Quasi-categories and Complete Segal Spaces: Quillen equivalences [thesis]

74 Pages·2010·0.56 MB·English

by  Daniela Egas Santander

F ACULTY OF SCIENCE
UNIVERSITY OF COPENHAGEN

Master thesis in Mathematics
Daniela Egas Santander
Quasi-categories and Complete Segal Spaces
Quillen equivalences
Advisor: Alexander Berglund
Handed in: September 30, 2010

Abstract
Quasi-categories and complete Segal spaces are models for homotopy the-
ories. On one hand, the category of simplicial sets admits a model structure
where the fibrant objects are exactly the quasi-categories referred to as the
Joyal model structure for quasi-categories. On the other hand, the category of
simplicial spaces admits a model category structure where the fibrant objects
are exactly the complete Segal spaces, referred to as the Rezk model structure
forcompleteSegalspaces.Thus,bothmodelcategoriesgiveahomotopytheory
of homotopy theories. This thesis shows that both of them are ”essentially the
same” in the sense that they are Quillen equivalent.
Resum´e
Kvasi-kategorierogfuldstændigeSegal-rumermodellerforhomotopiteorier.
P˚a den ene side tillader kategorien af simplicielle mængder en modelkategori-
struktur, hvor de fibrante objekter netop er kvasi-kategorierne, betegnet Joyal-
modelstrukturen for kvasi-kategorier. P˚a den anden side tillader kategorien af
simplicielle rum en modelstruktur, hvor de fibrante objekter er netop de fuld-
stændige Segal-rum, betegnet Rezk-modelstrukturen for fuldstændige Segal-
rum.S˚abeggemodelkategoriergiverenhomotopiteoriafhomotopiteorier.Dette
speciale viser at disse modelkategorier er ”essentielt den samme”, det vil sige
at de er Quillen ækvivalente.
Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Organization of the thesis and Notation . . . . . . . . . . . . . . . . 3
2 Model Categories 5
2.1 Basic Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The homotopy relation in a model category . . . . . . . . . . . . . . 12
2.3 The homotopy category . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Quillen Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Model Structures on the category of simplicial sets 32
3.1 Quasi-categories and the Category of simplicial sets . . . . . . . . . 32
3.2 Combinatorial results on saturated classes . . . . . . . . . . . . . . . 38
4 Model structures on S2 44
4.1 Basic constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Reedy structures on S(2) . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 The model structure for Segal spaces and complete Segal spaces . . . 56
5 Quillen Equivalences 64
1 1 Introduction
1 Introduction
1.1 Motivation
It is of interest to explore different possible models for homotopy theories such as
modelcategories,quasi-categories,completeSegalspaces,Segalcategoriesorsimpli-
cialcategories(categoriesenrichedoversimplicialsets),andhowthesecanberelated
to one another. Quillen introduced the notion of a model category as a category C
with three distinct classes of morphisms: weak equivalences (W), cofibrations (C)
and fibrations (F) together with certain axioms. The homotopy category Ho(C) is
the localization of C with respect to the weak equivalences W−1C i.e. the homotopy
category is obtained by formally inverting the weak equivalences in C. The model
category structure ensures that such localization exists. From this it is clear that
the homotopy category depends only on the class of weak equivalences and not on
the fibrations and cofibrations. Therefore, when passing from the original category
to the homotopy category the implicit information on the structure of C is lost.
Thus, comparing the homotopy categories of two model categories is not enough to
determine if these are ”essentially the same”. In order to solve this problem, the
notion of a Quillen equivalence is given. A Quillen equivalence between two model
categories, is an adjoint pair between such categories satisfying certain conditions
which induce and adjoint equivalence on the homotopy categories that respects the
additional structure.
Quasi-categories, where introduced by Boardman and Vogt in [4]. A simplicial
set X is a quasi-category if every inner horn Λk[n] → X can be filled. A quasi-
category X can be understood as a generalized category where the objects are
the elements in X and the morphisms are the elements of X . Boardman and
0 1
Vogt introduce the notion of homotopic morphisms in a quasi-category and show
that this relation is indeed an equivalence relation. Moreover, the inner horn filling
conditiongivesacompositionofmapsthatiswelldefineduptohomotopy.Withthis
machinery they construct the homotopy category of a quasi-category Ho(X), where
the objects are the 0-simplices X and the morphisms are given by the homotopy
0
classes of maps under the homotopy relation. Moreover, Joyal [10] defines a notion
of weak categorical equivalence between simplicial sets, which when restricted to
quasi-categories gives the notion of an equivalence of quasi-categories.
