Homotopy characters as a homotopy limit
aa r X i v : . [ m a t h . A T ] A ug HOMOTOPY CHARACTERS AS A HOMOTOPY LIMIT
SERGEY ARKHIPOV AND DARIA POLIAKOVA
Abstract.
For a Hopf DG-algebra corresponding to a derived algebraicgroup, we compute the homotopy limit of the associated cosimplicialsystem of DG-algebras given by the classifying space construction. Thehomotopy limit is taken in the model category of DG-categories. Theobjects of the resulting DG-category are Maurer-Cartan elements ofCobar( A ), or 1-dimensional A ∞ -comodules over A . These can be viewedas characters up to homotopy of the corresponding derived group. Theirtensor product is interpreted in terms of Kadeishvili’s multibraces. Wealso study the coderived category of DG-modules over this DG-category. Contents
1. Introduction 1Organization of the paper 3Acknowledgements 32. Preliminaries 32.1. Model categories involved 32.2. DG-modules 42.3. Cobar-constructions 43. The cosimplicial system 44. Maurer-Cartan elements in Cobar 75. CoMorita equivalences 86. Homotopy characters 117. Tensor products and multibraces 12Appendix A. Homotopy limit in DG-algebras 15A.1. Simplicial resolutions in
DGVect ( k ) 16A.2. Simplicial resolutions in DGAlg ( k ) 17A.3. Fat totalizations in DGVect ( k ) and DGAlg ( k ) 18A.4. Application to the cosimplicial system of a DG-bialgebra 20References 201. Introduction
The note is devoted to an explicit calculation of a homotopy limit for acertain cosimplicial diagram in the model category of DG-categories. Recallthat a general construction for representatives of such derived limits was given in the papers [BHW] and [AØ2]. Below we consider a baby examplewhere the answer appears to be both explicit and meaningful.Let us illustrate our answer in an important special case. Take the Hopfalgebra A of regular functions on an affine algebraic group G . The cosimpli-cial system we consider is given basically by the simplicial scheme X • real-izing BG . Notice that if we considered the DG-categories of quasicoherentsheaves on X n and passed to the homotopy limit, the resulting DG-categorywould have been a model for the derived category of quasicoherent sheaveson the classifying space BG which is known to be equivalent to the derivedcategory of representations of G .Our task is different: we treat the (DG)-algebras of regular functions on X n as DG-categories with one object and consider the corresponding ho-motopy limit. We prove that it is equivalent to an interesting subcategoryin the category of representations up to homotopy introduced earlier byAbad, Crainic, and Dherin (see [ACD]): the (non-additive) DG-category of characters up to homotopy of the group G also known as the DG-categoryof Maurer-Cartan elements in the Cobar construction for the coalgebra offunctions on G .The obtained answer illustrates a delicate issue: taking homotopy limit ofa diagram of DG-categories does not commute with the (infinity-) functor A DGMod ( A ). Namely, passing to the categories of modules levelwiseand then considering the homotopy limit would have produced the DG-category of quasicoherent sheaves on BG . Yet applying DGMod ( . . . ) to theDG-category of homotopy characters we get a different category.However, if we replace the derived categories of DG-modules by the coderivedones, this difference of the answers vanishes: the coderived category ofDG-modules over the DG-category of homotopy characters for G is quasi-equivalent to the coderived category of DG-modules over endomorphisms ofthe trivial character. By Positselski Koszul duality, the latter category isquasi-equivalent to the coderived category of representations for G .We conclude the paper by constructing an associative tensor product ofobjects in the DG-category of characters up to homotopy (in the generalityof a DG-Hopf algebra, since we never use commutativity of the algebra inour considerations). Recall that Abad, Crainic and Dherin also constructeda homotopy monoidal structure on their category of representations up tohomotopy (see [ACD]). Our answer agrees with theirs. We interpret thisanswer in terms of Kadeishvili’s multibraces.Notice that there is no expectation to produce a honest associative tensorproduct of morphisms before passing to the homotopy category. Instead we OMOTOPY CHARACTERS AS A HOMOTOPY LIMIT 3 plan to produce a homotopy coherent data descending to this structure aftertaking homology. This is work in progress.
Organization of the paper.
In Section 2 we give preliminaries on modelcategories, DG-modules and Cobar-constructions. In Section 3 we intro-duce the cosimplicial system of interest, state its homotopy limit in thecategory of DG-algebras, and give the first description of its homotopy limitin the category of DG-categories. In Section 4 we interpret this result interms of Maurer-Cartan elements in Cobar-construction. In Section 5 weexplain the coMorita equivalence between the homotopy limit taken in thecategory of DG-algebras and the homotopy limit taken in the category ofDG-categories. In Section 6 we reinterpret the homotopy limit category interms of representations up to homotopy in the sense of [AC]. In Section7 we discuss the monoidal structure (as in [ACD]) and how it is connectedto Kadeishvili’s multibraces. Finally in Appendix A we provide a detailedcomputation of the same homotopy limit in the category of DG-algebras, bymeans of simplicial resolutions.
Acknowledgements.
We are grateful to Leonid Positselski for many en-lightening comments, in particular for sharing the proof of Lemma 5.4 withus. The second author would like to thank Timothy Logvinenko for invitingher to present an early version of this project at GiC seminar in Cardiff, andRyszard Nest for useful discussions. The second author was supported bythe Danish National Research Foundation through the Centre for Symmetryand Deformation (DNRF92).2.
Preliminaries
Model categories involved.
