Simplicial and Cellular Structures in Algebraic Topology
Dai Tamaki
Abstract
This is a summary of my talk at Studio Phones seminar on February 22nd, 2012.

In this talk, I would like to explain major roles in homotopy theory played by simplicial and cellular
structures. Homotopy theory is a theory of deformations. According to Gel′ fand1 , therefore, homotopy
theory is one of the most fundamental ﬁelds in mathematics. Homotopy theory is, however, usually regarded as a part of topology. What is topology? From my viewpoint, topology has the following three
aspects:
1. treat continuous deformations seriously.
2. use invariants heavily.
3. provide global viewpoints.
From this point of view, the ﬁrst and the third play important roles in homotopy theory. Let me explain
how.

1

From Simplicial Complexes to Simplicial Sets and CW complexes

In an attempt to answer Heegaard's criticism on his ``Analysis Situs'' [Poi96], Poincaré decided to use
triangulations of manifolds. The target manifold is cut into pieces by smooth ``surfaces''. Later people
decided to abandon the smoothness requirement. Each piece is supposed to be homoeomorphic to aninterior of a convex polytope.
According to Dieudonné's book on the history of algebraic and diﬀerential topology [Die89], it was
Lefschetz [Lef08] who deﬁned the concept which is now called Euclidean simplicial complex.
Deﬁnition 1.1. A (Euclidean) polyhedral complex in Rn is a subspace K of Rn equipped with a familyof a ﬁnitely many number of maps {φi : Pi −→ K | i = 1, · · · , n} satisfying the following conditions:
1. Each Pi is a convex polytope;
2. Each φi is an aﬃne equivalence onto its image;∪n
3. K = i=1 φi (Pi );4. For i ̸= j, φi (Pi ) ∩ φj (Pj ) is a proper face of φi (Pi ) and φj (Pj ).
Pi 's or φi (Pi ) are called generating polytopes .
When all generating polytopes are simplices, (K, {φi }) is called a Euclidean simplicial complex.
1 See Introduction of the draft of the book by Kontsevich and Soibelman on deformation theory.

1The notion of Euclidean simplicial (polyhedral) complexes evolved into three structures: abstract simplicial complex, ordered simplicial complex, and cell complex.
Abstract simplicial complexes are one of the fundamental objects of study in modern combinatorics.
Deﬁnition 1.2. An abstract simplicial complex on the vertex set S is a collection X of ﬁnite subsets of S,
X ⊂ 2S ,satisfying the following condition: if σ ∈ X, τ ⊂ σ, then τ ∈ X. We usually assume X is essential, i.e.,∪
σ = S.
σ∈X

An element in X is called a face or a simplex of X. When the cardinality of a simplex σ ∈ X is n + 1,
we say σ is an n-dimensional simplex.
I do not know who introduced the current deﬁnition of abstract simplicial complexes. One of the
key facts which make abstract simplicial complexes so useful in combinatorics can be illustrated by thefollowing diagram:F/ Posets
ASC







Sd 
(1)
 ∆



 
OSC,
where ASC, OSC, and Posets are categories of abstract simplicial complexes, of ordered simplicial complexes, and of posets, respectively. Functors F , Sd, and ∆ are the face poset functor, the barycentric
subdivision functor, and the order complex functor, respectively.
Deﬁnition 1.3. An ordered (abstract) simplicial complex is an abstract simplicial complex K whose
vertex set S is partially ordered and the induced ordering on each simplex is a total order.
Deﬁnition 1.4. The set of all faces of an abstract simplicial complex K is denoted by F (K). It is regarded
as a poset under the inclusions and is called the face poset of K.
Deﬁnition 1.5. For a poset P , deﬁne an ordered simplicial complex ∆(P ) with vertex set P by∆(P ) = {σ ⊂ P | σ is totally ordered.} .
This is called the order complex of P .
Deﬁnition 1.6. For an abstract simplicial complex K, the ordered simplicial complex ∆(F (K)) is called
the barycentric subdivision and is denoted by Sd(K).
It was Henry Whitehead [Whi49] who introduced the closure ﬁniteness and weak topology conditions
on cell complexes and deﬁned CW complexes, which generalize Euclidean polyhedral complexes.
Deﬁnition 1.7. An n-cell in a topological space X is a subset e ⊂ X equipped with a continuous map
φ : Dn → e whose restriction φ|Int(Dn ) is a homeomorphism onto e.
A cell complex is a topological space X together with a family of cells {φλ : Dnλ → eλ }λ∈Λ withthe following properties:∪
1. X = λ∈Λ eλ .
2

