aa r X i v : . [ m a t h . A T ] F e b HOMOLOGY OF CATEGORIES VIA POLYGRAPHICRESOLUTIONS
LÉONARD GUETTA
Abstract.
In this paper, we prove that the polygraphic homology ofa small category, defined in terms of polygraphic resolutions in the cat-egory ω Cat of strict ω -categories, is naturally isomorphic to the ho-mology of its nerve, thereby extending a result of Lafont and Métayer.Along the way, we investigate homotopy colimits with respect to theFolk model structure and deduce a theorem which formally resemblesQuillen’s Theorem A. Introduction
In [Str87], Street defines a nerve functor N ω : ω Cat → b ∆from the category of strict ω -categories (that we shall simply call ω -cat-egories ) to the category of simplicial sets. This functor can be used totransfer the homotopy theory of simplicial sets to ω -categories as it is donein the articles [AM14, AM18, Gag18, Ara19, AM20a, AM20c, AM20b]. Inparticular, we can set the following definition: Definition.
Let C be an ω -category and k ∈ N . The k -th homology group H k ( C ) of C is the k -th homology group of its nerve N ω ( C ).On the other hand, in [Mét03] Métayer defines other homological invari-ants for ω -categories, which we call here polygraphic homology groups . Thedefinition of these homology groups is based on the notion of free ω -cate-gory on a polygraph , also known as free ω -category on a computad , whichare ω -categories obtained recursively from the empty ω -category by freelyadjoining cells. From now on, we will simply say free ω -category .Métayer observed [Mét03, Definition 4.1 and Proposition 4.3] that every ω -category C admits a polygraphic resolution , that is an arrow u : P → C of ω Cat , such that P is a free ω -category and u satisfies some propertiesbearing formal similarities with trivial fibrations of topological spaces (or ofsimplicial sets). Moreover, every free ω -category P can be “linearized” to achain complex λ ( P ) and Métayer proved [Mét03, Theorem 6.1] that given P → C and P ′ → C two polygraphic resolutions of the same ω -category,the chain complexes λ ( P ) and λ ( P ′ ) have the same homology groups. Thisleads to the following definition: Date : February 24, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Homology, Category, Polygraph, Computad.
Definition.
Let C be an ω -category and k ∈ N . The k -th polygraphichomology group H pol k ( C ) of C is the k -th homology group of λ ( P ) for anypolygraphic resolution P → C .In this article, we prove the following theorem: Theorem 1.
Let C be a small category. For every k ∈ N , we have H k ( C ) ≃ H pol k ( C ) . For the statement of this theorem to make sense, we have to consider smallcategories as particular cases of ω -categories. Namely, a (small) categorycan be defined as an ω -category with only trivial cells in dimension greaterthan 1. Beware that this last property doesn’t imply that the previoustheorem is trivial: given P → C , a polygraphic resolution of a small category C , P need not have only trivial cells in dimension greater than 1.The restriction of the previous theorem to the case of monoids seen assmall categories is exactly Corollary 3 of [LM09, Section 3.4]. As such,Theorem 1 is only a small generalization of Lafont and Métayer’s result.However, the novelty lies in the new proof we give, which is more conceptualthan the one in loc. cit. Note also that the actual result we shall obtain in this article (Theorem8.3) is more precise than Theorem 1. The first reason is that the homol-ogy of an ω -category (polygraphic or of the nerve) will be considered as achain complex up to quasi-isomorphism and not only a sequence of abeliangroups. The second and more important reason is that we will prove thatthe polygraphic homology and homology of the nerve of a small category are naturally isomorphic and even explicitly construct the natural isomorphism.This last point was not addressed at all in [LM09].We shall now give a sketch of the proof of Theorem 1. It is slightlysimpler than the proof of Theorem 8.3 but has the same key ingredients.The simplification mainly results from avoiding questions of naturality.The first step is to give a more abstract definition of the polygraphichomology. By a variation of the Dold–Kan equivalence (see for example[Bou90]), the category ω Cat ( Ab ) of ω -categories internal to abelian groupsis equivalent to the category Ch ≥ of chain complexes in non-negative de-gree. Thus, we have a forgetful functor Ch ≥ ≃ ω Cat ( Ab ) → ω Cat , which has a left adjoint λ : ω Cat → Ch ≥ . Moreover, when X is a free ω -category, λ ( X ) is exactly the linearization of X considered in the definitionof polygraphic homology by Métayer. Now, ω Cat admits a model structure,commonly referred to as the
Folk model structure [LMW10], with the equiv-alences of ω -categories (a generalization of the usual notion of equivalenceof categories) as weak equivalences and the free ω -categories as cofibrantobjects [Mét08]. As it turns out, if we equip ω Cat with the Folk modelstructure and Ch ≥ with the projective model structure, then λ is a leftQuillen functor and hence admits a left derived functor L λ : Ho( ω Cat ) → Ho( Ch ≥ ) . OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 3
The polygraphic homology groups of an ω -category X are exactly the ho-mology groups of L λ ( X ). We shall now simply write H pol ( X ) := L λ ( X ).Recall now that for every (small) category C , we have a canonical iso-morphism colim c ∈ C C/c ≃ C, where C/c is the slice over c . The realization that this colimit is a homotopycolimit with respect to the weak equivalences of the Folk model structure(Theorem 7.10) is at the origin of this paper. Since L λ is the left derivedfunctor of a left Quillen functor, it commutes with homotopy colimits. Inparticular, we have: H pol ( C ) ≃ hocolim c ∈ C H pol ( C/c ) . Then, we can show that the polygraphic homology of a small category witha final object is isomorphic to (the homology of) Z concentrated in degree 0(Lemma 6.3 and Proposition 6.6). Hence, we have H pol ( C ) ≃ hocolim c ∈ C Z . We conclude by remarking that the right-hand side of the previous equationis nothing but the homology of the nerve of C (see for example [GZ67,Appendix II, Proposition 3.3] or [Qui73, Section 1]). Note that when C is a monoid M , the category of functors M → Ch ≥ is isomorphic to thecategory Ch ≥ ( M ) of chain complexes of left Z M -modules and the colim M functor can be identified with the functor O Z M Z : Ch ≥ ( M ) → Ch ≥ . Hence, in that case we also recover the definition of homology of a monoidin terms of Tor functors.Let us end this introduction by mentioning that this paper is part of anongoing program carried out by the author, which aims at understandingfor which ω -categories C the following holds: H pol k ( C ) ≃ H k ( C ) for every k ≥ . Theorem 1 may lead us to think that all ω -categories satisfy this propertybut a counter-example discovered by Ara and Maltsiniotis shows that it isnot the case: Let C be the commutative monoid ( N , +) considered as a2-category with only one object and no non-trivial cells in dimension 1.This 2-category is free (as an ω -category) and a quick computation showsthat H pol k ( C ) ≃ ( Z for k = 0 ,
20 otherwise.But, as proved in [Ara19, Theorem 4.9 and Example 4.10], the nerve of C is a K ( Z ,
2) which has non-trivial homology groups in arbitrarily high evendimension. The description of polygraphic homology as a left derived functor has been around inthe folklore for quite some time and I claim no originality for this result. For example, itwill appear in [ABG + ]. LÉONARD GUETTA Generalities on homotopy colimits
The goal of this section is to provide a short summary on homotopy col-imits. The reader familiar with the subject may skip it and refer to it ifneeded. A localizer is a pair ( C , W ) where C is a category and W is a class ofarrows of C , which we usually refer to as the weak equivalences. We denoteby Ho( C ), the localization of C with respect to W and by γ : C →
Ho( C )the localization functor [GZ67, 1.1]. Recall the universal property of thelocalization: for every category D , the functor induced by pre-composition γ ∗ : Hom(Ho( C ) , D ) → Hom( C , D )is fully faithful and its essential image consists of the functors F : C → D that send the morphisms of W to isomorphisms of D .We shall always consider that C and Ho( C ) have the same class of objectsand implicitly use the equality γ ( X ) = X for every object X of C . Let ( C , W ) and ( C ′ , W ′ ) be two localizers and F : C → C ′ be a functor. If F preserves weak equivalences , i.e. F ( W ) ⊆ W ′ , then the universal propertyof the localization implies that there is a canonical functor F : Ho( C ) → Ho( C ′ )such that the square C C ′ Ho( C ) Ho( C ′ ) . Fγ γ ′ F is commutative. Remark 1.2.1.
