Gabriel-Zisman Cohomology and spectral sequences
aa r X i v : . [ m a t h . K T ] N ov GABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES
IMMA G ´ALVEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONKSA
BSTRACT . Extending constructions by Gabriel and Zisman, we develop a func-torial framework for the cohomology and homology of simplicial sets with verygeneral coefficient systems given by functors on simplex categories into abeliancategories. Furthermore we construct Leray type spectral sequences for any mapof simplicial sets. We also show that these constructions generalise and unify thevarious existing versions of cohomology and homology of small categories andas a bonus provide new insight into their functoriality. Cohomology of simplicialsets, cohomology of categories, Gabriel-Zisman cohomology, spectral sequences I NTRODUCTION
The purpose of this article is to investigate systematically the functoriality of Gabriel-Zisman cohomology and homology of simplicial sets. Gabriel-Zisman (co)homologywas introduced by the authors in [15] inspired by constructions originally due toThomason [33], Gabriel-Zisman [12] and Dress [9] in order to give a simplicialinterpretation of the various (co)homology theories for small categories includ-ing Baues-Wirsching and Hochschild-Mitchell (co)homology (compare [2, 14, 26]).Gabriel-Zisman (co)homology is defined for any simplicial set X with most gen-eral coefficient systems given by functors from the associated simplex category ∆ /X to a given abelian category A . More precisely, we will work here with gen-eral coefficient system functors from ∆ /X with values in arbitrary abelian cate-gories A , which are complete with exact products when considering cohomol-ogy and which are cocomplete with exact coproducts when considering homology.In particular, all constructions will work just fine when using coefficient systemsfunctors with values in the category A b of abelian groups. It turns out that thesegeneral coefficient systems, which we call Gabriel-Zisman natural systems, pro-vide a systematic framework to study the (co)homology of simplicial sets, espe-cially with respect to general naturality and functoriality properties. In particularwe will also show in a direct way how Thomason (co)homology of small categoriescan be interpreted as Gabriel-Zisman (co)homology using the nerve constructionand how its functoriality and naturality properties are just direct consequences ofthose of Gabriel-Zisman (co)homology. Another advantage of our approach is thatusing duality we get at once both cohomology and homology theories for smallcategories and simplicial sets. Furthermore, we will construct Leray type spectralsequences for Gabriel-Zisman cohomology and homology for any map f : X → Y of simplicial sets and identify the lower terms of these spectral sequences for par-ticular coefficient systems. The Leray-Serre spectral sequences in cohomology andhomology for Kan fibrations of simplicial sets are specialisations of these generalLeray type spectral sequences (compare [9, 12]). We aim to use these general Leray The first author was partially supported by Spanish Ministry of Science and Catalan govern-ment grants PID2019-103849GB-I00, 2017 SGR 932, MTM2017-90897-REDT, MTM2016-76453-C2-2-P(AEI/FEDER, UE), MTM2015-69135-P, and the third author by MTM2016-76453-C2-2-P (AEI/FEDER,UE) all of which are gratefully acknowledged. The second author thanks the Centre de RecercaMatem`atica (CRM) in Bellaterra, Spain for inviting him during the research programme HomotopyTheory and Higher Categories (HOCAT), where this work was initiated. type spectral sequences in the future for calculations in various different situationsand frameworks from algebraic geometry, algebraic topology and category theory.In a related homological context, Fimmel [11] developed a theory of Verdier du-ality for a particular class of cohomological coefficient systems on simplicial sets,corresponding via geometric realisation to sheaves on topological spaces. Such aduality theory was first conjectured by Beilinson and allows for interesting appli-cations for example to Beilinson’s theory of local adeles [3] and to buildings forrepresentations of reductive algebraic groups over finite fields. We expect that ourgeneral functorial formalism and the construction of Leray type spectral sequencesdeveloped here will give new insights and calculational tools in these algebraic sit-uations.Similar constructions as those considered here could also be made for cubicalinstead of simplicial sets as indicated by recent work of Husainov on the homologyof cubical sets [21, 22].The article is structured as follows: In the first section we will recall fundamen-tal constructions from the theory of simplicial sets and then introduce the generalconcepts of Gabriel-Zisman cohomology and homology of simplicial sets, studytheir functorial properties and show how these constructions unify and generaliseexisting notions of cohomology and homology of small categories. We also discussseveral interesting examples for future exploration and applications. In the secondsection we will construct Leray type spectral sequences in Gabriel-Zisman coho-mology and homology for any map of simplicial sets within our general frame-work. We will then specialise the coefficient systems for particular situations tobe able to identify the lower pages of these spectral sequences in more familiarterms. And finally, the classical Leray-Serre spectral sequences for cohomologyand homology of a Kan fibration of simplicial sets will be derived as special cases.1. G
ABRIEL -Z ISMAN ( CO ) HOMOLOGY OF SIMPLICIAL SETS
Categories of simplices and simplex categories.
We will collect in this sub-section several fundamental concepts from the theory of simplicial sets and smallcategories, which will be needed later (compare also the systematic accounts in [12,17, 23, 25, 16] and [30]).Let ∆ as usual be the category whose objects are the totally ordered finite sets [ m ] = { < < · · · < m } and whose morphisms are the order preserving functions θ : [ m ] → [ n ] between them. Alternatively, we can regard ∆ as a full subcategory ofthe category C at of small categories, whose objects are the categories [ m ] = (0 → → · · · → m ) .Among the morphisms of ∆ are the coface maps δ i : [ n − → [ n ] , ≤ i ≤ nδ i (0 → → · · · → n −
1) = (0 → → · · · → i − → i + 1 → · · · → n ) , composing the arrows i − → i → i + 1 , and the codegeneracy maps η j : [ n + 1] → [ n ] , ≤ j ≤ n,η j (0 → → · · · → n + 1) = (0 → → · · · → j → j → · · · → n ) inserting the identity morphism id j in the j -th position. These morphisms δ i and η j satisfy the usual cosimplicial identities and give a set of generators and relationsfor the category ∆ (compare [5], [17] and [25]).Let C be a category. A simplicial object in C is a functor X : ∆ op → C . Dually,a cosimplicial object in C is a functor X : ∆ → C . In particular, if C = S et is thecategory of sets a functor X : ∆ op → S et is called a simplicial set and a functor X : ∆ → S et a cosimplicial set . Simplicial objects in a category C form a category ABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES 3 ∆ op C , where the morphisms are natural transformations. Dually, we have thecategory of cosimplicial objects ∆ C .In the category ∆ op S et of simplicial sets we can consider for every integer n ≥ the representable simplicial set ∆[ n ] = Hom ∆ ( − , [ n ]) called the standard [ n ] -simplex . The n -simplices X n of a simplicial set X are given as X n = X ([ n ]) and wesometimes will also write X • = { X n } n ≥ to denote a simplicial set. As usual, wewill denote by d i : X n → X n − for ≤ i ≤ n the face maps and by s j : X n → X n +1 for ≤ j ≤ n the degeneracy maps .The Yoneda Lemma readily implies that the n -simplices of a simplicial set X arein bijective correspondence with the morphisms of simplicial sets from ∆[ n ] to X i.e., X n ∼ = Hom ∆ op S et (∆[ n ] , X ) . Thus morphisms of simplicial sets ∆[ n ] → ∆[ m ] can be identified with morphisms [ n ] → [ m ] of ∆ and vice versa. Definition 1.1.
