent complete Segal spaces can be shown by
considering the following chain of weak equivalences:
LCSS(nerve(C))← nerve(C)→ LSnerve(C)→ LC(C).
The map on the left is the localization functor in CSS and so is a weak equivalence
in CSS. The middle map is a weak equivalence in SeSp and therefore also a weak
equivalence in CSS, since the latter model category is a localization of the former.
The map on the right is Rezk’s completion, and it is a weak equivalence in CSS, as
given in the previous section. Therefore, the objects at the far left and right of this
14 J.E. BERGNER
zigzag, both of which are complete Segal spaces, are weakly equivalent as objects
of CSS. �
Now, we would like to compare either of these functors to the classifying diagram
construction for a model category M. In other words, we want to show that
N(M,W) is equivalent to LC(L
HM), where we first take the hammock localization
ofM to obtain a simplicial category, and then apply Rezk’s functor LC .
An initial problem here is that N(M,W) is not necessarily Reedy fibrant, and
so it is not necessarily a complete Segal space. We prove that a Reedy fibrant
replacement of it, denoted N(M,W)f , is in fact a complete Segal space in the
process of comparing the “mapping spaces” in this Reedy fibrant replacement to
the mapping spaces of the hammock localization LHM.
Theorem 6.2. Let M be a model category, and let W denote its subcategory of
weak equivalences. Then N(M,W)f , is a complete Segal space. Furthermore, for
any objects x, y of M, there is a weak equivalence of spaces mapN(M,W)f (x, y) ≃
MapLHM(x, y), and there is an equivalence of categories Ho(N(M,W)
f ) ≈ Ho(M).
This result was proved by Rezk in the case where M is a simplicial model
category [23, 8.3], namely, in the case where we do not need to pass to the simplicial
localization ofM to consider its function complexes. However, here we prove that,
as he conjectured [23, 8.4], the result continues to hold in this more general case.
We prove this theorem very similarly to the way Rezk proves it in the more
restricted case, using a proposition of Dwyer and Kan. To begin, we introduce
some terminology. LetM be a model category. A classification complex ofM, as
defined in [9, 1.2], is the nerve of any subcategory C ofM such that
(1) every map in C is a weak equivalence,
(2) if f : X → Y inM is a weak equivalence and either X or Y is in C, then f
is in C, and
(3) nerve(C) is homotopically small; i.e., each homotopy group of |nerve(C)| is
small [11, 2.2].
The special classification complex sc(X) of an object X in M is a connected
classification complex containing X .
LetM be a model category and X a fibrant-cofibrant object ofM. Denote by
Auth(X) the simplicial monoid of weak equivalences given by Auth
(X) in the
hammock localization LHM, and by BAuth(X) its classifying complex.
The following proposition was proved by Dwyer and Kan in [9, 2.3] in the case
thatM is a simplicial model category. However, the proof does not actually require
the simplicial structure; in fact, their proof is essentially the one given below, with
the extra step showing that the mapping spaces in the hammock localization are
equivalent to those given by the simplicial structure ofM [11, 4.8].
Proposition 6.3. Let X be an object of a model category M. The classifying
complex BAuth(X) is weakly equivalent to the special classification complex of X,
sc(X), and the two can be connected by a finite zig-zag of weak equivalences.
Proof. Let W be the subcategory of weak equivalences of M. Consider the con-
nected component of nerve(W) containing X . For the rest of this proof, we assume
that W is such that its nerve is connected. We further assume that nerve(W) is
homotopically small, taking an appropriate subcategory, as described in [11, 2.3],
if necessary.
COMPLETE SEGAL SPACES 15
In this case, by Proposition 3.3, the function complexes MapLW(X,X) are all iso-
morphic. Furthermore, by the same result, the classifying complex BMapLW(X,X)
has the homotopy type of nerve(W). Thus, we can take nerve(W) as sc(X).
Now, as in the statement of the proposition, we take Auth(X) to consist of the
components of MapLHM(X,X) which are invertible in π0mapL
HM(X,X). But,
by [11, 4.6(ii)], the map BMapLHW(X,X) → BAut
h(X) is a weak equivalence of
simplicial sets. Since LHW can be connected to LW by a finite string of weak
equivalences, it follows that so can MapLHW(X,X) and MapLW(X,X). Thus,
BMapLW(X,X) and BAut
h(X) can also be connected by such a string. It follows
that sc(X) has the same homotopy type as BAuth(X). �
Proof of Theorem 6.2. Consider the category M[n] of functors [n] → M. If M
is a model category, then M[n] can be given the structure of a model category
with the weak equivalences and fibrations given by levelwise weak equivalences and
fibrations inM. Given any map [m]→ [n], we obtain a functorM[n] →M[m].
