CCOMONAD COHOMOLOGY OF TRACK CATEGORIES
DAVID BLANC AND SIMONA PAOLI
Abstract.
We define a comonad cohomology of track categories, and show that itis related via a long exact sequence to the corresponding ( S , O )-cohomology. Undermild hypotheses, the comonad cohomology coincides, up to reindexing, with the( S , O )-cohomology, yielding an algebraic description of the latter. We also specializeto the case where the track category is a 2-groupoid. Introduction
One of several models for ( ∞ , simplicial categories ; that is, (small) categories Y enriched in simplicial sets. If the object set of Y is O , we say it is an ( S , O ) -category .One may analyze a topological space (or simplicial set) X by means of its Postnikovtower ( P n X ) ∞ n =0 , where the n -th Postnikov section P n X is an n -type (that is, hastrivial homotopy groups in dimension greater than n ). The successive sections arerelated through their k -invariants : cohomology classes in H n +1 ( P n − X ; π n X ).Since the Postnikov system is functorial (and preserves products), one can alsodefine it for a simplicial category Y : P n Y is then a category enriched in n -types,and its k -invariants are expressed in terms of the ( S , O )-cohomology of [DKS].A long-standing open problem is to find a purely “algebraic” description of Post-nikov systems, both for spaces and for simplicial categories. For the Postnikov sec-tions, there are various algebraic models of n -types – and thus of categories enrichedin n -types – in the literature, using a variety of higher categorical structures. How-ever, the problem of finding an algebraic model for the k -invariants is largely open.For this purpose, we need first an algebraic formulation of the cohomology theoriesused to define the k -invariants. This leads us to look for an algebraic description ofthe cohomology of a category enriched in a suitable algebraic model of n -types.We here realize the first step of this program, for track categories – that is, cate-gories enriched in groupoids. In the future we hope to extend this to the cohomologyof n -track categories – that is, those enriched in the n -fold groupoidal models of n -types developed by the authors in [BP2] and [P2].In [BP1] the authors introduced a cohomology theory for track categories (whichgeneralizes the Baues-Wirsching cohomology of categories – see [BW]), and showedthat it coincides, up to indexing with the corresponding ( S , O )-cohomology. This wasthen used to describe the first k -invariant for a 2-track category.A direct generalization of this approach is problematic, because of the difficultyof defining a full and faithful simplicial nerve of weak higher categorical structures.Instead, we use a version of Andr´e-Quillen cohomology, also known as comonad co-homology , since we use a comonad to produce a simplicial resolution of our track Date : April 16, 2019.2010
Mathematics Subject Classification.
Key words and phrases. track category, comonad cohomology, simplicial category. a r X i v : . [ m a t h . A T ] A p r DAVID BLANC AND SIMONA PAOLI category (see [BBe]). We envisage a generalization to higher dimensions, using the n -fold nature of the models of n -types in [BP2] and [P2].Our main result (see Corollaries 4.13 and 4.18) is that under mild hypotheses ona track category X (always satisfied up to 2-equivalence), the comonad cohomologyof X (Definition 4.11) coincides, up to a dimension shift, with its ( S , O )-cohomology.This follows from Theorem 4.12, which states that any track category X has a longexact sequence relating the comonad cohomology of X , its ( S , O )-cohomology and the( S , O )-cohomology of the category X of objects and 1-arrows of X . When the trackcategory X is a 2-groupoid, its ( S , O )-cohomology coincides with the cohomology ofits classifying space.0.1 . Notation and conventions. Denote by ∆ the category of finite ordered sets,so for any C , [∆ op , C ] is the category of simplicial objects in C , while [∆ , C ] isthe category of cosimplicial objects in C . In particular, we write S for the category[∆ op , Set ] of simplicial sets. We write c ( A ) ∈ [∆ op , C ] for the constant simplicialobject on A ∈ C .For any category C with finite limits, we write Gpd C for the category of groupoidsinternal to C – that is, diagrams in C of the form X × X X m (cid:47) (cid:47) X c (cid:8) (cid:8) d (cid:47) (cid:47) d (cid:47) (cid:47) X (cid:111) (cid:111) satisfying the obvious identities making the composition m associative and every‘1-cell’ in X invertible.For a fixed set O , we denote by Cat O the category of small categories with objectset O (and functors which are the identity on O ). In particular, a category Z enrichedin simplicial sets with object set O will be called an ( S , O )- category , and the categoryof all such will be denoted by ( S , O )- Cat . Equivalently, such a category Z can bethought of as a simplicial object in Cat O . This means C has a fixed object set O ineach dimension, and all face and degeneracy functors the identity on objects.More generally, if ( V , ⊗ ) is any monoidal category, a ( V , O )- category is a smallcategory C ∈
Cat O enriched over V . The category of all such categories will bedenoted by ( V , O )- Cat . Examples for ( V , ⊗ ) include S , Top , Gp , and Gpd , with ⊗ the Cartesian product. When V = Gpd , we call Z in Track O := ( Gpd , O )- Cat a trackcategory with object set O (see § V = S ∗ , with ⊗ = ∧ (smash product).We can identify an ( S ∗ , O )-category with a simplicial pointed O -category.0.2 . Organization. Section 1 provides some background material on the Bourneadjunction ( § § § S , O )-cohomology ( § § S , O )-cohomology (Theorem 3.29 and Corollary 3.34). Section 4 defines the comonad co-homology of track categories, and establishes the long exact sequence relating the( S , O )-cohomologies of X and of X and the comonad cohomology of X (Theorem4.12). Section 5 specializes to the case of a 2-groupoid, showing that in this case OMONAD COHOMOLOGY OF TRACK CATEGORIES 3 ( S , O )-cohomology coincides with that of the classifying space (Corollary 5.6). Thelong exact sequence of Corollary 5.8 recovers [P1, Theorem 13].0.3. Acknowledgements.
We would like to thank the referee for his or her pertinent andhelpful comments. The first author was supported by the Israel Science Foundationgrants 74/11 and 770/16. The second author would like to thank the Department ofMathematics of the University of Haifa for its hospitality during several visits.1.
Preliminaries
In this section we review some background material on the Bourne adjunction, theinternal arrow functor, modules, ( S , O )-cohomology, and simplicial model categories.1.1 . The Bourne adjunction. Let C be a category with finite limits and let Spl C be the category whose objects are the split epimorphisms with a given splitting:(1.2) ( A q (cid:47) (cid:47) B ) t (cid:111) (cid:111) Define R : Gpd
C →
Spl C by RX = ( X d (cid:47) (cid:47) X ) . s (cid:111) (cid:111) Let H : Spl
C →
Gpd C associate to Y = ( A q (cid:47) (cid:47) B ) t (cid:111) (cid:111) the object HY : A q × A q × A m (cid:47) (cid:47) A q × A pr (cid:47) (cid:47) pr (cid:47) (cid:47) A ∆ (cid:111) (cid:111) of Gpd C , where A q × A is the kernel pair of q , ∆ = (Id A , Id A ) is the diagonal map,and m = (pr , pr ) : A q × A q × A ∼ = ( A q × A ) × A ( A q × A ) → A q × A with pr i m = pr i π i ( i = 0 , π i , pr i are the two projections. Note that HY is an internal equivalence relation in C with Π ( HY ) = B . Consider the followingdiagram(1.3) Gpd C Ner (cid:47) (cid:47) R (cid:15) (cid:15) [∆ op , C ] Dec (cid:15) (cid:15)
Spl C n (cid:47) (cid:47) H (cid:79) (cid:79) Aug[∆ op , C ] + (cid:79) (cid:79) where Aug[∆ op , C ] is the category of augmented simplicial objects in C , + is thefunctor that forgets the augmentation, the d´ecalage functor Dec ( obtained by for-getting the last face operator) is its right adjoint, and nX is the nerve of the internalequivalence relation associated to X , augmented over itself, with Ner H = + n .Diagram (1.3) commutes up to isomorphism – that is, there is a naturalisomorphism α : Dec Ner ∼ = nR . Since + (cid:97) Dec, this implies that H (cid:97) R (see [Bo,Theorem 1]). DAVID BLANC AND SIMONA PAOLI
Given X ∈ Spl C as in (1.2), we have RHX = ( A q × A pr (cid:47) (cid:47) A ∆ (cid:111) (cid:111) ). The unit η : X → RHX of the adjunction H (cid:97) R is given by(1.4) A q (cid:47) (cid:47) t (cid:15) (cid:15) B t (cid:111) (cid:111) t (cid:15) (cid:15) A q × A pr (cid:47) (cid:47) A ∆ (cid:111) (cid:111) where t is determined by A Id (cid:37) (cid:37) t (cid:39) (cid:39) tq (cid:40) (cid:40) A q × A pr (cid:47) (cid:47) pr (cid:15) (cid:15) A q (cid:15) (cid:15) A q (cid:47) (cid:47) A so that(1.5) pr t = tq, pr t = Id , and t t = ( t, t ) = ∆ t . This show that η = ( t, t ) is a morphism in Spl C .Finally, if µ is the counit of the adjunction H (cid:97) R , and µ (cid:48) that of + (cid:97) Dec, thenfor any X ∈ Gpd C the following diagram commutes:Ner HRX
Ner µ (cid:47) (cid:47) Ner X + nRX (cid:111)(cid:107) + Dec Ner X µ (cid:48) Ner X (cid:56) (cid:56) Thus µ = P Ner µ where P (cid:97) Ner.1.6 . The internal arrow functor.
Let U : Gpd
C → C be the arrow functor,so
U Y = Y for Y ∈ Gpd C , and assume C is (co)complete with commuting finitecoproducts and pullbacks. For any X ∈ C let X s , X t be two copies of X , with F : X s (cid:96) X t → X the fold map and LX ∈ Gpd C the corresponding internalequivalence relation, so ( LX ) = X s (cid:96) X t and ( LX ) = ( X s (cid:96) X t ) F × ( X s (cid:96) X t ).Then X s (cid:96) X t F (cid:47) (cid:47) X i (cid:111) (cid:111) is an object of Spl C and(1.7) LX = H ( X s (cid:96) X t F (cid:47) (cid:47) X i (cid:111) (cid:111) )(cf. § i is the coproduct structure map. Therefore,( X s (cid:96) X t ) F × ( X s (cid:96) X t ) =( X s × X X s ) (cid:96) ( X s × X X t ) (cid:96) ( X t × X X s ) (cid:96) ( X t × X X t ) = X ss (cid:96) X st (cid:96) X ts (cid:96) X tt (1.8)where X ss = X s × X X s , X st = X s × X X t , X ts = X t × X X s , and X tt = X t × X X t . Underthe identification (1.8) the face and degeneracy maps of LX are as follows: OMONAD COHOMOLOGY OF TRACK CATEGORIES 5 s includes X s (cid:96) X t into X ss (cid:96) X tt ; d sends X ss and X st to X s , and X ts and X tt to X t ; and d sends X ss and X ts to X s , and X st and X tt to X t .To see that L is left adjoint to U , given f : X → Y = U Y , its adjoint ˜ f : LX → Y is given by ˜ f : X ss (cid:96) X st (cid:96) X ts (cid:96) X tt → Y (determined by s (cid:48) d (cid:48) f : X ss → Y , f : X st → Y , f ◦ τ : X ts → Y , and s d f : X tt → Y ), and ˜ f : X s (cid:96) X t → Y (determined by d (cid:48) f : X s → Y and d (cid:48) f : X t → Y ). Here τ : X ts → X st is the switchmap, with τ ◦ τ = Id.Conversely, given g : LX → Y with g : X ss (cid:96) X st (cid:96) X ts (cid:96) X tt → Y and g : X s (cid:96) X t → Y , its adjoint ˆ g : X → Y , has ˆ g determined by d f : X s → Y and d f : X t → Y , while ˆ g is determined by s d f : X ss → Y and s d f : X tt → Y (where f : X st → Y is the composite X st i −→ X ss (cid:96) X st (cid:96) X ts (cid:96) X tt g −→ Y ).1.9 . Modules. Recall that an abelian group object in a category D with finiteproducts is an object G equipped with a unit map σ : ∗ → G (where ∗ is the terminalobject), and inverse map i : G → G , and a multiplication map µ : G × G → G whichis associative, commutative and unital. We require further that(1.10) µ ◦ (Id , i ) ◦ ∆ = σ ◦ c ∗ , where ∆ : G → G × G is the diagonal, and c ∗ the map to ∗ .1.11. Definition.