Let X be a simplicial set. The category of simplices or simplex category of X is the comma category ∆ /X whose objects are the simplices x of X and whosemorphisms x → x ′ are morphisms θ of ∆ such that x = X ( θ )( x ′ ) . Alternatively,the objects are pairs ([ m ] , x ) , where x : ∆[ m ] → X , and morphisms are commutingtriangles, ∆[ m ] θ / / x " " ❊❊❊❊❊❊❊❊ ∆[ m ′ ] x ′ | | ①①①①①①①① X. Given a map f : X → Y of simplicial sets there is a functor ∆ /f : ∆ /X → ∆ /Y given by (∆ /f )( x ) = f x .The opposite category of the category of simplices ∆ /X of X can also be inter-preted as the Grothendieck construction for the functor X : ∆ op → S et , that is, asthe category (∆ /X ) op = Z ∆ op X. It also comes together with a natural projection functor P op : (∆ /X ) op = Z ∆ op X → ∆ op , which is a discrete Grothendieck fibration i.e., a Grothendieck fibration where allthe fibers are sets. In fact, every discrete Grothendieck fibration P : C → ∆ can be obtained as the Grothendieck construction R ∆ X of the functor X : ∆ op → S et, X ([ n ]) = X n = P − ([ n ]) . This gives an equivalence of categories D isc F ib (∆) ≃ ↔ ∆ op S et between the category of discrete Grothendieck fibrations over ∆ and the categoryof simplicial sets, which is a very special case of the equivalence of -categoriesbetween the -category F ib ( B ) of Grothendieck fibrations over a small category B and the -category of contravariant pseudofuncors PsdFun ( B op , C at ) from B into the -category C at of small categories (see [15, 2.3] and [19]). IMMA G ´ALVEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONKS
Any contravariant functor from a small category C into the category of sets is acolimit of representable functors Hom C ( − , c ) , and the Density Theorem [25, Chap.III, §7, Thm. 1] states that we can recover a simplicial set X via the isomorphism X ∼ = colim ([ m ] ,x ) ∈ ∆ /X ∆[ m ] . Let us here also recall that the nerve N ( C ) of a small category C is the simplicialset N ( C ) • = {N ( C ) n } whose n -simplices are given as N ( C ) n = Hom C at ([ n ] , C ) . In more concrete terms, an n -simplex σ of N ( C ) is just a string of n composablemorphisms γ i in C σ = ( C γ −→ C γ −→ · · · γ n −→ C n ) where C i = σ ( i ) are objects of C .The face and degeneracy maps d i and s j are then given by precomposition withthe coface and codegeneracy maps d i and s j . In other words, the value d i ( σ ) of theface map d i : N ( C ) n → N ( C ) n − is obtained from σ by omitting the object C i = σ ( i ) , and by omitting γ if i = 0 , composing γ i +1 and γ i if < i < n , or omitting γ n if i = n . Similarly, the value s j ( σ ) of the degeneracy map s j : N ( C ) n → N ( C ) n +1 is obtained from σ by repeating the object C j and inserting an identity morphism id C j .The nerve construction defines a functor N : C at → ∆ op S et from the categoryof small categories to the category of simplicial sets.We will finally define another simplex category, the simplex category of a smallcategory. Definition 1.2.
Let C be a small category. The simplex category ∆ / C of C is thecomma category whose objects are pairs ([ m ] , f ) , where [ m ] is an object of ∆ and f : [ m ] → C is a functor, and whose morphisms ([ m ] , f ) −→ ([ n ] , g ) are morphisms θ : [ m ] → [ n ] of ∆ with f = g ◦ θ .Thus objects ([ m ] , f ) of ∆ / C are elements of the simplicial nerve N ( C ) of C .We will often omit the [ m ] from the notation and regard objects as diagrams orstrings f = ( C f −→ C f −→ · · · f m −→ C m ) . The morphisms of ∆ / C are as usual generated by omitting or repeating objects C i in such diagrams.The simplex category ∆ / C of a small category C is therefore just the simplexcategory ∆ / N ( C ) of the nerve N ( C ) of C . It was shown by Illusie [23, VI.3] andLatch [24] that the functor ∆ / − : ∆ op S et → C at is in fact a weak homotopy in-verse to the nerve functor N : C at → ∆ op S et i.e., for any simplicial set X there isa weak equivalence of simplicial sets N (∆ /X ) ∼ → X. Another incarnation of the simplex category of C is given by the Grothendieckconstruction of the contravariant diagram of discrete categories given by the sim-plicial nerve, (∆ / C ) op ∼ = Z ∆ op N ( C ) where N ( C ) : ∆ op → S et → C at. Let F : C → D be a functor. If D is an object of D , then the fiber C D = F − ( D ) of F over D is the subcategory of C whose objects are the objects C of C such that F ( C ) = D and whose morphisms are the morphisms f : C → C ′ in C such that F ( f ) = id D . The left fiber C /D = F/D of F over D is the category of all pairs ( C, u ) ABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES 5 with C an object of C and u : F ( C ) → D a morphism in D and where a morphism ( C, u ) → ( C ′ , u ′ ) is given as a morphism v : C → C ′ in C such that u = u ′ ◦ F ( v ) .Dually, we have the notion of a right fiber D/ C = D/F of F over D .If C is a small category, then the simplex category ∆ / C is also given as the leftfiber over C of the embedding ∆ → C at .More generally, let C be any category and c an object of C . Given any cosimpli-cial object in C , that is, a functor F : ∆ → C , one can define the simplex category ∆ /c as the comma category whose objects are pairs ([ m ] , f ) , where [ m ] is an object of ∆ and f : F ([ m ]) → c is an arrow of C , and whose morphisms ([ m ] , f ) −→ ([ n ] , g ) are morphisms θ : [ m ] → [ n ] of ∆ with f = g ◦ F ( θ ) . The definitions of the simplexcategory above are for the obvious functors F : ∆ → ∆ op S et and F : ∆ → C at .Note that both of these are fully faithful functors.1.2. Gabriel-Zisman (co)homology of simplicial sets and its functorial proper-ties.
In this subsection we will present a systematic account of the constructionsand fundamental functorial properties of cohomology and homology of simpli-cial sets with general coefficient systems. The coefficient systems described herewere first introduced by Gabriel and Zisman [12, App. II.4] to analyse the homol-ogy of simplicial sets and were also discussed systematically by Dress [9]. Fimmel[11] also used these coefficient systems to construct a Verdier duality theory forsheaves on simplicial sets. As a particular application of our general frameworkwe will show how Thomason cohomology and homology of small categories asintroduced and studied by the authors in [15] fits into this picture.
Definition 1.3.
Let X be a simplicial set and M be a category. A functor T : ∆ /X → M is called a (covariant) Gabriel-Zisman natural system on X with values in M . Remark 1.4.
A Gabriel-Zisman natural system T will be termed a sheaf if T ( x θ → x ′ ) is an isomorphism in M whenever θ is a codegeneracy map s i : [ n + 1] → [ n ] in ∆ (or equivalently, whenever θ is surjective, cf. [11, Definition 3.2]).We now define a general cohomology theory for simplicial sets using theseGabriel-Zisman natural systems as coefficients. Definition 1.5.
Let X be a simplicial set and let T : ∆ /X → A be a Gabriel-Zismannatural system with values in a complete abelian category A with exact products.The Gabriel-Zisman cochain complex C ∗ GZ ( X, T ) of X is defined as C nGZ ( X, T ) := Y σ n ∈ X n T ( σ n ) , for each integer n ≥ , with differential d = n +1 X i =0 ( − i d i : Y σ n ∈ X n T ( σ n ) −→ Y σ n +1 ∈ X n +1 T ( σ n +1 ) . The components of these d i are the morphisms δ i : T ( σ n +1 ◦ δ i ) → T ( σ n +1 ) induced by the coface maps δ i : [ n ] → [ n + 1] . The n -th Gabriel-Zisman cohomology of X is the cohomology of this cochain complex, H nGZ ( X, T ) = H n ( C ∗ GZ ( X, T ) , d ) . IMMA G ´ALVEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONKS
Equivalently, C ∗ GZ ( X, T ) is the cochain complex associated to the cosimplicialobject Y σ ∈ X T ( σ ) d / / d / / Y σ ∈ X T ( σ ) s y y d / / d / / d / / Y σ ∈ X T ( σ ) s y y s y y / / / / / / / / Y σ ∈ X T ( σ ) y y y y y y . . . given as the cosimplicial replacement Q ∗ T of the functor T : ∆ /X → A (see [5,XI.5], [33]).For any simplicial set X and a complete abelian category A with exact prod-ucts, let NatS GZ be the category whose objects are the (covariant) Gabriel-Zismannatural systems T : ∆ /X → A with values in A . A morphism ( ϕ, τ ) : T X → T Y between Gabriel-Zisman natural systems ∆ /X T X −→ A consists of a morphism ϕ : Y → X of simplicial sets together with a natural transformation τ : T X ◦ ∆ /ϕ −→ T Y . The composition of morphisms is given by the following diagram ∆ /X T X & & ▼▼▼▼▼▼▼▼▼▼▼▼▼ ∆ /Y ∆ /ϕ o o τ = ⇒ T Y υ = ⇒ (cid:15) (cid:15) ∆ /Z ∆ /ψ o o T Z x x qqqqqqqqqqqqq A . The Gabriel-Zisman cochain complex defines in fact a functor C ∗ GZ : NatS GZ → coChn , C ∗ GZ ( T ) := C ∗ GZ ( X, T ) , from the category of Gabriel-Zisman natural systems with values in the abeliancategory A , to the category of cochain complexes in A . The functor C ∗ GZ is definedon objects as above, and on morphisms by C ∗ GZ ( ϕ, τ ) : C ∗ GZ ( X, T X ) −→ C ∗ GZ ( Y, T Y ) , ( a f ) [ n ] f → X ( τ g ( a ϕ ◦ g )) [ n ] g → Y . Gabriel-Zisman cohomology therefore becomes a functor from
NatS GZ to the cat-egory of graded objects in the category A . In fact, the correspondence T H ∗ GZ ( X, T ) is a cohomological ∂ -functor on the category NatS GZ of (covariant) Gabriel-Zismannatural systems.Dually, we define homology of simplicial sets with coefficients in contravariant Gabriel-Zisman natural systems. These coefficients are in fact the original onesused by Gabriel and Zisman [12, App. III.4] and Dress [9].