2. For an n-cell eλ , ∂eλ = eλ − eλ is a union of cells of dimension ≤ n − 1.
A cell complex X is called a CW complex if it satisﬁes the following two conditions:
1. it is closure ﬁnite i.e. eλ is covered by a ﬁnite number of cells for each λ, and
2. it has the weak topology with respect to the covering {eλ }λ∈Λ .
The triangle (1) does not generalize to CW complexes. We obtain, however, the following diagram by
restricting our attention to regular CW complexes:
CWreg



Sd 


CWreg o/ Posets



 ∆


OSC
|−|,
F

where CWreg is the category of regular CW complexes and | − | is the geometric realization functor.
Deﬁnition 1.8. A cell complex X is said to be regular if φλ : Dnλ → eλ is a homeomorphism.
Deﬁnition 1.9. For an ordered simplicial complex K, deﬁne a topological space |K| by
(
)
⨿
/
n
|K| =
Kn × ∆,n∼where ∆n is the standard n-simplex and ∼ is the equivalence relation generated by(di (x), s) ∼ (x, di (s)),
where di is the operation which removes the i-th vertex2 and di : ∆n−1 ,→ ∆n is the inclusion onto theface opposite to the i-th vertex. The space |K| is called the geometric realization of K.
Deﬁnition 1.10. For a regular cell complex X, the cell complex |∆(F (X))| is called the barycentric
subdivision of X and is denoted by Sd(X).
Another importance of the concept of ordered simplicial complex is that it was the origin of simplicial
sets introduced by Kan [Kan57] under the name ``semi-simplicial complexes''.
Deﬁnition 1.11. A simplicial set X consists of a sequence of sets
X0 , X1 , . . .
and mapsdisi: Xn+1 −→ Xn (0 ≤ i ≤ n + 1)
: Xn−1 −→ Xn (0 ≤ i ≤ n − 1)satisfying the following relations:di ◦ dj = dj−1 ◦ di ,





di ◦ sj = sj−1 ◦ di ,dj ◦ sj = 1 = dj+1 ◦ sjdi ◦ sj = sj ◦ di−1 ,s ◦ s = sijj+1 ◦ si ,

i<j
i<j
i>j+1
i ≤ j.

2 Note that since X is ordered the vertices of an n-dimensional simplex is numbered from 0 to n.

3

Before this work, Kan also investigated a possibility of using cubical complexes in [Kan55]. Simplicial
sets, however, have the following simple characterization.
Deﬁnition 1.12. The category of isomorphism classes of ﬁnite totally ordered sets and order preserving
maps is denoted by ∆.
Proposition 1.13. The category of simplicial sets is isomorphic to category of functors Funct(∆op , Sets).
In other words, the category of simplicial sets is the category of presheaves on ∆ with values in Sets.
op

Deﬁnition 1.14. The category of simplicial sets is denoted by Sets∆ .
We have seen the following developement of simplicial and cellular structures so far:.
.Euclidean simplicial (polyhedral)complex.Abstract simplicial complex.Ordered simplicial complex.CW complex.Simplicial set.Cubical set

2

Simplicial and Cellular Structures in Homotopy Theory

It was Kan who ﬁrst tried to develop general ``homotopy theory'' based on these ``well-behaved spaces''.
Kan ﬁrst tried to develop homotopy theory based on cubical complexes in a series of papers [Kan55;
Kan56a; Kan56b; Kan56c]. Then developed an abstract homotopy theory based on simplicial sets in[Kan57].
The concept of ``abstract homotopy theory'' was made explicit by Quillen in [Qui67] as model structures. The following simpliﬁed deﬁnition can be found in a paper [JT07] by Joyal and Tierney.
Deﬁnition 2.1. Let C be a category closed under ﬁnite limits and colimits. A model structure on C is a
triple (C, W, F ) of subcategories satisfying the following conditions:
1. W satisﬁes the ``two-out-of-three'' property.
2. The pair (C ∩ W, F ) is a weak factorization system.
3. The pair (C, W ∩ F ) is a weak factorization system.
A category with a model structure is called a model category. Morphisms in C ∩ W and W ∩ F arecalled acyclic coﬁbrations and acyclic ﬁbrations, respectively.
Weak factorization systems are characterized by lifting properties.
Deﬁnition 2.2. Let C be a category and consider a diagram
/ x
?

a
∃

u

f


b


/ y.