Given an object X in a category C , we denote by ( Gpd C , X )the subcategory of Gpd C consisting of those Y with Y = X (and groupoid mapswhich are the identity on X ). For X ∈ ( Gpd C , X ), an X -module is an abeliangroup object M in the slice category ( Gpd C , X ) /X . Since the terminal object of D = ( Gpd C , X ) /X is Id X : X → X , and the product of ρ : M → X with itselfin D is ρp = ρp : M ρ × M → X , a unit map for ρ : M → X is given by a section σ : X → M (with ρσ = Id), and the multiplication and inverse have the forms M ρ × M µ (cid:47) (cid:47) (cid:37) (cid:37) M ρ (cid:124) (cid:124) M i (cid:47) (cid:47) ρ (cid:35) (cid:35) M ρ (cid:123) (cid:123) X X, respectively. Note that (1.10) applied to ρ : M → X implies that(1.12) µ (Id , i ) ◦ ∆ M = σρ , for diagonal ∆ M : M → M ρ × M and σ : X → M the zero map of ρ : M → X .1.13. Remark.
Suppose that X = HY , for H : Spl
C →
Gpd C as in § Y = ( X q (cid:47) (cid:47) π ) t (cid:111) (cid:111) ∈ Spl C . Thus X = X q × X , and an X -module M → X is given by M ρ (cid:47) (cid:47) d (cid:15) (cid:15) d (cid:15) (cid:15) X q × X (cid:15) (cid:15) pr (cid:15) (cid:15) X (cid:47) (cid:47) X DAVID BLANC AND SIMONA PAOLI with ρ = ( d , d ). Note that the fiber M ( a, b ) of ρ over each ( a, b ) ∈ X q × X isan abelian group, with zero σ ( a, b ), and the zero map φ : X → M is given by X q × X φ (cid:47) (cid:47) pr (cid:15) (cid:15) pr (cid:15) (cid:15) M ρ (cid:47) (cid:47) d (cid:15) (cid:15) d (cid:15) (cid:15) X × X pr (cid:15) (cid:15) pr (cid:15) (cid:15) X (cid:91) (cid:91) X s (cid:91) (cid:91) X (cid:91) (cid:91) with ρ φ = Id. Thus for Y as in (1.14), the adjoint ˆ φ ∈ Hom
Spl ( C ) /RHY ( Y, R ( M ))is the composite X t (cid:47) (cid:47) q (cid:15) (cid:15) X × X φ (cid:47) (cid:47) pr (cid:15) (cid:15) M d (cid:15) (cid:15) π t (cid:47) (cid:47) t (cid:79) (cid:79) X (cid:79) (cid:79) X s (cid:79) (cid:79) where η = ( t , t ) is the unit of the adjunction H (cid:97) R , as in (1.4).1.15. Definition.
The idempotent map in
Spl C X tq (cid:47) (cid:47) q (cid:15) (cid:15) X q (cid:15) (cid:15) π (cid:47) (cid:47) t (cid:79) (cid:79) π t (cid:79) (cid:79) induces an idempotent e = H ( tq ) : X = H ( Y ) → X in Gpd C (for Y as in (1.14)).Note that e = tq and e = ( e , e ). We therefore obtain an idempotent operation e on Hom Gpd C /X ( X, M ), taking f : X → M to f e : X → M . We write f e = e ( f ).This sends ( a, b ) ∈ X × q X to f ( tqa, tqb ). Let e ∗ M be the pullback(1.16) e ∗ M r (cid:47) (cid:47) ρ (cid:48) (cid:15) (cid:15) M ρ (cid:15) (cid:15) X e (cid:47) (cid:47) X in Gpd C . If we denote the fiber of ρ (cid:48) at ( a, b ) ∈ X by ( e ∗ M ) ( a, b ), we have( e ∗ M ) ( a, b ) = ( a, b ) × ( ta,tb ) M ( ta, tb ), which is isomorphic under r ( a, b ) to M ( ta, tb ).The unit map σ (cid:48) of e ∗ M is given(1.17) X σe (cid:38) (cid:38) σ (cid:48) (cid:39) (cid:39) Id (cid:40) (cid:40) e ∗ M r (cid:47) (cid:47) ρ (cid:48) (cid:15) (cid:15) M ρ (cid:15) (cid:15) X e (cid:47) (cid:47) X where X σ (cid:47) (cid:47) Id (cid:32) (cid:32) M ρ (cid:125) (cid:125) X is the unit of ρ : M → X , so in particular(1.18) σ ∆ X = s : X → M . Thus for each ( a, b ) ∈ X we have σ (cid:48) ( a, b ) = (( a, b ) , ( σe )( a, b )) = (( a, b ) , σ ( tqa, tqb )). OMONAD COHOMOLOGY OF TRACK CATEGORIES 7
The multiplication ( e ∗ M ) × X ( e ∗ M ) µ (cid:48) −→ ( e ∗ M ) on (( a, b ) , m ) , (( a, b ) , m (cid:48) ) by(1.19) µ (cid:48) ((( a, b ) , m ) , (( a, b ) , m (cid:48) )) = (( a, b ) , µ ( m, m (cid:48) )) , so identifying ( e ∗ M ) × X ( e ∗ M ) with X × X ( M × X M ), we have µ (cid:48) = (Id , µ ).Finally, the zero map of e ∗ M is given on (( a, b ) , m ) ∈ ( e ∗ M ) by σ (cid:48) ρ (cid:48) (( a, b ) , m ) = σ (cid:48) ( a, b ) = { ( a, b ) , σ ( tqa, tqb ) } = { ( a, b ) , σρ ( m ) } since ρ ( m ) = ( tqa, tqb ). Thus O ( e ∗ M ) = σ (cid:48) ρ (cid:48) = (Id , σρ ) = (Id , O M ).1.20 . ( S , O )( S , O )( S , O ) -Categories. In [DK1, § S , O )- Cat , also valid for ( S ∗ , O )- Cat (see § f : X → Y is a fibration (respectively, a weak equivalence) if foreach a, b ∈ O , the induced map f ( a,b ) : X ( a, b ) → Y ( a, b ) is such.The cofibrations in ( S , O )- Cat or ( S ∗ , O )- Cat are not easy to describe. However, forany
K ∈
Cat O , the constant simplicial category c ( K ) ∈ [∆ op , Cat O ] ∼ = ( S , O )- Cat hasa cofibrant replacement defined as follows:Recall that a category Y ∈ Cat O is free if there exists a set S of non-identitymaps in Y (called generators) such that every non-identity map in Y can uniquely bewritten as a finite composite of maps in S . There is a forgetful functor U : Cat O → Graph O to the category of directed graphs, with left adjoint the free category functor F : Graph O → Cat O (see[Ha] and compare [DK1, § U and F are theidentity on objects.Similarly, an ( S , O )-category X ∈ [∆ op , Cat O ] = ( S , O )- Cat is free if for each k ∈ ∆, X k ∈ Cat O is free, and the degeneracy maps in X send generators to generators.Every free ( S , O )-category is cofibrant (cf. [DK1, § K ∈
Cat O ,a canonical cofibrant replacement F • K for c ( K ) in ( S , O )- Cat = [∆ op , Cat O ] ( § F U : Cat O → Cat O (so F n K := ( F U ) n +1 K ). Theaugmentation F • K → K induces a weak equivalence F • K (cid:39) c ( K ) in [∆ op , Cat O ] ∼ =( S , O )- Cat . If K is pointed, F • K is a ( S ∗ , O )-category.More generally, if X is any ( S , O )-category, thought of as a simplicial object in Cat O ,its standard Dwyer-Kan resolution is the cofibrant replacement given by the diagonalDiag F • X of the bisimplicial object F • X ∈ [∆ op , Cat O ] obtained by iterating F U in each simplicial dimension.1.21.
Definition.
The fundamental track category of an ( S , O )-category Z is obtainedby applying the fundamental groupoid functor ˆ π : S →
Gpd to each mapping space Z ( a, b ) (see [GJ, § I.8]. When Z is fibrant, Λ := ˆ π Z has a particularly simpledescription: for each a, b ∈ O , the set of objects of Λ( a, b ) is Z ( a, b ) , and for x, x (cid:48) ∈ Z ( a, b ) , the morphism set (Λ( a.b ))( x, x (cid:48) ) is { τ ∈ Z ( a, b ) : d τ = x, d τ = x (cid:48) } / ∼ ,where ∼ is determined by the 2-simplices of X . Since ˆ π commutes with cartesianproducts for Kan complexes, it extends to ( S , O )- Cat (after fibrant replacement).A module over a track category Λ ∈ ( Gpd , O )- Cat is an abelian group object M in( Gpd , O )- Cat / Λ (see § S , O )-category Z , for each n ≥ π Z -module by applying π n ( − ) to each mapping space of Z .For each track category Λ ∈ ( Gpd , O )- Cat , Λ-module M and n ≥
1, we have atwisted Eilenberg-Mac Lane ( S , O )-category E = E Λ ( M, n ) over Λ, with π n E ∼ = M and π i E ∼ = 0 for 2 ≤ i (cid:54) = n (see [DK2, §
1] and [DKS, § DAVID BLANC AND SIMONA PAOLI
Given Λ ∈ ( Gpd , O )- Cat , a Λ-module M , and an object Z ∈ ( S , O )- Cat equippedwith a twisting map p : ˆ π Z → Λ, the n -th ( S , O )- cohomology group of Z with coeffi-cients in M is H n SO ( Z/ Λ; M ) := [ Z, E Λ ( M, n )] ( S , O )- Cat /B Λ = π map ( S , O )- Cat /B Λ ( Z, E Λ ( M, n )) , where map ( S , O )- Cat /A ( Z, Y ) is the sub-simplicial set of map ( S , O )- Cat ( Z, Y ) consistingof maps over a fixed base A (cf. [DKS, § π Z , with p a weakequivalence; if in addition Z (cid:39) B Λ, we denote H n SO ( Z/ Λ; M ) simply by H n SO (Λ; M ).1.22 . Simplicial model categories. Recall that a simplicial model category M isa model category equipped with functors X (cid:55)→ X ⊗ K and X (cid:55)→ X K , natural in K ∈ S , satisfying appropriate axioms (cf. [Hi, Definition 9.1.6]).For example, S itself is a simplicial model category, with X ⊗ K := X × K and X K := map( K, X ), where map(
K, X ) ∈ S has map( K, X ) n := Hom S ( K × ∆[ n ] , X ).Similarly, ( S , O )- Cat is also a simplicial model category (see [DK1, Proposition 7.2]).1.23.
Definition.