Let Y = (y0 → y1 · · · → yn) be a fibrant-cofibrant object ofM
[n]. It restricts to
an object Y ′ = (y0 → y1 · · · → yn−1) inM
[n−1]. From this map, we obtain a map
of simplicial sets
BAuth
LHM[n](Y )→ BAut
LHM(yn)×BAut
LHM[n−1](Y
The homotopy fiber of this map is weakly equivalent to the union of those compo-
nents of MapLHM(yn−1, yn) containing the conjugates of the map fn−1 : yn−1 → yn,
or maps j ◦ fn−1 ◦ i, where i and j are self-homotopy equivalences.
Iterating this process, we can take the homotopy fiber of the map
BAuth
LHM[n](Y )→ BAut
LHM(yn)× · · · ×BAut
LHM(y0),
which is weakly equivalent to the union of the components of
MapLHM(yn−1, yn)× · · · ×MapLHM(y0, y1)
containing conjugates of the sequence of maps fi : yi → yi+1, 0 ≤ i ≤ n − 1.
However, applying Proposition 6.3 to the map in question shows that this simplicial
set is also the homotopy fiber of the map
sc(Y )→ sc(yn)× · · · × sc(y0).
Let U denote the simplicial space N(M,W) so that Un = nerve(we(M
[n])).
Then, let V be a Reedy fibrant replacement of U , from which we get weak equiva-
lences Un → Vn for all n ≥ 0.
For each n ≥ 0, there exists a map pn : Un → U
0 given by iterated face
maps to the “objects.” Then, for every (n+1)-tuple of objects (x0, x1, . . . , xn), the
homotopy fiber of pn over (x0, . . . , xn), given by
mapV (xn−1, xn)× · · · ×mapV (x0, x1),
is weakly equivalent to
MapLHM(x
n−1, x
n )× · · · ×MapLHM(x
0 , x
where xcf denotes a fibrant-cofibrant replacement of X inM. It follows that once
we take the Reedy fibrant replacement V of U , it is a Segal space.
Now, consider the set π0U0, which consists of the weak equivalence classes of
objects inM; it follows that π0V0 is an isomorphic set. Further, note that
HomHo(M)(x, y) = π0MapLHM(x
cf , ycf).
16 J.E. BERGNER
Thus, we have shown that Ho(M) is equivalent to Ho(V ).
It remains to show that V is a complete Segal space. Consider the space
Vhoequiv ⊆ V1, and define Uhoequiv to be the preimage of Vhoequiv under the natural
map U → V . Since V is a Reedy fibrant replacement for U , it suffices to show
that the complete Segal space condition holds, i.e., that U0 → Uhoequiv is a weak
equivalence of simplicial sets. Notice that Uhoequiv must consist precisely of the
components of U1 whose 0-simplices come from weak equivalences inM. In other
words, Uhoequiv = nerve(we(we(M))
[1]).
There is an adjoint pair of functors
F :M[1] ⇄M : G
for which F (x→ y) = x and G(x) = idx. This adjoint pair can be restricted to an
adjoint pair
F : nerve(we(we(M))[1]) ⇄ we(M) : G
which in turn induces a weak equivalence of simplicial sets on the nerves, Uhoequiv ≃
U0, which completes the proof. �
Now that we have proved that the mapping spaces and homotopy categories agree
for V and for LHM, it remains to show that they agree for LHM and LC(L
Theorem 6.4. Let M be a model category. For any x and y objects of LHM,
there is a weak equivalence of simplicial sets
MapLHM(x, y) ≃ mapLC(LHM)(x, y),
and there is an equivalence of categories π0L
HM≈ Ho(LC(L
HM)).
Note in particular that x and y are just objects of M, and that π0L
HM is
equivalent to the homotopy category Ho(M).
