On the functoriality of cohomology of categories

arXiv:math/0411478v1 [math.CT] 22 Nov 2004 ON THE FUNCTORIALITY OF COHOMOLOGY OF CATEGORIES FERNANDO MURO Abstract. In this paper we show that the Baues-Wirsching complex used to define cohomology of categories is a 2-functor from a certain 2-category of natural systems of abelian groups to the 2-category of chain complexes, chain homomorphism and relative homotopy classes of chain homotopies. As a consequence we derive (co)localization theorems for this cohomology. 1. Introduction Baues-Wirsching cohomology of a small category Cwith coefficients in a natural system Don Cwas defined in [2] as the cohomology of a certain cochain complex F∗(C, D). This cohomology generalizes some other cohomologies previously known, as for example •the Hochschild-Mitchell cohomology of Cwith coefficients in a functor D:Cop ×C→Ab ([10]), •the cohomology of the classifying space BCwith local coefficients D, •the Mac Lane cohomology of a ring ([8], [7]). The Baues-Wirsching complex F∗(C, D) as well as its cohomology H∗(C, D) are known to be functors on a certain category Nat of pairs (C, D). It is also known that equivalences of categories induce homotopy equivalences in Baues-Wirsching complexes and isomorphisms in cohomology groups, however this does not follow immediately from the fact that F∗and H∗are functors in Nat. More precisely, it is not known the behaviour of F∗(C, D) and H∗(C, D) with respect to natural transformations between functors in the first variable. The goal of this paper is to shed some light on that issue. We define a new 2-category NatFof pairs (C, D) containing Nat and prove that F∗is in fact a 2-functor in NatF(Theorem 5.1). The 2-morphisms in the category of cochain complexes will be homotopy classes of homotopies relative to the boundary of the cylinder. The category obtained from NatFby taking sets of connected components on morphism categories turns out to be a quotient of Nat (Proposition 5.2) and the cohomology functor H∗factors through this quotient category (Corollary 5.3). As an application of these results we obtain localization and colocalization theorems for Baues-Wirsching cohomology (Theorems 6.5 and 6.12). We also give some examples of how these (co)localization theorems can be used to carry out computations in cohomology of categories. The author was partially supported by the MCyT grant BFM2001-3195-C02-01 and the MECD FPU fellowship AP2000-3330. 1 2 FERNANDO MURO Pirashvili and Waldhausen defined in [12] the homology of a small category C with coefficients in a functor D:Cop ×C→Ab by using a complex F∗(C, D) which is similar, and in some sense dual, to the Baues-Wirsching complex. They also proved that this homology extends Mac Lane homology of rings ([8]), which is isomorphic to topological Hochschild homology in the sense of [3]. We claim that the definition of the Pirashvili-Waldhausen homology H∗(C, D) can be extended to natural systems Das coefficient objects and the functorial properties of F∗and H∗are similar to those described here for F∗and H∗, in particular (co)localization theorems should hold. 2. Notation and conventions Compelled by the necessarily intricate notation of this paper, we have decided to include at the beginning a section to fix