Notes on Higher Topos Theory

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Introduction

Let X be a topological space. We have the singular cohomology groups H n (X ; G) of X with coefficients in an abelian group G. The usual definition of H n (X ; G) in terms of singular G-valued cochains on X , but this does not go to a deep understanding of these groups. First we observe that H n (X ; G) is a representable functor of X ; that is, there exists an EilenbergMaclane space K(G, n) and a universal cohomology class η ∈ H n (K(G, n); G) which induces a bijection [X , K(G, n)] → H n (X ; G), where [X , K(G, n)] denotes the set of homotopy classes of maps from X to K(G, n). The space K(G, n) can be characterized up to homotopy equivalence by the fact that πk K(G, n) ' {∗} if k 6= n and πk K(G, n) ' G if k = n. For n = 1, an Eilenberg-Maclane space K(G, 1) is called classifying space for G and it is denoted by BG. Example 1. (i) If G = Z, then BG = K(G, 1) = S1 ; indeed the fundamental group of S1 is precisely Z; (ii) If G = Z2 , then K(G, 1) = RP∞ = S∞ /Z2 , where RP∞ is the infinite real projective space and S∞ is the infinite sphere. Note that S∞ is a double cover of RP∞ . The universal cover of a topological space X is a cover of X which is simply-connected. The universal cover of BG is a contractible space EG, which has a free action of the group G on itself. We have the canonical quotient map π : EG → BG. Each fiber of the map π is a discrete topological space on which the group G acts simply transitively (i.e. for any x, y ∈ π−1 (p), for some p ∈ BG, there exists a unique g ∈ G such that g x = y). This is to say that the space EG is a G-torsor over the classifying space BG. For every continuous map X → BG, the fiber product X˜ : EG ×BG X has the structure of a G-torsor on X , i.e. this space has a free action of G and a homeomorphism X˜ /G ' X . This construction determines a map from [X , BG] to the set of isomorphism classes of G-torsors on X . For instance, if X is a CW-complex, this map is 1

a bijection. We have three ways to interpret a cohomology class η ∈ H 1 (X ; G): (i) As G-valued singular cocycle on X ; (ii) As a continuous map X → BG; (iii) As a G-torsor on X , which is well-defined up to isomophism. These interpretations are very different from each other in general. The singular cohomology of a space X is constructed using continuous maps from simplices ∆k into X , which is not useful if there are not many maps into X ; for instance when every path in X is constant (e.g. every discrete space, Q). The second definition uses maps from X into the classifying space BG, which depends on the existence of real-valued functions on X . If X does not admit many real-valued functions, then the set of homotopy classes [X , BG] is not an useful invariant. If X˜ is a G-torsor on a topological space X , then the projection map X˜ → X is a local homeomorphism. We may identify X˜ with a sheaf of sets F in X . The action of G on X˜ determines an action of G on F . The sheaf F (with its G-action) and the space X˜ (with its G-action) determine each other up to canonical isomorphism. We can then formulate the definition of a G-torsor in terms of the category ShvSet (X ) of sheaves of sets on X without mentioning the topological space X . For a general space X , isormorphisms classes of G-torsors on X are classified not by the singular cohomology H 1sing (X ; G), but rather by the sheaf cohomology H 1 (X ; G ) of X with sheaf

coefficients in the constant sheaf G associated to G. The sheaf cohomology is defined for any sheaf of groups G on X . The cohomology H 1 (X ; G ) classifies G -torsors on X (i.e., sheaf sheaves F on X which carry an action of G and locally admit a G -equivariant isomorphism F ' G ) up to isomorphism. Let us consider a similar interpretation for cohomology classes η ∈ H 2 (X ; G). When X is a CW-complex we can identify η with a continuous map X → K(G, 2). We can think of K(G, 2) as the classifying space of the classifying space BG (which is equipped with a structure of a topological abelian group). This means that we can identify K(G, 2) with the quotient E/BG, where E is a contractible space with a free action of BG. Any cohomology class η ∈ H 2 (X ; G) determines a map X → K(G, 2) (well-defined up homotopy) and we can form the product X˜ = E ×BG X . Now X˜ is a torsor over X , not for the group G but for the classifying space BG. To have a complete analogy we need to interpret the map X˜ → X as defining some sheaf F on the space X . In this case this sheaf F should have the property that for each x ∈ X the stalk F x can be identified with the fiber at x of the projection map, which is homeomorphic to BG. (In this situation, the sheaf F is called gerbe on X .) Since the space BG is not discrete, the situation cannot be described by the sheaves of sets. In this case we have the notion of a groupoid-valued sheaf F on X , called stack on X . For larger n we need there should be a theory of n-stacks on X , which was proposed by Grothendieck. When n = 0, an n-stack on a topological space X is simply a sheaf of sets on X . The category ShvSet (X ) of sheaves of sets 2

on X is the very first example of Grothendieck topos. The main goal is then to understand the analog for n > 0.

