Cohomology of infinity-categories

Cohomology of infinity-categories Trabajo Fin de M´aster Doble M´aster MAES-MUM Author: Daniel Piera Gasque Advisor: Fernando Muro Jim´ enez Resumen En este trabajo definimos una teor´ıa de cohomolog´ıa para conjuntos simpliciales entendidos como ∞-categor´ıas. Tras su definici´on, estudiamos las propiedades elementales de la cohomolog´ıa y especialmente su relaci´on con cierta generalizaci´on de los espacios de Eilenberg-MacLane. Abstract In this memoir we define a cohomology theory for simplicial sets thought of as ∞-categories. After its definition, we study the elementary properties of the cohomology and especially its relation with a certain generalization of Eilenberg-MacLane spaces. Contents 1 Introduction 4 1.1 Motivation ....................................... 4 1.2 Notation......................................... 5 2 Preliminaries 6 2.1 Categorytheory .................................... 6 2.2 Simplicialobjects.................................... 7 2.3 Simplicial sets and categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 ∞-categorytheory ................................... 11 2.5 Naturalsystems..................................... 12 2.6 Baues-Wirching’s cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Cohomology of ∞-categories 15 3.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Simplicial properties of the cohomology . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Properties of L(D, n).................................. 23 3.4 Futuredirections .................................... 29 Bibliography 30 Chapter 1 Introduction 1.1 Motivation Mathematical concepts [...] are brought into being by a series of successive abstractions and generalizations, each resting on a combination of experience with preceding abstract concepts. A. D. Aleksandrov, “Mathematics: Its Content, Methods and Meaning” During the first half of the twentieth century, the blossoming of algebraic topology was marked by the development of homotopy, homology and cohomology. These functors-to-be were the motivation behind the birth of category theory, which in turn provided the language and necessary abstraction for their emancipation from algebraic topology. For instance, homology and cohomology engendered homological algebra, studying the properties of the involved functors and of (co)chain complexes. The emancipation from algebraic topology also brought with it the expansion of the scope of these tools, not being confined to classifying topological spaces, instead being applied to various geometric and algebraic structures. The example we are more interested in is the development of a cohomology theory for small categories. Meanwhile, category theory became a consolidated subject and thus began a process of abstraction of its own to higher category theory, mainly inspired precisely by homotopy theory and its requirements. The category of categories is a motivating example for n-categories as it not only has objects and morphisms, it also naturally has 2-morphisms between its (1- )morphisms, hence constituting a first example of a 2-category. This process can be repeated with 3-morphisms between 2-morphisms, 4-morphisms between 3-morphisms... up to ∞-categories, where there exist k-morphisms for each k≥1. However, there needs to be some conditions on the associativity and identity of these kmorphisms. The choice of these conditions is not trivial and, in order to account for some important motivating examples, the notion of associativity needs to be weakened from equality to a more subtle notion of equivalence. The vagueness of this definition made possible the appearance of multiple models for the notion of ∞-category. Joyal defined a homotopy theory (in technical terms, a model category structure) on simplicial sets which has proven to be the 4 CHAPTER 1. INTRODUCTION 5 most successful model for the theory of infinity-categories. Now, as ∞-category theory has grown and become more fruitful, it needs more tools. One of the goals of any theory is to classify its objects in some sense, and one tool that has proven useful in this task is cohomology. That is what this memoir aims to do, develop a cohomology theory for ∞-categories. This cohomology generalizes the cohomology of small categories presented by Baues and Wirsching in [1]. One possible application for this cohomology is to construct a Postnikov system interpolating between a simplicial set and its fundamental category. Every step of this interpolation would be classified by a cohomology class, analogously to the way in which Postnikov systems for topological spaces are constructed. This application exceeds the scope of this memoir. In chapter 2 of this memoir we explain the basic ingredients needed to develop our theory. We begin recalling some results from category theory, then present simplicial sets and their relation to categories, in order to define quasicategories. The last two sections of the chapter present the cohomology of small categories. In chapter 3 we define the cohomology of ∞-categories and study its properties. Then we study the relation of the cohomology with a generalization of Eilenberg-MacLane spaces and finish proving that this generalization is also a quasicategory. 1.2 Notation Although we will try to clarify the notation used in each section and result, there are some conventions assumed throughout this memoir that we set now. In a sequence of consecutive natural numbers with one of them missing we will denote the one missing with a circumflex, as in: (0,1, . . . , i −1, i + 1, . . . , n −1, n) = (0,1,...,bi, . . . , n −1, n). When talking about categories, an unspecified one will be denoted by C, its objects Obj(C) and its morphisms Mor(C). Morphisms between a pair of objects A, B of a category Cwill be denoted by MorC(A, B). Although in some cases we will drop the subscript if the context category is clear or irrelevant. The opposite category of Cwill be denoted by Cop. Any functor will be assumed to be covariant. The identity morphism for an object Cwill be denoted by idC, although in some cases we will drop the subscript if the context category is clear or irrelevant. Finally, we are going to use some known categories that will be denoted with boldface letters: •Set is the category of sets and maps. •Ab is the category of abelian groups and group homomorphisms. •Cat is the category of small categories and functors. Chapter 2 Preliminaries In this chapter we are going to list some known results and definitions that will be used throughout our memoir. 2.1 Category theory In order to talk about ∞-categories we are obviously going to need to use (1-)category theory. Although we will take most of it as known, there are a few definitions that we are going to reference explicitly. These results mainly follow [2] and their proofs are available there. Definition 2.1.1 (Natural isomorphism).A natural transformation α:F→Gwill be called a natural isomorphism if for every object Cin the source of Fand G, the component αC:F(C)→ G(C) is an isomorphism. We will denote it by α:F∼ =G. Definition 2.1.2 (Equivalence of categories).An equivalence of two categories Cand C0 consists of two functors together with two natural isomorphisms: C C0, F G η: idC∼ =GF, :FG ∼ =idC0. Definition 2.1.3 (Adjoint functors).An adjunction between two categories Cand C0is a pair of functors F:C→C0and G:C0→Cand a natural bijection for each C∈Obj(C) and C0∈Obj(C0): MorC0(F(C), C0)∼ =MorC(C, G(C0)). The functor Fis called the left adjoint and Gthe right adjoint. Lemma 2.1.4 (Yoneda lemma).Given a category Cand a functor F:C→Set. For any object Ain C, there exists a bijection: MorSet(MorC(A, −), F)∼ =F(A), associating to each natural transformation α: MorC(A, −)→Fthe element αA(idA) of the set F(A). The last result of this section explains the construction of a category from a given partially ordered set (poset), which we will use later: 6 CHAPTER 2. PRELIMINARIES 7 Definition 2.1.5 (Category of a poset).Given a poset S, it can be viewed as a category (which we will also denote by S) with: •Objects: elements of the set S. •Morphisms: there exists a unique morphism i→jif and only if i≥j, for any i, j ∈S. Any order-preserving map between posets, f:S→S0, can also be viewed as a functor between their associated categories. 