Since we always consider that localization functors are theidentity on objects, we have the equality F ( X ) = F ( X )for every object X of C . Let ( C , W ) and ( C ′ , W ′ ) be two localizers. A functor F : C → C ′ is totally left derivable when there exists a functor L F : Ho( C ) → Ho( C ′ )and a natural transformation α : L F ◦ γ ⇒ γ ′ ◦ F that makes L F the right Kan extension of γ ′ ◦ F along γ : C C ′ Ho( C ) Ho( C ′ ) . Fγ γ ′ L Fα OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 5
When this right Kan extension is absolute , we say that F is absolutely totallyleft derivable .Note that when a functor F is totally left derivable, the pair ( L F, α ) isunique up to a unique natural isomorphism and thus we shall refer to L F as the total left derived functor of F .The notion of (absolute) total right derivable functor is defined duallyand the notation R F is used. Remark 1.3.1. If F : C → C ′ preserves weak equivalences, then it followsfrom the universal property of the localization that F is absolutely totallyleft and right derivable and L F ≃ R F ≃ F . Let ( C , W ) be a localizer and A be a small category. We denote by C A the category of functors from A to C and natural transformations betweenthem. An arrow α : d → d ′ of C A is a pointwise weak equivalence when α a : d ( a ) → d ′ ( a ) belongs to W for every a ∈ A . We denote by W A the classof pointwise weak equivalences. This defines a localizer ( C A , W A ).Let k : C → C A be the diagonal functor, i.e. for an object X of C , k ( X ) : A → C is theconstant functor with value X . This functor preserves weak equivalences,whence a functor k : Ho( C ) → Ho( C A ) . Definition 1.5.
A localizer ( C , W ) has homotopy colimits when for everysmall category A , the functor k : Ho( C ) → Ho( C A )has a left adjoint. When a localizer ( C , W ) has homotopy colimits, we denote byhocolim A : Ho( C ) → Ho( C A )the left adjoint of k : Ho( C ) → Ho( C A ). For an object d of C A , the objecthocolim A ( d )of Ho( C ) is the homotopy colimit of d . For consistency, we also use thenotation hocolim a ∈ A d ( a ) . Note that when C has colimits and ( C , W ) has homotopy colimits, it fol-lows from Remark 1.3.1 and the dual of [Gon12, Theorem 3.4] that hocolim A is the total left derived functor of colim A . In particular, for every functor d : A → C , there is a canonical arrow of Ho( C )hocolim A ( d ) → colim A ( d ) . Let ( C , W ) and ( C ′ , W ′ ) be two localizers and F : C → C ′ be a functorthat preserves weak equivalences. For every small category A , the functorinduced by post-composition, which we abusively denote by F : C A → C ′ A , LÉONARD GUETTA again preserves weak equivalences and we have a commutative squareHo( C A ) Ho( C ′ A )Ho( C ) Ho( C ′ ) . Fk F k
Suppose now that ( C , W ) and ( C ′ , W ′ ) have homotopy colimits. Using theunit and co-unit of the adjunctions hocolim A ⊣ k ,Ho( C A ) Ho( C ′ A ) Ho( C ′ )Ho( C A ) Ho( C ) Ho( C ′ ) , F hocolim A hocolim A id k F k id η ǫ we obtain a natural transformationhocolim A ◦ F ⇒ F ◦ hocolim A . Hence, for every object d : A → C of C A , a canonical maphocolim A ( F ( d )) → F (hocolim A ( d )) . Homotopy colimits in combinatorial model categories
In this section, we recall a few useful results on homotopy colimits in thecontext of model categories. As for the previous section, it might be skippedat first reading and referred to when needed. We suppose however that thereader is familiar with the basics of model category theory. Recall that acombinatorial model category is a cofibrantly generated model category suchthat the underlying category is locally presentable.
Let ( C , W , Cof , Fib) and ( C ′ , W ′ , Cof ′ , Fib ′ ) be two model categoriesand F : C → C ′ be a functor. Recall that if F is a left Quillen functor (i.e.the left adjoint in a Quillen adjunction), then F is absolutely totally leftderivable and for every cofibrant object X of C , the canonical arrow L F ( X ) → F ( X )is an isomorphism of Ho( C ′ ). Remark 2.1.1.
Note that the definition of totally left derivable functoronly depends on the weak equivalences. In particular, if there are severalmodel structures for which a functor F is left Quillen, then for any cofibrantobject X of any such model structure, the canonical map L F ( X ) → F ( X )is an isomorphism. Proposition 2.2.
Let ( C , W , Cof , Fib) be a combinatorial model category.For every small category A , there exists:(1) A model structure on C A , called the projective model structure , withthe pointwise weak equivalences and the pointwise fibrations as weakequivalences and fibrations respectively. OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 7 (2) A model structure on C A , called the injective model structure , withthe pointwise weak equivalences and the pointwise cofibrations asweak equivalences and cofibrations respectively.Proof. See [Lur09, Proposition A.2.8.2] (cid:3)
Proposition 2.3.
Let ( C , W , Cof , Fib) be a combinatorial model categoryand A a small category. The adjunction colim A : C A C : k is a Quillen adjunction with respect to the projective model structure on C A .Proof. By definition of the projective model structure, k preserve weakequivalences and fibrations. (cid:3) We deduce from the previous proposition that if a localizer ( C , W ) canbe extended to a combinatorial model category ( C , W , Cof , Fib), then it hashomotopy colimits and hocolim A ≃ L colim A . Since colim A is left Quillen with respect to the projective model structure,it is particularly interesting to detect the cofibrant objects of this modelstructure in order to compute homotopy colimits. This is the goal of theparagraph below and the lemma that follows. Let C be a category with coproducts and A a small category. For everyobject X of C and every object a of A , we define X ⊗ a as the functor X ⊗ a : A → C b a Hom A ( a,b ) X. For every object a of A , this gives rise to a functor ⊗ a : C → C A X X ⊗ a. Lemma 2.6.
Let C = ( C , W , Cof , Fib) be a combinatorial model categoryand A a small category. For every object a of A and every cofibration f : X → Y of C , the arrow f ⊗ a : X ⊗ a → Y ⊗ a is a cofibration of the projective model structure on C A .Proof. We leave it to the reader to check that the functor ⊗ a is left adjointto ev a : C A → C F F ( a ) . Let α be a fibration of the projective model structure on C A . By definition, f has the left lifting property with respect to ev a ( α ). Hence, by adjunction, f ⊗ a has the left lifting property with respect to α , which is what we neededto prove. (cid:3) LÉONARD GUETTA
Let ( C , W , Cof , Fib) and ( C ′ , W ′ , Cof ′ , Fib ′ ) be two combinatorial modelcategories and let F : C → C ′ be a left Quillen functor. For every smallcategory A , the functor induced by post-composition F : C A → C ′ A is left Quillen both with respect to the projective model structure and the in-jective model structure. In particular, all arrows of the commutative square C A C ′ A C C ′ Fk F k are left Quillen functors when C A and C ′ A are equipped with the injectivemodel structure. Hence, by composition of left Quillen functors, we obtaina commutative square up to a canonical isomorphismHo( C A ) Ho( C ′ A )Ho( C ) Ho( C ′ ) . L Fk L F k ≃ In a similar fashion as in Paragraph 1.7, we obtain a natural transformationhocolim A ◦ L F ⇒ L F ◦ hocolim A . The next proposition tells us that left Quillen functors are “homotopy co-continuous”.