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An overview of Higher Category Theory

Let |∆n | = [0, 1, . . . , n] be a finite ordered set. This can be thought of as ordered n-simplex. Given an ordered n-simplex |∆n |, there are n + 1 face maps d j taking the ∆n to its jth (n − 1)face; there are n + 1 degeneracy maps s j sending ∆ to the jth degenerate (n + 1)-simplex living inside it. Consider the simplices |∆0 |, |∆1 |, . . . , |∆n |, . . . , for each n ∈ N. Then we have degeneracy and face maps d j : |∆n | → |∆n−1 | and s j : |∆n | → |∆n+1 | for each 0 ≤ j ≤ n, and for each n ∈ N. Consider a functor X sending |∆n | to a set X n for each n ∈ N, and sending the morphisms d j and s j to corresponding morphisms between sets. This is the prototype of simplicial set. Definition 1 (Simplicial sets). A simplicial set consists of a sequence of sets X 0 , X 1 , . . . , X n , . . . , for each n ∈ N, functions d j : X n → X n−1 and s j : X n → X n+1 for each 0 ≤ j ≤ n, and for each n ∈ N. Remark 1. Simplices of the form si y for some y ∈ X n−1 are called degenerate. Otherwise, if a simplex cannot be written in this way, it is called nondegenerate. Let ∆ denote the category of finite ordered sets [n], where the morphisms are orderpreserving (not necessarily strictly) functions [m] → [n]. Definition 2 (Categorical definition). A simplicial set is a contravariant functor X : ∆ → Set . Simplicial sets constitute a category, denoted by Set∆ . Morphisms in this category are the following. Definition 3 (Simplicial morphisms). If X and Y are simplicial sets, i.e. they are functors ∆ → Set, then a simplicial morphism f : X → Y is a natural transformation of these functors. Remark 2. To each ordered n-simplex [n] = |∆n |, we have an associated simplicial set ∆n . Example 2. Let X be a topological space. Let Sing n (X ) be the set of continuous functions from ∆n to X . Together with face and degeneracy maps, they constitute a simplicial set called the singular set of the topological space X . Let σ : ∆n → X be a continuous map representing a singular simplex. The singular simplex d j σ is defined as the restriction of σ to the jth face of ∆n . The singular simplex s j σ is defined to be the composition of σ : [0, . . . , n] → X with the geometric collapse represented by the degeneracy [0, . . . , n] → [0, . . . , j, j, . . . , n].

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Definition 4 (Geometric realization). Let X be a simplicial set. Let equip each X n with the discrete topology and let |∆n | be the n-simplex with its standard topology inhereted from the embedding in Rn+1 . The geometric realization |X | is given by |X | =

∞ Y

X n × |∆n |/ ∼,

n=0

where ∼ is the equivalence relation generated by the relations (x, D j (p)) ∼ (d j (x), p) for x ∈ X n+1 , p ∈ |∆n | and the relations (x, S j (p)) ∼ (s j (x), p) for x ∈ X n−1 , p ∈ |∆n |. The maps D j and S j are the face inclusions and collapses induced on the standard geometric simplices. (These maps are the geometric realization of face and degeneracy maps.)1 Theorem 2.1 (Milnor). If X is a simplicial set, then |X | is a CW -complex, with one n-cell for each nondegenerate n-simplex of X . Definition 5. A simplicial set X satisfies the Kan condition if any morphism of simplicial sets Λnj → X can be extended to a simplicial morphism ∆n → X . Such an X is called Kan complex, and it is referred to as being fibrant. Here Λnj is the simplicial set associated to the jth horn of the n-simplex |∆n |, which is the simplex |∆n | without the interior and the jth face. Remark 3. If X is a topologicalspace, then the simplicial set Sing(X ) is a Kan complex; this follows from the fact that the geometric realization |Λnj | of the horn is a retract of the simplex |∆n |. To any category C we can associate a simplicial set N (C ) called the nerve of C . Definition 6 (Nerve of a category). Let C be a category. For each n ≥ 0 we assign N (C )n = MapSet∆ (∆n , N (C )), which is the set of all functors from [n] (regarded as a category where the morphisms are given by the linear order) to the category C . The set N (C ) is precisely the set of all composible sequences of morphisms f1