Proof. Given the hammock localization LHM of the model categoryM, we have
the following composite map of simplicial spaces:
X = nerve(LHM)→ Xf → LC(L
Here, Xf denotes a Reedy fibrant replacement of X . This composite is just Rezk’s
method for assigning the complete Segal space LC(L
HM) to the simplicial category
On the left-hand side, the mapping spaces of X = nerve(LHM) are precisely
those of LHM, by the definition of the nerve functor. In the nerve, one of these
mapping spaces, say mapX(x, y) for some objects x and y ofM, is given by the fiber
over (x, y) of the map (d1, d0) : X1 → X0×X0. Although these mapping spaces can
be defined for X , there is no reason that they are homotopy invariant. When we
take a Reedy fibrant replacement Xf of X , however, this map becomes a fibration,
and hence this fiber is actually a homotopy fiber and so homotopy invariant. For a
general simplicial space, we cannot assume that the mapping spaces of the Reedy
fibrant replacement are equivalent to the original ones. However, if the 0-space of
the simplicial space in question is discrete in degree zero, then the map above is
a fibration. Using an argument similar to the one in [5, §5], we can find a Reedy
fibrant replacement functor which leaves the 0-space discrete. While the space in
degree one might be changed in this process of passing to X
1 , it will still be weakly
equivalent X1. In particular, the mapping spaces in X
f will be weakly equivalent
to those in X .
COMPLETE SEGAL SPACES 17
Since the objects of Xf are just the objects of LHM, or the objects ofM, this
equivalence of mapping spaces gives us also an equivalence of homotopy categories.
The right-most map is the one defined by Rezk, iXf : X
f → X̂f , which takes a
Segal space to a complete Segal space. But, he defines this map in such a way that it
is in fact a Dwyer-Kan equivalence. In other words, it induces weak equivalences on
mapping spaces and an equivalence of homotopy categories. Thus, the composite
map induces equivalences on mapping spaces and an equivalence on homotopy
categories. �
7. A Characterization of Complete Segal Spaces Arising from
Simplicial Categories
In this section, we give a thorough description of the weak equivalence type of
complete Segal spaces which occur as images of Rezk’s functor from the category
of simplicial categories. We consider several different cases, beginning with ones for
which we can use the classifying diagram construction, i.e., discrete categories, and
then proceed to more general simplicial categories.
It should be noted that we are characterizing these complete Segal spaces up to
weak equivalence, and so the resulting descriptions are of the homotopy type of the
spaces in each simplicial degree. For example, in the case of a discrete category,
we describe the corresponding complete Segal space in terms of the isomorphism
classes of objects, rather than in terms of individual objects, in order to simplify
the description. One could just take all objects, and generally get much larger
spaces, if the more precise description were needed for the complete Segal space
corresponding to a given category.
Furthermore, notice that determining the homotopy type of the spaces in degrees
zero and one are sufficient to determine the homotopy type of all the spaces, since we
are considering Segal spaces. Thus, we focus our attention on these spaces, adding
in a few comments about how to continue the process with the higher-degree spaces.
7.1. Case 1: C is a discrete groupoid. If C = G is a group, then applying
Rezk’s classifying diagram construction results in a complete Segal space equivalent
to BG, i.e., the constant simplicial space which is the simplicial set BG at each
level. In particular, since all morphisms are invertible, we obtain essentially no new
information at level 1 that we didn’t have already at level 0.
Example 7.1. Let G = Z/2. Then (NG)0 is just the nerve, or BZ/2. Then (NG)1
has two 0-simplices, given by the two morphisms (elements) of G. However, these
two objects of G[1] are isomorphic, and the automorphism group of either one of
them is Z/2. Thus, (NG)1 is also equivalent to BZ/2.
If C has more than one object but only one isomorphism class of objects, we get
instead a simplicial space weakly equivalent to the constant simplicial space which
is BAut(x) at each level, for a representative object x. If C has more than one
isomorphism class 〈x〉, then the result will be weakly equivalent to the constant
simplicial space
〈x〉 BAut(x).
7.2. Case 2: C is a discrete category. Since in the classifying diagram NC,
(NC)0 picks out the isomorphisms of C only, we still essentially have
〈x〉 BAut(x)
18 J.E. BERGNER
at level 0. However, if C is not a groupoid, then there is new information at level
1. It instead looks like ∐
〈x〉,〈y〉
BAut(
Hom(x, y)α)
where the α index the isomorphism classes of elements of Hom(x, y). The subspace
of (NC)1 corresponding to 〈x〉, 〈y〉, denoted (NC)1(x, y), fits into a fibration
Hom(x, y)→ (NC)1(x, y)→ BAut(x)×BAut(y).
The space in dimension 2 is determined, then, by the spaces at levels 0 and 1.