Let M be a simplicial model category. The realization | X | of X ∈ [∆ op , M ] is defined to be the coequalizer of the maps (cid:96) ( σ :[ n ] → [ k ]) ∈ ∆ X k ⊗ ∆[ n ] φ (cid:47) (cid:47) ψ (cid:47) (cid:47) (cid:96) n ≥ X n ⊗ ∆[ n ] , where on the summand indexed by σ : [ n ] → [ k ], φ is the composite of σ ∗ ⊗ ∆[ k ] : X k ⊗ ∆[ n ] → X n ⊗ ∆[ n ] with the inclusion into the coproduct, and ψ is the compositeof 1 X k ⊗ σ ∗ : X k ⊗ ∆[ n ] → X k ⊗ ∆[ k ] with the same inclusion (see [GJ, § VII.3]).Similarly, if X ∈ [∆ , M ] is a cosimplicial object in M , its total object Tot X isthe equalizer of (cid:81) [ n ] ∈ Ob ∆ ( X n ) ∆[ n ] φ (cid:47) (cid:47) ψ (cid:47) (cid:47) (cid:81) ( σ :[ n ] → [ k ]) ∈ ∆ ( X k ) ∆[ n ] with φ and ψ defined dually (cf. [Hi, Def. 18.6.3]).The following is a straightforward generalization of [BK, XII 4.3]:1.24. Lemma. If M = [∆ op , D ] is a simplicial model category and X ∈ [∆ op , M ] ∼ =[∆ op , D ] , then Diag X ∼ = | X | and map M (Diag X, K ) ∼ = Tot map M ( X, K ) . Short exact sequences
We now associate to any internal groupoid of the form X = HY (cf. § X -module M a certain short exact sequence of abelian groups (see Proposition 2.22).When C = Cat O , this can be rewritten in a more convenient form (see Proposition4.9); however, in this section we present it in a more general context, which may beuseful in future work. A similar short exact sequence appears in [VO, Theorem 3.5]for C an algebraic category (with a different description of the third term). When C = Gp , it reduces to [P1, Lemma 6], though the method of proof there is different. OMONAD COHOMOLOGY OF TRACK CATEGORIES 9
Definition.
Let X = H ( X q (cid:47) (cid:47) π ) t (cid:111) (cid:111) , with d X the discrete internal groupoidon X . We define j : d X → X to be the map X X (cid:47) (cid:47) Id (cid:15) (cid:15) Id (cid:15) (cid:15) X × q X pr (cid:15) (cid:15) pr (cid:15) (cid:15) X (cid:88) (cid:88) Id X (cid:47) (cid:47) X X (cid:90) (cid:90) in ( Gpd C , X ). Consider the pullback(2.2) j ∗ M k (cid:47) (cid:47) λ (cid:15) (cid:15) M ρ (cid:15) (cid:15) d X j (cid:47) (cid:47) X in Gpd C /X , where d = d = λ : ( j ∗ M ) → X , since d X is discrete. Because(2.2) induces(2.3) ( j ∗ M ) k (cid:47) (cid:47) λ (cid:15) (cid:15) M ρ (cid:15) (cid:15) X j =∆ X (cid:47) (cid:47) X × q X = X (a pullback in C ), we shall denote ( j ∗ M ) by j ∗ M .We shall use the following abbreviations for the relevant Hom groups:(2.4) Hom( π , t ∗ j ∗ M ) := Hom C /π (( π −→ π ) , ( t ∗ j ∗ M r −→ π )) , Hom( X , j ∗ M ) := Hom C /X (( X tq −→ X )) , ( j ∗ M λ −→ X )) , Hom( X , e ∗ M ) := Hom C /X (( X −→ X )) , ((( e ∗ M ) ρ −→ X ))) , Hom(
X, e ∗ M ) := Hom Gpd C /X (( X Id −→ X ) , ( e ∗ M ρ (cid:48) −→ X )) . Definition.
In the situation described in § X f (cid:47) (cid:47) e = tq (cid:35) (cid:35) j ∗ M λ (cid:121) (cid:121) X in C /X , we have ρ k f = ∆ X λ f = ∆ X e = e ∆ X , since the square(2.6) X e (cid:47) (cid:47) ∆ X (cid:15) (cid:15) X X (cid:15) (cid:15) X e (cid:47) (cid:47) X commutes. Now let v : X → ( e ∗ M ) be given by(2.7) X k f (cid:35) (cid:35) v (cid:35) (cid:35) ∆ X e = e ∆ X (cid:35) (cid:35) ( e ∗ M ) r (cid:47) (cid:47) ρ (cid:48) (cid:15) (cid:15) M ρ (cid:15) (cid:15) X e (cid:47) (cid:47) X where ρ k f = ∆ X λ f = ∆ X e = e ∆ X = e e ∆ X by (2.2) and (2.6).Since e = tq , the following diagram commutes:(2.8) X q (cid:47) (cid:47) v (cid:15) (cid:15) π t (cid:15) (cid:15) X X (cid:15) (cid:15) ( e ∗ M ) ρ (cid:48) (cid:47) (cid:47) X so v induces ( v, v ) : X = X × q X → ( e ∗ M ) × X ( e ∗ M ) . In the notation of (2.4),we may therefore define ϑ : Hom( X , j ∗ M ) → Hom( X , e ∗ M ) by letting ϑ ( f ) : X → ( e ∗ M ) be the composite(2.9) X v,v ) −−→ ( e ∗ M ) × X ( e ∗ M ) ,i ) −−−→ ( e ∗ M ) × X ( e ∗ M ) µ −→ ( e ∗ M ) where i : e ∗ M → e ∗ M is the inverse map for the abelian group structure on e ∗ M .2.10. Lemma.
The map ϑ = ( e , ϑ ) lands in Hom(
X, e ∗ M ) , for e = tq and ϑ asin (2.9) .Proof. By (1.12) we have σ (cid:48) ρ (cid:48) = µ (cid:48) (Id , i (cid:48) )∆ ( e ∗ M ) , so(2.11) X v (cid:47) (cid:47) e ∆ X (cid:35) (cid:35) ( e ∗ M ) ∆ ( e ∗ M )1 (cid:47) (cid:47) ρ (cid:48) (cid:15) (cid:15) ( e ∗ M ) × X ( e ∗ M ) (Id ,i ) (cid:47) (cid:47) ( e ∗ M ) × X ( e ∗ M ) µ (cid:47) (cid:47) ( e ∗ M ) X σ (cid:48) (cid:49) (cid:49) commutes, as does(2.12) X X (cid:47) (cid:47) v (cid:15) (cid:15) X v,v ) (cid:15) (cid:15) ( e ∗ M ) ( e ∗ M )1 (cid:47) (cid:47) ( e ∗ M ) × X ( e ∗ M ) so by (2.11), (2.12), (1.18), and the definition of ϑ ( f ) we see that(2.13) ϑ ( f )∆ X = µ (Id , i )( v, v )∆ X = σ (cid:48) e ∆ X = σ (cid:48) ∆ X e = s (cid:48) e By (1.19), for each ( a, b ) ∈ X × q X (with tqa = tqb ) we have(2.14) ϑ ( f )( a, b ) = (( tqa, tqa ) , µ ( k f a, i k f b )) , OMONAD COHOMOLOGY OF TRACK CATEGORIES 11 so d (cid:48) ϑ ( f )( a, b ) = tqa = e pr ( a, b ) and d (cid:48) ϑ ( f )( a, b ) = tqb = e pr ( a, b ). Thus(2.15) d i ϑ ( f ) = e pr i for i = 0 ,
1. Thus (2.13) and (2.15) show that(2.16) X ϑ ( f ) (cid:47) (cid:47) pr (cid:15) (cid:15) pr (cid:15) (cid:15) ( e ∗ M ) d (cid:48) (cid:15) (cid:15) d (cid:48) (cid:15) (cid:15) X X (cid:91) (cid:91) e (cid:47) (cid:47) X s (cid:48) (cid:87) (cid:87) commutes. Now, given ( a, b, c ) ∈ X × q X × q X (with tqa = tqb = tqc ), we have c (cid:48) ( ϑ ( f )( a, b ) × ϑ ( f )( b, c )) = (( tqa, tqa ) , µ ( k f a, i k f b ) ◦ µ ( k f b, i k f c ))while ϑ ( f ) c (cid:48)(cid:48) ( a, b, c ) = ϑ ( f )( a, c ) = (( tqa, tqa ) , µ ( k f a, i k f c )), where we denotedby c (cid:48)(cid:48) : X × X X → X and c (cid:48) : ( e ∗ M ) × X ( e ∗ M ) the compositions.Since µ is a map of groupoids, by the interchange rule we see that µ ( k f a, i k f b ) ◦ µ ( k f b, i k f c ) = µ ( k f a, i k f c )where ◦ the groupoid composition.Finally, ρ (cid:48) ◦ ϑ ( f ) = e , so ( e , ϑ ( f )) is indeed a morphism in Gpd C /X . (cid:3) Definition.
The pullback(2.18) t ∗ j ∗ M l (cid:47) (cid:47) r (cid:15) (cid:15) j ∗ M λ (cid:15) (cid:15) dπ t (cid:47) (cid:47) d X in Gpd C /dπ gives rise to a pullback(2.19) t ∗ j ∗ M l (cid:47) (cid:47) r (cid:15) (cid:15) j ∗ M λ (cid:15) (cid:15) π t (cid:47) (cid:47) X in C , so we may define(2.20) ξ : Hom( π , t ∗ j ∗ M ) → Hom( X , j ∗ M )by sending π f (cid:47) (cid:47) Id (cid:35) (cid:35) t ∗ j ∗ M r (cid:120) (cid:120) π , to the map ξ ( f ) given by(2.21) X l fq (cid:47) (cid:47) tq (cid:34) (cid:34) j ∗ M λ (cid:122) (cid:122) X Proposition.
Given X = H ( Y ) ∈ Gpd C as in (1.14) and M ∈ [( Gpd C , X ) /X ] ab ,there is a short exact sequence of abelian groups → Hom( π , t ∗ j ∗ M ) ξ −→ Hom( X , j ∗ M ) ϑ −→ Hom(
X, e ∗ M ) → , in the notation of (2.4) , where ξ is as in (2.20) and ϑ is as in Lemma 2.10. Proof.