fn

f2

C0 → C1 → . . . → C n , of length n. The degeneracy map si is the map sending the above n-simplex to the sequence idCi

fi

f1

f i+1

C0 → . . . → C i → C i → . . . ; the face map di sends the above n-simplex to f1

f i−1

f i+1 ◦ f i

f i+2

C0 → . . . → Ci−1 → Ci+1 → . . . . 1

The geometric realization gives a covariant functor from the category Set∆ of simplicial sets to the category Top of topological spaces. The geometric realization functor | · | and the singular set functor Sing (·) are adjoint functors.

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Remark 4. Note that the category C can be recovered (up to isomorphism) from its nerve N (C ). The objects of C are the elements of N (C )0 and a morphism from C0 to C1 in C is an element φ ∈ N (C )1 with d1 (φ) = C0 and d0 (φ) = C1 . Proposition 2.2 ([1], Proposition 1.1.2.2). Let X be a simplicial set. Then the following are equivalent (i) There exists a small category C and an isomorphism X ' N (C ); (ii) For each 0 < i < n, each map of simplicial sets Λni → X can be extented to a map ∆n → X in a unique way. Proof. See [1]. Definition 7 (∞-categories). An ∞-category is a simplicial set X which has the following property; For any 0 < i < n, any map f0 : Λni → X admits an extension f : ∆n → X . Remark 5. In previous works, ∞-categories were called weak Kan complexes. Example 3. Any Kan complex is an ∞-category. In particular, if X is a topological space, the its singular complex Sing(X ) is an ∞-category. Example 4. The nerve of any category C is an ∞-category. Definition 8 (Topological category). A topological category is a category which is enriched ofver C G , the category of compactly generated (and weakly Hausdorff) topological spaces. The category of topological categories is denoted by CatTop . In other words, for any pair of objects X , Y of C we have a compactly generated topological space MapC (X , Y ). We have two approaches to higher category theory. One is based on topological categories and the other one on simplicial sets. In order to show that they are equivalent to one another, we introduce a third approach based on simplicial categories. Definition 9 (Simplicial categories). A simplicial category is category which is enriched over the category Set∆ of simplicial sets. The category of simplicial categories is denoted by Cat∆ . Every simplicial category can be regarded as a simplicial object in the category Cat, the category of small categories. Conversely, a simplicial objects of Cat comes from a simplicial category if and only if the underlying simplicial set of objects is constant. The relationship between simplicial categories and topological categories is as follows. Let Set∆ denote the category of simplcial sets and C G the category of compactly generated Hausdorff spaces. There existss a pair of adjoint functors called geometric realization and singular complex functors. If C is a simplicial category, we can define a topological category |C | in the following way: 5

(i) The objects of |C | are the objects of C ; (ii) If X , Y ∈ C , then Map|C | (X , Y ) = |M apC (X , Y )|; (iii) The composition law for morphisms in |C | is obtained from the composition law on C by applying the geometric realization functor. Similarly, if C is a topological category, we can obtain a simplicial category applying the functor Sing. The geometric realization and singular complex functors determine ad adjunction between Cat∆ and CatTop . This is essentially equivalent to the fact that the geometric realization and singular complex functors determine and equivalence between the homotopy theory of topological spaces and the homotopy theory of simplicial sets. More precisely, we recall that a map f : S → T of simplicial sets is said to be weak homotopy equivalence if the induced map |S| → |T | of topological spaces is a weak homotopy equivalence. A theorem of Quillen says that the morphisms S → Sing(S) and |Sing(X )| → X are weak homotopoy equivalences for every compactly generated topological space X and every simplicial set S, in the respective categories. From this it follows that the category C G with the inversion of weak homotopy equivalences is equivalent to the category obtained from Set∆ inverting the weak homotopy equivalences. We denote either of these equivalent categories with H . If C is a simplicial category, we let hC denote the H -enriched category obtained by applying the functor Set∆ → H to each of the morphism spaces MapSet∆ of C . We will refer to hC as the homotopy category of C . The same construction of homotopy categories is given for topological categories as well. Let C be a topological category. We define the homotopy category hC of C as follows: (i) The objects of hC are the objects of C ; (ii) For X , Y ∈ C , we have M aphC (X , Y ) = [M apC (X , Y )]; (iii) The composition law on hC is obtained from the composition law on C by applying the functor θ : C G → H , given by X 7→ [X ]; here [X ] denotes the weak homotopy equivalence class. From this it follows that the adjoint functors induce isomorphisms between homotopy categories of topological and simplicial categories. We want to show that the theory of simplicial categories is closely related to the category of ∞-categories. This can be done relating simplicial categories with simplicial sets be means of the simplicial nerve functor N : Cat∆ → Set∆ . Recalle that the nerve of an ordinary category C is characterized by HomSet∆ (∆n , N (C )) = H omCat ([n], C ). 6