the meaning of some symbols. Additional notation will also appear in the development of the paper, but it will not contradict in any case that introduced here. symbols meaning C,D,Esmall categories. X, Y objects in those categories. f, g, h, k, σ morphisms in those categories. ϕ, ψ, ξ, ζ functors between those categories. α, β, γ, ε, t, s natural transformations. →arrow for morphisms in ordinary categories, 1-morphisms in 2-categories, functors, and 2-functors. ⇒arrow for 2-morphisms and natural transformations. 1 identity morphism or identity 2-morphism, a subscript will clarify the meaning in ambiguous cases. Ab the category of objects: abelian groups, morphisms: homomorphisms. Cat the category of objects: small categories, morphisms: functors. Cat2the standard 2-category of objects: small categories, 1-morphisms: functors, 2-morphisms: natural transformations. D, E, G natural systems, see Definition 4.1. ccochain in cohomology of categories. A∗, B∗, C∗cochain complexes of abelian groups. dthe differential in all cochain complexes. p, q, h, r graded morphisms between graded abelian groups. These symbols can be altered by adding superscripts or subscripts. In 2-categories the word “morphism” will be synonym of “1-morphism”. All categories can be regarded as 2-categories with only the trivial 2-morphisms. ON THE FUNCTORIALITY OF COHOMOLOGY OF CATEGORIES 3 The symbol •stands for an unspecified object in an arbitrary category. It can appear several times in a diagram, however in general it will stand for a different object each time. The composite of morphisms, say •f → • g → •, or functors will be indicated by juxtaposition gf, as well as the vertical composition of 2-morphisms or natural transformations βα:ϕ⇒ξas in the diagram ϕ α  • ϕ  ψ// ξ ??•, α  β  or equivalently ψ. β  ξ We will use the symbol ∗for the horizontal composition of 2-morphisms or natural transformations β∗α:ξϕ ⇒ζψ as in the following diagram • ϕ %% ψ 99• α  ξ %% ζ 99• β  3. Factorization categories Factorization categories are the source of the coefficient objects for the Baues- Wirsching cohomology of small categories, see Definition 4.1. A thorough study of their properties is essential to study in depth the functoriality of the Baues- Wirsching complex in Section 5. Definition 3.1. The factorization category FCof a small category Chas objects: morphisms in C, morphisms: (h, k): f→gare pairs of morphisms in Csuch that kfh =g, that is commutative diagrams in C •k//• • f OO • g OO h oo and composition is defined by (h′, k′)(h, k) = (hh′, k′k). One can easily check that factorization categories define a functor F:Cat −→ Cat. This functor is defined on morphisms as follows: a functor ϕ:C→Dis sent to another one F(ϕ): FC−→ FD, which is given on objects: by F(ϕ)(f) = ϕ(f), on morphisms: by F(ϕ)(h, k) = (ϕ(h), ϕ(k)). Notice that the functor Fpreserves products, therefore we can consider the 2-category CatFobtained from Cat2by applying the functor Fto morphism categories. Let us make explicit the structure of CatF: 4 FERNANDO MURO (3.A) objects: are small categories; 1-morphisms: α:C→Dare actually natural transformations α:ϕ⇒ψ between functors ϕ, ψ :C→D, and composition βα in CatFis horizontal composition β∗αof natural transformations; 2-morphisms: (ε, γ): α⇒βare