Definition 1.6.
Let X be a simplicial set and M be a category. A functor T : (∆ /X ) op → M is called a (contravariant) Gabriel-Zisman natural system on X with values in M .Using these general coefficient systems, we define now the Gabriel-Zisman ho-mology of a simplicial set X . Definition 1.7.
Let X be a simplicial set and let T : (∆ /X ) op → A be a contravari-ant Gabriel-Zisman natural system with values in a cocomplete abelian category A with exact coproducts. The Gabriel-Zisman chain complex C GZ ∗ ( X, T ) of X isdefined as C GZn ( X, T ) := M σ n ∈ X n T ( σ n ) , ABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES 7 for each integer n ≥ , with differentials d n : M σ n +1 ∈ X n +1 T ( σ n +1 ) −→ M σ n ∈ X n T ( σ n ) a f n +1 X i =0 ( − i ( δ i ) ( a f ) , where ( δ i ) : T ( σ n +1 ) → T ( σ n +1 ◦ δ i ) is induced by the coface map δ i : [ n ] → [ n +1] .The n -th Gabriel-Zisman homology of X is defined as the homology of this chaincomplex, H GZn ( X, T ) := H n ( C GZ ∗ ( X, T ) , d ) . Again, the Gabriel-Zisman chain complex is just the chain complex correspond-ing to a certain simplicial object in A , given by the simplicial replacement of T .Let NatS GZ be the category with objects the contravariant Gabriel-Zisman nat-ural systems T : (∆ /X ) op → A , and in which a morphism ( ϕ, τ ) : X → Y is givenby a functor ϕ : X → Y together with a natural transformation τ : T X → T Y ◦ ∆ /ϕ .The composition of morphisms is described by the following diagram, (∆ /X ) op T X ' ' PPPPPPPPPPPPPPP ∆ /ϕ / / (∆ /Y ) op τ = ⇒ T Y υ = ⇒ (cid:15) (cid:15) ∆ /ψ / / (∆ /Z ) opT Z w w ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ A . The Gabriel-Zisman chain complex defines a functor C GZ ∗ : NatS GZ → Chn , C GZ ∗ ( T ) := C GZ ∗ ( X, T ) , where for morphisms we define C GZ ∗ ( ϕ, τ ) : C GZ ∗ ( X, T X ) −→ C GZ ∗ ( Y, T Y ) using the maps τ f : T X ( f ) −→ T Y ( ϕ ◦ f ) . Dually, Gabriel-Zisman homology therefore defines a functor from
NatS GZ to thecategory of graded objects in A .We will give now another interpretation of Gabriel-Zisman (co)homology, whichis useful for analyzing its functorial properties.Given a cosimplicial object in an abelian category A i.e., a functor F : ∆ → A we have the associated cochain complex ( C ∗ ( F ) , d ) of F defined as C n ( F ) : = F ([ n ]) for each integer n ≥ , with differential d = n +1 X i =0 ( − i d i : F ([ n ]) → F ([ n + 1]) , where d i = F ( δ i ) and δ i : [ n ] → [ n + 1] for ≤ i ≤ n + 1 are the respective cofacemaps. We can now define the n -th cohomology of the cosimplicial object F as thecohomology of the associated cochain complex H n ( F ) := H n ( C ∗ ( F ) , d ) . We therefore get a sequence of functors H ∗ = ( H n ) n ∈ N : F un (∆ , A ) → A , F H n ( F ) . IMMA G ´ALVEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONKS
Now we consider the following general situation. Let C be a small category and A a complete abelian category. Given two functors P : C → ∆ and T : C → A ,we have the right Kan extension Ran P ( T ) of the functor T along P (see [25, Chap.X]): C T (cid:15) (cid:15) P / / ∆ Ran P ( T ) | | ② ② ② ② ② ② ② A It is an object of the functor category F un (∆ , A ) , in other words a cosimplicialobject of the abelian category A and we define: Definition 1.8.
Let C be a small category and A a complete abelian category.Given two functors P : C → ∆ and T : C → A the n -th cohomology of P withcoefficients in T is defined as H n ( P, T ) := H n ( Ran P ( T )) . Dually, given now a simplicial object of an abelian category A i.e., a functor F : ∆ op → A we have the associated chain complex ( C ∗ ( F ) , d ) of F defined as C n ( F ) := F ([ n ]) for each integer n ≥ , with differential d = n +1 X i =0 ( − i d i : F ([ n + 1]) → F ([ n ]) , where d i = F ( δ i ) and δ i : [ n ] → [ n + 1] for ≤ i ≤ n + 1 are the respectivecoface maps. So we can define the n -th homology of the simplicial object F as thehomology of the associated chain complex H n ( F ) := H n ( C ∗ ( F ) , d ) . We therefore get a sequence of functors H ∗ = ( H n ) n ∈ N : F un (∆ op , A ) → A , F H n ( F ) . Now we consider the following general situation. Let C be a small category and A a cocomplete abelian category. Given two functors P : C → ∆ and T : C op → A ,we have the left Kan extension Lan P ( T ) of the functor T along P op (see [25, Chap.X]): C opT (cid:15) (cid:15) P op / / ∆ opLan P ( T ) { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ A It is an object of the functor category F un (∆ op , A ) , in other words a simplicialobject of the abelian category A and we define: Definition 1.9.
Let C be a small category and A a cocomplete abelian category.Given two functors P : C → ∆ and T : C op → A the n -th homology of P withcoefficients in T is defined as H n ( P, T ) := H n ( Lan P ( T )) . ABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES 9
Now we would like to interpret this general cohomology and homology as acertain Ext and Tor construction, and in order to do so recall the following con-structions (compare [15, 1.3, Remark 1.7] and [25, X.4], ):
Definition 1.10.
Let A b be the category of abelian groups, and A an additivecategory. Then(1) The category A is cotensored over A b if there is a functor Hom : A b op × A → A , satisfying the natural exponential lawHom A ( a, Hom ( A, b )) ∼ = Hom A b ( A, Hom A ( a, b )) . (2) The category A is tensored over A b if there is a functor ⊗ : A b × A → A satisfying the natural exponential lawHom A ( A ⊗ a, b ) ∼ = Hom A b ( A, Hom A ( a, b )) . Let F : C → A b , T : C → A be diagrams over C . The symbolic hom Hom C ( F, T ) as an object of A is determined by natural isomorphismsHom A ( a, Hom C ( F, T )) ∼ = N at ( F, Hom A ( a, T ( − ))) . Dually, for diagrams F : C → A b , T : C op → A , the symbolic tensor product F ⊗ C T as an object of A is determined by natural isomorphismsHom A ( F ⊗ C T, b ) ∼ = N at ( F, Hom A ( T ( − ) , b )) . Now let Z : C → A b be the constant diagram with value Z . Then following thearguments and their duals in [15, 1.3] we have for any diagram T : C → A thatHom C ( Z , T ) ∼ = lim C T and for any diagram T : C op → A we have dually Z ⊗ C T ∼ = colim C op T. Recall that a resolution of Z is a functor B ∗ : ∆ op → F un ( C , A b ) such that, foreach object c of C , the reduced homology groups of the complexes B ∗ ( c ) are triv-ial. A resolution is free if for each n the functor B n : C → A b is a coproduct ofrepresentable functors Z Hom ( c, − ) .We now express the general cohomology and homology constructions intro-duced above as derived functors of lim and colim. Suppose that A is a completeabelian category, with exact products. Let B ∗ be a free resolution of Z . Then thederived functors of Hom C ( Z , − ) ∼ = lim C ( − ) are given by the cohomology of thefollowing Ext complex, Ext ∗ C ( Z , − ) := Hom C ( B ∗ , − ) . Dually, suppose that A is a cocomplete abelian category, with exact coproducts.Then the derived functors of Z ⊗ C ( − ) ∼ = colim C op ( − ) are given by the homologyof the following Tor complex,Tor C ∗ ( Z , − ) := B ∗ ⊗ C ( − ) . Theorem 1.11.