ASegalspaceisasimplicialspacewhichsatisfiescertainconditions.Morespecif-
ically, let I be the n-chain in the n-th simplex ∆[n]. There is a natural inclusion
n
i : I (cid:44)→ ∆[n]. A simplicial space is said to satisfy the Segal condition if the map
n n
induced by the inclusion
i \X : X → X × X × X .....X × X
n n 1 X0 1 X0 1 1 X0 1
is a weak homotopy equivalence for all n ≥ 2 where the codomain is the colimit of
the diagram
X −d→0 X ←d1− X −d→0 X ....X ←d1− X
1 0 1 0 0 1
A Segal space is a vertically Reedy fibrant simplicial space, that satisfies the Segal
condition. Given a fixed Segal space X in [17] Rezk defines the set of objects of X
1 Introduction 2
to be the 0-simplices of X and for two such objects he associates a mapping space
0
map (x,y)tobethefiberover(x,y)ofthemap(d ,d ) : X → X ×X .Twopoints
X 0 1 1 0 0
f,g ∈ map (x,y) are said to be homotopic if they belong to the same component in
X
map (x,y).Thisconstructiongivesacompositionbetweenpointsinmappingspaces
X
which is both well defined and associative up to homotopy. Using this relation Rezk
defines the homotopy category Ho(X) of a Segal space X. Moreover, he defines the
notionofacompleteSegalspace,whichisaSegalspacethatsatisfiesacompleteness
condition. A notion of equivalence between simplicial spaces is given, namely the
Rezk weak equivalences.
A Segal precategory is a simplicial space such that X is a discrete simplicial set
0
and a Segal category is a Segal precategory that satisfies the Segal condition. Then,
one can associate to a given Segal category X a homotopy category Ho(X) in the
samewayasforSegalspaces.ASegalcategorycanberegardedasageneralizationof
simplicial categories (categories enriched over simplicial sets) in which composition
is well defined up to homotopy. In [8] a notion of weak equivalence between Segal
precategories is given and it is referred to as a DK-equivalence.
Finally, in [5] Dwyer and Kan construct from a model category C a simplicial
category LH(C) via the Hammock localization, which is referred to as the simplicial
homotopycategoryofC.Thisconstructiondependsonlyontheclassofweakequiv-
alences of C and unlike the construction of the homotopy category, the simplicial
localizationofC retainstheimplicitstructureoftheoriginalmodelcategory.Dwyer
and Kan also develop a notion of equivalence between simplicial categories, namely
the DK-equivalences, which gave the underlying ideas for defining the equivalences
between Segal precategories and simplicial spaces mentioned above. The homotopy
category of a simplicial category C is given by the category of components of C i.e.
the category whose objects are the objects of C and the morphisms are given by the
components of the internal hom-sets. Moreover, two Quillen equivalent model cate-
gories give equivalent simplicial localizations and the category of components of the
simplicial homotopy category of a model category C is equivalent to its homotopy
category Ho(C). Finally, they show that every simplicial category, can be obtained,
up to DK-equivalence by a simplicial localization of some model category with a
given class of weak equivalences. Thus, all simplicial categories can be regarded as
homotopy theories.
These four models: quasi-categories, complete Segal spaces, Segal categories and
simplicial categories can be regarded also as objects of a suitable category. Thus one
might ask if these categories, together with the distinct classes of maps described,
give rise to a homotopy theory, which can then be thought of as a homotopy theory
of homotopy theories. This holds in all four cases as follows. In [3] Bergner shows
that the category of small simplicial categories admits a model category structure
where the weak equivalences are given by the DK-equivalences. In [10] Joyal shows
that the category of simplicial sets admits a model category structure, where the
weak equivalences are the categorical weak equivalences and the fibrant objects
are the quasi-categories. In [17] Rezk shows that the category of simplicial spaces
admits a model category structure, where the weak equivalences are the Rezk weak
equivalences and the fibrant objects are the complete Segal spaces. In [8, 2, 13] it is
shown that the category of Segal precategories admits a model category structure,
3 1 Introduction
where the weak equivalences are the DK-equivalences and the fibrant objects are
the Segal categories.
Thus, these four categories give rise to a homotopy theory of homotopy theories.