We say f has the right lifting property with respect to u if, for any commutative square as above, there
exists a morphism b → x ﬁlling in the diagram. In this case we also say that u has the left lifting property
with respect to f .
4

Deﬁnition 2.3. Let C be a category. A weak factorization system is a pair of subcategories (A, B) satisfying the following conditions:
1. Any morphism f in C can be factored as f = p ◦ i with i in A and p in B, i.e. A has (A, B)factorizations.
2. A morphism f has the right lifting property with respect to every morphism in A if and only if f
belongs to B.
3. A morphism g has the left lifting property with respect to every morphism in B if and only if f
belongs to A.
Remark 2.4. Hovey requires that the weak factorizations in his deﬁnition of model category to be functorial. There are, however, examples of model categories whose weak factorizations are not functorial.
See Chorny's paper [Cho03].
One of the most important discovery of Henry Whitehead is that CW complexes behave extremely wellwith respect to homotopy. For example, any pair (X, A) of a CW complex X and its subcomplex A has
the homotopy extension property (HEP). On the other hand, covering homotopy property of ﬁber bundles
evolved into the deﬁnition of ﬁbrations by Hurewicz [Hur55] and Serre [Ser51]. Quillen's deﬁnition ofmodel structure is based on these structures.
op
Quillen constructed model structures on the category Sets∆ of simplicial sets and the category Spacesof topological spaces. He invented so-called the small object argument for this purpose. The point isthat the model structures on these categories are generated by ``cellular structures''. We use transﬁnite
compositions to deﬁne ``cell complexes'' based on a class of morphisms I.
Deﬁnition 2.5. Let C be a category and I be a small subcategory of C closed under colimits. Deﬁne theclass I-cell of relative I-cell complexes in C by a transﬁnite induction as follows: Morphisms in I belong
to I-cell. A morphism f : A → B is in I-cell if there exists an ordinal λ and a colimit-preseving functor
X : λ −→ Csatisfying the following properties:
1. f is the composition of X.
2. For each β with β + 1 < λ, there is a pushout square
/ Xβ






Cβ


gβ 



/ Xβ+1Dβsuch that gβ ∈ I1 .

We say an object A in C is an I-cell complex if the canonical morphism∅ −→ Abelongs to I-cell, where ∅ is the initial object in C.
Deﬁnition 2.6. Let C be a category closed under small colimits and κ be a cardinal.
51. An object x is κ-small relative to a subcategory D if for every regular cardinal λ ≥ κ and every
λ-sequence
a0 → a1 → a2 → · · · → aβ → · · ·with β < λ in C such that aβ → aβ+1 is in D for every cardinal β satisfying β + 1 < λ, the mapof sets()
colim C(x, aβ ) −→ C x, colim aβ
β<λ

β<λ

is an isomorphism.
2. An object x is κ-small relative to a class of morphisms I if it is κ-small relative to the subcategoryof relative I-cell complexes.
An object x is said to be small relative to I if x is κ-small relative to I for some cardinal κ.
There is a canonical way of producing a weak factorization system from such a class of morphisms ora subcategory I.
Theorem 2.7 (The Small Object Argument). Let C be a category closed under arbitrary small colimits
and I be a small subcategory. Suppose the domains of morphisms in I are small relative to I. Then there
exists a functorial weak factorization system (γ, δ) on C such that γ(f ) is in I-cell and δ(f ) is in I-inj forany morphism f in C.
The class I-inj used in the above theorem is deﬁned as follows.
Deﬁnition 2.8. Let I be a class of morphisms (i.e. a subcategory) in a category C. A morphism f is
1. I-injective if it has the right lifting property with respect to morphisms in I,
2. I-projective if it has the left lifting property with respect to morphisms in I,
3. I-coﬁbration if it has the left lifting property with respect to every I-injective morphism,
4. I-ﬁbration if it has the right lifting property with respect to every I-projective morphism.
The subcategories of I-injectives, I-projectives, I-coﬁbrations, and I-ﬁbrations are denoted by I-inj,
I-proj, I-cof, and I-ﬁb, respectively.
Deﬁnition 2.9. Let C be a category. We say a class of morphisms I permits the small object argument if
the domains of elements in I are small relative to I.
Deﬁnition 2.10. A coﬁbrantly generated model category is a model category C equipped with two classesof morphisms I and J satisfying the following conditions:
1. I permits the small object argument.
2. A morphism is a acyclic ﬁbration if and only if it has the right lifting property with respect to allelements in I.
3. J permits the small object argument.
4. A morphism is a ﬁbration if and only if it has the right lifting property with respect to all elementsof J.
Elements in I and J are called generating coﬁbrations and generating acyclic coﬁbrations, respectively.
6