Proposition 2.8.
Let ( C , W , Cof , Fib) and ( C ′ , W ′ , Cof ′ , Fib ′ ) be two com-binatorial model categories, F : C → C ′ be a left Quillen functor and A be asmall category. For every object d : A → C of C A , the canonical map hocolim A ( L F ( d )) → L F (hocolim A ( d )) is an isomorphism of Ho( C ′ ) .Proof. We use here the projective model structures on C A and C ′ A . Sinceevery object d of C A is isomorphic in Ho( C A ) to a cofibrant one, it sufficesto show that the maphocolim A ( L F ( d )) → L F (hocolim A ( d ))is an isomorphism when d is cofibrant. But in this case, the previous mapcan be identified with the image of the mapcolim A ( F ( d )) → F (colim A ( d ))in the localization of C ′ A . Since F is a left adjoint, this map is indeed anisomorphism. (cid:3) OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 9 ω -categories We denote by ω Cat the category of (small) strict ω -categories and strict ω -functors. Since we shall never deal with non-strict ω -categories, weomit the word “strict” and simply say ω -categories and ω -functors. For an ω -category C , we denote by C n the set of n -cells of C . For x ∈ C n , we referto the integer n as the dimension of x .For an n -cell x with n > s ( x ) and t ( x ) are respectively the sourceand target of x (which are ( n k
When n = 1, we have a canonical functor1 Cat → Cat from the category of 1-categories to the category of small categories, whichsimply forgets the k -cells for k >
1. Since this functor is an isomorphismof categories, we usually identify the categories
Cat and 1
Cat and considerthat the terms “1-category” and “(small) category” are synonyms.
Let n ∈ N . The functor ω Cat → Set C C n is representable and we define the n -globe D n as the ω -category representingthis functor. ( D n is in fact an n -category.) For an n -cell x of an ω -category C , we denote by h x i : D n → C the associated morphism of ω Cat . Here are some pictures of D n in lowdimension: D = • , D = • • , D = • • , D = • • ⇛ . For an ω -category C , let us denote by Par n ( C ) the set of pair of parallel n -cells of C . The functor ω Cat → Set C Par n ( C )is representable and we define the n -sphere S n as the ω -category representingthis functor. ( S n is in fact an n -category.) If x and y are parallel n -cells of C , we denote by h x, y i : S n → C the associated morphism of ω Cat . Here are some pictures of S n in lowdimension: S = • • , S = • • , S = • • . Suppose now that n >
0, and let C be an ω -category. The canonical mor-phism C n → Par n − ( C ) x ( s ( x ) , t ( x )) , is natural in C . Hence, a canonical morphism i n : S n − → D n . OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 11
We also define S − to be the empty ω -category (which is the initial objectof ω Cat ), and i to be the unique morphism i : S − → D . A basis of an ω -category C is a graded set of cells of C Σ = (Σ n ⊆ C n ) n ∈ N such that for every n ∈ N , the commutative square a x ∈ Σ n S n − C ≤ n − a x ∈ Σ n D n C ≤ n , a x ∈ Σ n i n h s ( x ) ,t ( x ) ih x i where the anonymous arrow is the canonical inclusion, is a pushout square.That is, C ≤ n is obtained from C ≤ n − by freely adjoining the n -cells thatbelongs to Σ n . An ω -category is free when it has a basis. Note that afree ω -category has a unique basis (see [Mak05, Section 4, Proposition 8.3]).This allows us to speak of the basis of a free ω -category. Cells that belongto Σ n are referred to as generating n -cells of C .Recall from [Gue20] the following definition: Definition 3.6. An ω -functor f : C → D is a discrete Conduché ω -functor if for every n ≥
0, for every n -cell x of C and for every pair ( y , y ) of k -composable n -cells of D with k < n such that f ( x ) = y ∗ k y , there exists a unique pair ( x , x ) of k -composable n -cells of C such that(1) x = x ∗ k x (2) f ( x ) = y and f ( x ) = y . Lemma 3.7.
Let C ′ CD ′ D uf ′ fv be a pullback square in ω Cat . If f is a discrete Conduché ω -functor, thenso is f ′ .Proof. Left to the reader. See [Gue20, remark 4.5]. (cid:3)
Proposition 3.8.
Let f : C → D be a discrete Conduché ω -functor. If D is a free ω -category, then so is C .More precisely, if we denote by Σ Dn the set of generating n -cells of D , thenthe set of generating n -cells of C is Σ Cn = { x ∈ C n | f ( x ) ∈ Σ Dn } . Proof.
This is Theorem 5.12(1) from [Gue20]. (cid:3)
Let C be an ω -category. We define the equivalence relation ∼ ω on theset C n by co-induction on n ∈ N . Let x, y ∈ C n , then x ∼ ω y when:- x and y are parallel,- there exist r, s ∈ C n +1 such that r : x → y , s : y → x , r ∗ n s ∼ ω y and s ∗ n r ∼ ω x . For details on this definition and the proof that it is an equivalence relation,see [LMW10, section 4.2]. An ω -functor f : C → D is an equivalence of ω -categories when:- for every y ∈ D , there exists a x ∈ C such that f ( x ) ∼ ω y, - for every x, y ∈ C n that are parallel and every β ∈ D n +1 such that β : f ( x ) → f ( y ) , there exists α ∈ C n +1 such that α : x → y and f ( α ) ∼ ω β. Theorem 3.11.
There exists a combinatorial model structure on ω Cat whose weak equivalences are the equivalences of ω -categories, and such thatthe set { i n : S n − → D n | n ∈ N } is a set of generating cofibrations.Proof. This is the main result of [LMW10]. (cid:3)
We refer to the model structure of the previous theorem as the
Folkmodel structure on ω Cat . Data of this model structure will often be referredto by using the adjective folk, e.g. folk cofibration . From now on, unless oth-erwise explicitly specified, we will always consider that ω Cat is equippedwith this model structure. In particular, Ho( ω Cat ) will always be the lo-calization of ω Cat with respect to the class of equivalences of ω -categories. Proposition 3.13. An ω -category is folk cofibrant if and only if it is free.Proof. The fact that every free ω -category is cofibrant follows immediatelyfrom the fact that the i n : S n − → D n are cofibrations and that every ω -category C is the colimit of the canonical diagram ∅ = C ≤− → C ≤ → · · · → C ≤ n → C ≤ n +1 · · · For the converse, see [Mét08]. (cid:3)
OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 13 Polygraphic homology
Let C be an ω -category. We define a chain complex in non-negativedegree λ ( C ) in the following way:- for n ∈ N , λ ( C ) n is the abelian group obtained by quotienting thefree abelian group Z C n by the congruence generated by the relations x ∗ k y ∼ x + y for all x, y ∈ C n that are k -composable,- the differential ∂ : λ ( C ) n → λ ( C ) n − is induced by the map Z C n → Z C n − x ∈ C n t ( x ) − s ( x ) . The axioms of ω -categories imply that ∂ ◦ ∂ = 0. Now let f : C → D be an ω -functor. The map Z C n → Z D n x ∈ C n f ( x )induces a map λ ( f ) n : λ ( C ) n → λ ( D ) n . Since f commutes with source and target, we obtain a morphism of chaincomplexes ( λ ( f ) n ) n ∈ N . This defines a functor λ : ω Cat → Ch ≥ , where Ch ≥ is the category of chain complexes in non-negative degree, whichwe call the abelianization functor . Lemma 4.2.