When C is itself a simplicial category we want to use its simplicial structure introducing a thickening of the ordinary category [n], called C [∆n ]. This construction leads to the following. Proposition 2.3 ([1], Proposition 1.1.5.10). Let C be a simplicial category having the property that, for every pair of objects X , Y in C , the simplicial set MapC (X , Y ) is a Kan complex. Then the simplicial nerve N (C ) is an ∞-category. Corollary 2.4 ([1], Corollary 1.1.5.12). Let C be a topological category. Then the topolgical nerve N (C ) is an ∞-category. Proof. The singular complex of any topological space is a Kan complex.

For any ordinary category C we have an opposite category C op , defined as follows: (i) the objects of C op are the objects of C ; (ii) for X , Y ∈ C , we have HomC op (X , Y ) = HomC (Y, X ). Remark 6. A simplicial category is an ∞-category if and only its opposite is. Let C be a category. In higher category theory, for every pair of objects X , Y of C we have a morphism space MapC (X , Y ). In the setting of topological or simplicial categories, this morphism space is a topological space or a simplicial set respectively. In the setting of ∞-categories, it is not obvious how to define MapC (X , Y ). Definition 10. Let S be a simplicial set containing vertices x and y and let H denote the homotopy category of spaces. We define MapS (x, y) = MaphS (x, y) ∈ H to be the object of H representing the space of maps from x to y in S. Here hS denotes the homotopy category of S regarded as a H -enriched category. Definition 11. In order to compute MaphS (x, y) ∈ H when S is an ∞-category, we define a new simplicial set HomRS (x, y), the space of right morphisms from x to y. This simplicial set is such that HomSet∆ denotes the set of all z : ∆n+1 → S such that z|∆{n+1} = y and z|∆{0,...,n} is a constant simplex at the vertex x. The face and degeneracy maps on HomRS (x, y)n are defined to coincide with corresponding operations on Sn+1 . Proposition 2.5 ([1], Proposition 1.2.2.3). Let C be an ∞-category containing a pair of objects x and y. The simplicial set HomRS (x, y) is a Kan complex. Proof. Remark 7. The definition of HomRS (x, y) is not self-dual: HomRS (x, y) 6= HomRSop (x, y) in general.