Theorem 2.11. Let C be a category closed under small limits and colimits and W be a subcategory thatis closed under retracts and satisﬁes the two-out-of-three axiom.
Suppose that I and J are classes of morphisms of C satisfying the following properties:
1. I and J permit the small object argument.
2. J−cof ⊂ I−cof ∩ W .
3. I−inj ⊂ J−inj ∩ W .
4. One of the following conditions holds:(a) I−cof ∩ W ⊂ J−cof,(b) J−inj ∩ W ⊂ I−inj.
Then there exists a coﬁbrantly generated model structure on C in which W is the subcategory of weakequivalences, I is the class of generating coﬁbrations, and J is the class of generating acyclic coﬁbrations.
In a coﬁbrantly generated model category, ``cells'' are generators of coﬁbrations. In particular, in the
Quillen model structure of topological spaces, coﬁbrations are retracts of relative CW complexes.
Quillen also proved that, under his model structures, the geometric realization functor | − | and thesingular simplicial set functor S deﬁne a Quillen equivalenceop| − | : Sets∆ ←→ Spaces : S.
Deﬁnition 2.12. Let C and D be model categories.
• We say a functor f : C → D is a left Quillen functor if f is a left adjoint and preserves coﬁbrations
and trivial coﬁbrations.
• Dually, we say a functor g : D → C is a right Quillen functor if g is a right adjoint and preservesﬁbrations and trivial ﬁbrations.
• A Quillen adjunction is an adjunction∼=

φ : D(f (x), y) −→ C(x, g(y))
such that f is a left Quillen functor. We denote a Quillen adjunction by(f, g, φ) : C −→ D.
• We say a Quillen adjunction(f, g, φ) : C −→ D
is a Quillen equivalence if, for all coﬁbrant x in C and ﬁbrant y in D, we have
WD ∩ D(f (x), y) = φ−1 (WC ∩ C(x, g(y))).
Theorem 2.13. The geometricrealization and the singular simplicial set functors deﬁne a Quillen equivop
alence between Sets∆ and Spaces.

7

In particular, the geometric realization |X| of any simplicial set X has a canonical structure of CW
complex and we have a weak homotopy equivalence≃|S(Y )| −→ Yw

for any topological space Y . In other words, any topological space can be replaced by a CW complex up
to a weak homotopy equivalence.
After Quillen's discovery, model categories have been used in many ﬁelds. We can reﬁne classicalhomological algebra based on triangulated categories by using stable model structures.
Why is the concept of model categories so useful?
One of the answers to this question is the following.
Fact 2.14. Triangulated categories suﬀer the well-known defect that the cone construcion and other``colimit-type'' constructions are not functorial. By lifting to model categories, we may deﬁne homotopy
colimits of any diagrams under a mild condition.
One of the conditions on a model category under which we may perform homotopy limits and colimits
is the existence of a simplicial enrichment. This is one of main topics in Hirschhorn's book [Hir03].
It should be also noted that triangulated categories are often constructed as a homotopy category of acategory with richer structures such as (stable) dg categories, A∞ -categories, and model categories.
Dwyer and Kan [DK80c; DK80a; DK80b] found that simplicial structures can be used to extend theconstruction of the homotopy category of a model category to categories with a designated class of ``weak
equivalences''.
Theorem 2.15. Given a category C and a subcategory W with C0 = W0 , there exists a simplicial
category LW C with
π0 (LW C(x, y)) ∼= C[W −1 ](x, y).
Recall that, for a model category C, we deﬁne the homotopy category Ho(C) by formaly invertingthe class W of weak equivalences, i.e. Ho(C) = C[W −1 ]. This naive deﬁnition, however, suﬀers from a
set theoretical diﬃculty. The discovery of Dwyer and Kan suggests the usefulness of simplicial structure
in localizations of categories.
An interesting and important fact is that the existence of simplicial enrichment also leads to anotherapproach to homotopy categories, i.e. the theory of (∞, 1)-categories.