Let A be an additive category. For any functor P : C → ∆ there exists aresolution B P ∗ of the constant functor Z such that (1) If A is complete and cotensored over A b , then Ran P ( T ) ∼ = Hom C ( B P ∗ , T ) : ∆ → A natural in T : C → A . (2) If A is cocomplete and tensored over A b , then Lan P ( T ) ∼ = B P ∗ ⊗ C T : ∆ → A natural in T : C op → A . (3) If P : C → ∆ is a discrete fibration over ∆ , then there is a natural isomorphism B Pn ∼ = M c ∈ C : P ( c )= n Z Hom C ( c, d ) and hence B P ∗ is a free resolution of Z .Proof. We set B P ∗ = Z Hom ∆ ( − , P ( − )) : ∆ op → F un ( C , A b ) and observe that thefunctor B P ∗ ( d ) = Z Hom ∆ ( − , P ( d )) : ∆ op → A b is contractible since it is the stan-dard simplex of dimension P ( d ) . We therefore have a resolution of Z . The naturalisomorphisms of (1) and (2) now follow by expressing the Kan extensions andsymbolic hom and tensor functors in terms of (co)ends: Ran P ( T ) ∼ = Z d ∈ C Hom ( Z Hom ∆ ( − , P ( d )) , T ( d )) = Z d ∈ C Hom ( B P ∗ ( d ) , T ( d )) ∼ = Hom C ( B P ∗ , T ) ,Lan P ( T ) ∼ = Z d ∈ C op Z Hom ∆ op ( P op ( d ) , − ) ⊗ T ( d ) = Z d ∈ C op B P ∗ ( d ) ⊗ T ( d ) ∼ = B P ∗ ⊗ C T. If P : C → ∆ is a discrete fibration over ∆ there is a natural bijectionHom ∆ ( n, P ( d )) ∼ = a c ∈ C : P ( c )= n Hom C ( c, d ) for each n ≥ and each object d of C . Thus B Pn ( d ) = Z Hom ∆ ( n, P ( d )) ∼ = M c ∈ C : P ( c )= n Z Hom C ( c, d ) and therefore the resolution B P ∗ : ∆ op → F un ( C , A b ) of Z is free. (cid:3) The following is then immediate:
Corollary 1.12.
Let C be a small category and A an additive category, and let P : C → ∆ be a discrete fibration. (1) If A is complete, with exact products, and T : C → A a functor, then the coho-mology groups of P with coefficients in T are derived functors, H n ( P, T ) = H n ( Ran P ( T )) ∼ = Ext n C ( Z , T ) ∼ = lim n C T = H n ( C , T ) . (2) If A is cocomplete, with exact coproducts, and T : C op → A a functor, then thehomology groups of P with coefficients in T are derived functors, H n ( P, T ) = H n ( Lan P ( T )) ∼ = Tor n C ( Z , T ) ∼ = colim C n T = H n ( C , T ) . As noted earlier, the discrete fibrations P : C → ∆ are just given as the projec-tions P X : ∆ /X → ∆ from the simplex category of a simplicial set X . Theorem 1.13.
Let X be a simplicial set and let T : ∆ /X → A be a Gabriel-Zismannatural system with values in a complete abelian category A with exact products. Thecohomology of P X : ∆ /X → ∆ coincides with the Gabriel-Zisman cohomology of X , C ∗ ( P X , T ) ∼ = C ∗ GZ ( X, T ) , H ∗ ( P X , T ) ∼ = H ∗ GZ ( X, T ) , and Gabriel-Zisman cohomology may be identified as a derived functor, H nGZ ( X, T ) ∼ = Ext n ∆ /X ( Z , T ) ∼ = lim n ∆ /X T = H n (∆ /X, T ) . ABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES 11
Proof.
From [12, Appendix II.4], it follows that the right Kan extension
Ran P X ( T ) of T along the forgetful functor P X : ∆ /X → ∆ , ∆ /X T (cid:15) (cid:15) P X / / ∆ Ran PX ( T )= Q ∗ T { { ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ A is precisely the cosimplicial replacement Q ∗ T of the functor T : ∆ /X → A .So we apply the above Theorem 1.11 to the right Kan extension Ran P X ( T ) and use the identification of the cosimplicial replacement Q ∗ T with the Gabriel-Zisman cochain complex C ∗ GZ ( X, T ) as constructed above to get the desired iso-morphisms.The last isomorphism is just the usual identification of the derived functors ofthe limit functor lim n ∆ /X T with the cohomology of the category ∆ /X with coeffi-cients in T (see [29, 31] or [14]). (cid:3) Dually, we also have a similar isomorphism for Gabriel-Zisman homology ofsimplicial sets.
Theorem 1.14.
Let X be a simplicial set and let T : (∆ /X ) op → A be a Gabriel-Zismannatural system with values in a cocomplete abelian category A with exact coproducts. Thehomology of P X : ∆ /X → X coincides with the Gabriel-Zisman homology of X , C ∗ ( P X , T ) ∼ = C GZ ∗ ( X, T ) , H ∗ ( P X , T ) ∼ = H GZ ∗ ( X, T ) , and Gabriel-Zisman homology may be identified as a derived functor, H GZn ( X, T ) ∼ = Tor (∆ /X ) op n ( Z , T ) ∼ = colim (∆ /X ) op n T = H n ((∆ /X ) op , T ) . Proof.
This is basically [12, Proposition 4.2]. Alternatively, we can argue duallyalong the same lines as in the proof of Theorem 1.13 using the resolution of theconstant functor Z involving the dual notions, namely the symbolic tensor prod-uct functor and its derived Tor-functor for contravariant Gabriel-Zisman naturalsystems T : (∆ / C ) op → A . (cid:3) Let us now look at several examples to illustrate the broad realm of applica-tions and the necessity for the use of general Gabriel-Zisman natural systems ascohomological coefficient systems in contrast to more specialised coeffcients. TheLeray type spectral sequences constructed in the following sections will then pro-vide useful computational tools in all these frameworks of examples.
Example 1.15 (Thomason (co)homology of categories) . We can interpret Thoma-son (co)homology of categories as introduced by the authors in [15] both in termsof Gabriel-Zisman (co)homology of simplicial sets and via (co)homology of Kanextensions.Let C be a (small) category, A be a complete abelian category with exact prod-ucts and T : ∆ / C → A a (covariant) Thomason natural system. From the generaldiscussions above we see immediately that there are natural isomorphisms H ∗ T h ( C , T ) ∼ = H ∗ ( P C , T ) ∼ = H ∗ GZ ( N ( C ) , T ) , where P = P C : ∆ / C → ∆ is the forgetful functor and by identifying the cate-gories of simplices ∆ / C = ∆ / N ( C ) , where N ( C ) is the nerve of the category C .Here the notion of a Gabriel-Zisman natural system T : ∆ / N ( C ) → A coincides with that of a Thomason natural system as we can readily identify the simplex cat-egory ∆ / C of C with the category of simplices ∆ / N ( C ) over the simplicial nerveof C (see [15]).Dually, if A is a cocomplete abelian category with exact coproducts and givena (contravariant) Thomason natural system T : (∆ / C ) op → A we have naturalisomorphisms H T h ∗ ( C , T ) ∼ = H ∗ ( P C , T ) ∼ = H GZ ∗ ( N ( C ) , T ) . As discussed in detail in [14] and [15], Thomason (co)homology generalises all theother (co)homology theories for small categories in the literature, including Baues-Wirsching and Hochschild-Mitchell (co)homology (compare for example [2, 7, 26,28, 29]). Therefore the functoriality properties of these (co)homology theories aredirect consequences of those of Gabriel-Zisman (co)homology as discussed above.