Let C be a free ω -category and let Σ = (Σ n ) n N be its basis.Then for every n ∈ N , λ ( C ) n is isomorphic to the free abelian group Z Σ n .Proof. Let G be an abelian group. For every n ∈ N , we define an n -category B n G with:- ( B n G ) k is a singleton set for every k < n ,- ( B n G ) n = G - for all x and y in G and i < n , x ∗ i y := x + y. It is straightforward to check that this defines an n -category. Note that when n = 1, the previous definition would still make sense without the hypothesisthat G be abelian, but for n ≥ n = 0, we only needed that G was a set.This defines a functor B n : Ab → n Cat G B n G, which is easily seen to be right adjoint to the functor n Cat → Ab C λ ( C ) n . Now, if C is an ω -category then λ ( C ≤ n ) n = λ ( C ) n and if C is free with basisΣ = (Σ n ) n ∈ N , then for every abelian group G there is a natural isomorphismHom n Cat ( C ≤ n , B n G ) ≃ Hom
Set (Σ n , | G | ) , where | G | is the underlying set of G . Altogether, we haveHom Ab ( λ ( C ) n , G ) ≃ Hom Ab ( λ ( C ≤ n ) n , G ) ≃ Hom n Cat ( C ≤ n , B n G ) ≃ Hom
Set (Σ n , | G | ) ≃ Hom Ab ( Z Σ n , G ) . (cid:3) Lemma 4.3.
The functor λ is a left adjoint.Proof. The category of chain complexes is equivalent to the category ω Cat ( Ab )of ω -categories internal to abelian groups (see [Bou90, Theorem 3.3]) andwith this identification, the functor λ : ω Cat → ω Cat ( Ab ) is nothing butthe left adjoint of the canonical forgetful functor ω Cat ( Ab ) → ω Cat . (cid:3) Let u, v : C → D be two ω -functors. An oplax transformation α from u to v consists of the following data:- for every 0-cell x of C , a 1-cell of Dα x : u ( x ) → v ( x ) , - for every n -cell of x of C with n >
0, an ( n + 1)-cell of Dα x : α t n − ( x ) ∗ n − · · · ∗ α t ( x ) ∗ u ( x ) → v ( x ) ∗ α s ( x ) ∗ · · · ∗ n − α s n − ( x ) subject to the following axioms:(1) for every n -cell x , α x = 1 α x , (2) for all 0 ≤ k < n , for all n -cells x and y that are k -composable, α x ∗ k y = (cid:16) v ( t k +1 ( x )) ∗ α s ( x ) ∗ · · · ∗ n − α s n − ( x ) ∗ k α y (cid:17) ∗ k +1 (cid:16) α t n − ( x ) ∗ n − · · · ∗ α t ( x ) ∗ u ( s k +1 ( y )) (cid:17) . We use the notation α : u ⇒ v to say that α is an oplax transformationfrom u to v . Let
B C D E f uv g be a diagram in ω Cat and α : u ⇒ v be an oplax transformation. The dataof ( g ⋆ α ) x := g ( α x )for each cell x of C (resp. ( α ⋆ f ) x := α f ( x ) for each cell x of B ) defines an oplax transformation from gu to gv (resp. uf to vf ) which we denote by g ⋆ α (resp. α ⋆ f ). Lemma 4.6.
Let u, v : C → D be two ω -functors. If there is an oplaxtransformation α : u ⇒ v , then there is a homotopy of chain complexes from λ ( u ) to λ ( v ) . OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 15
Proof.
For every n -cell x of C (resp. D ), let us use the notation [ x ] for theimage of x in λ ( C ) n (resp. λ ( D ) n ).Let h n be the map h n : λ ( C ) n → λ ( D ) n +1 [ x ] [ α x ] . The definition of oplax transformations implies that h n is linear and thatfor every n -cell x of C , ∂ ( h n ( x )) + h n − ( ∂ ( x )) = [ v ( x )] − [ u ( x )] . Details are left to the reader. (cid:3)
Recall that the category of chain complexes in non-negative degree Ch ≥ has a cofibrantly generated model structure where:- the weak equivalences are the quasi-isomorphisms, i.e. the morphismsof chain complexes that induce an isomorphism on homology groups,- the cofibrations are the morphisms of chain complexes f : X → Y such that for every n ≥ f n : X n → Y n is a monomorphism withprojective cokernel,- the fibrations are the morphisms of chain complexes f : X → Y suchthat for every n > f n : X n → Y n is an epimorphism.(See for example [DS95, Section 7].) From now on, we will implicitly considerthat the category Ch ≥ is equipped with this model structure. Proposition 4.8.
The functor λ : ω Cat → Ch ≥ is left Quillen.Proof. The fact that λ is a left adjoint is Lemma 4.3.A simple computation using Lemma 4.2 shows that for every n ∈ N , λ ( i n ) : λ ( S n − ) → λ ( D n )is a monomorphism with projective cokernel. This shows that λ preservescofibrations.Then, we know from [LMW10, Sections 4.6 and 4.7] and [AM20b, Re-marque B.1.16] (see also [AL20, Paragraph 3.11]) that there exists a set ofgenerating trivial cofibrations J of the Folk model structure on ω Cat suchthat every j : X → Y in J satisfies the following conditions:- there exists r : Y → X such that r ◦ j = 1 X ,- there exists an oplax transformation α : j ◦ r ⇒ Y .From Lemma 4.6, we conclude that λ preserves trivial cofibrations. (cid:3) The previous proposition leads the following definition:
Definition 4.9.
We define the polygraphic homology functor H pol : Ho( ω Cat ) → Ho( Ch ≥ )as the total left derived functor of λ : ω Cat → Ch ≥ . Nerve of ω -categories and the comparison map We denote by ∆ the category whose objects are the finite non-emptytotally ordered sets [ n ] = { < · · · < n } and whose morphisms are thenon-decreasing maps. For n ∈ N and 0 ≤ i ≤ n , we denote by δ i : [ n − → [ n ]the only injective increasing map whose image does not contain i .The category b ∆ of simplicial sets is the category of presheaves on ∆. Fora simplicial set X , we use the notations X n := X ([ n ]) ∂ i := X ( δ i ) : X n → X n . Elements of X n are referred to as n -simplices of X . From now on, we will consider that the category b ∆ is equipped withthe model structure defined by Quillen in [Qui67]. A weak equivalence ofsimplicial sets is a weak equivalence for this model structure. We denote by O : ∆ → ω Cat the cosimplicial object introduced byStreet in [Str87]. The ω -category O n is the n -oriental . For a definition andbasic properties of this cosimplicial object we refer to op. cit. , [Ste04] and[AM20b, Chapitre 7].For every n ∈ N , the ω -category O n is free and the set of generating k -cells is canonically isomorphic to the set of increasing sequences0 ≤ i < i < · · · < i k ≤ n. We will denote such a generating cell by h i i · · · i k i . In particular, O n is an n -category and it has a unique generating n -cell, namely h · · · n i , whichwe call the principal cell of O n .Here are some pictures in low dimension: O = h i , O = h i h i , h i O = h ih i h i . h ih i h ih i For every ω -category X , the nerve of X is the simplicial set N ω ( X )defined as N ω ( X ) : ∆ op → Set [ n ] Hom ω Cat ( O n , X ) . By post-composition, this yields a functor N ω : ω Cat → b ∆ X N ω ( X ) . OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 17
Note that when X is a 1-category, N ω ( X ) is canonically isomorphic to theusual nerve of X , that is, the simplicial set∆ op → Set [ n ] Hom
Cat ([ n ] , X ) , where [ n ] is seen as a 1-category.By the usual Kan extension technique, O : ∆ → ω Cat can be extendedto a functor c ω : b ∆ → ω Cat , which is left adjoint to the nerve functor N ω . Lemma 5.5.
The nerve functor N ω : ω Cat → b ∆ sends the equivalences of ω -categories to weak equivalences of simplicial sets. In particular, this means that the nerve functor induces a functor N ω : Ho( ω Cat ) → Ho( b ∆) . Proof.