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For every ordinary category C , the nerve N (C ) is an ∞-category: the nerve functor is fully faithful. Proposition 2.6 ([1], Proposition 1.2.3.1). The nerve functor Cat → Set∆ is right adjoint to the functor h : Set∆ → Cat, which associates to every simplicial set S its homotopy category hS. Remark 8. If C is a simplicial category, then we do not have in general that hC ' hN (C ). This is the case when C is fibrant, i.e. every simplicial set MapC (X , Y ) is a Kan complex. As in ordinary category theory, we can speak of objects and morphisms in a higher category C . If S is a simplicial set, then the objects of S are the vertices ∆0 → S, and the morphisms of S are the edges ∆1 → S. A morphism φ : ∆1 → S is said to have a source X = φ(0) and a target Y = φ(1). We refer to a morphism φ with source X and target Y as φ : X → Y . If f , g : X → Y are two morphisms in a higher category C , then f and g are homotopic if they determine the same morphism in the homotopy category hC . A morphism f : X → Y in an ∞-category C is said to be an equivalence if it determines an isomorphism in the homotopy category hC . In this situation, X and Y are isomorphic as objects of hC . Example 5. If C is a topological category, then the requirement on a morphism f to be an equivalence is weaker than the requirement to be an isomorphism. For instance, suppose that C is the category of CW -complexes which is a topological category by endowing the sets HomC (X , Y ) with the compactly generated compact open topology. Two objects X and Y in C are equivalenc if and only if they are homotopy equivalent. In the setting of ∞-categories, there is a useful characterization of equivalences which is due to Joyal. Proposition 2.7 (Joyal). Let C be an ∞-category and φ : ∆1 → C a morphism of C . Then φ is an equivalence if and only if, for every n ≥ 2, and every map f0 : Λ0n → C such that f0 |∆{0,1} = φ, there exists an extension of f0 to ∆n . Definition 12 (∞-groupoid). Let C be an ∞-category. We say that C is an ∞-groupoid if the homotopy category hC is a groupoid; this is precisely equivalent to saying that every morphism in C is an equivalence. Proposition 2.8 (Joyal). Let C be a simplicial set. The following conditions are equivalent: (i) The simplicial set C is an ∞-category, and its homotopy category hC is a groupoid. (ii) The simplicial set C satisfies the extension condition for all horn inclusions Λni ⊂ ∆n for 0 ≤ i < n; (iii) The simplicial set C satisfies the extension condition for all horn inclusions Λni ⊂ ∆n for 0 < i ≤ n.

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(iv) The simplicial set C is a Kan complex; in other words, it satisfies the extension condition for all horn inclusions Λni ⊂ ∆n for 0 ≤ i ≤ n. Remark 9. For any ∞-groupoid C , there exists a topological space X such that C = π≤∞ X , the fundamental groupoid of X . Proposition 2.9 ([1], Proposition 1.2.5.3). Let C be an ∞-category. Let C 0 ⊂ C be the largest simplicial subset of C having the property that every edge of C 0 is an equivalence in C . Then C 0 is a Kan complex. It may be characterized by the following universal property: for any Kan complex K, the induced map HomSet∆ (K, C 0 ) → HomSet∆ (K, C ) is a bijection. Let C be an ∞-category and hC its homotopy category. The notion of commutative diagram in hC is given in terms of homotopy in C , i.e. the usual commutative diagrams with morphisms up to homotopy. We replace this notion with the more refined notion of a homotopy coherent diagram in C . Let us assume that F : J → H is a functor from an ordinary category J into the homotopy category of spaces H . Hence F assigns to each object X in J a space (for instance, a CW complex) F (X ), and to each morphism φ : X → Y in J a continuous map of spaces F (φ) : F (X ) → F (Y ) (well-defined up to homotopy), such that F (φ ◦ ψ) is homotopic to F (φ) ◦ F (ψ) for any pair of composable morphisms φ, ψ in J . In this situation, it may not be possible to extend the functor F to a functor F from J to the category Top of topological spaces; in other words, an F such that induces a functor J → H naturally isomorphic to F may not exist. A homotopy coherent diagram in C is a functor F : J → hC together with all additional data needed to guarantee the existence of a lifting F of F as above. Definition 13 (Functors between higher categories). If C and D are ∞-categories, then a functor from C to D is a map from C to D as simplicial sets. We will refer to Fun(C , D) as the ∞-category if functors from C to D.(Let K be an arbitrary simplicial set. Then for every ∞-category C , the simplicial set Fun(K, C ) is an ∞-category.) Remark 10 (Set-theoretical technicalities). In either ordinary or higher category theory, we deal with set theoretical issues; for instance, when the collection of objects or morphisms we have is too large to form a set. The basic issues are resolved if we can decide about the size of collections. The first may be given by the follwing: (i) Assuming the existence of sufficiently many Grothendieck universes; (ii) Working with classes (when the collections of sets are too large to be regarded as sets themselves); (iii) Working in ZF (Zermelo-Frankel) with the incorporation of the theory of classes. Alternatively, another approach is working exclusively with small categories. In the rest it is used the first approach, where we assume that for every cardinal κ0 there exists a strongly inaccessible cardinal κ ≥ κ0 . 9

Definition 14 (∞-category of spaces). Let Kan denote the full subcategory of Set∆ spanned by the collection of Kan complexes. We will regard Kan as a simplicial category. Let S = N (Kan) denote the (simplicial) nerve of Kan. The nerve S is the ∞-category of spaces.

References [1] J. Lurie, Higher Topos Theory.

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