Moreover all of these are Quillen equivalent because they are interconnected with
each other via a series of Quillen equivalences. This result is significant, since dif-
ferent models present different advantages and disadvantages. For example, taken
apart some set-theoretic issues, one can develop the simplicial localization of any
small category with a distinct class of weak equivalences. However, the weak equiva-
lencesinthecategoryofsimplicialcategoriesarehardtoidentify.Ontheotherhand,
the weak equivalences between complete Segal spaces are easily identified since they
are given by level-wise weak equivalences. Moreover, quasi-categories are objects in
the category of simplicial sets, thus calculations may become more straight forward
if one would like to obtain a specific result on a given homotopy theory.
The present work explores in detail the equivalence between the model struc-
ture for complete Segal spaces in the category of simplicial spaces (S(2)) and the
model structure for quasi-categories in the category of simplicial sets (S) which are
presented in [11]. For this consider the functor i∗ : S(2) → S which sends a simpli-
1
cial space X to its first row and the functor t : S(2) → S which sends a simplicial
!
space to its total simplicial set. The main results are presented in Theorem 5.11 and
Theorem 5.18.
Theorem. 5.11 The adjoint pair
p∗ : S (cid:28) S(2) : i∗
1 1
is a Quillen equivalence between the model category structure of quasi-categories and
the model category structure of complete Segal spaces.
Theorem. 5.18 The adjoint pair
t : S(2) (cid:28) S : t!
!
is a Quillen equivalence between the model structure of complete Segal spaces and
the model structure for quasi-categories.
1.2 Organization of the thesis and Notation
Insection§2therequiredresultsonmodelcategoriesandweakfactorizationsystems
are given. In section §3 quasi-categories are briefly introduced. The classical model
structure for simplicial sets and the Joyal model structure for quasi-categories are
presented. Additionally some combinatorial results are shown, which will be used
repeatedlyinlatersections.Insection§4Reedymodelstructuresoversimplicialsets
are introduced. Segal spaces and complete Segal spaces are defined and the model
structures for both are given. The main properties of Segal spaces and complete
Segal spaces are proved in this section. In section §5 the two main theorems are
proved.
The following notation will be used in the present work. For a category C we
will write Ob(C) for the objects of the category and HomC(A,B) for the set of
1 Introduction 4
morphisms from A to B. Given functors F : C → D and G : D → C we will write
(F,G) or F : C (cid:28) D : G to denote an adjunction where F is the left adjoint. We
denote by CI the arrow category of a category C i.e. the functor category from I
to C where I is the category with two objects and exactly one morphism between
them {0 → 1}. The category of sets will be denoted by Set and the category of
small categories by Cat. The category ∆ is the category whose objects are non-
empty finite ordinals and the morphisms are order preserving maps. A simplicial
set X is a functor X : ∆op → Set. The n-simplex of simplicial set X is denoted by
X and is given by X = X[n]. The standard n-simplex is the representable functor
n n
∆[n] = Hom (−,n). The co-face maps are the maps di : [n−1] → [n] that skip i
∆
and the co-degeneracy maps are maps si : [n+1] → [n] that repeat i. To these we
associate the face maps on a simplicial set d : X → X where d = Xdi and the
i n n−1 i
degeneracymapss : X → X wheres = Xsi.Thei-thfaceof∆[n]isthesubset
i n n+1 i
given by the image of d : ∆[n−1] → ∆[n] and is denoted by ∂ ∆[n]. The n-sphere
i i
of ∆[n] is the subset obtained from the union of all the faces of ∆[n] and is denoted
by ∂∆[n]. The canonical inclusion of the n-sphere into the n-simplex is denoted
by δ : ∂∆[n] (cid:44)→ ∆[n]. The subset Λk[n] of ∆[n] is Λk[n] = ∂∆[n]\∂ ∆[n] and the
n k
inclusionintothen-simplexwillbedenotedbyhk : Λk[n] (cid:44)→ ∆[n].Ak-horn,denotes
n
a map Λk[n] → X. A simplicial set is a Kan complex if all horns can be filled i.e. if
they can be extended along the inclusion into the standard simplex. The category
of simplicial sets is denoted by S and Kan denotes the full subcategory of S whose
objectsareKancomplexes.AbisimplicialspaceX isafunctorX : ∆op×∆op → Set.