2.2 Simplicial objects After category theory, the other ingredient we need to define ∞-categories are simplicial sets. In this section we list some basic definitions for them. Definition 2.2.1 (Simplex category).The simplex category, denoted by ∆, is given by: •Objects: finite totally ordered sets [n] = {0<1< . . . < n}for n≥0. •Morphisms: weakly order-preserving maps between those sets. More explicitly, given a morphism α: [m]→[n], for each 0 ≤i<j≤m, it satisfies 0 ≤α(i)≤α(j)≤n. There are some special morphisms in ∆, the cofaces and codegeneracies, denoted respectively by diand sifor each n≥0 and for each 0 ≤i≤n. di: [n−1] →[n], di(j) =    jif j < i, j+ 1 if j≥i, (2.2.2) si: [n+ 1] →[n], si(j) =    jif j≤i, j−1 if j > i. Any morphism α: [m]→[n] in ∆can be factorized as a composition of cofaces and codegeneracies, so we will usually only consider these instead of all morphisms. The objects of ∆can also be viewed as categories, as in Definition 2.1.5, and their morphisms as functors. Definition 2.2.3 (Simplicial object).A simplicial object in a category Cis a functor ∆op →C. A particular type of simplicial objects are simplicial sets, which we can describe explicitly: Definition 2.2.4 (Simplicial set).A simplicial set Xis a family of sets and maps between them: · · · XnXn−1· · · X2X1X0. d0,...,dn s0,...,sn−1 d0,d1,d2d0,d1 s0,s1s0 More precisely, we have the following ingredients: CHAPTER 2. PRELIMINARIES 8 •Sets Xnfor each n≥0, called sets of n-simplices. •Face and degeneracy maps di:Xn→Xn−1and si:Xn→Xn+1, for 0 ≤i≤n, satisfying the following conditions, called simplicial identities: didj=dj−1diif i < j, disj=sj−1diif i < j, djsj=dj+1sj= id,(2.2.5) disj=sjdi−1if i>j+ 1, sisj=sj+1siif i≤j. Considering Definition 2.2.3, this construction defines a functor X:∆op →Set with X([n]) = Xn,X(di) = diand X(si) = si. Definition 2.2.6 (Simplicial map).Given two simplicial sets Xand Y, a simplicial map between them, f:X→Y, is a collection of maps between sets fn:Xn→Ynfor n≥0, satisfying the following condition for any α: [m]→[n]: fm◦X(α) = Y(α)◦fn. where we are considering Xand Yas functors. We know that any morphism α: [m]→[n] can be factorized as a composition of cofaces and codegeneracies, thus for f:X→Yto be a simplicial map it suffices that: fndi=difn+1, for each 0 ≤i≤n+ 1, fnsi=sifn−1, for each 0 ≤i≤n−1. From now on, we will not use the subscript in fn, instead we will distinguish each of the maps by context. We will denote the category of simplicial sets and simplicial maps by sSet. There is a particular family of simplicial sets which we are going to use extensively in this memoir: Definition 2.2.7 (n-simplex).The standard n-simplex, which we will refer to as n-simplex, is the simplicial set ∆[n]:= Mor∆(−,[n]). The set of n-simplices of ∆[n], ∆[n]n, is defined as Mor∆([n],[n]) and thus it has a distinguished element id[n]. The element di(id[n])∈∆[n]n−1will be called the i-th face of the n-simplex. We can depict the first few simplices as: ∆[0] : ∆[1] : ∆[2] : 0 0 1 0 2 1 CHAPTER 2. PRELIMINARIES 9 Definition 2.2.8 (Horns of a simplex).For n≥1 and 0 ≤i≤n, the i-horn of the n-simplex, Λi[n], is, intuitively, the sub-simplicial set of ∆[n] given by the union of its faces except the i-th one. More explicitly: Λi[n]m={α: [m]→[n]|([n]\ {i})6⊂ α([m])}. The i-horns for 0 < i < n will be called inner horns. For example, the 2-simplex has three horns, but only one of them, Λ1[2], is an inner horn: Λ0[2] : Λ1[2] : Λ2[2] : 0 2 1 0 2 1 0 2 1 Corollary 2.2.9. For any simplicial set X, applying the Yoneda Lemma we obtain that its n-simplices are in bijection with the simplicial maps from the standard n-simplex to X: Xn∼ =MorsSet(∆[n], X). This means that any a∈Xncan be thought of as a simplicial map a: ∆[n]→X, satisfying precisely a(id[n]) = a∈Xn. We will denote both elements simply by aand will distinguish between them only by context. 