Since every ω -category is fibrant for the Folk model structure [LMW10,Proposition 9], it follows from Ken Brown’s Lemma [Hov07, Lemma 1.1.12]that it suffices to show that the nerve sends the folk trivial fibrations toweak equivalences of simplicial sets. In particular, it suffices to show thestronger condition that the nerve sends the folk trivial fibrations to trivialfibrations of simplicial sets.By adjunction, this is equivalent to showing that the functor c ω : b ∆ → ω Cat sends the cofibrations of simplicial sets to folk cofibrations. Since c ω is cocontinuous and the cofibrations of simplicial sets are generated by thecanonical inclusions ∂ ∆ n → ∆ n for n ∈ N , it suffices to show that c ω sends these inclusions to folk cofibra-tions.Now, it follows from any reference on orientals previously cited that theimage of the inclusion ∂ ∆ n → ∆ n by c ω can be identified with the canonicalinclusion ( O n ) ≤ n − → O n . Since O n is free, this last morphism is by definition a pushout of a coproductof folk cofibrations, hence a folk cofibration. (cid:3) Let X be a simplicial set. We denote by K n ( X ) the abelian group of n -chains of X , that is the free abelian group on X n , and by ∂ : K n ( X ) → K n ( X ) the linear map defined for x ∈ X n by ∂ ( x ) = X ≤ i ≤ n ( i ∂ i ( x ) . It follows from the simplicial identities [GJ09, Section I.1] that ∂ ◦ ∂ = 0and thus, the previous data defines a chain complex K ( X ). This canonicallydefines a functor K : b ∆ → Ch ≥ in the expected way. Recall that an n -simplex x of a simplicial set X is degenerate if thereexists a surjective non-decreasing map ϕ : [ n ] → [ k ] with k < n and a k -simplex x ′ of X such that X ( ϕ )( x ′ ) = x . For every n ∈ N , we define D n ( X ) as the subgroup of K n ( X ) spanned by the degenerate n -simplices.We denote by κ n ( X ) the abelian group of normalized chain complex , κ n ( X ) = K n ( X ) /D n ( X ) . It follows from the simplicial identities that ∂ ( D n ( X )) ⊆ D n ( X ) and wedenote by ∂ : κ n ( X ) → κ n ( X )the map induced by the differential of K ( X ). This data defines a chaincomplex κ ( X ) that we call the normalized chain complex of X . This yieldsa functor κ : b ∆ → Ch ≥ . Lemma 5.8.
The functor κ : b ∆ → Ch ≥ is left Quillen and sends the weakequivalences of simplicial sets to quasi-isomorphisms.Proof. From the Dold–Kan equivalence, we know that Ch ≥ is equivalentto the category Ab (∆) of simplicial abelian groups. With this identificationthe functor κ : b ∆ → Ch ≥ is left adjoint of the canonical forgetful functor U : Ch ≥ ≃ Ab (∆) → b ∆induced by the forgetful functor from abelian groups to sets. It follows from[GJ09, Lemma 2.9 and Corollary 2.10] that U is right Quillen, hence κ isleft Quillen. The fact that κ preserves weak equivalences follows from KenBrown’s Lemma [Hov07, Lemma 1.1.12] and the fact that all simplicial setsare cofibrant. (cid:3) Lemma 5.9.
The triangle of functors b ∆ ω CatCh ≥ c ω κ λ is commutative (up to a canonical isomorphism).Proof. Since all the functors involved are cocontinuous, it suffices to showthat the triangle is commutative when we pre-compose it by the Yonedaembedding ∆ → b ∆. This property follows straightforwardly from the de-scription of the orientals in [Ste04]. (cid:3) From Lemma 5.9, the co-unit of the adjunction c ω ⊣ N ω induces anatural transformation κN ω ≃ λc ω N ω ⇒ λ. From Lemma 5.5, Lemma 5.8, Remark 1.3.1 and the universal property ofleft derivable functors, we obtain a natural transformation κN ω ⇒ H pol , which we refer to as the canonical comparison map . Following Remark 1.2.1,for every ω -category C this canonical comparison map reads κN ω ( C ) → H pol ( C ) . OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 19 The case of contractible ω -categories For every ω -category C , we denote by p C : C → D the unique morphism from C to D (which is a terminal object of ω Cat ). Definition 6.2. An ω -category C is contractible if there exists a 0-cell x of C and an oplax transformation C D C. id C p C h x i α For later reference, we put here the following lemma.
Lemma 6.3.
Let C be a -category. If C has a terminal object, then it iscontractible.Proof. Let x be the terminal object of C . For every object y of C , thecanonical arrow y → x of C defines a natural transformation C D C. p C id C h x i The result follows then from the fact that any natural transformation be-tween 1-functors obviously defines an oplax transformation. (cid:3)
For the next lemma, notice that an immediate computation using Lemma4.2 shows that λ ( D ) is canonically isomorphic to Z considered as a chaincomplex concentrated in degree 0. Lemma 6.4.
Let C be a contractible ω -category. The morphism of chaincomplexes λ ( p C ) : λ ( C ) → λ ( D ) ≃ Z is a quasi-isomorphism.Proof. This follows immediately from Lemma 4.6. (cid:3)
Lemma 6.5.
Let C ′ D ′ C D f ′ ǫ u vf ǫ be commutative squares in ω Cat for ǫ ∈ { , } .If C ′ is a free ω -category and v is a folk trivial fibration, then for everyoplax transformation α : f ⇒ f , there is an oplax transformation α ′ : f ′ ⇒ f ′
10 LÉONARD GUETTA such that v ⋆ α ′ = α ⋆ u. Proof.
We denote by ⊗ the Gray monoidal product (see for example [AM20b,Appendice A]) on the category ω Cat . Recall that the unit of this monoidalproduct is the ω -category D .From [AM20b, Appendice B], we know that given two ω -functors u, v : C → D , the set of oplax transformations from u to v is in bijection with theset of functors α : D ⊗ C → D such that the diagram( D ∐ D ) ⊗ C ≃ C ∐ C D ⊗ C D, i ⊗ C h u,v i α where i : D ∐ D ≃ S → D is the morphism introduced in 3.4, is commu-tative. We use the same letter to denote an oplax transformation and thefunctor D ⊗ C → D associated to it.Moreover, for an ω -functor f : B → C , the oplax transformation α ⋆ f isrepresented by the functor D ⊗ B D ⊗ C D D ⊗ f α and for an ω -functor g : D → E , the oplax transformation g⋆α is representedby the functor D ⊗ C D E. α g
Using this way of representing oplax transformations, the hypotheses ofthe present lemma yield the following commutative square( D ∐ D ) ⊗ C ′ D ′ D ⊗ C ′ D ⊗ C D. i ⊗ C ′ h f ′ ,f ′ i v D ⊗ u α Since i is a folk cofibration and C ′ is cofibrant, it follows that the leftvertical morphism of the previous square is a folk cofibration (see [Luc17,Proposition 5.1.2.7] or [AL20]). By hypothesis, v is a folk trivial fibrationand thus the above square admits a lift α ′ : D ⊗ C ′ → D ′ . The commutativity of the two induced triangle shows what we needed toprove. (cid:3)
Proposition 6.6.
Let C be an ω -category. If C is contractible, then thecanonical comparison map κN ω ( C ) → H pol ( C ) is an isomorphism of Ho( Ch ≥ ) . More precisely, this morphism can beidentified with the identity morphism id Z : Z → Z (where Z is seen as anobject of Ho( Ch ≥ ) concentrated in degree ). OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 21
Proof.