3

Simplicial and Cellular Structures in Higher Category Theoryop

Deﬁnition 3.1. A simplicial category is aop category enriched over the category Sets∆ . The category ofsimplicial categories is denoted by Sets∆ -Cats.
op

The work of Dwyer and Kan suggests that the category Sets∆ -Cats can be regarded as the categoryof categories whose homotopy categories can be deﬁned. It turns out there are several other candidates
for such models for ``homotopy theory of homotopy theories''.
One of them also originates in an old work of Kan. Recall that an object x in a model category C issaid to be ﬁbrant if the canonical morphism x → ∗ to the terminal object is a ﬁbration.
op

Deﬁnition 3.2. Fibrant objects in Sets∆ are called Kan complexes.

8

Proposition 3.3. A simplicial set X is a Kan complex if and only if, for n ≥ 0, 0 ≤ i ≤ n, and any
morphism of simplicial sets
φ : Λni −→ X,
there exists an extension φ̃ : ∆n → X, where Λni is the simplicial subset of ∆n generated by all faces but
the i-th one.
The nerve of a small category has an analogous property.
Proposition 3.4. For a simplicial set K, the following conditions are equivalent:
1. There exists a small category X with K ∼= N (X).
2. For any n ≥ 0, 0 < i < n and a morphism φ : Λi → K, there exists a unique extensionφ̃ : ∆n → K of φ.
As a class containing these two classes of simplicial sets, Boardman and Vogt [BV73] introduced andstudied simplicial sets satisfying the restricted Kan condition, called weak Kan complexes or quasicategories these days.
Deﬁnition 3.5. A quasicategory is a simplicial set K satisfying the condition that, for any n ≥ 0, 0 <i < n and a morphism φ : Λi → K, there exists an extension φ̃ : ∆n → K of φ.
It was Joyal [Joy02] who ﬁrst formulated the theory of quasicategories. Then Lurie [Lur09a] used it
as a model for higher categories, more precisely categories whose higher morphisms are invertible.
It is well-known that the nerve functorop

N : Cats −→ Sets∆

is an embedding. By Proposition 3.4, the nerve functor embeds Cats into the category QCats of quaiscategories. It is also well-known that the nerve functor has a left adjoint
opτ1 : Sets∆ −→ Cats,
which can be used to deﬁne the homotopy category construction for quasicategories.
Deﬁnition 3.6. The restriction of τ1 to QCats is denoted byh : QCats −→ Cats .
For a quaiscategory X, h(X) is called the homotopy category of X.
Thus the category QCats of quasicategories is another model for the category of categories whose homotopy categories can be deﬁned. Lurie found quasicategories are useful for developing derived algebraic
geometry and other ``higher geometries and algebras'' [Lurc; Lurb; Lura; Lurd; Lure; Lur09b].
It turns out there are many more models for homotopy theories of homotopy theories other than simplicial categories and quasicategories:• Segal categories. [HS]• complete Segal spaces. [Rez01]• relative categories. [BK]
It is known that all these theories are equivalent to each other [Ber10]. And simplicial structures playessential roles in all of these theories.
9References[Ber10]Julia E. Bergner. ``A survey of (∞, 1)-categories''. In: Towards higher categories. Vol. 152.
IMA Vol. Math. Appl. New York: Springer, 2010, pp. 69–83. : 10.1007/978-1-44191524-5_2. arXiv:math/0610239. : http://dx.doi.org/10.1007/978-1-44191524-5_2 (cit. on p. 9).

[BK]

C. Barwick and D. M. Kan. Relative categories: Another model for the homotopy theory ofhomotopy theories. eprint: \href{http://arxiv.org/abs/1011.1691}{arXiv:1011.
1691} (cit. on p. 9).

[BV73]J. M. Boardman and R. M. Vogt. Homotopy invariant algebraic structures on topologicalspaces. Lecture Notes in Mathematics, Vol. 347. Berlin: Springer-Verlag, 1973, pp. x+257(cit. on p. 9).