natural transformations such that γαε =β, that is commutative diagrams of natural transformations ψγ+3ζ ϕ α KS ξ β KS ε ks vertical composition of 2-morphisms is given by (ε′, γ′)(ε, γ) = (εε′, γ′γ), and the horizontal composition of (ε, γ) and (ε′, γ′) as in the following diagram in CatF C α && β 88D (ε,γ)  α′ %% β′ 99E (ε′,γ′)  is (ε′, γ′)∗(ε, γ) = (ε′∗ε, γ′∗γ). There is a unique 2-functor Cat −→ Cat2 which is the identity on objects and 1-morphisms. Moreover, there is also a unique 2-functor Cat −→ CatF which is the identity on objects and sends a functor ϕ:C→Dto the identity natural transformation 1ϕ:ϕ⇒ϕregarded as a morphism 1ϕ:C→Din CatF. Proposition 3.2. There is defined a 2-functor F:CatF→Cat2fitting into a commutative diagram Cat F//  Cat  CatFF//Cat2 where the vertical arrows are the 2-functors previously defined. Proof. The new 2-functor Fis defined in the following way, we use the notation in Definition 3.1 and (3.A): on objects: FCis the factorization category, on 1-morphisms: the functor F(α): FC→ FDis defined on objects: given an object fin FC, which is a morphism f:X→Y in C,F(α)(f) = αYϕ(f) = ψ(f)αX; on morphisms: F(α)(h, k) = (ϕ(h), ψ(k)). ON THE FUNCTORIALITY OF COHOMOLOGY OF CATEGORIES 5 on 2-morphisms: F(ε, γ): F(α)⇒ F(β) is the natural transformation which evaluated on fas above is the morphism F(ε, γ)f= (εX, γY) in FD. It is a straightforward exercise to check that this definition is consistent and Fis indeed a 2-functor. Moreover, the diagram in the statement commutes because all 2- functors are the identity on objects, F(1ϕ)(f) = ϕ(f) = F(ϕ)(f) and F(1ϕ)(h, k) = (ϕ(h), ϕ(k)) = F(ϕ)(h, k).  4. Baues-Wirsching cohomology of categories Definition 4.1. Recall from [2] that a natural system on Cis a functor D:FC→ Ab. The Baues-Wirsching complex F∗(C, D) of a small category Cwith coefficients in a natural system Don Cis a cochain complex of abelian groups concentrated in non-negative dimensions. In dimension nthis complex is given by the following product indexed by all sequences of morphisms of length n−1 in C Fn(C, D) = Y •σ1 ←···σn ←• D(σ1···σn). In this formula we assume that a sequence of length 0 is an object Xin Cwhich we also identify with the identity morphism 1X. The coordinate of c∈Fn(C, D) in •σ1 ← · · · σn ← • will be denoted by c(σ1,...,σn). The differential dis defined as d(c)(σ1,...,σn+1) = D(1, σ1)c(σ2,...,σn+1) + n X i=1 (−1)ic(σ1,...,σiσi+1, . . . , σn+1) +(−1)n+1D(σn+1,1)c(σ1,...,σn). over an n-cochain cfor n≥1, and d(c)(σ) = D(1, σ)c(X)−D(σ, 1)c(Y) for n= 0 and σ:X→Y. The cohomology of Cwith coefficients in Dis the cohomology of the complex F∗(C, D), it is denoted by H∗(C, D). Baues and Wirsching noticed that H∗and F∗are functors in the category Nat defined as follows: objects: are pairs (C, D) where Dis a natural system on C, morphisms: (ϕ, t): (C, D)→(D, E) are pairs given by a functor ϕ:D→C and a natural transformation t:DF(ϕ)⇒E, and composition is given by the formula (ψ, s)(ϕ, t) = (ϕψ, s(t∗1F(ψ))). Let Cochain be the category of cochain complexes of abelian groups and cochain homomorphisms. As a functor F∗:Nat −→ Cochain is defined as follows on objects: F∗(C, D) is the Baues-Wirsching complex; on morphisms: Fn(ϕ, t)(c)(σ1,...,σn) = tσ(c(ϕ(σ1), . . . , ϕ(σn))), where σ=σ1···σn; and