We first show that Im ξ ⊆ ker ϑ : Given f (cid:48) : π → t ∗ j ∗ M in Hom( π , t ∗ j ∗ M ),the map ξ ( f (cid:48) ) ∈ Hom( X , j ∗ M ) is given by the composite X q −→ π f (cid:48) −→ t ∗ j ∗ M l −→ j ∗ M By (2.14) and (2.21), for each ( a, b ) ∈ X we have ϑ ( ξ ( f (cid:48) ))( a, b ) = { ( tqa, tqb ) , µ ( k l f (cid:48) q ( a ) , i k l f (cid:48) q ( b )) } . Since q ( a ) = q ( b ), we have ϑ ( ξ ( f (cid:48) ))( a, b ) = (( tqa, tqb ) , Gpd C /X ( X, e ∗ M ). This shows that Im ξ ⊆ ker ϑ .Given g (cid:48) : X → j ∗ M (as in (2.21)) in ker ϑ , for all ( a, b ) ∈ X we have ϑ ( g (cid:48) )( a, b ) = { ( tqa, tqa ) , µ ( k g (cid:48) ( a ) , i k g (cid:48) ( b ) } = { ( tqa, tqb ) , } . Thus k g (cid:48) pr ( a, b ) = k g (cid:48) a = k g (cid:48) b = k g (cid:48) pr ( a, b ), so that(2.23) k g (cid:48) pr = k g (cid:48) pr . Since X pr (cid:47) (cid:47) pr (cid:47) (cid:47) X q (cid:47) (cid:47) π is a coequalizer, it follows from (2.23) that there is amap f : π → M with f q = k g (cid:48) , and thus f = f qt = k g (cid:48) t , so ρ f = ρ k g (cid:48) t =∆ X λ g (cid:48) t = ∆ X tqt = ∆ X t . Hence there is f : π → j ∗ M defined by f (cid:62) t : π → M × X into the pullback (2.3). Since λ f = t , there is also a map f (cid:48) : π → t ∗ j ∗ M defined by f (cid:62) Id : π → j ∗ M × π into the pullback (2.19).By (2.23) and the above we have k g (cid:48) = f q = k f q . Since k is monic, thisimplies that g (cid:48) = f q , and since f = l f (cid:48) , also g (cid:48) = l f (cid:48) q = ϑ ( f (cid:48) ). This shows thatker ϑ ⊆ Im ξ . In conclusion, ker ϑ = Im ξ .To show that ξ is monic, assume given f, g ∈ Hom( π , t ∗ j ∗ M ) with ξf = ξg .Then l f q = l gq , which implies that(2.24) l f = l g , since q is epic. Also, ρ k l f = ∆ X ts f and ρ k l g = ∆ X ts g , so by (2.24) wehave ∆ X ts f = ∆ X ts g and therefore s f = qpr ∆ X ts f = qpr ∆ X ts g = s g .By the definition of t ∗ j ∗ M as the pullback (2.19), together with (2.24) thisimplies that f = g . Thus ξ is a monomorphism.To show that ϑ is onto, assume given φ ∈ Hom
Gpd C /X ( X, e ∗ M ) (so that ρ (cid:48) ◦ φ = e ). The adjoint of φ in Hom Spl C /Y ( Y, R ( e ∗ M )), for Y as in (1.14), is given by acommuting triangle in Spl C :(2.25) X g (cid:47) (cid:47) q (cid:15) (cid:15) ( t,tq ) (cid:32) (cid:32) ( e ∗ M ) d (cid:15) (cid:15) ρ (cid:48) (cid:124) (cid:124) π t (cid:79) (cid:79) t (cid:47) (cid:47) t (cid:32) (cid:32) X s (cid:79) (cid:79) Id (cid:124) (cid:124) X pr (cid:15) (cid:15) X X (cid:79) (cid:79) OMONAD COHOMOLOGY OF TRACK CATEGORIES 13
The adjoint φ of ( g, t ) is given by postcomposing with the counit of H (cid:97) R , so φ isthe horizontal composite in:(2.26) X × q X g,g ) (cid:47) (cid:47) pr (cid:15) (cid:15) pr (cid:15) (cid:15) g (cid:37) (cid:37) ( e ∗ M ) × d (cid:48) ( e ∗ M ) µ (cid:47) (cid:47) pr (cid:15) (cid:15) pr (cid:15) (cid:15) ( e ∗ M ) d (cid:48) (cid:15) (cid:15) d (cid:48) (cid:15) (cid:15) ρ (cid:48) (cid:121) (cid:121) X X (cid:71) (cid:71) g (cid:47) (cid:47) e (cid:38) (cid:38) ( e ∗ M ) d (cid:48) (cid:47) (cid:47) ∆ ( e ∗ M )1 (cid:86) (cid:86) X s (cid:48) (cid:87) (cid:87) Id (cid:121) (cid:121) X × q X pr (cid:15) (cid:15) pr (cid:15) (cid:15) X X (cid:87) (cid:87) By § µ is µ { (( a, b ) , m ) , (( a, b ) , m (cid:48) ) } = (( a, b ) , m ◦ m (cid:48) ). Since t tq = ∆ tq by (1.5), ) by (2.25) we have ρ (cid:48) g = t tq = ∆ tq = ∆ e .We have a map g : X → ( e ∗ M ) into the pullback square of (2.7) given by σ j (cid:62) t tq : X → M × X . This in turn defines g (cid:48) : X → j ∗ ( e ∗ M ) , given by anotherpullback: X g (cid:39) (cid:39) g (cid:48) (cid:39) (cid:39) tq (cid:39) (cid:39) j ∗ ( e ∗ M ) k (cid:47) (cid:47) (cid:15) (cid:15) ( e ∗ M ) ρ (cid:48) (cid:15) (cid:15) X X (cid:47) (cid:47) X We define g (cid:48) M : X → j ∗ M into the pullback (2.3) by σ j (cid:62) tq : X → M × X ,since ∆ X tq = ρ σ j . By the definitions of g , g (cid:48) M , and ϑ , for each ( a, b ) ∈ X wehave ϑ ( g (cid:48) M )( a, b ) = { ( tqa, tqb ) , µ ( g (cid:48) M ( a ) , i k g (cid:48) M ( b )) } , while by (2.26) we have: φ ( a, b ) = { ( tqa, tqb ) , η (( tqa, tqb ) , k g (cid:48) M ( a ) , ( tqa, tqb ) , k (cid:48) g (cid:48) M ( b ) } ==(( tqa, tqb ) , g (cid:48) M ( a ) ◦ i k g (cid:48) M ( b )) , with g (cid:48) M ( a ) ◦ i k g (cid:48) M ( b ) ∈ M ( tqa, tqb ). By the Eckmann-Hilton argument, theabelian group structure on M ( tqa, tqb ) is the same as the groupoid structure, so g (cid:48) M ( a ) ◦ i k g (cid:48) M ( b ) = µ ( k g (cid:48) M ( a ) , i k g (cid:48) M ( b )). We conclude that ϑ ( g (cid:48) )( a, b ) = φ ( a, b )for each ( a, b ) ∈ X , so ϑ ( g (cid:48) ) = φ , as required. (cid:3) We now rewrite the map ϑ of Proposition 2.22 in a different form (see Lemma 2.32).This will be used in Proposition 4.9 in the case C = Cat O :2.27. Definition.
Given X = H ( Y ) ∈ Gpd C as in (1.14), we define a map(2.28) ϑ (cid:48) : Hom C /X ( X , j ∗ M ) → Hom
Gpd C /X ( X, M )as follows: the pullback square (2.3) implies that a map X → j ∗ M is given by f : X → M with ρ f = ∆ X . We then define ¯ ϑ ( f ) : X → M by ¯ ϑ ( f )( a, b ) = σ ( a, b ) ◦ f ( b ) − f ( a ) ◦ σ ( a, b ), where ◦ is the groupoid composition in M and the subtraction is that of the abelian group object M . This gives rise to a map X ϑ ( f ) (cid:47) (cid:47) pr (cid:15) (cid:15) M d (cid:15) (cid:15) X (cid:47) (cid:47) ∆ X (cid:79) (cid:79) X s (cid:79) (cid:79) in Spl C , where for each ( a, b ) ∈ X , with d ¯ ϑ ( f )( a, b ) = (Id ◦ p )( a, b ) = a we have¯ ϑ ( f )∆ X ( a ) = ¯ ϑ ( f )( a, a ) = σ ( a, a ) f ( a ) − f ( a ) σ ( a, a ) = O M = σ ( a, a ) = s ( a ) . Here σ ∆ X = s : X → M because ( σ, Id) : X → M is a map of groupoids. Thuswe have a map in Hom Spl C /RHY ( Y, R ( M )) given by the composite X t (cid:47) (cid:47) q (cid:15) (cid:15) X ϑ ( f ) (cid:47) (cid:47) pr (cid:15) (cid:15) M d (cid:15) (cid:15) π t (cid:47) (cid:47) t (cid:79) (cid:79) X (cid:47) (cid:47) ∆ X (cid:79) (cid:79) X s (cid:79) (cid:79) By § ϑ (cid:48) ( f ) in Hom Gpd C /X ( X, M ) with ϑ (cid:48) ( f ) = Id.and ϑ (cid:48) ( f ) = ¯ ϑ ( f ). We define(2.29) φ : Hom Gpd C /X (( X Id −→ X ) , ( M ρ −→ X )) → Hom
Gpd C /X (( X e −→ X ) , ( e ∗ M ρ (cid:48) −→ X ))as follows. By § Gpd C /X (( X Id −→ X ) , ( M ρ −→ X )) is determinedby the map(2.30) X g (cid:47) (cid:47) pr (cid:15) (cid:15) M d (cid:15) (cid:15) X (cid:47) (cid:47) ∆ X (cid:79) (cid:79) X s (cid:79) (cid:79) in Spl C . We associate to this another map in Spl C :(2.31) X (cid:101) φ ( g ) (cid:47) (cid:47) pr (cid:15) (cid:15) e ∗ M d (cid:48) (cid:15) (cid:15) X e (cid:47) (cid:47) ∆ X (cid:79) (cid:79) X s (cid:48) (cid:79) (cid:79) where (cid:101) φ ( g ) into the pullback square of (2.7) is given by ψ g (cid:62) e : X → M × X ,with ψ g ( a, b ) = σ ( tqa, a ) g ( a, b ) σ ( b, tqb ). Note that, since ρ ( m ) = ( ∂ m, ∂ m ): ρ ψ g ( a, b ) = ρ σ ( tqa, a ) ρ g ( a, b ) ρ σ ( b, tqb ) = ( tqa, a )( a, b )( b, tqb ) = e ( a, b ) . We may check the commutativity of (2.31), using (1.17). The map ( e , (cid:101) φ ( g )) in(2.31) then defines an element φ ( g ) in Hom Gpd C /X (( X e −→ X ) , ( e ∗ M ρ (cid:48) −→ X )).2.32. Lemma.
For ϑ (cid:48) as in (2.28) and φ as in (2.29) , the diagram (2.33) Hom C /X (( X Id −→ X ) , ( j ∗ M λ −→ X ) ϑ (cid:48) (cid:47) (cid:47) ( tq ) ∗ (cid:15) (cid:15) Hom
Gpd C /X ( X, M ) φ (cid:15) (cid:15) Hom C /X (( X tq −→ X ) , ( j ∗ M λ −→ X ) ϑ (cid:47) (cid:47) Hom
Gpd C /X ( X, e ∗ M ) OMONAD COHOMOLOGY OF TRACK CATEGORIES 15 commutes, with vertical isomorphisms, where ( tq ) ∗ (Id , f ) = ( tq, ˆ f ) for (Id , f ) ∈ Hom C /X ( X , j ∗ M ) and ˆ f ( a ) = σ ( tqa, a ) f ( a ) σ ( a, tqa ) .Proof. For each ( a, b ) ∈ X we have θ (cid:48) (Id , f )( a, b ) = σ ( a, b ) f ( b ) − f ( a ) σ ( a, b ), so φϑ (cid:48) (Id , f )( a, b ) = σ ( tqa, a ) { σ ( a, b ) f ( b ) − f ( a ) σ ( a, b ) } σ ( tqb, b ) == σ ( tqa, a ) σ ( a, b ) f ( b ) σ ( b, tqb ) − σ ( tqa, a ) f ( a ) σ ( a, b ) σ ( b, tqb ) == ˆ f ( b ) − ˆ f ( a ) = ϑ ( ˆ f )( a, b ) = ϑ ( tq ) ∗ (Id , f )( a, b )Thus (2.33) commutes. The map ( tq ) ∗ is an isomorphism, with inverse sending( tq, g ) : X → e ∗ j ∗ M to (Id , (cid:101) g ) : X → j ∗ M , where (cid:101) g ( a ) = σ ( a, tqa ) g ( a ) σ ( tqa, a ).The map φ is an isomorphism by construction. (cid:3) Comonad resolutions for ( S , O ) -cohomology We now define the comonad on track categories which is used to construct functorialcofibrant replacements, yielding a formula for computing the ( S , O )-cohomology of atrack category. This will provide crucial ingredients (Theorem 3.29 and Corollary3.34) for our main result, Theorem 4.12.3.1 . Track categories. Track categories, the objects of
Track O = Gpd ( Cat O ) of § X with object set O (and functors which are identity onobjects) such that for each a, b ∈ O the category X ( a, b ) is a groupoid. The doublenerve functor provides an embedding N (2) : Track O (cid:44) → [∆ op , Set ]The lower right corner of N (2) X (omitting degeneracies), appears as follows, withthe vertical direction groupoidal, and the horizontal categorical: · · · (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) X × X X (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:111) (cid:111) (cid:111) (cid:111) O (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:111) (cid:111) X × O X (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:79) (cid:79) (cid:79) (cid:79) X (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:111) (cid:111) (cid:111) (cid:111) (cid:79) (cid:79) (cid:79) (cid:79) O (cid:15) (cid:15) (cid:15) (cid:15) (cid:111) (cid:111) (cid:79) (cid:79) (cid:79) (cid:79) X × O X (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:79) (cid:79) X (cid:47) (cid:47) (cid:47) (cid:47) (cid:111) (cid:111) (cid:111) (cid:111) (cid:79) (cid:79) O (cid:111) (cid:111) (cid:79) (cid:79) There is a functor Π : Track O → Cat O given by dividing out by the 2-cells: thatis,(Π X ) = O and (Π X ) = qX , where X is the groupoid of 1- and 2-cells in X and q : Gpd → Set is the connected component functor.3.2.