The subspace corresponding to isomorphism classes of objects 〈x〉, 〈y〉, 〈z〉, denoted
(NC)2(x, y, z), fits into a fibration
Hom(x, y)×Hom(y, z)→ (NC)2(x, y, z)→ BAut(x) ×BAut(y)×BAut(z).
The whole space (NC)2, up to homotopy, looks like
〈x〉,〈y〉,〈z〉
〈α〉,〈β〉
Hom(x, y)α ×Hom(y, z)β
We could describe each (NC)n analogously.
Example 7.2. Let C denote the category with two objects and one nontrivial
morphism between them (· → ·). If {e} denotes the trivial group, then (NC)0 ≃
B{e} ∐ B{e} and (NC)1 ≃ B{e} ∐ B{e} ∐ B{e}. In particular, NC is not equiv-
alent to the classifying diagram of the trivial category with one object and one
morphism, which would be the constant simplicial space B{e}. However, note that
the nerves of these two categories are homotopy equivalent. Thus, we can see that
the classifying diagram is more refined than the nerve in distinguishing between
these two categories.
7.3. Case 3: C is a simplicial groupoid. First, consider the case where we have
a simplicial group G. Let Gn denote the group of n-simplices of G. Then
hocolim∆op(nerve(Gn)
t) = nerve(G).
Let LC denote Rezk’s completion functor which makes the nerve into a complete
Segal space. We claim that
LC(hocolim∆op(nerve(Gn)
t)) ≃ LC(hocolim∆opLC(nerve(Gn)
We actually prove the more general statement that, for any X = hocolim∆opXn,
LC(hocolim∆opXn) ≃ LC(hocolim∆opLCXn).
To prove this claim, first note that we have Rezk’s completion map
i : hocolim∆opXn → LC(hocolim∆opXn)
which is a weak equivalence. Furthermore, since in CSS any complete Segal space
Y is a local object and every object is cofibrant, we have a weak equivalence of
spaces
Map(LC(hocolim∆opXn), Y ) ≃Map(hocolim∆opXn, Y ).
COMPLETE SEGAL SPACES 19
So, for any complete Segal space Y , we have that
Map(LChocolim∆op(LCXn), Y ) ≃Map(hocolim∆op(LCXn), Y )
≃ holim∆Map(LCXn, Y )
≃ holim∆Map(Xn, Y )
≃Map(hocolim∆opXn, Y )
≃Map(LChocolim∆opXn, Y ).
Note that the above calculation depends on the fact that,
Map(hocolim∆opXn, Y ) ≃ holim∆Map(Xn, Y ),
which follows from working levelwise on simplicial sets.
Then, since Gn is a discrete group, completing its nerve is the same as taking
the classifying diagram NGn which, by case 1, is weakly equivalent to the constant
simplicial space BGn, denoted here cBGn. Thus we have:
LC(nerve(Gn)) ≃ LC [hocolim∆op(nerve(Gn))]
≃ LC [hocolim∆op(LC(nerve(Gn)))]
≃ LC [hocolim∆op(cBGn)]
≃ LC(BG)
≃ BG.
So, we obtain a simplicial space weakly equivalent to the constant simplicial
space with BG at each level. (Recall, however, that BG here is obtained by taking
the diagonal of the simplicial nerve, so it is not quite the identical case.) If we
have a simplicial groupoid, rather than a simplicial group, we obtain the analogous
result, replacing BG with ∐
BAut(x).
7.4. Case 4: C is a simplicial category with every morphism invertible up
to homotopy. Alternatively stated, this case covers the situation in which π0(C)
is a groupoid.
Recall that we have a model structure SCO on the category of categories with a
fixed object set O, in which the cofibrant objects are retracts of free objects. So,
taking a cofibrant replacement of C in this model category structure SCO essen-
tially gives a free replacement of C, denoted F (C), which is weakly equivalent to C.
(This cofibrant category can be obtained by taking a simplicial resolution F∗C and
then taking a diagonal [12, 6.1].) Now, taking the localization with respect to all
morphisms results in a simplicial groupoid. So, we have Dwyer-Kan equivalences
F (C)−1F (C) F (C)
≃oo ≃ //C
But, now F (C)−1F (C) is a simplicial groupoid weakly equivalent to C, so we have
now reduced this situation to case 3.
Note that, to write down a description of this complete Segal space in terms of the
original category C, we need to take isomorphism classes of objects in π0(C), or weak
equivalence classes, as well as self-maps which are invertible up to homotopy rather
than strict automorphisms. While we will still use 〈x〉 to denote the equivalence
class of a given object, we will use Auth(x) to signify homotopy automorphisms of
20 J.E. BERGNER
x. Thus, the complete Segal space corresponding to C in this case essentially looks
like ∐
BAuth(x)
at each level.