Consider first the case when C is cofibrant for the Folk model struc-ture. It follows respectively from Lemma 6.4 and [AM20c, Corollaire A.13]that the morphisms H pol ( C ) → H pol ( D )and κN ω ( C ) → κN ω ( D ) , induced by the canonical morphism p C : C → D , are isomorphisms ofHo( Ch ≥ ). Moreover, it is straightforward to check that the canonical com-parison map κN ω ( D ) → H pol ( D )is an isomorphism Ho( Ch ≥ ) (and can be identified with the identity mor-phism id Z : Z → Z , where Z is seen as a chain complex concentrated indegree 0). From the naturality square κN ω ( C ) H pol ( C ) κN ω ( D ) H pol ( D )we deduce that the top arrow is an isomorphism of Ho( Ch ≥ ).In the general case, we know by definition of contractible ω -categoriesthat there exist a 0-cell x of C and an oplax transformation α of the form C D C. id C p C h x i α Now let us choose any folk trivial fibration u : P → C with P cofibrant. Bydefinition, folk trivial fibrations have the right lifting property to i : S − → D , which means exactly that they are surjective on 0-cells. In particular,there exists a 0-cell x ′ of P such that u ( x ′ ) = x . It follows then from Lemma6.5 that there exists an oplax transformation P D P, id P p P h x ′ i α ′ which proves that P is also contractible.Finally, consider the naturality square κN ω ( P ) H pol ( P ) κN ω ( C ) H pol ( C )induced by u . Since P is contractible and folk cofibrant, we have alreadyproved that the top arrow is an isomorphism. Since u is a weak equivalencefor the Folk model structure, the right vertical morphism is an isomorphismand it follows from Lemma 5.5 and Lemma 5.8 that the left vertical arrowis also an isomorphism. Hence, the bottom arrow is an isomorphism. (cid:3) The folk homotopy colimit theorem
For every object a of a 1-category A , we denote by A/a the slice cate-gory over a . That is, an object of A/a is a pair ( b, p : b → a ) where b is anobject of A and p is an arrow of A , and an arrow ( b, p ) → ( b ′ , p ′ ) of A/a isan arrow q : b → b ′ of A such that p ′ ◦ q = p .We denote by π a : A/a → A ( b, p ) b the canonical forgetful functor. Definition 7.2.
Let A be a 1-category, a be an object of A , X be an ω -category and f : X → A be an ω -functor. We define the ω -category X/a and the ω -functor f /a : X/a → A/a as the following pullback
X/a XA/a A. y f/a fπ a More explicitly, an n -arrow of X/a is a pair ( x, p ) where x is an n -arrowof X and p is an arrow of A of the form p : f ( x ) → a if n = 0and p : f ( t ( x )) → a if n > . From now on, we adopt the convention that t ( x ) = x for every -cell x of X . This unifies the cases n = 0 and n > above. For n >
0, the source and target of an n -arrow ( x, p ) of X/a are given by s (( x, p )) = ( s ( x ) , p ) and t (( x, p )) = ( t ( x ) , p ) . For ( x, p ) an n -arrow of X/a , we have( f /a )( x, p ) = ( f ( x ) , p ) , and the canonical arrow X/a → X is simply expressed as( x, p ) x. Let f : X → A be an ω -functor with A a 1-category. Every arrow β : a → a ′ of A induces an ω -functor X/β : X/a → X/a ′ ( x, p ) ( x, β ◦ p ) . This defines a functor X/ : A → ω Cat a X/a.
OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 23
Moreover, for every arrow β : a → a ′ of A , the triangle X/a X/a ′ X X/β is commutative. By universal property, this induces a canonical arrowcolim a ∈ A X/a → X. Now, let
X X ′ A gf f ′ be a commutative triangle in ω Cat with A a 1-category. For every a ∈ Ob( A ),we define the ω -functor g/a as g/a : X/a → X ′ /a ( x, p ) ( g ( x ) , p ) . This construction is natural in a , which means that for every arrow β : a → a ′ of A , the following square is commutative X/a X ′ /aX/a ′ X ′ /a ′ . g/aX/β X ′ /βg/a ′ In particular, g induces an ω -functorcolim a ∈ A X/a → colim a ∈ A X ′ /a and the square colim a ∈ A X/a X colim a ∈ A X ′ /a X ′ g is commutative. Lemma 7.5.
Let f : X → A be an ω -functor with A is a -category. Thecanonical arrow colim a ∈ A X/a → X is an isomorphism.Proof. We have to show that the cocone(
X/a → X ) a ∈ Ob( A ) is colimiting. Let ( g a : X/a → C ) a ∈ Ob( A )4 LÉONARD GUETTA be another cocone and let x be an n -arrow of X . Notice that the pair( x, f ( t ( x )) )is an n -arrow of X/f ( t ( x )). We leave it to the reader to prove that theformula φ : X → Cx g f ( t ( x )) ( x, f ( t ( x )) )defines an ω -functor. This proves the existence part of the universal prop-erty.It is straightforward to check that for every a ∈ Ob( A ) the triangle X/a XC g a φ is commutative. Now let φ ′ : X → C be another ω -functor that makes theprevious triangles commute and let x be an n -arrow of X . Since the triangle X/f ( t ( x )) XC g f ( t x )) φ ′ is commutative, we necessarily have φ ′ ( x ) = g f ( t ( x )) ( x, f ( t ( x )) )which proves that φ ′ = φ . (cid:3) Lemma 7.6. If X is free, then for every a ∈ Ob( A ) the ω -category X/a isfree. More precisely, if we denote by Σ Xn the set of generating n -cells of X ,then the set of generating n -cells of X/a is Σ X/an = { ( x, p ) ∈ ( X/a ) n | x ∈ Σ Xn } . Proof.
Remark first that for every a ∈ Ob( A ), the map π a : A/a → A is a discrete Conduché ω -functor (Definition 3.6). Hence, from Lemma 3.7,we deduce that the canonical map X/a → X is also a discrete Conduché ω -functor. We conclude with Proposition 3.8. (cid:3) Let f : X → A be as before and suppose that X is free. Every arrow β : a → a ′ of A induces a map:Σ X/a ′ n → Σ X/an ( x, p ) ( x, β ◦ p ) . This defines a functor Σ X/ n : A → Set a Σ X/an . OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 25
Lemma 7.8. If X is free, then there is a natural isomorphism Σ X/ n ≃ a x ∈ Σ Xn Hom A ( f ( t ( x )) , ) . Proof.
Let a be an object of A . For each x ∈ Σ Xn , there is a canonical mapHom A ( f ( t ( x )) , a ) → Σ X/an p ( x, p )that induces by universal property a map a x ∈ Σ Xn Hom A ( f ( t ( x )) , a ) → Σ X/an . The naturality in a and the fact that this map is an isomorphism are obvious. (cid:3) Proposition 7.9.
Let f : X → A be an ω -functor with A a 1-category and X a free ω -category. The functor X/ : A → ω Cat a X/a is a cofibrant object for the projective model structure on ω Cat A induced bythe Folk model structure on ω Cat .Proof.
Recall that the set { i n : S n − → D n | n ∈ N } is a set a generating cofibrations for the Folk model structure on ω Cat .From Lemmas 7.6 and 7.8 we deduce that for every a ∈ Ob( A ) and every n ∈ N , the canonical square a x ∈ Σ Xn a Hom A ( f ( t ( x )) ,a ) S n − ( X/a ) ≤ n − a x ∈ Σ Xn a Hom A ( f ( t ( x )) ,a ) D n ( X/a ) ≤ n is a pushout square. It is straightforward to check that this square is naturalin a , which means that for every n ∈ N we have a pushout square of ω Cat A a x ∈ Σ Xn S n − ⊗ f ( t ( x )) ( X/ ) ≤ n − a x ∈ Σ Xn D n ⊗ f ( t ( x )) ( X/ ) ≤ n (see 2.5 for notations). Hence, from Lemma 2.6, for every n ≥ X/ − ) ≤ n − → ( X/ − ) ≤ n is a cofibration for the projective model structure on ω Cat A . The resultfollows then from the fact that the colimit of ∅ = ( X/ ) ≤− → ( X/ ) ≤ → · · · → ( X/ ) ≤ n → · · · is X/ . (cid:3) Theorem 7.10.
Let X be an ω -category, A be a -category and f : X → A be an ω -functor. The canonical arrow of Ho( ω Cat )hocolim a ∈ A X/a → colim a ∈ A X/a (see 1.6) is an isomorphism.
NB.
Note that in the previous theorem, we did not suppose that X wasfolk cofibrant. Proof of Theorem 7.10.