Definition.
We say X ∈ Track O is homotopically discrete if, for each a, b ∈ O ,the groupoid X ( a, b ) is an equivalence relation, that is, a groupoid with no non-trivialloops.3.3. Remark.
By taking nerves in the groupoid direction we define(3.4) I : Track O = Gpd ( Cat O ) → [∆ op , Cat O ] = ( S , O )- Cat If F : X → Y in Track O is a 2-equivalence (so for each a, b ∈ O , F ( a, b ) isan equivalence of groupoids), IF is a Dwyer-Kan equivalence of the corresponding( S , O )-categories. In particular, if X ∈ Track O is homotopically discrete, and d Π X isa track category with only identity 2-cells), this holds for the obvious F : X → dΠ X . . The comonad KKK . Taking C = Cat O in § L : Cat O (cid:28) Gpd ( Cat O ) = Track O : U .
Let
Graph O be the category of reflexive graphs with object set O and morphismswhich are identity on objects (where a reflexive graph is a diagram X d (cid:47) (cid:47) d (cid:47) (cid:47) X s (cid:111) (cid:111) with d s = d s = Id). There are adjoint functors F : Graph O (cid:29) Cat O : V , where V is the forgetful functor and F is the free category functor. By composition, we obtaina pair of adjoint functors(3.6) LF : Graph O (cid:29) Track O : V U , and therefore a comonad ( K , ε, δ ) , K = LF V U : Track O → Track O , where ε is thecounit of the adjunction (3.6), δ = LF ( η ) V U , and η the unit of the adjunction (3.6).For each X ∈ Track O we obtain a simplicial object K • X ∈ [∆ op , Track O ] with K n X = K n +1 X and face and degeneracy maps given by ∂ i = K i ε K n − i : K n +1 X → K n X and σ i = K i δ K n − i : K n +1 X → K n +2 X .
The simplicial object K • X is augmented over X via ε : K • X → X , and K • X is asimplicial resolution of X (see [W]).3.7. Remark.
The augmented simplicial object
V U ( K • X ) V Uε −−→
V U X is aspherical(see for instance [W, Proposition 8.6.10]).3.8.
Remark.
Given X ∈ Cat O by (1.7) we see that LX is a homotopically discretetrack category (Definition 3.2), with Π LX = X . There are two canonical splittingsΠ LX = X → ( LX ) = X s (cid:96) X t , given by the inclusion in the s or in the t copy of X . Since K Y = L ( F V U Y ) for each Y ∈ Track O , the same holds for K Y .Furthermore, since F V U Y is a free category, so is Π K Y = F V U Y . Since F pre-serves coproducts (being a left adjoint), ( K Y ) = F V U Y (cid:96)
F V U Y = F ( V U Y (cid:96)
V U Y )and, using (1.8), ( K Y ) = F ( V U Y (cid:96)
V U Y (cid:96)
V U Y (cid:96)
V U Y ). Thus both ( K Y ) and ( K Y ) are free categories. Similarly, ( K Y ) r is a free category for each r > . The comonad resolution. Let I : [∆ op , Track O ] → [∆ op , ( S , O )- Cat ] be thefunctor obtained by applying the internal nerve functor I of (3.4) levelwise ineach simplicial dimension – so for X ∈ Track O , I K • X ∈ [∆ op , ( S , O )- Cat ]. Similarly, N : ( S , O )- Cat → [∆ op , Set ] is obtained by applying the nerve functor in the categorydirection; applying this levelwise to I K • X yields(3.10) W = N I K • X ∈ [∆ op , Set ] . Below is a picture of the corner of W , in which the horizontal simplicial direction isgiven by the comonad resolution, the vertical is given by the nerve of the groupoidin each track category, and the diagonal is given by the nerve of the category in each OMONAD COHOMOLOGY OF TRACK CATEGORIES 17 track category.(3.11) •• • •• • ( K X ) × O ( K X ) ( K X ) × O ( K X ) ( K X ) × O ( K X ) ( K X ) × O ( K X ) ( K X ) × O ( K X ) ( K X ) × O ( K X ) ( K X ) × O ( K X ) ( K X ) × O ( K X ) ( K X ) × O ( K X ) ( K X ) × O ( K X ) ( K X ) × O ( K X ) ( K X ) × O ( K X ) ( K X ) × ( K X ) ( K X ) ( K X ) × ( K X ) ( K X ) ( K X ) × ( K X ) ( K X ) ( K X ) × ( K X ) ( K X ) ( K X ) × ( K X ) ( K X ) ( K X ) × ( K X ) ( K X ) ( K X ) ( K X ) ( K X ) ( K X ) ( K X ) ( K X ) ( K X ) ( K X ) ( K X ) ( K X ) ( K X ) ( K X ) O OO OO O
Note that the augmentation (cid:15) : K • X → X induces a map in [∆ op , Set ]:(3.12) W → N IcX.
Now let Z = W (2) be W thought of as a simplicial object in [∆ op , Set ] along thedirection appearing diagonal in the picture, that is(3.13) Z ∈ [∆ op , [∆ op , Set ]] , with Z the constant bisimplicial set at O , Z given by... ( K X ) × ( K X ) ( K X ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) ( K X ) × ( K X ) ( K X ) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) ( K X ) × ( K X ) ( K X ) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) · · · ( K X ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) ( K X ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) ( K X ) (cid:15) (cid:15) (cid:15) (cid:15) ( K X ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) ( K X ) (cid:47) (cid:47) (cid:47) (cid:47) ( K X ) with(3.14) Z k ∼ = Z × Z k · · ·× Z Z , for each k ≥ op , Set ] → [∆ op , Set ] dimensionwise to Z (viewed as in (3.13)), we obtain Diag Z ∈ [∆ op , [∆ op , Set ]].To show that Diag Z is an ( S , O )-category, we must show that it behaves like thenerve of a category object in simplicial sets in the outward simplicial direction: thismeans that (Diag Z ) is the “simplicial set of objects” – and indeed it is the constantsimplicial set at O . Similarly, (Diag Z ) is the “simplicial set of arrows”. Since Diag preserves limits, by (3.14) we have(Diag Z ) k ∼ = Diag Z k ∼ = Diag Z × Diag Z k · · ·× Diag Z Diag Z =(Diag Z ) × (Diag Z ) k · · ·× (Diag Z ) (Diag Z ) . for each k ≥
2. Thus Diag Z is 2-coskeletal, with unique fill-ins for inner 2-horns(the composite) so it is indeed in ( S , O )- Cat , and the map (3.12) induces a map α : Diag Z → IX for I as in (3.4).3.15. Lemma.
For Z and Diag Z as above, α : Diag Z → IX is a Dwyer-Kanequivalence in ( S , O ) - Cat .Proof.
We need to show that, for each a, b ∈ O (3.16) (Diag Z )( a, b ) → X ( a, b )is a weak homotopy equivalence. Note that (Diag Z )( a, b ) is the diagonal of(3.17) ... ... ... ··· ( K X ) ( a, b ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) ( K X ) ( a, b ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) ( K X ) ( a, b ) (cid:15) (cid:15) (cid:15) (cid:15) ··· ( K X ) ( a, b ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) ( K X ) ( a, b ) (cid:47) (cid:47) (cid:47) (cid:47) ( K X ) ( a, b ) , and we have a map from (3.17) to the horizontally constant bisimplicial set(3.18) ... ... ... ··· X ( a, b ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) X ( a, b ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) X ( a, b ) (cid:15) (cid:15) (cid:15) (cid:15) ··· X ( a, b ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) X ( a, b ) (cid:47) (cid:47) (cid:47) (cid:47) X ( a, b )inducing the map (3.16) on diagonals.We shall show that this map of bisimplicial sets from (3.17) to (3.18) is a weakequivalence of simplicial sets in each vertical dimension. The corresponding map ofdiagonals (3.16) is then a weak equivalence by [GJ, Prop.1.7].Consider first vertical dimension 1. We must show that the map of simplicial sets(3.19) ( K • X ) ( a, b ) → cX ( a, b )is a weak equivalence, where cX ( a, b ) denotes the constant simplicial set at X ( a, b ).Note that W = ( V U W ) for each W ∈ Track O , where U is the internal arrowfunctor and V is the underlying graph functor (with ( V Y ) = Y for each Y ∈ Cat O ),so W ( a, b ) = ( V U W ) ( a, b ) and thus ( K • X ) ( a, b ) = ( V U K • X ) ( a, b ) for each a, b ∈ O .By Remark 3.7, the simplicial object V U K • X is aspherical, so ( V U K • X ) ( a, b )is, too, and is thus weakly equivalent to c ( V U X ) ( a, b ) = cX ( a, b ). Thus (3.19) isa weak equivalence.In vertical dimension 0, from the vertical simplicial structure of (3.17) we see(3.20) ( K • X ) ( a, b ) → cX ( a, b )is a retract of (3.19), so it is also a weak equivalence. OMONAD COHOMOLOGY OF TRACK CATEGORIES 19
In vertical dimension 2, we must show that(3.21) ( K • X ) ( a, b ) × ( K • X ) ( a,b ) ( K • X ) ( a, b ) → cX ( a, b ) × cX ( a,b ) cX ( a, b )is a weak equivalence. This is the induced map of pullbacks of the diagram(3.22) ( K • X ) ( a, b ) ∂ • (cid:47) (cid:47) (cid:15) (cid:15) ( K • X ) ( a, b ) (cid:15) (cid:15) ( K • X ) ( a, b ) ∂ • (cid:111) (cid:111) (cid:15) (cid:15) cX ( a, b ) ∂ (cid:47) (cid:47) cX ( a, b ) cX ( a, b ) ∂ (cid:111) (cid:111) in S . By the above discussion, the vertical maps in (3.22) are weak equivalences.By definition of K there is a pullback(3.23) ( K • X ) ( a, b ) ∂ • (cid:47) (cid:47) ∂ • (cid:15) (cid:15) ( K • X ) ( a, b ) ∇ (cid:15) (cid:15) ( K • X ) ( a, b ) ∇ (cid:47) (cid:47) Π ( K • X ) ( a, b )in S , where ∇ : ( K • X ) ( a, b ) = Π ( K • X ) ( a, b ) (cid:96) Π ( K • X ) ( a, b ) → Π ( K • X ) ( a, b ) . is the fold map. To see that ∇ is a fibration, let Y • = Π ( K • X ) ( a, b ). For anycommuting diagram Λ k [ n ] α (cid:47) (cid:47) (cid:127) (cid:95) j (cid:15) (cid:15) Y • (cid:96) Y •∇ (cid:15) (cid:15) ∆[ n ] β (cid:47) (cid:47) Y • in S , α factors through i t : Y • (cid:44) → Y • (cid:96) Y • ( t = 1 , k [ n ] is connected, soΛ k [ n ] α (cid:47) (cid:47) α (cid:48) (cid:37) (cid:37) Y • (cid:96) Y • Y • i t (cid:56) (cid:56) commutes, and thus ∇ i t β = β and i t βj = i t ∇ α = i t ∇ i t α (cid:48) = i t α (cid:48) = α .The maps ∂ • and ∂ • are fibrations, since they are pullbacks of such by (3.23).The bottom horizontal maps in (3.22) are fibrations since their target is discrete.We conclude that the induced map of pullbacks (3.21) is a weak equivalence.Vertical dimension i > (cid:3) Lemma.