2.3 Simplicial sets and categories There are two important functors between categories and simplicial sets which are going to play a central role in this memoir. Definition 2.3.1 (Nerve of a category).Given a small category C, its nerve N•(C) is a simplicial set with Nn(C) defined as the set of functors from [n] to C: Nn(C) = MorCat([n],C). Given a coface map of ∆,di: [n−1] →[n], it induces face maps on the nerve, di: Nn(C)−→ Nn−1(C), defined by precomposition with di: [n−1] [n]C. di di(f) f Codegeneracies si: [n]→[n−1] analogously induce degeneracies si: Nn−1(C)→Nn(C). An element of Nn(C) (a functor [n]→C) can be viewed as an n-tuple of composable morphisms in C: CnCn−1· · · C1C0. fnfn−1f2f1 CHAPTER 3. COHOMOLOGY OF ∞-CATEGORIES 16 However, using the simplicial identities and the relations of Mor(τ1X) we get: {dn−1 2a◦dn−2 1d0a}={d2dn−2 2a◦d0dn−2 2a}={d1dn−2 2a}={dn−1 1a}, where we are using the relation d1a∼d2a◦d0a,a∈X2, for the 2-simplex dn−2 2a∈X2. All the terms of the summation from (3.1.2) satisfy f(dia)∈D({dn−2 1dia}) = D({dn−1 1a}), because i6= 0, n. The last summand is analogous to the first one: D(id,{dn−1 0a})f(dna)∈D({dn−2 1dna◦dn−1 0a}) with {dn−2 1dna◦dn−1 0a}={d2dn−2 1a◦d0dn−2 1a}={dn−1 1a} In the case n= 1 it is clear that ∂f(a)∈D({a}), because a∈X1, which implies f(d0a)∈ D({idd0a}) and f(d1a)∈D({idd1a}), so D({a},id)f(d0a) and D(id,{a})f(d1a) are in D({a}). Lemma 3.1.5. The coboundary ∂, satisfies the cochain complex condition, ∂2= 0. Proof. For f∈Fn−2(X, D) and a∈Xn: (∂2f)(a) =D({dn−1 2a},id)(∂f)(d0a) + n−1 X i=1 (−1)i(∂f)(dia)+(−1)nD(id,{dn−1 0a})(∂f)(dna) = =D({dn−1 2a},id) D({dn−2 2d0a},id)f(d2 0a) |{z } (1) + n−2 X i=1 (−1)if(did0a) | {z } (2) + (−1)n−1D(id,{dn−1 0a})f(dn−1d0a) | {z } (3) !+ n−1 X i=1 (−1)i D({dn−2 2dia},id)f(d0dia) | {z } (4) + n−2 X j=1 (−1)jf(djdia) | {z } (5) + (−1)n−1D(id,{dn−2 0dia})f(dn−1dia) | {z } (6) ! + (−1)nD(id,{dn−1 0a}) D({dn−2 2dna},id)f(d0dna) | {z } (7) + n−2 X i=1 (−1)if(didna) | {z } (8) + (−1)n−1D(id,{dn−2 0dna})f(dn−1dna) | {z } (9) ! From this point everything cancels out using the simplicial identities, the relations of the fundamental category and the composition of the maps D(−,−). First of all dn−2 2dn=dn−1 2 and dn−1d0=d0dn, so (3) and (7) directly cancel out. Then, looking at (1), we can use that: {dn−1 2a◦dn−2 2d0a}={d2dn−2 3a◦d0dn−2 3a}={d1dn−2 3a}={dn−2 2d1a}, CHAPTER 3. COHOMOLOGY OF ∞-CATEGORIES 17 which, recalling d0d1=d2 0, makes it coincide with the case i= 1 from (4) (except the sign), cancelling out. The rest of the i’s from (4) can be cancelled out with the summation (2): to do so we use dn−2 2di=dn−1 2(for any 2 ≤i≤n−1) and taking the term i=kfrom (2) and the term i=k+ 1 from (4) we have dkd0=d0dk+1 for 1 ≤k≤n−2. The term (5) cancels out with itself. Using the first simplicial identity we have that the cases j= 1, i = 2 and j= 1, i = 1 cancel each other out. The same can be said for j= 1, i = 3 and j= 2, i = 1 and, more generally, for j= 1, i =kand j=k−1, i = 1, where 2 ≤k≤n−1. We can repeat the same argument for j= 2, i =kand j=k−1, i = 2, 3 ≤k≤n−1. Analogously, we can generalize this to j=l, i =kand j=k−1, i =l, which cancel each other out for 1≤l≤n−2 and l+ 1 ≤k≤n−1. The rest of the terms are (6), (8) and (9), and they can be done in the same way we have done (1), (2) and (4). First, we can cancel out (9) and the case i=n−1 from (6), to do so we use: {dn−2 0dna◦dn−1 0a}={d2dn−2 0a◦d0dn−2 0a}={d1dn−2 0a}={dn−2 0dna}. Lastly, we can cancel out the cases i=kfrom (6) with the cases i=kfrom (8) for 1 ≤k≤n−2, taking into account dn−2 0dk=dn−1 0and dkdn=dn−1dkwhen k≤n−2. Now we want to check that our cochain complex extends the one developed by Baues and Wirsching, explained in Definition 2.6.1. Proposition 3.1.6. Given a small category Cand a natural system D:FC→Ab, which we can also consider as D:Fτ1NC→Ab, we have: F•(NC, D) = F• BW (C, D). Proof. As we know, τ1NC=C, which means we do not need to consider the equivalence classes associated to the morphisms of the fundamental category. For any n > 0, Fn(NC, D) is the abelian group of maps: f: NnC−→ G g∈Mor(τ1NC) D(g), satisfying f(a)∈D(dn−1 1a). Using that a∈Nn(C), we can write a= (a1, . . . , an) and therefore dn−1 1a=a1◦ · · · ◦ an: f: NnC−→ G g∈Mor(C) D(g), satisfying f(a1, . . . , an)∈D(a1◦ · · · ◦ an), which is exactly the definition of the elements of Fn BW (C, D). For the case n= 0, F0(NC, D) is the abelian group of maps: f: N0C−→ G a∈Obj(τ1NC) D(ida), satisfying f(a)∈D(ida), which, taking into account τ1NC=C, coincides with the definition of F0 BW (C, D). CHAPTER 3. COHOMOLOGY OF ∞-CATEGORIES 18 We now need to check that the coboundaries are equal. For the case n > 1 that means comparing (2.6.2) and (3.1.2). The definition of the coboundary of Fn(NC, D) without equivalence classes is: ∂f(a) = D(dn−1 2a, id)f(d0a) + n−1 X i=1 (−1)if(dia)+(−1)nD(id, dn−1 0a)f(dna). We can write more explicitly the faces of a= (a1, . . . , an) using the identities (2.3.2) and (2.3.3): ∂f(a) =D(a1,id)f(a2, . . . , an) + n−1 X i=1 (−1)if(a1, . . . , aiai+1, . . . , an) + (−1)nD(id, an)f(a1, . . . , an−1), which is the same definition given for ∂BW . For the case n= 1, working with (3.1.3) and removing equivalence classes we get: ∂f(a) = D(a, id)f(d0a)−D(id, a)f(d1a), for a:d0a→d1a, meaning ∂coincides with the ∂BW defined in (2.6.3). With this proposition proven, we can denote both the Baues-Wirsching’s cochain complex and the cochain complex defined in Definition 3.1.1 by F•(−,−). We claim that cochains define a functor Fn:Nat →Ab. We know how it behaves on objects, let us see it on morphisms and then prove its functoriality. For that, we recall that any morphism (λ, µ): (X, D)→(X0, D0) in Nat can be decomposed as (λ, µ) = (id, µ)(λ, id). We are going to study these morphisms separately: Lemma 3.1.7. A morphism (λ, id): (X, D)→(X0,(τ1λ)∗D) induces a homomorphism: Fn(λ, id) :=λ∗:Fn(X, D)−→ Fn(X0,(τ1λ)∗D), defined for f∈Fn(X, D) and a∈X0 nby precomposition with λ: λ∗f(a) = f(λ(a)). Proof. We have to check that λ∗f∈Fn(X0,(τ1λ)∗D), which means checking if λ∗f(a)∈ (τ1λ)∗D({dn−1 1a}). We know f∈Fn(X, D), so: λ∗f(a) = f(λ(a)) ∈D({dn−1 1λ(a)}). Using the fact that λis a simplicial map, the definition of the fundamental category functor and the construction in (2.5.3), we get: D({dn−1 1λ(a)}) = D({λ(dn−1 1a)}) = D((τ1λ)({dn−1 1a})) = (τ1λ)∗D({dn−1 1a}). Now, we have to prove the compatibility with the coboundary, i.e. ∂λ∗=λ∗∂. Given f∈Fn−1(X, D) and a∈X0 nand using the fact that λis a simplicial map, we have: λ∗(∂f)(a) =(∂f)(λa) = D({dn−1 2λa},id)f(d0λa) + n−1 X i=1 (−1)if(diλa)+(−1)nD(id,{dn−1 0λa})f(dnλa) = =D({λdn−1 2a},id)f(λd0a) + n−1 X i=1 (−1)if(λdia)+(−1)nD(id,{λdn−1 0a})f(λdna). CHAPTER 3. COHOMOLOGY OF ∞-CATEGORIES 19 Focusing on the first term and using the definitions of λ∗f,τ1λand (τ1λ)∗Dwe have: D({λdn−1 2a},id)f(λd0a) = D((τ1λ)({dn−1 2a}),id)f(λd0a)=(τ1λ)∗D({dn−1 2a},id)(λ∗f)(d0a). Then, doing analogous calculations for each term in the sum, we arrive at: λ∗(∂f)(a) =(τ1λ)∗D({dn−1 2a},id)(λ∗f)(d0a) + n−1 X i=1 (−1)i(λ∗f)(dia) + (−1)n(τ1λ)∗D(id,{dn−1 0a})(λ∗f)(dna) = ∂(λ∗f)(a). Lemma 3.1.8. A morphism (id, µ): (X, D)→(X, D0) induces a homomorphism: Fn(id, µ):=µ∗:Fn(X, D)−→ Fn(X, D0), defined for f∈Fn(X, D) and a∈Xnby postcomposition with µ: µ∗f(a) = µ(f(a)). Proof. We have to check that µ∗f∈Fn(X, D0), i.e. µ∗f(a)∈D0({dn−1 1a}). We have f(a)∈ D({dn−1 1a}) and µ:D→D0, so: µ∗f(a) = µ(f(a)) ∈D0({dn−1 1a}). Now, we have to prove the compatibility