Example 1.16 (Sheaves on topological spaces) . Let X • be a simplicial set. We havethe geometric realisation functor | | : ∆ op S et → T op given on objects as a coend or colimit as follows (see [12, 5]) | X • | = Z [ n ] X n × ∆ n = colim (cid:18) a [ n ] → [ m ] X m × ∆ n ⇒ a [ n ] X n × ∆ n (cid:19) where ∆ n is the topological standard n -simplex in R n +1 . We have ∆ n = | ∆[ n ] | .It turns out that | X • | is a compactly generated Hausdorff topological space. Let R be a noetherian ring and Shv ( | X • | ) be the abelian category of sheaves of R -modules over | X • | . Let σ ∈ X n i.e., σ ∈ Hom ∆ op S et (∆[ n ] , X • ) . We get an inducedcontinuous map | σ | : ∆ n → | X • | . Let
F ∈
Shv ( | X • | ) be a sheaf on | X • | and assume that the inverse image sheaf | σ | ∗ F is constant on the subset inn( ∆ n ) of inner points of the topological space ∆ n for every simplex σ ∈ X/ ∆ . Let F σ denote the stalk of F at such an inner point.Then F : ∆ /X • → R − Mod , F ( σ ) = F σ is a Gabriel-Zisman (covariant) natural system, which in fact is a sheaf and Gabriel-Zisman cohomology H ∗ GZ ( X • , F ) gives sheaf cohomology H ∗ ( | X • | , F ) of the topo-logical space | X • | . In fact, when starting with a general Gabriel-Zisman (covariant)natural system on ∆ /X • , geometric realisation always produces a sheaf on | X • | and defines a left exact functor from the category NatS GZ of (covariant) naturalsystems to the category Shv ( | X • | ) of sheaves on the topological space | X • | (see[11, Prop. 3.1]). Example 1.17 (Parshin-Beilinson adeles of schemes) . Let X be a noetherian schemeand Qcoh ( X ) denote the abelian category of quasi-coherent O X -modules. Further-more let P ( X ) be the set of points of the scheme X . Let S • ( X ) be the associatedsimplicial set of flags of irreducible closed subschemes of X , ordered by inclusion,given as follows: consider the set of points P ( X ) of X with the partial order ≥ on P ( X ) defined by η ≥ ν if ν ∈ { η } . Then S • ( X ) is the simplicial nerve of thepartially ordered set ( P ( X ) , ≥ ) , with the set of n -simplices S ( X ) n = { ( ν , ν , . . . , ν n ) | ν i ∈ P ( X ); ν i ≥ ν i +1 } and the usual face and degeneracy maps d i and s i for ≤ i ≤ n induced fromthe partially ordered set structure. If X is an affine scheme, the flags of S ( X ) • arejust sequences of prime ideals ordered by inclusions. Beilinson [3] (see also [21]for more details) constructed for any K ⊂ S ( X ) • and any quasi-coherent sheaf F ABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES 13 on X a space of adeles A ( K, F ) which is an abelian group functorial in F . Thenthe groups of local adeles A ( { σ } , F ) , for any simplex σ ∈ S ( X ) • , give rise to aGabriel-Zisman (covariant) natural system by setting (compare [21, 11]) F : ∆ /S ( X ) • → A b, F ( σ ) = A ( { σ } , F ) , which actually is a sheaf and we have that (see [21, Prop. 2.1.4]) A ( K, F ) ⊂ Y σ ∈ K A ( { σ } , F ) . In particular, we can consider the abelian group of n -dimensional adeles of X withcoefficients in F defined as A n ( X, F ) = A ( S ( X ) n , F ) . It turns out that the sequence of groups of global adeles A n ( X, F ) on X gives acosimplicial abelian group A • ( X, F ) and therefore a cochain complex. Its coho-mology, which is the Gabriel-Zisman cohomology H ∗ GZ ( S ( X ) • , F ) for S • ( X ) cal-culates sheaf cohomology i.e., if F is a quasi-coherent O X -module, then we havean isomorphism [21, Thm 4.2.3] H ∗ ( A • ( X, F )) ∼ = H ∗ ( X, F ) . Parshin [27] gave first a definition of adeles for smooth proper algebraic surfacesover a perfect field, which was later extended by Beilinson [3] for arbitrary noe-therian schemes.
Example 1.18 (Buildings of reductive algebraic groups) . Let G be a reductive al-gebraic group over the finite field F q and Rep ( G ) be the category of finite dimen-sional representations of the finite group of F q -rational points G ( F q ) . Associated to G is a simplicial set ∆( G ) • , the combinatorial building of G consisting of inclusionchains in the poset of subgroups of G given by parabolic subgroups. For any sim-plex σ ∈ ∆( G ) • we have a parabolic subgroup P σ ⊂ G . Let R u ( P ) be the unipotentradical of a parabolic subgroup P and R u ( P )( F q ) its group of F q -rational points.Let M ∈ Rep ( G ) , then we obtain a (covariant) Gabriel-Zisman natural system bysetting F : ∆ / ∆( G ) • → Rep ( G ) , F ( σ ) = M R u ( P σ ) and inclusion maps for different simplices. Here M R u ( P σ ) ⊂ M and F turns out tobe again a sheaf (see [11]) and Gabriel-Zisman cohomology H ∗ GZ (∆( X ) • , F ) givesthe cohomology of the building with coefficients being representations as sheaveson the building (compare [6, 32]). Remark 1.19 (Higher categories) . As mentioned above, Gabriel-Zisman cohomol-ogy extends and unifies many notions of cohomology of categories. Recall thatthe factorisation category (also known as the twisted arrow category) F act ( C ) ofa category C has objects the morphisms f : x → y of C and arrows ( h, k ) : f → f ′ ,where f ′ = kf h in C . Then Baues-Wirsching cohomology H ∗ ( C , D ) was de-fined in [2], for natural systems of coefficients D : F act ( C ) → A b . The relationto Thomason cohomology arises from the existence of a functor η C : ∆ / C → F act ( C ) from the category of simplices to the factorisation category of C , see [15]. Weremark that analogous notions will provide extensions of Thomason and Baues-Wirsching cohomologies to: • One can define a category of simplices ∆ / D of a 2-category D , with objects a given by the lax functors a : [ m ] → D and arrows a → b given by morphisms σ : [ m ] → [ n ] of ∆ , where a = bσ . One can also define a factorisation category F act ( D ) , with objects the 1-morphisms f : x → y of D and arrows ( h, k, ξ ) : f → f ′ , where ξ : kf h → f ′ is a 2-morphism of D . Furthermore we can give a natural transformation η D : ∆ / D → F act ( D ) . We can define notions of Thomason and Baues-Wirsching cohomologiesfor 2-categories D , with coefficient systems on ∆ / D and on F act ( D ) re-spectively. • , also known as decomposition spaces [10, 13]. The 2-Segalcondition specifies a particular class of simplicial sets more general thannerves of ordinary categories, which are characterised by the 1-Segal con-dition. It was shown recently, in [4], that a simplicial set X is 2-Segal if andonly if its edgewise subdivision is 1-Segal, and we denote the category de-fined by this edgewise subdivision by F act ( X ) . If X is 1-Segal this agreeswith the definition of the category of factorisations above. We can de-fine notions of Thomason and Baues-Wirsching cohomologies for 2-Segalspaces X , with coefficient systems on the categories ∆ /X and F act ( X ) re-spectively, related once more by a natural transformation η X : ∆ /X → F act ( X ) . There is also an obvious notion of cohomology of ∞ -categories: if we model an ∞ -category by a quasi-category, that is, by an inner-Kan simplicial set, then wecan take its Gabriel-Zisman cohomology. We do not see an analogue of Baues-Wirsching cohomology for ∞ -categories.2. S PECTRAL SEQUENCES FOR G ABRIEL -Z ISMAN ( CO ) HOMOLOGY (Co)homology spectral sequences for maps of simplicial sets.
In this sub-section we will derive Leray type Gabriel-Zisman (co)homology spectral sequencesfor a given map of simplicial sets. In order to do so, we will work first in a moresuitable general categorical setting.Let C and D be small categories, A a complete abelian category and T : C → A be a functor. Now let us assume that we also have a functor u : C → D togetherwith functors P : C → ∆ and Q : D → ∆ such that P = Q ◦ u i.e., we have acommutative diagram of the form C u / / P ❅❅❅❅❅❅❅❅ D Q ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ ∆ inducing a commutative diagram between functor categories, where the respec-tive functors are given by precomposition and right Kan extensions F un ( C , A ) Ran u / / Ran P % % ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ F un ( D , A ) u ∗ o o Ran Q y y tttttttttttttttttttt F un (∆ , A ) Q ∗ tttttttttttttttttttt P ∗ e e ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ It follows immediately from Definition 1.8 and the above that we have an iso-morphism H ∗ ( P, T ) ∼ = H ∗ ( Q, Ran u ( T )) . ABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES 15
From the previous diagram we get now the following Grothendieck compositefunctor spectral sequence [18] (compare also [1, 14, 15]).
Theorem 2.1.
Let C and D be small categories and T : C → A be a functor to a completeabelian category. Let u : C → D be a functor together with functors P : C → ∆ and Q : D → ∆ such that P = Q ◦ u . Then there is a spectral sequence: E p,q ∼ = H p ( Q, Ran qu ( T )) ⇒ H p + q ( P, T ) , which is natural in u and T and where Ran qu ( T ) denotes the q -th right satellite of Ran u ( T ) . Dually, using analogue constructions as just described, we obtain also a homol-ogy version of the above spectral sequence
Theorem 2.2.
Let C and D be small categories and T : C op → A be a functor to acocomplete abelian category. Let u : C → D be a functor together with functors P : C → ∆ and Q : D → ∆ such that P = Q ◦ u . Then there is a spectral sequence: E p,q ∼ = H p ( Q, Lan uq ( T )) ⇒ H p + q ( P, T ) , which is natural in u and T and where Lan uq ( T ) denotes the q -th left satellite of Lan u ( T ) . We will now derive general Leray type spectral sequences for Gabriel-Zisman(co)homology for any map of simplicial sets using the machinery developed above.In special cases, we can in addition also simplify them by using concrete fiber data.Let us first introduce the following general constructions:
Definition 2.3.