For any X ∈ Track O , Diag
N I K • X is a free ( S , O ) -category (see § By Remark 3.8, for each track category K r X the nerve in the groupoiddirection is a free category in each simplicial degree. Thus for Z := N I K • X ,Diag Z is also a free category in each simplicial degree. By § σ i : K n +1 X → K n +2 X are given by σ i = K i LF ( η ) V U K n − i , where η : Id → V U LF is the unit of the adjunction. Therefore, σ i sends generators to generators and so thesame holds for the degeneracies of Diag Z , so it is a free ( S , O )-category. (cid:3) Corollary.
For any X ∈ Track O , Diag
N I K • X is a cofibrant replacement of IX in ( S , O ) - Cat . Proof.
By Lemma 3.15, the map Diag Z → IX is a weak equivalence. (cid:3) We now show how to use the comonad resolution of a track category to computeits ( S , O )-cohomology:3.26. Proposition.
For X ∈ Track O , let Z = N I K • X and let M be a Dwyer-Kanmodule over X . Then (3.27) H n − i SO ( IX ; M ) = π i map ( S , O ) - Cat /IX (Diag Z, K X ( M, n )) . Proof.
Let φ : Diag F • X → IX be the Dwyer-Kan standard free resolution of § H n − i SO ( IX ; M ) = π i map ( S , O )- Cat /IX (Diag F • X, K X ( M, n )) , where K X ( M, n ) is the twisted Eilenberg-Mac Lane ( S , O )-category of § α : Diag Z → IX is a cofibrant replacement for IX , so given acommuting diagram ∗ (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) Diag F • X φ (cid:15) (cid:15) Diag Z α (cid:47) (cid:47) IX, there is a lift ψ : Diag Z → Diag F • X with φψ = α and ψ a weak equivalence. Hence π i map ( S , O )- Cat /IX (Diag Z, K X ( M, n )) ∼ = π i map ( S , O )- Cat /IX (Diag F • X, K X ( M, n ))Thus (3.27) follows from (3.28). (cid:3)
Theorem.
Let X ∈ Track O , and M a module over X ; then for each s ≥ , H s SO ( IX ; M ) = π s C • for the cosimplicial abelian group (3.30) C • := π map ( S , O ) - Cat /X ( I K • X, M ) . Proof.
Since Diag Z = Diag I K • X , by Proposition 3.26 and Lemma 1.24 we have H n − i SO ( IX ; M ) = π i map ( S , O )- Cat /IX (Diag Z, K X ( M, n )= π i Tot map [∆ , Cat O ] /IX ( I K • X, K X ( M, n )) . (3.31)the homotopy spectral sequence of the cosimplicial space W • = map ( S , O )- Cat ( I K • X, K X ( M, n ))(see [BK, X6]) has E s,t = π s π t W • ⇒ π t − s Tot W • with E s,t = π t map( I K s X, K X ( M, n )) = H n − t SO ( I K s X ; M ) . Here we used the fact that I K s X is a cofibrant ( S , O )-category, since it is free in eachdimension and the degeneracy maps take generators to generators.By Remark 3.8, I K s X is homotopically discrete, so I K s X → I dΠ K s X is a weakequivalence. Hence I K s X is a cofibrant replacement of I dΠ K s X , and so(3.32) H n − t SO ( I K s X ; M ) = H n − t SO ( I dΠ K s X ; M ) . Recall from [BBl, Theorem 3.10] that H s SO (d C ; M ) = H s +1BW ( C , M ) for any category C and s >
0, where H • BW is the Baues-Wirsching cohomology of [BW]. Hence if C isfree, H s SO (d C ; M ) = 0 for each s >
0, by [BW, Theorem 6.3].
OMONAD COHOMOLOGY OF TRACK CATEGORIES 21 If n (cid:54) = t , it follows from by (3.32) that E s,t = H n − t SO ( I K s X ; M ) = 0, since Π I K s X is a free category, so the spectral sequence collapses at the E -term, and(3.33) H s SO ( IX ; M ) = π n − s Tot W • = π s π n map( I K • X, K X ( M, n )) . Since π n map( I K • X, K X ( M, n )) = H S , O ) ( I dΠ K • X, M ), this is independent of n . Wededuce from (3.33) that H s SO ( IX ; M ) = π s C • for C • as in (3.30). (cid:3) Corollary.
For any X ∈ Track O , X -module M , and s ≥ we have (3.35) H s SO ( I d X ; j ∗ M ) ∼ = π s (cid:98) C • , where (cid:98) C • is the cosimplicial abelian group π map ( S , O ) - Cat /X ( I ( K • X ) , j ∗ M ) .Proof. Let Z = N I d( K • X ) . Then Diag Z → IX is a cofibrant replacement, byCorollary 3.25. Therefore, by Lemma 1.24: H n − i SO (( I d X ; j ∗ M ) = π i map ( S , O )- Cat /I d X (Diag Z, K X ( j ∗ M, n )) == π i Tot map [∆ , Cat O ] ( Id ( K • X ) , j ∗ M ) . The homotopy spectral sequence for W • = map ( S , O )- Cat ( I d( K • X ) , K X ( j ∗ M, n )) againcollapses at the E -term, since ( K s X ) is a free category, yielding (3.35). (cid:3)
4. ( S , O ) -cohomology of track categories and comonad cohomology We now use the comonad of § C = Cat O , in termsof mapping spaces, and use it to prove our main result, Theorem 4.12.4.1 . Mapping spaces. Given maps f : A → B and g : M → B in a simplicialmodel category C , we let map C /B ( A, M ) be the homotopy pullbackmap C /B ( A, M ) (cid:47) (cid:47) (cid:15) (cid:15) map C ( A, M ) g ∗ (cid:15) (cid:15) { f } (cid:47) (cid:47) map C ( A, B )If A is cofibrant and g is a fibration, so is g ∗ . Moreover, if h : B → D is a trivialfibration, so is h ∗ : map( A, B ) → map( A, D ).Thus we obtain a map ¯ h ∗ : map C /B ( A, M ) → map C /D ( A, M ) defined bymap C /B ( A, M ) (cid:47) (cid:47) (cid:15) (cid:15) ¯ h ∗ (cid:36) (cid:36) map C ( A, M ) (cid:15) (cid:15) Id (cid:34) (cid:34) { f } (cid:47) (cid:47) (cid:36) (cid:36) map C ( A, B ) h ∗ (cid:34) (cid:34) map C /D ( A, M ) (cid:47) (cid:47) (cid:15) (cid:15) map C ( A, M ) (cid:15) (cid:15) { hf } (cid:47) (cid:47) map C ( A, D )Since h ∗ is a weak equivalence, so is ¯ h ∗ .When C = Track O , we will use this construction several times for X = HY as in(1.14) and e ∗ M , j ∗ M , and t ∗ j ∗ M be as in (1.16), (2.2), and (2.18), respectively: (i) The diagram { Id X } (cid:47) (cid:47) (cid:15) (cid:15) map( X, X ) q ∗ (cid:15) (cid:15) map( X, M ) ρ ∗ (cid:111) (cid:111) Id (cid:15) (cid:15) { q } (cid:47) (cid:47) map( X, dπ ) map( X, M ) q ∗ ρ ∗ (cid:111) (cid:111) induces a map ¯ q ∗ : map Track O /X ( X, M ) → map Track O / d π ( X, M ), where M is a track category over d π via the map q : X → d π , which is a weakequivalence (since X is homotopically discrete) and a fibration (since d π isdiscrete). Hence ¯ q ∗ is a weak equivalence.(ii) The diagram { ∆ X } (cid:47) (cid:47) (cid:15) (cid:15) map(d X , X ) t ∗ (cid:15) (cid:15) map(d X , j ∗ M ) (cid:111) (cid:111) t ∗ (cid:15) (cid:15) { ∆ X t } (cid:47) (cid:47) map(d π , X ) map(d π , t ∗ j ∗ M ) (cid:111) (cid:111) induces ¯ t ∗ : map Track O / d X (d X , j ∗ M ) → map Track O / d π (d π , t ∗ j ∗ M ).Since q ∗ t ∗ = e ∗ , we have a diagram { ∆ X t } (cid:47) (cid:47) (cid:15) (cid:15) map( dπ , X ) q ∗ (cid:15) (cid:15) map( dπ , t ∗ j ∗ M ) ( ρke ) ∗ (cid:111) (cid:111) q ∗ (cid:15) (cid:15) { ∆ X t } (cid:47) (cid:47) map(d X , X ) map(d π , e ∗ j ∗ M ) (cid:111) (cid:111) inducing ¯ q ∗ : map Track O / d π (d π , t ∗ j ∗ M ) → map Track O / d π (d π , e ∗ j ∗ M ).(iii) The diagram { Id X } (cid:47) (cid:47) (cid:15) (cid:15) map( X, X ) j ∗ (cid:15) (cid:15) map( X, M ) ρ ∗ (cid:111) (cid:111) j ∗ (cid:15) (cid:15) { j } (cid:47) (cid:47) map(d X , X ) map(d X , j ∗ M ) λ ∗ (cid:111) (cid:111) induces ¯ ∗ : map Track O /X ( X, M ) → map Track O / d X (d X , j ∗ M ).(iv) The diagram { q } (cid:47) (cid:47) (cid:15) (cid:15) map( X, dπ ) t ∗ (cid:15) (cid:15) map( X, M ) q ∗ ρ ∗ (cid:111) (cid:111) t ∗ j ∗ (cid:15) (cid:15) { qjt } = { Id dπ } (cid:47) (cid:47) map( dπ , dπ ) map(d π , t ∗ j ∗ M ) q ∗ ρ ∗ (cid:111) (cid:111) induces ¯ t ∗ : map Track O /dπ ( X, M ) → map Track O /dπ ( dπ , t ∗ j ∗ M ). Since t ∗ is atrivial fibration, ¯ t ∗ is a weak equivalence.(v) The diagram { Id X } (cid:47) (cid:47) (cid:15) (cid:15) map( X, X ) e ∗ (cid:15) (cid:15) map( X, M ) ρ ∗ (cid:111) (cid:111) e ∗ (cid:15) (cid:15) { e } (cid:47) (cid:47) map( X, X ) map(
X, e ∗ M ) ρ (cid:48)∗ (cid:111) (cid:111) induces ¯ e ∗ : map Track O /X ( X, M ) → map Track O /X ( X, e ∗ M ). Moreover, e : X → X is a weak equivalence (since X is homotopically discrete), so ¯ e ∗ is, too. OMONAD COHOMOLOGY OF TRACK CATEGORIES 23 (vi) Using the fact that q ∗ e ∗ = q ∗ (since qe = qtq = q ), we have { e } (cid:47) (cid:47) (cid:15) (cid:15) map( X, X ) q ∗ (cid:15) (cid:15) map( X, e ∗ M ) ρ (cid:48)∗ (cid:111) (cid:111) r ∗ (cid:15) (cid:15) { q } (cid:47) (cid:47) map( X, d π ) map( X, M ) (cid:111) (cid:111) which induces ¯ q (cid:48)∗ : map Track O /X ( X, e ∗ M ) → map Track O / d π ( X, M ), and(4.2) map
Track O /X ( X, M ) e ∗ (cid:47) (cid:47) ¯ q ∗ (cid:42) (cid:42) map Track O /X ( X, e ∗ M ) ¯ q (cid:48)∗ (cid:116) (cid:116) map Track O /dπ ( X, M )commutes, since ¯ q (cid:48)∗ e ∗ = q ∗ .(vii) Finally, the diagram { e } (cid:47) (cid:47) (cid:15) (cid:15) map( X, X ) j ∗ (cid:15) (cid:15) map( X, e ∗ M ) ρ (cid:48)∗ (cid:111) (cid:111) j ∗ (cid:15) (cid:15) { jtq } (cid:47) (cid:47) map(d X , X ) map(d X , e ∗ j ∗ M ) ρ ∗ (cid:111) (cid:111) induces (¯ (cid:48) ) ∗ : map Track O /X ( X, e ∗ M ) → map Track O / d X (d X , e ∗ j ∗ M ).4.3. Lemma.