with the coboundary, i.e. ∂µ∗=µ∗∂. Given f∈Fn−1(X, D) and a∈Xnand using the fact that λis a simplicial map, we have: ∂(µ∗f)(a) =D0({dn−1 2a},id)(µ∗f)(d0a) + n−1 X i=1 (−1)i(µ∗f)(dia)+(−1)nD0(id,{dn−1 0a})(µ∗f)(dna) = =D0({dn−1 2a},id)µ(f(d0a)) + n−1 X i=1 (−1)iµ(f(dia)) + (−1)nD0(id,{dn−1 0a})µ(f(dna)). From there we can use the fact that µ:D→D0is a natural transformation, which implies D0(−,−)µ=µD(−,−), giving us: ∂(µ∗f)(a) =µD({dn−1 2a},id)(f(d0a)) + n−1 X i=1 (−1)iµ(f(dia)) + (−1)nµD(id,{dn−1 0a})(f(dna)) = =µ D({dn−1 2a},id)f(d0a) + n−1 X i=1 (−1)if(dia)+(−1)nD(id,{dn−1 0a})f(dna)!= =µ(∂f(a)) = µ∗∂f(a). Proposition 3.1.9. The cochains define a functor Fn:Nat →Ab. Any morphism (λ, µ): (X, D)→ (X0, D0) induces the following homomorphism on cochains: Fn(λ, µ):=µ∗λ∗:Fn(X, D)−→ Fn(X0, D0). CHAPTER 3. COHOMOLOGY OF ∞-CATEGORIES 20 Proof. It is straightforward to see that Fn(id,id) = idFn(X,D). In the case of compositions, given (λ, µ): (X, D)→(X0, D0) and (λ0, µ0): (X0, D0)→(X00, D00) we have: Fn((λ0, µ0)(λ, µ)) = Fn(λλ0, µ0◦(τ1λ0)∗µ) = µ0 ∗((τ1λ0)∗µ)∗λ0∗λ∗. But we can write ((τ1λ0)∗µ)∗λ0∗ =λ0∗µ∗, because for f∈Fn(X0,(τ1λ)∗D) and a∈X0 n: λ0∗µ∗f(a) = ((τ1λ0)∗µ)∗f(λ0(a)) = ((τ1λ0)∗µ)∗λ0∗f(a), and that gives us: Fn((λ0, µ0)(λ, µ)) = µ0 ∗λ0∗µ∗λ∗=Fn(λ0, µ0)Fn(λ, µ). Having studied these basic properties of our cochains, we can now define our cohomology: Definition 3.1.10. The cohomology of a simplicial set Xwith coefficients on a natural system Don τ1X,Hn(X, D), is the cohomology of the cochain complex Fn(X, D) with the coboundary ∂. Therefore, the cohomology defines functors Hn:Nat →Ab. 3.2 Simplicial properties of the cohomology We explore further properties of our cohomology by studying its relation with a generalization of Eilenberg-MacLane spaces. The theorems in this section follow closely [4, §17], with some important changes to adapt them to our situation. Definition 3.2.1. Fixing n≥0 we define the simplicial set L•(D, n) with: Lq(D, n) = {(a, b)|a∈Nqτ1X, b ∈Fn([q], a∗D)}. In the second component, we are considering [q] as a category and taking a∈Nqτ1Xas a functor a: [q]→τ1X, so a∗Dmakes sense as a natural system on [q]. We define the face and degeneracy maps of this simplicial set as: di:Lq(D, n)−→ Lq−1(D, n), (a, b)7−→ di(a, b)=(dia, (di)∗b), si:Lq−1(D, n)−→ Lq(D, n), (a, b)7−→ si(a, b)=(sia, (si)∗b). With di: [q−1] →[q] and si: [q]→[q−1] as defined in (2.2.2), but thinking of them as functors. In the case of the coface map, it induces di: Nn([q−1]) →Nn([q]), which in turn by Lemma 3.1.7 induces: (di)∗:Fn([q], a∗D)−→ Fn([q−1],(di)∗a∗D). This face map is well-defined because we know that (di)∗a∗D= (adi)∗D= (dia)∗Dby the definition of face map of the nerve, which makes Fn([q−1],(di)∗a∗D) = Fn([q−1],(dia)∗D) as desired. The degeneracies work analogously. Considering the simplicial set as a functor, L(D, n): ∆op →Set, any simplicial map α: [q]→ [m] induces: L(D, n)(α): Lm(D, n)−→ Lq(D, n), (a, b)7−→ L(D, n)(α) (a, b) = (Nτ1X(α)a, α∗b).(3.2.2) CHAPTER 3. COHOMOLOGY OF ∞-CATEGORIES 21 This simplicial set comes equipped with a projection map onto the first component: p:L(D, n)−→ Nτ1X, (a, b)7−→ a. Definition 3.2.3. We want to consider the set of simplicial maps between Xand L(D, n) which make the following triangle commute: X L(D, n), Nτ1X Ψp with Ψ as defined in (2.3.8) and pbeing the projection onto the first component. This set will be denoted by homNτ1X(X, L(D, n)) and any element gcan be described as: g:Xq−→ Lq(D, n), a7−→ g(a) = (Ψ(a), g(a)2). This set carries an abelian group structure defined as follows, for g, h ∈homNτ1X(X, L(D, n)): (g+h)(a) = (Ψ(a), g(a)2+h(a)2). Having defined this abelian group, our next goal is to find an isomorphism: homNτ1X(X, L(D, n)) Fn(X, D). φ ϕ Lemma 3.2.4. We define the morphism φ: homNτ1X(X, L(D, n)) →Fn(X, D) as follows, given g∈homNτ1X(X, L(D, n)) and a∈Xn: φ(g)(a) = g(a)2(id[n]), where id[n]∈Nn([n]) and (g(a))2∈Fn([n],(Ψ(a))∗D). Proof. We