Given a map of simplicial sets f : X → Y , the fiber functor F ( − ) : ∆ /Y → ∆ op S et is defined as follows:For each object y : ∆[ n ] → Y of the simplex category ∆ /Y , let F y be the fiber of f over y , which is the simplicial set F y = ∆[ n ] × Y X = { ( σ, x ) ∈ ∆[ n ] × X : y ◦ σ = f ( x ) } , given by the pullback(1) F y ❴✤ ¯ y / / (cid:15) (cid:15) X f (cid:15) (cid:15) ∆[ n ] y / / Y. For each morphism from y : ∆[ n ] → Y to y ′ : ∆[ n ′ ] → Y , given by θ : ∆[ n ] → ∆[ n ′ ] and satisfying y = y ′ ◦ θ , let F θ be the simplicial map given as: θ × Y X : F y → F y ′ , ( σ, x ) ( θ ◦ σ, x ) . Remark 2.4.
Given a simplicial map f : X → Y and a Gabriel-Zisman naturalsystem T : ∆ /X → A we have induced (covariant) natural systems T y on thefibers F y , for each object y ∈ ∆ /Y , defined by T y = T ◦ ∆ / ¯ y : ∆ /F y → ∆ /X → A . For each q ≥ we get functors H qGZ ( F ( − ) , T ( − ) ) : (∆ /Y ) op → A defined on objects by y H qGZ ( F y , T y ) and on morphisms θ from y to y ′ by θ ∗ : H qGZ ( F y ′ , T y ′ ) → H qGZ ( F y , T y ) . since T y = θ ∗ T y ′ . Dually, given a Gabriel-Zisman natural system T : (∆ /X ) op → A we have in-duced (contravariant) natural systems T y , and for each q ≥ get functors H GZq ( F ( − ) , T ( − ) ) : ∆ /Y → A defined on objects by y H GZq ( F y , T y ) and on morphisms θ from y to y ′ by θ ∗ : H GZq ( F y , T y ) → H GZq ( F y ′ , T y ′ ) . We now make the following definition:
Definition 2.5.
Let f : X → Y be a map of simplicial sets and T : ∆ /Y → A a (co-variant) Gabriel-Zisman natural system. The map f is called locally cohomologicallyconstant if for each morphism θ : y → y ′ of the simplex category ∆ /Y the inducedmap in cohomology θ ∗ : H qGZ ( F y ′ , T y ′ ) ∼ = → H qGZ ( F y , T y ) is an isomorphism.Let f : X → Y be a map of simplicial sets and T : ∆ /Y → A a (covariant)Gabriel-Zisman natural system. From the pullback square (1) and functoriality ofGabriel-Zisman cohomology we get an induced map in cohomology H ∗ GZ (∆[ n ] , T ∆[ n ] ) → H ∗ GZ ( F y , ( f ∗ T ) y ) , where for a given simplex y : ∆[ n ] → Y of Y we let T ∆[ n ] : ∆ / ∆[ n ] → ∆ /Y → A bethe restricted Gabriel-Zisman natural system and ( f ∗ T ) y = ( f ∗ T ) ◦ ∆ / ¯ y : ∆ /F y → ∆ /X → A the induced Gabriel-Zisman natural system. We make the followingdefinition: Definition 2.6.
Let f : X → Y be a map of simplicial sets and T : ∆ /Y → A a (co-variant) Gabriel-Zisman natural system. The map f is called locally cohomologicallytrivial if for every simplex y : ∆[ n ] → Y of Y the induced map in cohomology H ∗ GZ (∆[ n ] , T ∆[ n ] ) ∼ = → H ∗ GZ ( F y , ( f ∗ T ) y ) is an isomorphism.The following useful lemma gives an alternative description of the fiber of ageneral map of simplicial sets. Lemma 2.7.
Let f : X → Y be a map of simplicial sets. The simplex category of a fiber F y is naturally isomorphic to the left fiber of ∆ /f : ∆ /X → ∆ /Y over the object y , ∆ /F y ∼ = (∆ /f ) /y. Proof.
An object of the left fiber of ∆ /f over y is just a map σ : (∆ /f )( x ) → y in thecomma category ∆ /Y , for some x ∈ ∆ /X , as in the following diagram: ∆[ m ] x / / (∆ /f )( x ) ❍❍❍❍❍ $ $ ❍❍❍❍❍❍ σ (cid:15) (cid:15) X f (cid:15) (cid:15) ∆[ n ] y / / Y. Such a diagram may alternatively be interpreted as a map ( σ, x ) : ∆[ m ] → ∆[ n ] × Y X , and hence as an object of the category ∆ /F y . ABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES 17
Now a morphism in (∆ /f ) /y is just a map θ : ∆[ m ] → ∆[ m ′ ] which fits into adiagram of the form ∆[ m ] θ ❙❙❙❙❙ ) ) ❙❙❙❙❙ x ( ( σ " " ∆[ m ′ ] x ′ / / σ ′ (cid:15) (cid:15) X f (cid:15) (cid:15) ∆[ n ] y / / Y. This may be interpreted as a morphism θ : ( σ, x ) → ( σ ′ , x ′ ) in ∆ /F y . (cid:3) Now given any map f : X → Y of simplicial sets we can derive a general coho-mology spectral sequence, which compares the Gabriel-Zisman cohomology of X and Y . Theorem 2.8.
Let X and Y be simplicial sets and f : X → Y be a map of simplicialsets. Let A be a complete abelian category with exact products. Given a Gabriel-Zismannatural system T : ∆ /X → A on X , there is a cohomology spectral sequence E p,q ∼ = H pGZ ( Y, ( R q (∆ /f ) ∗ )( T )) ⇒ H p + qGZ ( X, T ) which is natural in f and T and where R q (∆ /f ) ∗ = Ran q ∆ /f is the q -th right satellite ofthe right Kan extension Ran ∆ /f along the induced functor ∆ /f : ∆ /X → ∆ /Y betweenthe simplex categories.Proof. Let X and Y be simplicial sets, f : X → Y be a map of simplicial sets and A be a complete abelian category with exact products. With the categories C = ∆ /X and D = ∆ /Y and the functors P = P X : ∆ /X → ∆ , Q = Q Y : ∆ /Y → ∆ and u = ∆ /f : ∆ /X → ∆ /Y we are exactly in the situation of Theorem 2.1, with P = Q ◦ u and we get the following commutative diagram: F un (∆ /X, A ) Ran ∆ /f / / Ran PX % % ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ F un (∆ /Y, A ) (∆ /f ) ∗ o o Ran QY y y sssssssssssssssssssss F un (∆ , A ) Q ∗ Y sssssssssssssssssssss P ∗ X e e ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ Therefore, Theorem 2.1 gives a spectral sequence of the form E p,q ∼ = H p ( Q Y , Ran qu ( T )) ⇒ H p + q ( P X , T ) , which is natural in u and T .Identifying the above cohomologies of the functors Q Y and P X as Gabriel-Zisman cohomology following Theorem 1.13 we get the desired spectral sequenceof the form E p,q ∼ = H pGZ ( Y, ( R q (∆ /f ) ∗ )( T )) ⇒ H p + qGZ ( X, T ) . and the naturality of the spectral sequence with respect to f and T follows directlyfrom the above identifications. (cid:3) Remark 2.9.
If we start with a Gabriel-Zisman natural system which is actually asheaf (see Remark 1.4), then the above spectral sequence corresponds to the Leray spectral sequence for sheaf cohomology, in fact, if applying the geometric realisa-tion functor as in Example 1.16 we will obtain the classical Leray spectral sequencefor sheaf cohomology of a continuous map of topological spaces.We can identify the E -term of the spectral sequence further by relating thesatellites of the right Kan extension to derived limit data of the fiber of the simpli-cial map f . Corollary 2.10.
Let X and Y be simplicial sets and f : X → Y be a map of simplicialsets. Let A be a complete abelian category with exact products. Let T : ∆ /X → A be aGabriel-Zisman natural system on X . Then there is a cohomology spectral sequence of theform E p,q ∼ = H pGZ ( Y, H qGZ ( − / (∆ /f ) , T ◦ Q ( − ) )) ⇒ H p + qGZ ( X, T ) which is natural in f and T and where H qGZ ( − / (∆ /f ) , T ◦ Q ( − ) ) = lim q − / (∆ /f ) ( T ◦ Q ( − ) ) : ∆ /Y → A . Proof.
For each simplex y of ∆ /Y , let Q ( y ) : y/ (∆ /f ) → ∆ /X be the forgetful func-tor and denote by H qGZ ( y/ (∆ /f ) , T ◦ Q ( y ) ) the derived limit lim q (cid:18) y/ (∆ /f ) Q ( y ) −→ ∆ /X T −→ A (cid:19) . Using [14, Corollary 1.3] allows us to identify the terms in the E -page of the spec-tral sequence in Theorem 2.8 as E p,q ∼ = H pGZ ( Y, H qGZ ( − / (∆ /f ) , T ◦ Q ( − ) )) . and in addition gives us the desired abutment. (cid:3) As a direct consequence, we also have the following general statement for lo-cally cohomologically trivial maps of simplicial sets. This can be seen as a coho-mological analogue of Quillen’s Theorem A (see also [8, 29]) for Gabriel-Zismancohomology
Proposition 2.11.