For any map f : A → B in Cat O , with A free, there is an isomorphism π map Track O /B ( A, M ) = H ( A ; M ) ∼ = Hom Cat O /B ( A, M ) . Proof.
Since A is free, c ( A ) ∈ s O - Cat = ( S , O )- Cat is cofibrant, and is its ownfundamental track category. A module M , as an abelian group object in Track O /B ,is an Eilenberg-Mac Lane object E B ( M , § n -simplices of the cosimplicial abelian group map Track O /B ( A, M ) are mapsof categories over B of the form σ : A ⊗ ∆[ n ] → M n where A ⊗ ∆[ n ] is the coproductin Cat O of one copy of A for each n -simplex of ∆[ n ].Since M = B , map Track O /B ( A, M ) is the singleton { f } . The non-degenerate partof map Track O /B ( A, M ) is Hom Cat O ( A, M ), so the 1-cycles are Hom Cat O /B ( A, M ).Since M , and thus map Track O /B ( A, M ), is a 1-Postnikov section, the 1-cycles areequal to π map Track O /B ( A, M ). (cid:3) Lemma.
For X ∈ Track O of the form HY and M an X -module, the diagram (4.5) π map Track O /X ( X, M ) (cid:107)(cid:111) (cid:15) (cid:15) j ∗ (cid:47) (cid:47) Hom
Cat O /X (( X −→ X ) , ( j ∗ M → X ) (cid:107)(cid:111) (cid:15) (cid:15) Hom
Cat O /π ( π , t ∗ j ∗ M ) q ∗ (cid:47) (cid:47) Hom
Cat O /X (( X e −→ X ) , ( e ∗ j ∗ M → X )) commutes, and there is an isomorphism ω : Hom Cat O /X (( X e −→ X ) , ( e ∗ j ∗ M λ −→ X )) → Hom
Cat O /X ( X e −→ X ) , j ∗ M λ −→ X )) such that ωq ∗ = ξ , for ξ as in (2.20) . Proof.
We have a commuting diagram(4.6) map
Track O /X ( X, M ) ¯ ∗ (cid:47) (cid:47) ¯ e ∗ (cid:15) (cid:15) ¯ q ∗ (cid:23) (cid:23) map Track O / d X (d X , j ∗ M ) ˜ e ∗ (cid:15) (cid:15) map Track O /X ( X, e ∗ M ) (¯ (cid:48) ) ∗ (cid:47) (cid:47) ¯ q (cid:48)∗ (cid:15) (cid:15) map Track O / d X (d X , e ∗ j ∗ M )map Track O /dπ ( X, M ) ¯ t ∗ (cid:15) (cid:15) map Track O /dπ ( dπ , t ∗ j ∗ M ) q ∗ (cid:54) (cid:54) in which the vertical maps are weak equivalences by § π yields(4.7) π map Track O /X ( X, M ) (cid:107)(cid:111) (cid:15) (cid:15) j ∗ (cid:47) (cid:47) π map Track O / d X (d X , j ∗ M ) (cid:107)(cid:111) χ (cid:15) (cid:15) π map Track O /dπ ( dπ , t ∗ j ∗ M ) q ∗ (cid:47) (cid:47) π map Track O / d X ((d X e −−→ d X ) , e ∗ j ∗ M )with vertical isomorphisms. By Lemma 4.3, π map Track O / d X (d X , j ∗ M ) ∼ = Hom Cat O /X ((d X −→ d X ) , ( j ∗ M λ −→ d X )) π map Track O /dπ ( dπ , t ∗ j ∗ M ) ∼ = Hom Cat O /π ( π , t ∗ j ∗ M ) ,π map Track O / d X (d X , e ∗ j ∗ M ) ∼ = Hom Cat O /X ( X , q ∗ t ∗ j ∗ M ) . Hence from (4.7) we obtain π map Track O /X ( X, M ) (cid:107)(cid:111) (cid:15) (cid:15) j ∗ (cid:47) (cid:47) Hom
Cat O /X (( X −→ X ) , ( j ∗ M → X )) χ (cid:15) (cid:15) Hom
Cat O /π ( π , t ∗ j ∗ M ) q ∗ (cid:47) (cid:47) Hom
Cat O /X (( X −→ X ) , ( e ∗ j ∗ M → X ))The right vertical map χ in (4.7) sends ¯ f : X → j ∗ M to ¯ f e : X → e ∗ j ∗ M ,where ¯ f is given by Id (cid:62) f into the pullback (2.3) defining j ∗ M . We now rewrite χ : Hom Cat O /X (( X −→ X ) , ( j ∗ M → X )) → Hom
Cat O /X (( X e −→ X ) , ( e ∗ j ∗ M λ −→ X ))in a different form, in order to show that it is an isomorphism. Note that the map ofgroupoids ρ = ( ρ , Id) : M → X satisfies ρ ( m ) = ( ∂ ( m ) , ∂ ( m )) for all m ∈ M ,where M = M ∂ (cid:47) (cid:47) ∂ (cid:47) (cid:47) X . Thus the map ¯ f = Id (cid:62) f into (2.3) has ρ f = j = ∆ X so ρf ( a ) = ( a, a ) = ( ∂ f ( a ) , ∂ f ( a )), and thus f takes X to (cid:96) a ∈ X M ( a, a ). Similarly,a map ¯ g : X → e ∗ j ∗ M into the pullback defining e ∗ j ∗ M is given by g (cid:62) e : X → M × X with ∆ X e e = ∆ X e = ρ g , so g : X → (cid:96) a ∈ X M ( tqa, tqa ). For each a ∈ X there is an isomorphism ω : M ( a, a ) → M ( tqa, tqa ) taking m ∈ M ( a, a ) to σ ( tqa, a ) ◦ m ◦ σ ( a, tqa ), whose inverse ω − : M ( tqa, tqa ) → M ( a, a ) takes m (cid:48) to σ ( a, tqa ) ◦ m (cid:48) ◦ σ ( tqa, a ). OMONAD COHOMOLOGY OF TRACK CATEGORIES 25
Given f : X → (cid:96) a M ( a, a ), there is a commuting diagram X f (cid:47) (cid:47) e (cid:15) (cid:15) (cid:96) a M ( a, a ) ω (cid:15) (cid:15) X f (cid:47) (cid:47) (cid:96) a M ( tqa, tqa )so that χ ( ¯ f ) = ( e , f e ) = ( e , ωf ). Thus χ is an isomorphism with inverse given by χ − ( g ) = (Id , ω − g ).Finally, the isomorphism ω : Hom Cat O /X ((Id X ) , ( q ∗ t ∗ j ∗ M λ (cid:48)(cid:48) −→ X )) → Hom
Cat O /X (( tq ) , ( j ∗ M λ −→ X ))sends h to ω ( h ) = l v h , where the maps l and v are given by(4.8) q ∗ t ∗ j ∗ M λ (cid:48)(cid:48) (cid:15) (cid:15) v (cid:47) (cid:47) t ∗ j ∗ M λ (cid:48) (cid:15) (cid:15) l (cid:47) (cid:47) j ∗ M λ (cid:15) (cid:15) k (cid:47) (cid:47) M ρ (cid:15) (cid:15) X q (cid:47) (cid:47) π t (cid:47) (cid:47) X X (cid:47) (cid:47) X Define f (cid:48) in Hom Cat O /X (( X tq −→ X ) , ( j ∗ M λ −→ X )) by tq (cid:62) f into the pullback(2.3). Thus ρ f = ∆ X tq , which also implies ρ f = ∆ X ( tq )( tq ). It follows that thereis a map f (cid:48)(cid:48) making the following diagram commute X f (cid:40) (cid:40) f (cid:48)(cid:48) (cid:40) (cid:40) tq (cid:41) (cid:41) q ∗ t ∗ j ∗ M k l v (cid:47) (cid:47) λ (cid:48)(cid:48) (cid:15) (cid:15) M ρ (cid:15) (cid:15) X X tq (cid:47) (cid:47) X We claim that f (cid:48) = l v f (cid:48)(cid:48) = ω ( f (cid:48)(cid:48) ). In fact, k f (cid:48) = f = k l v f (cid:48)(cid:48) , while by (4.8) wehave λ f (cid:48) = tq = tq tq = tqλ (cid:48)(cid:48) f (cid:48)(cid:48) = tλ (cid:48) v f (cid:48)(cid:48) = λ l v f (cid:48)(cid:48) . Together these imply that f (cid:48) = l v f (cid:48)(cid:48) , as claimed. Thus ω is surjective.Assume given h, g ∈ Hom
Cat O /X ( X , q ∗ t ∗ j ∗ M ) with ω ( h ) = ω ( g ), (i.e., l v h = l v g ). Then k l v h = k l v g and λ (cid:48)(cid:48) h = tq = λ (cid:48)(cid:48) g , so h = g . This shows that ω isinjective, and thus an isomorphism.To see that ωq ∗ = ξ , Let h ∈ Hom
Cat O /X ( π , t ∗ j ∗ M ) be given by f (cid:62) Id : π → M × π in the pullback (2.18). Then k l h = f and λ (cid:48) h = Id, and so ρ f q = ρ k l hq = ∆ X tλ (cid:48) hq = ∆ X tq = ∆ X ( tq )( tq ). Hence there is a map h (cid:48) making X fq (cid:38) (cid:38) h (cid:48) (cid:39) (cid:39) tq (cid:40) (cid:40) q ∗ t ∗ j ∗ M k l v (cid:47) (cid:47) λ (cid:48)(cid:48) (cid:15) (cid:15) M ρ (cid:15) (cid:15) X X tq (cid:47) (cid:47) X commute, and h (cid:48) = q ∗ h , so that ω ( q ∗ h ) = ω ( h (cid:48) ) = l v h (cid:48) . Moreover, k l v h (cid:48) = f q = k l hq and∆ X tλ (cid:48) v h (cid:48) = ∆ X λ l v h (cid:48) = ρ k l v h (cid:48) = ρ f q = ∆ X tq tq = ∆ X tq so λ (cid:48) v h (cid:48) = q ∆ X tλ (cid:48) v h (cid:48) = q ∆ X tq = q = λ (cid:48) hq , since q ∆ X t = Id and λ (cid:48) h = Id.This implies that v h (cid:48) = hq . We deduce that ωq ∗ ( h ) = l hq = ξ ( h ), so ωq ∗ = ξ . (cid:3) Proposition.