Hn=HnF∗:Nat −→ Ab, n ∈Z. 6 FERNANDO MURO 5. The Baues-Wirsching complex as a 2-functor This section is the core of the paper. Its main goal is to extend F∗to a 2-functor from an adequate 2-category NatFwith the same objects as Nat to the following 2-category Cochain2: objects: cochain complexes of abelian groups; 1-morphisms: cochain homomorphisms, that is graded homomorphisms p:A∗→B∗of degree 0 such that dp =pd; 2-morphisms: [h]: p⇒qare relative homotopy classes of homotopies between pand q, that is [h] is represented by a degree −1 homomorphism h:A∗→B∗such that dh +hd =−p+qand [h] = [h′] if there exists r:A∗→B∗of degree −2 such that dr −rd =−h+h′; vertical composition of 2-morphisms is given by [h′][h] = [h′+h], and the horizontal composition of [h] and [h′] in the following diagram A∗ p '' q 77B∗ h  p′ '' q′ 77C∗ h′  is [h′]∗[h] = [h′p+q′h] = [p′h+h′q]; one can use the degree −2 homomorphism h′h:A∗→C∗to check the last equality. Notice that morphism categories in Cochain2are in fact groupoids. Moreover, there is a unique 2-functor (5.A) ı:Cochain −→ Cochain2 which is the identity on objects and 1-morphisms. Let us define the 2-category NatF: (5.B) objects: are pairs (C, D) where Dis a natural system on C; 1-morphisms: (α, t): (C, D)→(D, E) are pairs given by a natural transformation α:ϕ⇒ψbetween functors ϕ, ψ:D→C, or equivalently a morphism α:D→Cin CatF, see (3.A), and a natural transformation t:DF(α)⇒E, where Fis the functor defined in Proposition 3.2, and composition is defined as (β, s)(α, t) = (α∗β, s(t∗1F(β))); 2-morphisms: (ε, γ): (α, t)⇒(β, s) are 2-morphisms (ε, γ): α⇒βin CatFsuch that t=s(1D∗F(ε, γ)), that is the following diagram of natural transformations commutes (5.C) DF(α) t !) L L L L L L L L L L L L L L 1D∗F(ε,γ)  E DF(β) s 5= r r r r r r r r r r r r r r ON THE FUNCTORIALITY OF COHOMOLOGY OF CATEGORIES 7 vertical and horizontal compositions of 2-morphisms in NatFare defined as in CatF, that is (ε′, γ′)(ε, γ) = (εε′, γ′γ) and given a diagram in NatF (C, D) (α,t) (( (β,s) 66(D, E) (ε,γ)  (α′,t′) '' (β′,s′) 77(E, G) (ε′,γ′)  the horizontal composition (ε′, γ′)∗(ε, γ) = (ε∗ε′, γ ∗γ′) in NatFcoincides with the horizontal composition of the following diagram in CatF Cyy α ff β D (ε,γ)  xx α′ ff β′ E (ε′,γ′)  It is tedious but straightforward to check that NatFis indeed a well-defined 2-category. Moreover, there is a unique 2-functor :Nat −→ NatF which is the identity on objects and sends a morphism (ϕ, t) to (1ϕ, t). This makes sense because of the commutativity of the diagram in Proposition 3.2. Theorem 5.1. There is defined a 2-functor F∗:NatF→Cochain2fitting into a commutative diagram Nat F∗ //   Cochain ı  NatFF∗//Cochain2 Proof. The new 2-functor F∗is defined as follows, we use the notation in (3.B) and (5.B): on objects: F∗(C, D) is the Baues-Wirsching complex; on 1-morphisms: F∗(α, t)(c)(σ1,...,σn) = tσ(1D∗ F(1ϕ, α))σc(ϕ(σ1),...,ϕ(σn)), here σ=σ1···σnand the formula makes sense because c(ϕ(σ1),...,ϕ(σn)) ∈D(ϕ(σ)) = (DF(1ϕ))(σ); on 2-morphisms: F∗(ε, γ) = [h(ε,γ)]: F∗(α, t)→F∗(β, s) where for an (n+ 1)-dimensional cochain cif n > 0h(ε,γ)(c) is defined as h(ε,γ)(c)(σ1,...,σn) = sσ(1D∗ F(1ξ, γα))σPn i=0(−1)ic(ϕ(σ1),...,ϕ(σi), εXi, ξ(σi+1),...,ξ(σn)), where Xiis the source of σiand/or the target of σi+1, notice that c(ϕ(σ1),...,ϕ(σi), εXi, ξ(σi+1),...,ξ(σn)) ∈(DF(ε))(σ); and if n= 0 h(ε,γ)(c)(X) = s1X(1D∗ F(1ξ, γα))1Xc(εX). 