Let P be a free ω -category and g : P → X bea trivial fibration for the Folk model structure. We have a commutativediagram in Ho( ω Cat )(1) hocolim a ∈ A P/a colim a ∈ A P/a hocolim a ∈ A X/a colim a ∈ A X/a where the vertical arrows are induced by the arrows g/a : P/a → X/a.
Since trivial fibrations are stable by pullback, g/a is a trivial fibration. Thisproves that the left vertical arrow of square (1) is an isomorphism.Moreover, from paragraph 7.4 and Lemma 7.5, we deduce that the rightvertical arrow of (1) can be identified with the image of g : P → X inHo( ω Cat ) and hence is an isomorphism.Finally, from Proposition 7.9, we deduce that the top horizontal arrow of(1) is an isomorphism. This proves the theorem. (cid:3) Homology of 1-categories
For a functor d : A → Cat with A a small category, we denoteby R A d the Grothendieck construction for d . We refer to [Mal05, Section3.1] for a definition and basic properties of this construction. Recall thatthere is a canonical morphism R A d → A as well a canonical morphism R A d → colim A ( d ), and that if d is the functor a A/a then these twomorphisms can be identified via the isomorphism colim a ∈ A A/a ≃ A . Lemma 8.2.
Let A be a small category and consider the functor A → ω Cat a A/a.
The canonical map hocolim a ∈ A N ω ( A/a ) → N ω (hocolim a ∈ A A/a ) is an isomorphism of Ho( b ∆) . OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 27
Proof.
The proof is long and we divide it in several parts. Recall that weconsider
Cat as a full subcategory of ω Cat (see Remark 3.3). Given afunctor d : A → Cat , we still denote by d the functor obtained by post-composition A Cat ω Cat . d Preliminaries:
We say that a morphism f : X → Y of ω Cat is a
Thomasonequivalence when N ω ( f ) is a weak equivalence of simplicial sets. Wedenote by W Th ω the class of Thomason equivalences. In order to avoidany confusion, we use the notation Ho Th ( ω Cat ) for the localizationof ω Cat with respect to the Thomason equivalences and the notationHo folk ( ω Cat ) the localization of ω Cat with respect to W folk ω , theweak equivalences of the folk model structure (3.12).Similarly, we denote by W Th1 the class of arrows of
Cat whoseelements are the Thomason equivalences (seen as arrows of ω Cat )and by Ho Th ( Cat ) the localization of
Cat with respect to W Th1 .Note that W Th1 is indeed the class of weak equivalences of the modelstructure on
Cat considered by Thomason in [Tho80].Thus, we have defined three localizers: (
Cat , W Th1 ), ( ω Cat , W Th ω )and ( ω Cat , W folk ω ). We have already seen that ( ω Cat , W folk ω ) hashomotopy colimits and from the existence of the Thomason modelstructure on Cat [Tho80], we deduce that (
Cat , W Th1 ) has homotopycolimits. Although the existence of a model structure on ω Cat withthe Thomason equivalences as weak equivalences is not established(see [AM14] though), we will see later in this proof that ( ω Cat , W Th ω )has homotopy colimits. In order to explicitly distinguish the homo-topy colimits in these three localizers, we use the self-explanatorynotations: Cat , Th hocolim A , ω Cat , Th hocolim A and ω Cat , folk hocolim A . Similarly, we use the notations
Cat colim A and ω Cat colim A to distinguish colimits in Cat and ω Cat . Thomason’s homotopy colimit theorem:
Let us recall an important re-sult due to Thomason [Tho79] (see also [Mal05, section 3.1]): thefunctor Z A : Cat A → Cat preserves the Thomason equivalences and the induced functor Z A : Ho Th ( Cat A ) → Ho Th ( Cat )is a left adjoint to the diagonal functor (1.4) k : Ho Th ( Cat ) → Ho Th ( Cat A ) . Hence, there is a canonical isomorphism of functors Z A ≃ Cat , Th hocolim A . Now, since for every object a of A the category A/a has a ter-minal object, it follows from [Qui73, Section 1,Corollary 2] that thecanonical morphism p A/a : A/a → D to the terminal category is a Thomason equivalence. In particular,if d : A → Cat is the functor a A/a , the induced map(2) Z A d → Z A k ( D ) , where k ( D ) is the constant diagram with value D , is an isomor-phism of Ho Th ( Cat ). A quick computation left to the reader showsthat Z A k ( D ) ≃ A and that the map (2) can be identified with the canonical map Cat , Th hocolim a ∈ A A/a → Cat colim a ∈ A A/a, which is thus an isomorphism of Ho Th ( Cat ). Preservation of Thomason homotopy colimits:
From [Gag18, Theo-rem 2.4 and 6.9], it follows that N ω : ω Cat → b ∆ induces an equiv-alence of prederivators (here ω Cat is equipped with the Thomasonequivalences). Concretely to us, this implies that ( ω Cat , W Th ω ) hashomotopy colimits and that for every functor d : A → ω Cat , thecanonical map b ∆ hocolim A ( N ω ( d )) → N ω ( ω Cat , Th hocolim A ( d ))is an isomorphism of Ho( b ∆). (For the reader unfamiliar with thetheory of prederivators, the above result follows also from the factthat Gagna’s result together with [BK12, Theorem 1.8] imply that N ω induces an equivalence between the associated ∞ -categories inthe sense of Lurie [Lur09].)Similarly, the usual nerve functor for 1-categories N : Cat → b ∆induces an equivalence of prederivators and from the commutativityof the triangle Cat ω Cat b ∆ , iN N ω and the fact that i preserves the Thomason equivalences (by defini-tion), we deduce that i also induces an equivalence of prederivators.Hence, for every functor d : A → Cat , the canonical map ω Cat , Th hocolim A ( d ) → Cat , Th hocolim A ( d ) OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 29 is an isomorphism of Ho Th ( ω Cat ). Consider now the commutativesquare in Ho Th ( ω Cat ): Cat , Th hocolim a ∈ A A/a ω Cat , Th hocolim a ∈ A A/a
Cat colim a ∈ A A/a ω Cat colim a ∈ A A/a.
So far we have proved that the top horizontal arrow and the left ver-tical arrows are isomorphisms. Since the inclusion functor
Cat → ω Cat preserves colimits, the bottom horizontal arrow is also an iso-morphism. This implies that the right vertical arrow is an isomor-phism.
Comparing Folk and Thomason homotopy colimits:
From Lemma 5.5,we have that W folk ω ⊆ W Th ω . In particular, the identity functor ω Cat → ω Cat induces a functorHo folk ( ω Cat ) → Ho Th ( ω Cat ) , and for every functor d : A → ω Cat , we have a commutative trianglein Ho Th ( ω Cat ): ω Cat , folk hocolim A ( d ) ω Cat , Th hocolim A ( d ) ω Cat colim A ( d ) . In the case d is the functor a A/a , we have already proved thatthe two slanted arrows of the previous triangle were isomorphisms.Hence, the canonical map ω Cat , folk hocolim a ∈ A A/a → ω Cat , Th hocolim a ∈ A A/a is an isomorphism. Consider now the commutative triangle inducedby the nerve functor: b ∆ hocolim A ( N ω ( d )) N ω ( ω Cat , folk hocolim A ( d )) N ω ( ω Cat , Th hocolim A ( d )) . We have already seen that the slanted arrow on the right is an iso-morphism. In the case that d is the functor a A/a , it follows fromwhat we have proved that the horizontal arrow is an isomorphism.
Hence, the canonical map b ∆ hocolim a ∈ A N ω ( A/a ) → N ω ( ω Cat , folk hocolim a ∈ A A/a )is an isomorphism. (cid:3)
We can now prove the main theorem of this paper.
Theorem 8.3.