[Cho03]

Boris Chorny. ``The model category of maps of spaces is not coﬁbrantly generated''. In: Proc.
Amer. Math. Soc. 131.7 (2003), 2255–2259 (electronic). : 0002-9939. : 10 . 1090 /S0002- 9939- 03- 06901- 6. : http://dx.doi.org/10.1090/S0002- 9939- 0306901-6 (cit. on p. 5).

[Die89]Jean Dieudonné. A history of algebraic and diﬀerential topology. 1900--1960. Boston, MA:
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[DK80a]W. G. Dwyer and D. M. Kan. ``Calculating simplicial localizations''. In: J. Pure Appl. Algebra
18.1 (1980), pp. 17–35. : 0022-4049. : 10.1016/0022- 4049(80)90113- 9. :http://dx.doi.org/10.1016/0022-4049(80)90113-9 (cit. on p. 8).

[DK80b]W. G. Dwyer and D. M. Kan. ``Function complexes in homotopical algebra''. In: Topology
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[DK80c]W. G. Dwyer and D. M. Kan. ``Simplicial localizations of categories''. In: J. Pure Appl. Algebra 17.3 (1980), pp. 267–284. : 0022-4049. : 10.1016/0022-4049(80)90049-3.
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[Hir03]

Philip S. Hirschhorn. Model categories and their localizations. Vol. 99. Mathematical Surveys
and Monographs. Providence, RI: American Mathematical Society, 2003, pp. xvi+457. :0-8218-3279-4 (cit. on p. 8).

[HS]

André Hirschowitz and Carlos Simpson. Descente pour les n-champs (Descent for n-stacks).
arXiv:math/9807049 (cit. on p. 9).

[Hur55]

Witold Hurewicz. ``On the concept of ﬁber space''. In: Proc. Nat. Acad. Sci. U. S. A. 41 (1955),
pp. 956–961. : 0027-8424 (cit. on p. 5).

[Joy02]

A. Joyal. ``Quasi-categories and Kan complexes''. In: J. Pure Appl. Algebra 175.1-3 (2002).
Special volume celebrating the 70th birthday of Professor Max Kelly, pp. 207–222. : 00224049. : 10.1016/S0022-4049(02)00135-4. : http://dx.doi.org/10.1016/
S0022-4049(02)00135-4 (cit. on p. 9).

[JT07]

André Joyal and Myles Tierney. ``Quasi-categories vs Segal spaces''. In: Categories in algebra,
geometry and mathematical physics. Vol. 431. Contemp. Math. Providence, RI: Amer. Math.
Soc., 2007, pp. 277–326. arXiv:math/0607820 (cit. on p. 4).

[Kan55]

Daniel M. Kan. ``Abstract homotopy. I''. In: Proc. Nat. Acad. Sci. U.S.A. 41 (1955), pp. 1092–
1096. : 0027-8424 (cit. on p. 4).

[Kan56a]Daniel M. Kan. ``Abstract homotopy. II''. In: Proc. Nat. Acad. Sci. U.S.A. 42 (1956), pp. 255–
258. : 0027-8424 (cit. on p. 4).
10[Kan56b] Daniel M. Kan. ``Abstract homotopy. III''. In: Proc. Nat. Acad. Sci. U.S.A. 42 (1956), pp. 419–
421. : 0027-8424 (cit. on p. 4).
[Kan56c]Daniel M. Kan. ``Abstract homotopy. IV''. In: Proc. Nat. Acad. Sci. U.S.A. 42 (1956), pp. 542–
544. : 0027-8424 (cit. on p. 4).

[Kan57]

Daniel M. Kan. ``On c. s. s. complexes''. In: Amer. J. Math. 79 (1957), pp. 449–476. :0002-9327 (cit. on pp. 3, 4).

[Lef08]

S. Lefschetz. Topology. Colloquium Publications. Reprint. The original book was publishedin 1930. American Mathematical Society, 2008, pp. xi+413. : 9780821846032 (cit. onp. 1).

[Lura]

Jacob Lurie. Derived Algebraic Geometry III: Commutative Algebra. arXiv:math/0703204(cit. on p. 9).

[Lurb]

Jacob Lurie. Derived Algebraic Geometry II: Noncommutative Algebra. arXiv:math/0702299(cit. on p. 9).

[Lurc]

Jacob Lurie. Derived Algebraic Geometry I: Stable Inﬁnity Categories. arXiv:math/0608228(cit. on p. 9).

[Lurd]

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