8 FERNANDO MURO A tedious but straightforward computation shows that indeed dh(ε,γ)+h(ε,γ)d=−F∗(α, t) + F∗(β, s). For this, essentially, one only needs to use the naturality property of natural transformations and the commutativity of (5.C). It is easy to see that F∗preserves composition of 1-morphisms. In order to check that F∗preserves vertical composition of 2-morphisms we consider a diagram in NatF (C, D) (α,t) !! (α′,t′)// (β,s) == (D, E) (ε,γ)  (ε′,γ′)  where ψγ+3ψ′γ′ +3ζ ϕ α KS ϕ′ α′ KS ε ksξ β KS ε′ ks is a commutative diagram of natural transformations between functors ϕ, ϕ′, ψ, ψ′, ξ, ζ :D→C. We define a degree −2 homomorphism r(ε′,γ′);(ε,γ):F∗(C, D)−→ F∗(D, E) in the following way, if cis an (n+ 2)-cochain with n > 0 then r(ε′,γ′);(ε,γ)(c)(σ1,...,σn) = sσ(1D∗ F(1ξ, γ′γα))σPn i=0 Pn j=i(−1)i+jc(ϕ(σ1),..., ϕ(σi), εXi, ϕ′(σi+1),...,ϕ′(σj), ε′ Xj, ξ(σj+1),...,ξ(σn)) and for n= 0 r(ε′,γ′);(ε,γ)(c)(X) = s1X(1D∗ F(1ξ, γ′γα))1Xc(εX, ε′ X). It is hard but straightforward to check that dr(ε′,γ′);(ε,γ)−r(ε′,γ′);(ε,γ)d=−h(ε,γ)−h(ε′,γ′)+h(εε′,γ′γ), therefore F∗(ε′, γ′)F∗(ε, γ) = F∗(εε′, γ′γ). Let us see that F∗preserves horizontal composition of 2-morphisms. Consider a diagram in NatF (C, D) (α,t) (( (β,s) 66(D, E) (ε,γ)  (α′,t′) '' (β′,s′) 77(E, G) (ε′,γ′)  ON THE FUNCTORIALITY OF COHOMOLOGY OF CATEGORIES 9 Here ψγ+3ζ ϕ α KS ξ β KS ε ks and ψ′γ′ +3ζ′ ϕ′ α′ KS ξ′ β′ KS ε′ ks are commutative diagrams of natural transformations between functors ϕ, ψ, ξ, ζ :D→Cand ϕ′, ψ′, ξ′, ζ′:E→D. We define a degree −2 homomorphism r′ (ε′,γ′);(ε,γ):F∗(C, D)−→ F∗(E, G) over an (n+ 2)-cochain cwith n > 0 as r′ (ε′,γ′);(ε,γ)(c)(σ1,...,σn) = s′ σ(s∗1F(β′))σ(1D∗ F(1ξξ′,(γ∗γ′)(α∗α′)))σPn i=0 Pn j=i(−1)i+jc(ϕϕ′(σ1),..., ϕϕ′(σi), ϕ(ε′ Xi), ϕξ′(σi+1),...,ϕξ′(σj), εξ′(Xj), ξξ′(σj+1),...,ξξ′(σn)) and for n= 0 r′ (ε′,γ′);(ε,γ)(c)(X) = s′ 1X(s∗1F(β′))1X(1D∗F(1ξξ′,(γ∗γ′)(α∗α′)))1Xc(ϕ(ε′ X), εξ′(X)). After a laborious computation one can check that dr′ (ε′,γ′);(ε,γ)−r′ (ε′,γ′);(ε,γ)d=−h(ε′,γ′)F∗(α, t)−F∗(β′, s′)h(ε,γ)+h(ε∗ε′,γ∗γ′), hence F∗(ε′, γ′)∗F∗(ε, γ) = F∗(ε∗ε′, γ ∗γ′). The commutativity of the diagram in the statement follows easily from the commutativity of the diagram in Proposition 3.2.  The set π0Cof connected components of a small category Cis formed by equivalence classes {X}of objects in C. Two objects X, Y are equivalent {X}={Y}if there exists a sequence of (non-composable) morphisms in Cconnecting them X→ • ← · · · → • ← Y. This defines a product-preserving functor from small categories to sets π0:Cat −→ Set with π0(ϕ){X}={ϕ(X)}. Moreover, one can obtain an ordinary category M0 from a 2-category Mby taking π0on morphism categories and also an ordinary functor ρ0:M0→N0from a 2-functor ρ:M→N. If Mis a category regarded as a 2-category with only the trivial 2-morphisms then M0=M. The homotopy category of cochain complexes Cochain/≃coincides with Cochain0 2and the 2-functor ı:Cochain →Cochain2in (5.A) induces the natural projection ı0:Cochain →Cochain/≃onto the quotient category. By Theorem 5.1 there is a commutative diagram of functors (5.D) Nat F∗ // 0  Cochain ı0  Nat0 F(F∗)0//Cochain/≃ Proposition 5.2. The functor 0:Nat →Nat0 Fis full.