should check if it is correctly defined. That means proving φ(g) is an n-cochain, i.e. it satisfies φ(g)(a)∈D({dn−1 1a}): φ(g)(a) = g(a)2(id[n])∈(Ψ(a))∗D(dn−1 1id[n]). This natural system makes sense as Ψ(a)∈Nnτ1X, thus Ψ(a): [n]→τ1X. This means (Ψ(a))∗D(dn−1 1id[n]) = D(Ψ(adn−1 1id[n])) and from this point we can use the fact that ais a simplicial map and we get: D(Ψ(adn−1 1id[n])) = D(Ψ(dn−1a(id[n]))) = D(Ψ(dn−1 1a)). Then, considering Ψ as a functor and using the fact that dn−1 1a∈X1, we have: D(Ψ(dn−1 1a)) = D({dn−1 1a}) as desired. CHAPTER 3. COHOMOLOGY OF ∞-CATEGORIES 22 Lemma 3.2.5. We define the morphism ϕ:Fn(X, D)→homNτ1X(X, L(D, n)), for f∈Fn(X, D) and a∈Xq, as: ϕ(f)(a) = (Ψ(a), a∗f), where a: N([q]) →Xinduces a∗:Fn(X, D)→Fn([q],(τ1(a))∗D). Proof. First, we need to prove that Fn([q],(τ1a)∗D) is equal to Fn([q],(Ψ(a))∗D). Given v∈ Nn[q], we have: a∗f(v) = f(a(v)) ∈D({dn−1 1a(v)}) = D({a(dn−1 1v)}), and, as a(dn−1 1v)∈X1, we can use the fact that Ψ restricted to X1, Ψ: X1→N1τ1X= Mor(τ1X), sends each 1-simplex to its equivalence class, the same that τ1does. (Ψ(a)) ∗D({dn−1 1v}) = D({a(dn−1 1v)})=(τ1a)∗D({dn−1 1v}). We also need to check that ϕ(f) is a simplicial map. We will only check it for faces, as degeneracies are analogous: di(ϕ(f)(a)) = di(Ψ(a), a∗f)=(diΨ(a),(di)∗a∗f) = = (Ψ(dia),(adi)∗f) = (Ψ(dia),(dia)∗f) = ϕ(f)(dia). With these morphisms defined and verified, we now turn to proving that they constitute an isomorphism. Theorem 3.2.6. ϕand φare isomorphisms and inverses to each other. Proof. We will prove that both φϕ and ϕφ are identities. First, for f∈Fn(X, D) and a∈Xn: (φϕ)(f)(a) = φ(ϕ(f))(a) = ϕ(f)(a)2(id[n]) = a∗f(id[n]) = f(a(id[n])) = f(a). Conversely, for g∈homNτ1X(X, L(D, n)) and a∈Xq: (ϕφ)(g)(a) = ϕ(φ(g))(a) = (Ψ(a), a∗(φ(g))). To prove that ϕφ = id, the only thing left to be checked is that a∗(φ(g)) = g(a)2, so we will study only the second component. Taking α∈Nn([q]), we can consider α: [n]→[q]. With that, and knowing that both aand gare simplicial maps, we get: a∗(φ(g))(α) = φ(g)(a(α)) = g(a(α))2(id[n]) = g(X(α)a)2(id[n]) = = (L(D, n)(α)g(a))2(id[n]) = α∗(g(a))2(id[n]) = g(a)2(αid[n]) = g(a)2(α), where we are considering the simplicial sets Xand L(D, n) as functors and using the definition of L(D, n) on morphisms as in (3.2.2). Now, we want to restrict this isomorphism to cocycles: CHAPTER 3. COHOMOLOGY OF ∞-CATEGORIES 23 Definition 3.2.7. Fixing n≥0 we define the simplicial set K•(D, n) as the following subsimplicial set of L(D, n): Kq(D, n) = {(a, b)|a∈Nqτ1X, b ∈Zn([q], a∗D)}, which is correctly defined because for (a, b)∈Kq(D, n), (di)∗band (si)∗bare also cocycles due to the functoriality of Fn(−,−) with respect to ∂. Theorem 3.2.8. The restrictions of ϕand φdefine an isomorphism: homNτ1X(X, K(D, n)) Zn(X, D). φ ϕ Proof. Taking f∈Zn(X, D) we want to check that for any a∈Xq,ϕ(f)(a)∈Kq(D, n), which means we need to study the second component to see if it is a cocycle: ∂(ϕ(f)(a))2=∂a∗(f) = a∗(∂(f)) = a∗(0) = 0, where we have used the definition of ϕ, the functoriality of a∗and the fact that fis a cocycle. Conversely, given ϕ(f)∈homNτ1X(X, K(D, n)), we want to check f∈Zn(X, D). Taking a∈Xn+1, we can think of it as a map a: Nn+1 →Xwith a(id[n+1], so we have: ∂f(a) = ∂f(a(id[n+1])) = ∂(a∗f)(id[n+1]) = ∂(ϕ(f)(a))2(id[n+1]) = 0. 3.3 Properties of L(D, n) We prove that both L(D, n) and K(D, n) are quasicategories. This result could be used to prove the invariance by quasiequivalences of the cohomology. The first result below is from [3, Lemma 0032] and the second one is adapted from [5, Section 08NT]. Lemma 3.3.1. The nerve of a category, NC, is a quasicategory. Moreover, each inner horn admits a unique filler morphism: Λi[n] NC ∆[n] ∃!for 0 < i < n. Theorem 3.3.2. L(D, n) is a quasicategory. Proof. For some 0 < k < q, a map σfrom Λk[q] is defined by the images of the (q−1)-simplices diid[q]for i= 0,...,bk, . . . q. We will denote their images by: σ: Λk[q]−→ L(D, n), diid[q]7−→ (ui, vi)∈Lq−1(D, n), which satisfy the relations of the simplices of the horn, namely di(uj, vj) = diσ(djid[q]) = dj−1σ(diid[q]) = dj−1(ui, vi) for i<jand i, j 6=k. CHAPTER 3. COHOMOLOGY OF ∞-CATEGORIES 24 To prove this theorem we need to find a q-simplex (u, v)∈Lq(D, n) such that di(u, v) = (ui, vi) for each i= 0,...,bk, . . . q. The component u∈Nqτ1Xof (u, v) can be defined by Lemma 3.3.1 and satisfies diu=ui. So the only thing missing is finding a cochain v∈Fn([q], u∗D) such that (di)∗v=vifor all i6=k. We will construct this cochain in two steps. The first step is to define c∈Fn([q], u∗D) satisfying (di)∗c=vifor all i<k. We will do so by induction on r, 0 ≤r < k, to define a cr∈Fn([q], u∗D) such that (di)∗cr=vifor i≤r. To define the initial case, c0∈Fn([q], u∗D), we are going to study the cochain (s0)∗v0∈ Fn([q],(s0u0)∗D) = Fn([q],(s0d0u)∗D). Taking τ∈Nn[q], we have: c0(τ)∈u∗D(dn−1 1τ) = D({udn−1 1τ}) = D({uτ(d1)n−1}), (s0)∗v0(τ)∈(s0d0u)∗D(dn−1 1τ) = D{s0d0udn−1 1τ}) = D({ud0s0τ(d1)n−1}). We want to study when those two abelian groups are the same, for that we need to study and compare τ(d1)n−1and d0s0τ(d1)n−1. As τ∈Nn[q], it can be viewed as morphism τ: [n]→[q], thus: [1] [n] [q], (d1)n−1τ [1] [n] [q] [q−1] [q]. (d1)n−1τ s0d0 The category [1] is the category with two objects and one morphism between them, so in both cases we are mapping that morphism, 1 ≥0, to a morphism in [q]. Taking into account the definitions of the cofaces and codegeneracies we have (d1)n−1(0) = 0, (d1)n−1(1) = n,d0s0(0) = 1 and d0s0(l) = lfor any l > 0. This means that the source (image of 1) and target (image of 0) of the morphisms in [q] are: τ(d1)n−1(1) = τ(n), τ(d1)n−1(0) = τ(0), d0s0τ(d1)n−1(1) = 1 if τ(n)=0, τ(n) if τ(n)>0, d0s0τ(d1)n−1(0) = 1 if τ(0) = 0, τ(0) if τ(0) >0. Thus, the images of the morphism 1 ≥0 are u(d1)n−1(1 ≥0) = τ(n)≥τ(0) and d0s0τ(d1)n−1(1 ≥ 0) = max{τ(n),1} ≥ max{τ(0),1}. With all of this explained we can define c0∈Fn([q], u∗D) with τ∈Nn[q]: c0(τ) =    0 if τ(n)=0, (s0)∗v0(τ) if τ(n)>0, τ(0) >0, D(u(1 ≥0),id)((s0)∗v0(τ)) if τ(n)>0, τ(0) = 0. CHAPTER 3. COHOMOLOGY OF ∞-CATEGORIES 25 In any of the three cases, c0satisfies the cochain condition. The first case is obvious. For the second one we need to use the previous explanation: (s0)∗v0(τ)∈D({ud0s0τ(d1)n−1}) = D({u(τ(n)≥τ(0))}) = D({uτ(d1)n−1}). Finally, in the third one we have: (s0)∗v0(τ)∈D({ud0s0τ(d1)n−1}) = D({u(τ(n)≥1)}), then applying the map D(u(1 ≥0),id): D({u(τ(n)≥1)})→D({u(1 ≥0) ◦u(τ(n)≥1)}) = D({u(τ(n)≥0)}) we get what we wanted. Now, we want to prove that (d0)∗c0=v0. Taking τ∈Nn[q−1]: (d0)∗c0(τ) = c0(d0τ), and using the definition of d0(j) = j+ 1 for all 0 ≤j≤q−1, we know that (d0τ)(0) >0, which means we are in the second case of the definition of c0, so: c0(d0τ) = ((s0)∗v0)(d0τ) = v0(s0d0τ) = v0(τ). Now, for the inductive step, we suppose there exists a cras desired for each r < k −1, and we define a cochain yr∈Fn([q], u∗D) such that cr+1 =cr+yrsatisfies (di)∗(cr+yr) = vifor all i≤r+ 1. Analogously to the construction of c0, before we define yrwe need to study the cochain: (y0)r= (sr+1)∗(vr+1 −(dr+1)∗cr). By hypothesis cr∈Fn([q], u∗D), thus (dr+1)∗cr∈Fn([q−1],(dr+1u)∗D) = Fn([q−1], u∗ r+1D), which means (y0)r∈Fn([q],(sr+1dr+1u)∗D). Taking τ∈Nn[q], we compare yrand (y0)r: yr(τ)∈u∗D(dn−1 1τ) = D({udn−1 1τ}) = D({u(τ(n)≥τ(0))}), (y0)r(τ)∈(sr+1dr+1u)∗D(dn−1 1τ) = D{sr+1dr+1udn−1 1τ}) = D({udr+1sr+1(τ(n)≥τ(0))}). Using the definition of the cochains and codegeneracies we know: dr+1sr+1(j) =    jif j < r + 1, r+ 2 if j=r+ 1, jif j > r + 1, which means: dr+1sr+1(τ(n)≥τ(0)) =        idr+1 if τ(n) = τ(0) = r+ 1, τ(n)≥τ(0) if τ(n)6=r+ 1, τ(0) 6=r+ 1, r+ 2 ≥τ(0) if τ(n) = r+ 1, τ(0) < r + 1, τ(n)≥r+ 2 if τ(n)> r + 1, τ(0) = r+ 1. Now, we can define yr∈Fn([q], u∗D) for any τ∈Nn[q]: yr(τ) =    0 if τ(n) = r+ 1, (y0)rif τ(n)6=r+ 1, τ(0) 6=r+ 1, D(u(r+ 2 ≥r+ 1),id)(y0)rif τ(n)> r + 1, τ(0) = r+ 1.