Let f : X → Y be a map of simplicial sets and T : ∆ /Y → A a(covariant) Gabriel-Zisman natural system. If f is locally cohomologically trivial, then f induces an isomorphism in cohomology: H ∗ GZ ( Y, T ) ∼ = → H ∗ GZ ( X, f ∗ T ) . Proof.
For every simplex y : ∆[ n ] → Y of the simplex category ∆ /Y we have thefollowing commutative diagram F y ❴✤ ¯ y / / (cid:15) (cid:15) X f (cid:15) (cid:15) f / / Y id (cid:15) (cid:15) ∆[ n ] y / / Y id / / Y. The naturality of the spectral sequence of Theorem 2.8 gives a morphism ofspectral sequences E ∗ , ∗ r ( id, T ) → E ∗ , ∗ r ( f, T ) . Because f is locally cohomolog-ical trivial, we get an isomorphism of E -pages i.e., E ∗ , ∗ ( id, T ) ∼ = → E ∗ , ∗ ( f, T ) .Therefore we also get an isomorphism of the abutments, which implies the state-ment. (cid:3) Dually, we can derive a homology spectral sequence computing the Gabriel-Zisman homologies for a simplicial map f : X → Y , which gives the dual versionof Theorem 2.8. ABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES 19
Theorem 2.12.
Let X and Y be simplicial sets and f : X → Y be a map of simplicialsets. Let A be a cocomplete abelian category with exact coproducts. Given a contravariantGabriel-Zisman natural system T : (∆ /X ) op → A on X , there is a homology spectralsequence E p,q ∼ = H GZp ( Y, ( L q ((∆ /f ) op ) ∗ )( T )) ⇒ H GZp + q ( X, T ) which is natural in f and T and where L q ((∆ /f ) op ) ∗ = Lan (∆ /f ) op q is the q -th left satel-lite of Lan (∆ /f ) op , the left Kan extension along the induced functor (∆ /f ) op : (∆ /X ) op → (∆ /Y ) op between the simplex categories.Proof. Let X and Y be simplicial sets and A be a cocomplete abelian category withexact coproducts. Given a map f : X → Y of simplicial sets we have the followingcommutative diagram: F un ((∆ /X ) op , A ) Lan (∆ /f ) op / / colim (∆ /X ) op $ $ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ F un ((∆ /Y ) op , A ) ((∆ /f ) op ) ∗ o o colim (∆ /Y ) op z z tttttttttttttttttttt A c : : tttttttttttttttttttt c d d ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ Here, c denotes the respective constant diagram functors and ((∆ /f ) op ) ∗ is pre-composition with (∆ /f ) op , the induced functor between the simplex categories of X and Y . The other functors in the diagram are the left adjoints of these, givenby the limits colim (∆ /X ) op , colim (∆ /Y ) op and by Lan (∆ /f ) op , which is the left Kanextension along the functor (∆ /f ) op .We obtain a Grothendieck spectral sequence [18] for the derived functors of thecomposite functorcolim (∆ /X ) op ( − ) = colim (∆ /Y ) op Lan (∆ /f ) op ( − ) which can be interpreted as an Andr´e spectral sequence as constructed in general-ity in [14, Section 1.1] (see also [1] and [8]).In our situation here it converges to the homology of the simplex category ∆ /X of the simplicial set X with coefficients being a contravariant Gabriel-Zisman nat-ural system T of F un ((∆ /X ) op , A ) . Therefore, [14, Theorem 1.4] gives a cohomol-ogy spectral sequence of the form: E p,q ∼ = H p ((∆ /Y ) op , ( L q ((∆ /f ) op ) ∗ )( T )) ⇒ H p + q ((∆ /X ) op , T ) where L q ((∆ /f ) op ) ∗ is the q -th left satellite of Lan (∆ /f ) op .Identifying the homologies of the involved simplex categories (∆ /X ) op and (∆ /Y ) op with the Gabriel-Zisman homologies of the given simplicial sets X and Y using Proposition 1.14 finally gives us the homology spectral sequence E p,q ∼ = H GZp ( Y, ( L q ((∆ /f ) op ) ∗ )( T )) ⇒ H GZp + q ( X, T ) . The naturality of the spectral sequence with respect to f and T follows directlyfrom the construction. (cid:3) Again, we can identify the E -term of the spectral sequence by relating thesatellites of the left Kan extension to derived colimit data of the fiber of the simpli-cial map f . Corollary 2.13.
Let X and Y be simplicial sets and f : X → Y be a map of simplicial sets.Let A be a cocomplete abelian category with exact coproducts. Let T : (∆ /X ) op → A be a contravariant Gabriel-Zisman natural system on X . Then there is a homology spectralsequence of the form E p,q ∼ = H GZp ( Y, H GZq ((∆ /f ) / − , T ◦ Q ( − ) )) ⇒ H GZp + q ( X, T ) which is natural in f and T and where H GZq ((∆ /f ) / − , T ◦ Q ( − ) ) = colim (∆ /f ) op / − q ( T ◦ Q op ( − ) ) : (∆ /Y ) op → A . Proof.
For each simplex y of (∆ /Y ) op , let Q op ( y ) : (∆ /f ) op /y → (∆ /X ) op be the for-getful functor and denote by H GZq ((∆ /f ) /y ) , T ◦ Q ( y ) ) the derived colimitcolim q (cid:18) (∆ /f ) op /y Q op ( y ) −→ (∆ /X ) op T −→ A (cid:19) . Using [14, Corollary 1.5] allows us now to identify the terms in the E -page of theabove spectral sequence as E p,q ∼ = H GZp ( Y, H GZq ((∆ /f ) / − , T ◦ Q ( − ) )) . while the spectral sequence converges to the same abutment. (cid:3) Specialisation of coefficient systems and spectral sequences.
In this finalsubsection we will specialise the general coefficient systems in order to identify the E -terms of the (co)homology spectral sequence further. The classical Leray-Serrespectral sequences for Kan fibrations of simplicial sets will appear as a special case.Let us start by introducing some useful special Gabriel-Zisman natural systems inorder to simplify our Leray type spectral sequences in various situations. Definition 2.14.
Let X be a simplicial set and M be a category. A (covariant)Gabriel-Zisman natural system T : ∆ /X → M on X is called invertible or a (covari-ant) local system if it sends all morphism of ∆ /X to isomorphisms of M .Dually, a (contravariant) Gabriel-Zisman natural system T : (∆ /X ) op → M on X is called invertible or a (contravariant) local system if it sends all morphism of (∆ /X ) op to isomorphisms of M .Let X be a simplicial set and M be a category. Let T : ∆ /X → M be a (co-variant) invertible Gabriel-Zisman natural system on X . Then we can define thefunctor T − : (∆ /X ) op → M , whose value on objects is the same as for the functor T and whose value on a morphism α of ∆ /X is T − ( α ) = T ( α ) − . Dually, givena (contravariant) invertible Gabriel-Zisman natural system T : (∆ /X ) op → M on X , we can define similarly the functor T − : ∆ /X → M , whose value on objectsis the same as for the functor T and whose value on a morphism α of (∆ /X ) op is T − ( α ) = T ( α ) − .The following proposition gives an alternative description of Gabriel-Zisman(co)homology for invertible coefficient functors (compare also [12, App. II. 4.4]). Proposition 2.15.
Let X be a simplicial set and A be a complete and cocomplete abeliancategory with exact products and coproducts.Given a (covariant) local system T : ∆ /X → A on X , there is an isomorphism H ∗ GZ ( X, T ) = H ∗ (∆ /X, T ) ∼ = H ∗ ((∆ /X ) op , T − ) , natural in X and T .Dually, given a (contravariant) local system T : (∆ /X ) op → A on X , there is anisomorphism H GZ ∗ ( X, T ) = H ∗ ((∆ /X ) op , T ) ∼ = H ∗ (∆ /X, T − ) , natural in X and T . ABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES 21
Proof.
In the case of homology, this follows verbatim as in the proof of the proposi-tion in [12, App. II.4.4] by interpreting Gabriel-Zisman homology of simplicial setsas Thomason homology of small categories applied to the respective categories ofsimplicies (see [15]). The case for cohomology follows analogous from the dualarguments using ∆ /X instead of (∆ /X ) op . (cid:3) Now let f : X → Y be a map of simplicial sets, which is locally cohomologicallyconstant and let T : ∆ /X → A be a Gabriel-Zisman natural system. Then weobtain an induced covariant functor for each q ≥ H qGZ ( F ( − ) , T ( − ) ) − : ∆ /Y → A defined on objects by y H qGZ ( F y , T y ) and which maps morphisms θ from y to y ′ in ∆ /X to the induced inverse mor-phism ( θ ∗ ) − : H qGZ ( F y , T y ) → H qGZ ( F y ′ , T y ′ ) . This allows us to derive the following cohomology spectral sequence for locallycohomologically constant maps of simplicial sets
Proposition 2.16.