For any X = HY as in (1.14) and X -module M , there is ashort exact sequence → π map Track O /X ( X, M ) j ∗ −→ π map Track O / d X ( dX , j ∗ M ) ϑ (cid:48) −→ Hom
Track O /X ( X, M ) → Proof.
This follows by taking C = Cat O in Proposition 2.22, together with (2.33)and the top right corner of (4.5) identified with π map Track O / d X ( dX , j ∗ M ). (cid:3) Let X ∈ Track O and M ∈ [( Track O , X ) /X ] ab and¯ I K • X ∈ [∆ op , ( S , O )- Cat /IX ]be as in Section 3.9. Note that the augmentation ε : K • X → X can be thoughtof as a map to the constant simplicial object, so a compatible sequence of maps ε n : K n X → X in Track O allowing us to pull back M to ε ∗ n M .4.11. Definition.
For each X ∈ Track O and M ∈ [( Track O , X ) /X ] ab , the comonadcohomology of X with coefficients in M is defined by H s C ( X, M ) = π s Hom
Track O /X ( K • X, M ) . Theorem.
Assume given X ∈ Track O and M ∈ [( Track O , X ) /X ] ab . Thenthere is a long exact sequence of abelian groups · · · → H n SO ( IX ; M ) → H n SO ( I d X ; j ∗ M ) → H n C ( X, M ) →→ H n +1SO ( IX ; M ) → · · · Proof.
By Remark 3.8. K n X is homotopically discrete for each n , and we can choosea splitting Π K n X t n −→ K n X to K n X q n −→ Π K n X , because K n X = LA with A = F U V X n − and q n : A s (cid:96) A t → A (see § → π map Track O /X ( I K • X, M ) j ∗ −→ π map Track O /dX ( I d( K • X ) , j ∗ M ) →→ Hom
Track O /X ( K • X, M ) → ε • : K • X → X to pull back M to ε • M .We therefore obtain a corresponding long exact sequence → π s π map Track O /X ( K • X, M ) j ∗ −→ π s π map Track O /dX (d( K • X ) , j ∗ M ) →→ π s Hom
Track O /X ( K • X, M ) → · · · . By definition, π s Hom
Track O /X ( K • X, M ) = H sC ( X, M ), with π s π map Track O /X ( ¯ I K • X, M ) = H s SO ( IX ; M )and π s π map Track O / d X (d( K • X ) , j ∗ M ) = H s SO ( I d X ; j ∗ M ) by Theorem 3.29 andCorollary 3.34. (cid:3) OMONAD COHOMOLOGY OF TRACK CATEGORIES 27
Corollary.
For X ∈ Track O with X a free category and M ∈ [( Track O , X ) /X ] ab (4.14) H n +1SO ( IX ; M ) ∼ = H n C ( X, M ) for each n ≥ .Proof. Recall from [BBl, Theorem 3.10] that H n SO ( I d X ; j ∗ M ) ∼ = H n +1BW ( IX , j ∗ M ),and, since X is free, H n +1BW ( IX , j ∗ M ) = 0 by [BW, Theorem 6.3]. Thus the longexact sequence of Theorem 4.12 yields (4.14) for each n ≥ (cid:3) Lemma.
There is a functor S : Track O → Track O such that ( s X ) is a freecategory, for each X ∈ Track O , with a natural -equivalence s X : S ( X ) → X Proof.
Given X ∈ Gpd C and a map f : Y → X in C , consider the pullback(4.16) Y (cid:47) (cid:47) f (cid:15) (cid:15) Y × Y f × f (cid:15) (cid:15) X ∂ ,∂ ) (cid:47) (cid:47) X × X in C . Then there is X ( f ) ∈ Gpd C with ( X ( f )) = Y and X ( f )) = Y , such that( f , f ) : X ( f ) → X is an internal functor.Now let ε X : F V X → X be the counit of the adjunction F (cid:97) V of § SX := X ( ε X ), where ( ε X ) ∈ Track O is the construction (4.16). Then( SX ) = F V X is a free category, and there is a map s X : SX → X in Track O . Wewish to show that it is a 2-equivalence.Since s X is the identity on objects, to it suffices to show that for each a, b ∈ O , themap s X ( a, b ) : S ( X )( a, b ) → X ( a, b ) is an equivalence of categories. The pullback(4.17) ( S ( X )) (cid:47) (cid:47) s X (cid:15) (cid:15) F V X × F V X ε X × ε X (cid:15) (cid:15) X (cid:47) (cid:47) X × X in Cat induces a pullback of sets { S ( X )( a, b ) } (cid:47) (cid:47) { s X ( a,b ) } (cid:15) (cid:15) { F V X ( a, b ) × F V X ( a, b ) } ε X × ε X (cid:15) (cid:15) { X ( a, b ) } (cid:47) (cid:47) { X ( a, b ) × X ( a, b ) } Thus for each ( c, d ) ∈ { F V X ( a, b ) × F V X ( a, b ) } , the map { s X ( a, b ) } ( c, d ) is abijection. Thus s X ( a, b ) is fully faithful. Since ( F V X ) → X is surjective, as is( F V X )( a, b ) → X ( a, b ), s X ( a, b ) is surjective on objects, so it is an equivalence ofcategories. (cid:3) We finally use our previous results to conclude that the ( S , O )-cohomology of atrack category can always be calculated from a comonad cohomology.4.18. Corollary.
For X ∈ Track O , and M an X -module, H n +1SO ( IX ; M ) ∼ = H nC ( S ( X ) , M ) for each n > , where S ( X ) is as in Lemma 4.15.Proof. By Lemma 4.15 the map s X : S ( X ) → X is a 2-equivalence in Track O , since( s X ) = ε X is bijective on objects. Hence Is X is a Dwyer-Kan equivalence in( S , O )- Cat , so H n SO ( IX ; M ) ∼ = H n SO ( IS ( X ); M ). By Lemma 4.15, SX satisfies thehypotheses of Corollary 4.13, so also H n +1SO ( IS ( X ); M ) ∼ = H n C ( S ( X ) , M ). (cid:3) The groupoidal case
A 2-groupoid is a special case of a track category in which all cells are (strictly) in-vertible. The category of such is denoted by 2-
Gpd O = Gpd ( Gpd O ), with i : 2- Gpd O (cid:44) → Track O the full and faithful inclusion.For X ∈ Gpd O and W = N I K • X ∈ [∆ op , Set ] as in § S = W (1) ∈ [∆ op , [∆ op , Set ]] be W thought of as a simplicial object along the horizontal direction.Thus for each i ≥ S i is the bisimplicial set · · · (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) ( K i +1 X ) × ( K i +1 X ) ( K i +1 X ) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) O (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) ( K i +1 X ) × O ( K i +1 X ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) ( K i +1 X ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) O (cid:15) (cid:15) (cid:15) (cid:15) ( K i +1 X ) × O ( K i +1 X ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) ( K i +1 X ) (cid:47) (cid:47) (cid:47) (cid:47) O Applying Diag : [∆ op , Set ] → [∆ op , Set ] = S dimensionwise to W (1) we obtainDiag S ∈ [∆ op , [∆ op , Set ]], with(5.1) Diag (3) W = Diag Diag S = Diag Diag Z for Z := W (2) ∈ [∆ op , [∆ op , Set ]] as in (3.13).5.2.
Definition.
The classifying space of X ∈ Gpd O is BX = Diag N (2) X , where N (2) : 2- Gpd O → [∆ op , Set ] is the double nerve functor.5.3.
Remark.
By Lemma 3.15, Diag Z → IX is a Dwyer-Kan equivalence, wherewe think of Diag Z as an ( S , O )-category by the discussion preceding the Lemma3.15. Conversely, we may also think of IX as a bisimplicial set (implicitly, byapplying the nerve functor in the category direction), with (Diag Z ) = ( IX ) = c ( O )(cf. § Z ) j → ( IX ) j is a weak homotopy equivalence for all j ≥
0. Hence Diag Z → IX induces a weak homotopy equivalence of diagonalsDiag Diag Z (cid:39) Diag IX . Since Diag IX = Diag N (2) X = BX , by (5.1) we have(5.4) BX (cid:39) Diag (3)
W .
The cohomology groups of X ∈ Gpd O with coefficients in an X -module M aredefined to be H n − t ( BX, M ) = π t map [∆ op , Set ] ( BX, K ( M, n )). By (5.4) this can bewritten as π t map [∆ op , Set ] (Diag (3) W, K ( M, n )).5.5.
Lemma.
Given
C ∈
Cat O , with N C ∈ S viewed as a discrete ( S , O ) -category,and M be a C -module, we have H n ( B C , M ) ∼ = H n SO ( N C ; M ) for each n ≥ .Proof. Since N C is a discrete ( S , O )-category, we see that map ( S , O )- Cat ( N C , K ( M, n )),is map S ( N C , K ( M, n )), so H n SO ( C ; M ) = π map S ( N C , K ( M, n )) = H n ( B C , M ). (cid:3) Proposition.
For X ∈ - Gpd O and M an X -module, H s ( BX, M ) = H s SO ( X ; M ) for any s ≥ Proof.
By (5.1) and Lemma 1.24map S (Diag (3) W, K ( M, n )) = map S (Diag Diag S, K ( M, n )) ∼ = ∼ = Tot map S (Diag S, K ( M, n )) . OMONAD COHOMOLOGY OF TRACK CATEGORIES 29
Therefore, the homotopy spectral sequence for W • = map [∆ op , Set ] (Diag S, K ( M, n )),with E s,t = π s π t W • ⇒ π t − s Tot W • , has E s,t = π t map((Diag S ) s , K ( M, n )). But(Diag S ) s = Diag I K s +1 X (cid:39) I Π K s +1 X , since K s +1 X is homotopically discrete, so E s,t = H n − t ( BI Π K s +1 X, M ) = H n − t SO ( I Π K s +1 X ; M ) , by Lemma 5.5. Since K s +1 X is free, by [BBl, Theorem 3.10] we have E s,t = H n − t SO ( I Π K s +1 X ; M ) = H n − t +1BW ( I Π K s +1 X, M ) = 0for n (cid:54) = t . Thus the spectral sequence collapses at the E -term and(5.7) H s ( BX, M ) = π n − s Tot W • = π s π n W • = π s π n map S (Diag S, K ( M, n )) . Since (Diag S ) s → I Π K s +1 X is a weak equivalence for all s ≥ π n map [∆ op , Set ] (Diag S, K ( M, n )) = H ( B Diag
S, M ) ∼ = H ( B I Π K • X, M )Thus H s ( BX, M ) = π s H ( B I Π K • X, M ) by (5.7). On the other hand, in theproof of Theorem 3.29 we showed that H s SO ( X ; M ) = π s H ( I Π K • X ; M ), while byLemma 5.5 we have H ( B Π K • X, M ) = H ( I Π K • X ; M ).It follows that H s ( BX, M ) = H s SO ( X ; M ). (cid:3) From Theorem 4.12 and Proposition 5.6 we deduce:5.8.
Corollary.
Any X ∈ - Gpd O and X -module M have a long exact sequence → H n ( BX, M ) → H n ( BX , j ∗ M ) → H nC ( X, M ) → H n +1 ( BX, M ) → · · · . Remark.
A 2-groupoid with a single object is an internal groupoid in the categoryof groups, equivalent to a crossed module. It can be shown that in this case the longexact sequence of Corollary 5.8 recovers [P1, Theorem 13].
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