Let A be a small category seen as an object of ω Cat . Thecanonical comparison map κN ω ( A ) → H pol ( A ) is an isomorphism of Ho( Ch ≥ ) .Proof. We have a commutative diagramhocolim a ∈ A κN ω ( A/a ) κN ω (hocolim a ∈ A A/a ) κN ω (colim a ∈ A A/a ) κN ω ( A )hocolim a ∈ A H pol ( A/a ) H pol (hocolim a ∈ A A/a ) H pol (colim a ∈ A A/a ) H pol ( A ) , ( A ) ( B ) ( C )where the vertical arrows are induced by the canonical comparison map. Thegoal is to show that the right vertical map of square ( C ) is an isomorphism.By Lemma 7.5 the horizontal arrows of square ( C ) are isomorphisms. ByTheorem 7.10, the horizontal arrows of square ( B ) are isomorphisms. ByLemma 8.2, Proposition 2.8 and Lemma 5.8, the top horizontal arrow ofsquare ( A ) is an isomorphism, and by Proposition 2.8 the bottom horizontalarrow of the same square is an isomorphism. Finally, from Proposition 6.6,Lemma 6.3 and the fact that for every object a of A the category A/a hasa terminal object, we deduce that the left vertical arrow of square ( A ) is anisomorphism. By 2-out-of-3 property for isomorphisms, this shows what wewanted. (cid:3) Complement: a folk Theorem A
As a corollary of Theorem 7.10, we obtain Proposition 9.2 below . It isto be compare with Theorem A of Quillen [Qui73, Theorem A] and its gen-eralization for ω -categories by Ara and Maltsiniotis [AM18] and [AM20c].However, note that in the pre-cited references, the weak equivalences con-sidered are the ones induced by the nerve functor N ω (which we called Thomason equivalences in the proof of Lemma 8.2), whereas in the proposi-tion below we work with the weak equivalences of the Folk model structure(3.10).
Proposition 9.2.
Let
X YA uv w Since the time of writing of this article, the author has proved a generalization of thisresult, which can be found in [Gue21]. However, the proof of this new result is completelydifferent from the one used here and both are interesting in their own right.
OMOLOGY OF CATEGORIES VIA POLYGRAPHIC RESOLUTIONS 31 be a commutative triangle in ω Cat and suppose that A is a -category. Iffor every a ∈ Ob( A ) , the induced arrow u/a : X/a → Y /a is an equivalence of ω -categories, then u is also an equivalence of ω -cate-gories.Proof. Consider the commutative square in Ho( ω Cat ):hocolim a ∈ A X/a colim a ∈ A X/a hocolim a ∈ A Y /a colim a ∈ A Y /a where the vertical arrows are induced by the arrows u/a : X/a → Y /a.
Since we supposed that these arrows were weak equivalences of the Folkmodel structure, it follows that the left vertical arrow of the previous squareis an isomorphism. From Theorem 7.10, the horizontal arrows are isomor-phisms.This proves that the right vertical arrow is also an isomorphism but itfollows from 7.4 and Lemma 7.5 that this arrow can be identified with theimage of u : X → Y in Ho( ω Cat ). (cid:3) Acknowledgment
I am infinitely grateful to Georges Maltsiniotis for explaining homotopycolimits to me. His point of view on the subject has infused into me, andinto this article.
References [ABG + ] Dimitri Ara, Albert Burroni, Yves Guiraud, Philippe Malbos, François Métayer,and Samuel Mimram. Polygraphs: From Rewriting to Higher Categories. Inpreparation.[AL20] Dimitri Ara and Maxime Lucas. The folk model category structure on strict ω -categories is monoidal. Theory and Applications of Categories , 35:742–808,2020.[AM14] Dimitri Ara and Georges Maltsiniotis. Vers une structure de catégorie de mod-èles à la Thomason sur la catégorie des n -catégories strictes. Advances in Math-ematics , 259:557–654, 2014.[AM18] Dimitri Ara and Georges Maltsiniotis. Un théorème A de Quillen pour les ∞ -catégories strictes I : la preuve simpliciale. Advances in Mathematics ,328:446–500, 2018.[AM20a] Dimitri Ara and Georges Maltsiniotis. Comparaison des nerfs n -catégoriques. arXiv preprint arXiv:2010.00266 , 2020.[AM20b] Dimitri Ara and Georges Maltsiniotis. Joint et tranches pour les ∞ -catégoriesstrictes. Mémoires de la Société Mathématique de France , 165, 2020.[AM20c] Dimitri Ara and Georges Maltsiniotis. Un théorème A de Quillen pour les ∞ -catégories strictes II : la preuve ∞ -catégorique. Higher Structures , 4(1):284–388, 2020.[Ara19] Dimitri Ara. A Quillen theorem B for strict ∞ -categories. Journal of the LondonMathematical Society , 100(2):470–497, 2019. [BK12] Clark Barwick and Daniel M Kan. A characterization of simplicial localizationfunctors and a discussion of DK equivalences.
Indagationes Mathematicae , 23(1-2):69–79, 2012.[Bou90] Dominique Bourn. Another denormalization theorem for abelian chain com-plexes.
Journal of Pure and Applied Algebra , 66(3):229–249, 1990.[DS95] William G. Dwyer and Jan Spalinski. Homotopy theories and model categories.In
Handbook of Algebraic Topology , pages 73–126. ScienceDirect, 1995.[Gag18] Andrea Gagna. Strict n-categories and augmented directed complexes modelhomotopy types.
Advances in Mathematics , 331:542–564, 2018.[GJ09] Paul G. Goerss and John F. Jardine.
Simplicial homotopy theory . Birkhäuser,2009.[Gon12] Beatriz Rodríguez González. A derivability criterion based on the existence ofadjunctions. arXiv preprint arXiv:1202.3359 , 2012.[Gue20] Léonard Guetta. Polygraphs and discrete Conduché ω -functors. Higher Struc-tures , 4(2):134–166, 2020.[Gue21] Léonard Guetta.
Homology of strict ω -categories . PhD thesis,Université de Paris, 2021. Available on the author’s webpage .[GZ67] Peter Gabriel and Michel Zisman. Calculus of fractions and homotopy theory .Springer, 1967.[Hov07] Mark Hovey.
Model categories . American Mathematical Society, 2007.[LM09] Yves Lafont and François Métayer. Polygraphic resolutions and homology ofmonoids.
Journal of Pure and Applied Algebra , 213(6):947–968, 2009.[LMW10] Yves Lafont, François Métayer, and Krzysztof Worytkiewicz. A folk modelstructure on omega-cat.
Advances in Mathematics , 224(3):1183–1231, 2010.[Luc17] Maxime Lucas.
Cubical categories for homotopy and rewriting . PhD thesis, Uni-versité Paris Diderot, 2017.[Lur09] Jacob Lurie.
Higher topos theory . Princeton University Press, 2009.[Mak05] Michael Makkai. The word problem for computads.
Available on the author’sweb page , 2005.[Mal05] Georges Maltsiniotis.
La théorie de l’homotopie de Grothendieck . Société Math-ématique de France, 2005.[Mét03] François Métayer. Resolutions by polygraphs.
Theory and Applications of Cat-egories , 11(7):148–184, 2003.[Mét08] François Métayer. Cofibrant objects among higher-dimensional categories.
Ho-mology, Homotopy and Applications , 10(1):181–203, 2008.[Qui67] Daniel G. Quillen.
Homotopical algebra . Springer, 1967.[Qui73] Daniel G. Quillen. Higher algebraic K-theory: I. In
Higher K-theories , pages85–147. Springer, 1973.[Ste04] Richard Steiner. Omega-categories and chain complexes.
Homology, Homotopyand Applications , 6(1):175–200, 2004.[Str87] Ross Street. The algebra of oriented simplexes.
Journal of Pure and AppliedAlgebra , 49(3):283–335, 1987.[Tho79] Robert W. Thomason. Homotopy colimits in the category of small categories.
Mathematical Proceedings of the Cambridge University Press , 85(1):91–109,1979.[Tho80] Robert W. Thomason. Cat as a closed model category.
Cahiers de topologie etgéométrie différentielle catégoriques , 21(3):305–324, 1980.
Email address : [email protected]@irif.fr