Let X and Y be simplicial sets and f : X → Y be a map of simplicialsets, which is locally cohomologically constant. Let A be a complete abelian category withexact products and T : ∆ /X → A be a Gabriel-Zisman natural system on X . Then thereis a cohomology spectral sequence of the form E p,q ∼ = H pGZ ( Y, H qGZ ( F ( − ) , T ( − ) ) − ) ⇒ H p + qGZ ( X, T ) which is natural in f and T .Proof. This follows from the identification of the E -page of the general Leray typespectral sequence in Corollary 2.10 for the particular case of a given locally coho-mologically constant map of simplicial sets using the natural isomorphism H qGZ ( − / (∆ /f ) , T ◦ Q ( − ) ) ∼ = H qGZ ( F ( − ) , T ( − ) ) − ) . The abutment of the spectral sequence does not change and it is again natural in f and T . (cid:3) Finally, we will derive the Leray-Serre spectral sequences of a Kan fibrationof simplicial sets in cohomology and homology with local coefficients from ourgeneral setting (compare also [12, App. II.4.4], [9]).Let f : X → Y first be any map of simplicial sets. Furthermore, let A be acomplete abelian category with exact products and T : ∆ /X → A a covariantlocal system on X . Then Lemma 2.7 and Proposition 2.15 imply ( R n (∆ /f ) ∗ )( T − )( y ) = H n ( F y , T | F y ) , where T | F y is given as the composition ∆ /F y ∆ /pr −→ ∆ /X T −→ A If in addition f : X → Y is a Kan fibration of simplicial sets, then ( R n (∆ /f ) ∗ )( T − ) : y H n ( F y , T | F y ) induces a covariant local system H qGZ ( f, T ) : ∆ /X → A . Dually, we can makesimilar considerations starting with a cocomplete abelian category with exact co-products and a contravariant local system T : (∆ /X ) op → A on X . We then obtaina contravariant local system H GZq ( f, T ) : (∆ /X ) op → A induced by ( L n (∆ /f ) ∗ )( T − ) : y H n ( F y , T | F y ) , where T | F y is now given as the composition (∆ /F y ) op (∆ /pr ) op −→ (∆ /X ) op T −→ A . The following follows now from Theorem 2.8 and Theorem 2.12 and recovers theLeray-Serre spectral sequence of a Kan fibration (compare also [9]).
Proposition 2.17 (Leray-Serre spectral sequence) . Let f : X → Y be a map of simpli-cial sets, which is a Kan fibration. Let A be a complete abelian category with exact productsand T : ∆ /X → A be a covariant local system on X . Then there is a cohomology spectralsequence of the form E p,q ∼ = H pGZ ( Y, H qGZ ( f, T )) ⇒ H p + qGZ ( X, T ) which is natural in f and T .Dually, let A be a cocomplete abelian category with exact coproducts and T : (∆ /X ) op → A be a contravariant local system on X . Then there is a homology spectral sequence of theform E p,q ∼ = H GZp ( Y, H GZq ( f, T )) ⇒ H GZp + q ( X, T ) which is natural in f and T . Let us finally remark that when taking geometric realisation again we will re-cover the Leray-Serre spectral sequence for fibrations of topological spaces.R
EFERENCES[1] M. Andr´e, Limites et fibr´es,
C. R. Acad. Sci. Paris, S´er. A. (1965), 756–759.[2] H. J. Baues, G. Wirsching, Cohomology of small categories,
J. Pure Appl. Algebra (1985), 187–211.[3] A. A. Beilinson, Residues and Adeles, Funct. Anal. Pril. (1) (1980), 44–45; English transl. in: Funct. Anal. Appl. (1) (1980), 34–35.[4] J. E. Bergner, A. M. Osorno, V. Ozornova, M. Rovelli, C. I. Scheimbauer, The edgewise subdivisioncriterion for 2-Segal objects, Proc. Amer. Math. Soc. (2020), 71–82.[5] A. K. Bousfield, D. M. Kan,
Homotopy Limits, Completions and Localizations , Lecture Notes in Math. , Springer-Verlag, Berlin 1972.[6] K. S. Brown,
Buildings , Springer Verlag, Berlin-Heidelberg-New York 1989.[7] C. Ciblis, M. J. Redondo, Cartan-Leray spectral sequence for Galois coverings of linear categories,
J. Algebra (2005), 310–325.[8] D.-C. Cisinski, Images directes cohomologiques dans les cat´egories de mod`eles,
Ann. Math. BlaisePascal (2003), no. 2, 195–244.[9] A. Dress, Zur Spektralsequenz von Faserungen, Invent. Math. (1967), 172–178.[10] T. Dyckerhoff, M. Kapranov, Higher Segal Spaces , Lecture Notes in Math. , Springer, Berlin2019.[11] T. Fimmel, Verdier Duality for Systems of Coefficients over Simplicial Sets,
Math. Nachr. (1998),51–122.[12] P. Gabriel, M. Zisman,
Calculus of fractions and homotopy theory , Ergebnisse der Mathematik undihrer Grenzgebiete, Band 35 Springer-Verlag New York, Inc., New York 1967.[13] I. G´alvez-Carrillo, J. Kock, A. Tonks, Decomposition spaces, incidence algebras and M¨obius inver-sion I: basic theory,
Adv. Math. (2018), 952–1015.[14] I. G´alvez-Carrillo, F. Neumann, A. Tonks, Andr´e spectral sequences for Baues-Wirsching coho-mology of categories,
J. Pure Appl. Algebra (2012), 2549–2561.[15] I. G´alvez-Carrillo, F. Neumann, A. Tonks, Thomason cohomology of categories,
J. Pure Appl. Alge-bra (2013), 2163–2179.[16] S. I. Gelfand, Yu. I. Manin,
Methods of Homological Algebra , Second Edition, Springer Monographsin Math., Springer, Berlin 2003.[17] P. G. Goerss, J. F. Jardine,
Simplicial Homotopy Theory , Progress in Math. Vol. , Birkh¨auser Verlag,Basel 1999.[18] A. Grothendieck, Sur quelques points d’alg`ebre homologique,
Tˆohoku Math. J. (2) (1957), 119–221.[19] A. Grothendieck, Revˆetements ´etales et groupe fondamental. Fasc. II: Expos´es 6, 8 `a 11. S´eminairede G´eom´etrie Alg´ebrique (SGA1), 1960/61. Troisi`eme ´edition, corrig´ee Publ. Institut des Hautes´Etudes Scientifiques , Paris 1963.[20] A. Huber, On the Parshin-Beilinson adeles for schemes,
Abh. Math. Sem. Univ. Hamburg (1991),249–273. ABRIEL-ZISMAN COHOMOLOGY AND SPECTRAL SEQUENCES 23 [21] A. A. Husainov, Homological dimension theory of small categories,
J. Math. Sci. , Vol. , No. 1,(2002), 2273–2321.[22] A. A. Husainov, Homology Groups of Cubical Sets,
Appl. Cat. Struct. , (2019), 199–216.[23] L. Illusie, Complexe Cotangent et D´eformations II , Lecture Notes in Math. , Springer-Verlag, Berlin1972.[24] D. M. Latch, The uniqueness of homology for the category of small categories,
J. Pure Appl. Algebra (1977), 221–237.[25] S. MacLane, Categories for the Working Mathematician , Second Edition, Graduate Texts in Math. .Springer-Verlag, New York 1998.[26] B. Mitchell, Rings with several objects, Advances in Math. (1972), 1–161.[27] A. N. Parshin, On the arithmetic of two-dimensional schemes I, repartitions and residues, Izv.Akad. Nauk SSSR Ser. Mat. (4) (1976), 736–773; English transl. in: Math. USSR Izv. (4) (1976),695–729.[28] T. Pirashvili, M. J. Redondo, Cohomology of the Grothendieck construction, Manuscr. Math. (2006), 151–162.[29] D. Quillen, Higher algebraic K -theory. I. Algebraic K -theory, I: Higher K -theories (Proc. Conf.,Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85–147. Lecture Notes in Math. , Springer,Berlin 1973.[30] B. Richter, From Categories to Homotopy Theory , Cambridge Studies in Advanced Mathematics ,Cambridge University Press, Cambridge 2020.[31] J. E. Roos, Sur les foncteurs d´eriv´es de lim . Applications,
C. R. Acad. Sci. Paris, S´er. A. (1961),3702–3704.[32] P. Schneider, U. Stuhler, Representation theory and sheaves on the Bruhat-Tits building, P ubl.Math. IH ´ES, Vol. (1997), 97–191.[33] C. Weibel, Homotopy ends and Thomason model categories, Selecta Mathematica , New ser. (2001), 533–564.D EPARTAMENT DE M ATEM ` ATIQUES , U
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