The m-th row of X is denoted by X and the n-th column by X . A simplicial
m∗ ∗n
space X is a functor ∆op → S. Both notions are equivalent. One can understand
a simplicial space as a bisimplicial set by putting X = (X ) and a bisimpicial
mn m n
set as a simplicial space by putting X = X . The category of bisimplicial sets is
m m∗
denoted by S(2).
5 2 Model Categories
2 Model Categories
2.1 Basic Constructions
The following concepts will be used in the definition of a model category.
Definition 2.1. Consider the following diagram in a category C.
A B
h
i p
C D
If the outer square is commutative, a lift of such square is a map h : C → B
making the whole diagram commute.
If such a lift exists for any commutative square, the map i is said to have the
left lifting property with respect to p and the map p is said to have the right lifting
property with respect to i. This will be denoted by i (cid:116) p.
Proposition 2.2. Consider the adjoint pair
F : C (cid:28) D : G
Then F(f) (cid:116) g if and only if f (cid:116) G(g) for every morphism f in C and g in D.
Proof. Let f : X → Y be a morphism in C and g : A → B be a morphism in D.
The result follows directly from the adjunction, since it implies a bijection between
diagrams in C and D of the following form
F(X) A X G(A)
F(f) g f G(g)
F(Y) B Y G(B)
Definition 2.3. Let M be a class of morphisms in C then define the following
classes of morphisms
• (cid:116)M = {f ∈ Hom(C)|f (cid:116) m ∀m ∈ M}
• M(cid:116) = {f ∈ Hom(C)|m (cid:116) f ∀m ∈ M}
Moreover, let M and M denote two classes of morphisms in C. We say, M (cid:116) M
1 2 1 2
if all maps in M have the left lifting property with respect to all maps in M .
1 2
Remark 2.4. Observe that from the definition it follows that
M ⊆ (cid:116)M ⇔ M (cid:116) M ⇔ M ⊆ M(cid:116)
1 2 1 2 2 1
2 Model Categories 6
Definition 2.5. Let f : A → B and g : A(cid:48) → B(cid:48) be maps in a category C; f is
said to be a retract of g if there are maps i, j, r, s such that the following diagram
commutes
i r
A A(cid:48) A
f g f
j s
B B(cid:48) B
(2.1)
and ri = id , sj = id .
A B
The following properties will be useful in future constructions
Lemma 2.6. Let f be a morphism and M be a class of morphisms in a category
C. The following hold
i If f (cid:116) f then f is an isomorphism.
ii The classes (cid:116)M and M(cid:116) contain all isomorphisms.
iii The classes (cid:116)M and M(cid:116) are closed under compositions and retracts
Proof. (i) If f (cid:116) f then the following diagram has a lift g.
id
X
X X
g
f f
Y Y
id
Y
The map g defines an inverse of f. Condition (ii) holds trivially. (iii) Let f,g be
morphisms in (cid:116)M and consider the commutative diagram
j
X A
f
Y m
g
Z B
k
(2.2)
with m ∈ M. The outer square of the following commutative diagram has a lift h
1
since f ∈(cid:116) M
j
X A
h
1
f m
Y B
kg
7 2 Model Categories
Then, the outer square of the following diagram commutes and has a lift h since
2
g ∈(cid:116) M
h
1
Y A
h
g 2 m
Z B
k
The morphism h defines a lift of the diagram 2.2, so (cid:116)M is closed under composi-
2
tion. The result for M(cid:116) follows similarly. Now, let g belong to (cid:116)M and let f be a
retract of g as in diagram 2.1. Consider the following commutative diagram where
m belongs to M.
α
A C
f m
B D
β
(2.3)
The outer square of the following diagram commutes and has a lift h since g belongs
to M
r α
A(cid:48) A C
h
g m
B(cid:48) B D
s β
Then the map hj defines a lift for diagram 2.3 showing that (cid:116)M is closed under
retracts. The result for M(cid:116) follows similarly.
Definition 2.7. A model category, is a category C with three distinguished classes
of maps:
- weak equivalences (→˜, W)
- cofibrations ((cid:44)→, C)
- fibrations ((cid:16), F)
allofwhichareclosedundercompositionandcontainallidentitymaps.Acofibration
whichisalsoaweakequivalenceiscalledanacycliccofibration.Likewise,afibration
which is also a weak equivalence is called an acyclic fibration. The category C is
subject to the following axioms:
MC1 C has all finite limits and colimits.
MC2 Let f and g be two maps in C such that gf is defined. Then if two of the
three are weak equivalences, then so is the third. This axiom is also referred
to as the two out of three property.