7.5. Case 5: C is any simplicial category. First consider the subcategory of C
containing all the objects of C and only the morphisms of C which are invertible up
to homotopy. Apply case 5 to get a complete Segal space, but take only the 0-space
of it, to be the 0-space of the desired complete Segal space.
To find the 1-space, first recall the definition of the completion functor as applied
to a Segal space W :
LC(W ) = LCSS(diag([m] 7→ (W
E(m))n)).
Recall further that (WE(m))n = Map(E(m)×∆[n]
t,W ). Thus, the Segal space we
obtain (before applying the functor LCSS) looks like
Map(E(0)×∆[0]t,W )⇐ Map(E(1)×∆[1]t,W ) ⇚ Map(E(2)×∆[2]t,W ) · · · .
If the Segal space W is a fibrant replacement of nerve(C), then the space at level 1
consists of diagrams
x′ // y
with the maps in the appropriate simplicial level.
For simplicity, we restrict to a given pair of objects x and y, representing given
equivalence classes. Consider the homotopy automorphisms of x and y. If they are
not all invertible, we take a cofibrant replacement and group completion as in case
4. So, without loss of generality, assume that Aut(x) and Aut(y) are simplicial
groups. Note that we have
Aut(x) = hocolim∆opAut(x)n
Aut(y) = hocolim∆opAut(y)n.
Now look at
Map(x, y) = hocolim∆opMap(x, y)n.
Consider for each n ≥ 0 the discrete category C(x, y)n which has as objects Map(x, y)n
and as morphisms pairs (α, β) of automorphisms in Aut(x)n × Aut(y)n making a
commutative square
with f, f ′ ∈Map(x, y)n.
Thus, the 1-space that we are interested in is also the 1-space of the complete
Segal space given by
LCSS(hocolim∆op(nerve(C(x, y)n))).
COMPLETE SEGAL SPACES 21
Using a straightforward argument about localization functors similar to the one in
case 3 (which can be found in [3, 4.1]), we can also apply the functor LCSS on the
inside to get an equivalent simplicial space
LCSS(hocolim∆op(LCSSnerve(C(x, y)n))).
But, since C(x, y)n is a discrete category, this space is just
LCSShocolim∆op(NC(x, y)n) ≃ hocolim∆op(NC(x, y)n).
Now, we restrict to the 1-space here, which is
hocolim∆op
〈x〉,〈y〉
Mapn(x, y)α
〈x〉,〈y〉
Map(x, y)α
As with the previous case, we can then go back and weaken to homotopy auto-
morphisms and equivalence classes of objects to consider categories before taking
a group completion, so our space looks like
〈x〉,〈y〉
BAuth
Map(x, y)α
We could then obtain the 2-space of our complete Segal space by considering
categories C(x, y, z)n defined similarly, and the description of the 2-space of the
classifying diagram of a discrete category as given in case 2.
We can summarize these results in the following theorem. For an object x of
a simplicial category C, let 〈x〉 denote the weak equivalence class of x in C, and
for a morphism α : x → y, let 〈α〉 denote the weak equivalence class of α in the
morphism category C[1]. Let Auth(x) denote the space of self-maps of x which are
invertible in π0C.
Theorem 7.3. Let C be a simplicial category. The complete Segal space corre-
sponding to C has the form
BAuth(x)⇐
〈x〉,〈y〉
BAuth
Map(x, y)α
 ⇚ · · · .
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Kansas State University, 138 Cardwell Hall Manhattan, KS 66506
E-mail address: bergnerj@member.ams.org
http://www-math.mit.edu/~psh
	1. Introduction and Overview
	2. Background on Model Categories and Simplicial Objects
	3. Simplicial Categories and Simplicial Localizations
	4. Complete Segal Spaces
	5. Obtaining Complete Segal Spaces from Simplicial Categories and Model Categories
	6. Comparison of Functors from SC to CSS
	7. A Characterization of Complete Segal Spaces Arising from Simplicial Categories
	7.1. Case 1: C is a discrete groupoid
	7.2. Case 2: C is a discrete category
	7.3. Case 3: C is a simplicial groupoid
	7.4. Case 4: C is a simplicial category with every morphism invertible up to homotopy
	7.5. Case 5: C is any simplicial category
	References