The category of DG-algebras
DGAlg ( k )is equipped with projective model structure which is right-transferred fromthe category of chain complexes along the adjunction “tensor algebra func-tor/forgetful functor”. The weak equivalences are the quasiisomorphisms,the fibrations are the surjections, and the cofibrations are defined by the leftlifting property.In this paper, we mostly work with more general objects. Recall thata DG-category is by definition a category enriched over the monoidal cat-egory of complexes of vector spaces, denoted by DGVect ( k ). Every DG-algebra is a DG-category with one object. We denote the category of smallDG-categories and DG-functors over a field k by DGCat ( k ). Tabuada con-structed a model category structure on DGCat ( k ), with weak equivalencesbeing quasi-equivalences of DG-categories (see [Tab]).For an arbitrary model category C , the category C ∆ opp is equipped withReedy model structure (see [Hov] or [Hir]). SERGEY ARKHIPOV AND DARIA POLIAKOVA
DG-modules.
A DG-functor from a DG-category A with values in theDG-category DGVect ( k ) is called an A -DG module. Notice that this agreeswith the definition of a DG-module over a DG-algebra. The DG-categoryof A -DG-modules is denoted by DGMod ( A ).2.3. Cobar-constructions.
In our paper we will be dealing with two sortsof Cobar-construction for DG-coalgebras. In the first construction, the com-plex happens to be acyclic whenever the coalgebra is counital; conceptuallyit is a cofree resolution of the coalgebra as a comodule over itself. In the sec-ond construction, the coaugmentation of the coalgebra provides boundaryterms for the differential; the resulting complex is quasiisomorphic to whatis known as r educed Cobar-construction, and it is similar to the standardcomplex computing Cotor C ( k , k ). Note however, that in this note we areusing products not sums. Let us give the definitions and the notation. Definition 2.1.
Let C be (not necessarily counital or coaugmented) DG-coalgebra. As a graded vector space,Cobar( C ) = b T ( C [ − ∞ Y i =0 C [ − ⊗ i The multiplication is that of a complete tensor algebra. The differential isgiven by d = d C + ∆ on generators and extends to the rest of the algebraby Leinbiz rule. Remark 2.2. If C is counital, this Cobar construction is actually acyclic,with counit giving rise to a contraction.If C is coaugemented with coaugmentation 1 : k → C , then there is thefollowing modification. Definition 2.3.
As a graded algebra, Cobar coaug ( C ) ≃ b T ( C [ − d = d C + ∆ + 1 ⊗ id − id ⊗ Remark 2.4.
In the coaugmented case, 1 C is a Maurer-Cartan element inCobar( C ), and the differential in the later construction is the differential inthe former construction twisted by this Maurer-Cartan element.3. The cosimplicial system
Let (
A, m, , ∆ , ǫ ) be a (unital, counital) DG-bialgebra. Informally, in thecase when A is commutative we should view it as the algebra of functionson a derived affine algebraic group scheme. Notice however that we neveruse commutativity of A in our main statements.Consider the cosimplicial system A • of DG-algebras corresponding to theclassifying space construction:(1) k / / / / A / / / / / / A ⊗ · · · OMOTOPY CHARACTERS AS A HOMOTOPY LIMIT 5
Let ∂ in denote the face map A ⊗ n → A ⊗ n +1 and s in denote the degeneracymap A ⊗ n → A ⊗ n − . Then in the system above with faces and degeneraciesgiven by ∂ in = ⊗ id ⊗ n i = 0id ⊗ i − ⊗ ∆ ⊗ id ⊗ n − i < i < n + 1id ⊗ n ⊗ i = n + 1 s in = id ⊗ i ⊗ ǫ ⊗ id ⊗ n − i − There are several homotopy limit computations that can be done in rela-tion to system (1):(a) One can compute the homotopy limit in the category of DG-algebras(b) One can view every DG-algebra as a DG-category with one objectand compute the homotopy limit in the category of DG-categories(c) One can apply DG-Mod functor and compute the homotopy limit ofthis new system of DG-categories.The answer to (a) is folklore. The homotopy limit of the cosimplicialsystem is given by reduced Cobar-construction of the corresponding coaug-mented DG-coalgebra. We were not able to locate the proof of this statementin the literature, thus we reproduce it in Appendix A.In this paper we mainly discuss the answer to (b). The comparison be-tween (b) and (c) is discussed in Section 5.In the papers [BHW] and [AØ2] the authors realized homotopy limits in
DGCat ( k ) ∆ op as derived totalizations. Below we cite Prop. 4.0.2 from [AØ2],with formulas written in their most explicit form. To simplify the notation,we denote by ∂ ( i ...i k ) an inclusion with image i , . . . , i k . Theorem 3.1.
For C • a cosimplicial system of DG categories, an objectof holim C is the data of ( X, a = { a i } i ≥ ), where X is an object of C and a i ∈ Hom − i C i ( d (0) X, d ( n ) X ) with a homotopy invertible and subject to d ( a n ) = − n − X k =1 ( − n − k ∂ ( k...n ) ( a n − k ) ◦ ∂ (0 ...k ) ( a k )+ n − X k =1 ( − n − k ∂ (0 ... ˆ k...n ) ( a n − ) . (2)The complex of morphisms between ( X, a ) and (
Y, b ) in degree m is givenby Hom m (( X, a ) , ( Y, b )) = ∞ Y i =0 Hom m − i C i ( ∂ (0) ( X ) , ∂ ( i ) ( Y ))where we read ∂ (0) : C → C as id C . For f = { f i } ∈ Hom m (( X, a ) , ( Y, b ))its differential is given by
SERGEY ARKHIPOV AND DARIA POLIAKOVA d ( f ) n = d ( f n ) + X ( − n − k ∂ ( k...n ) ( f n − k ) ◦ ∂ (0 ...k ) ( a k ) − n − X k =1 ( − m ( n − k +1) ∂ ( k...n ) ( b n − k ) ◦ ∂ (0 ...k ) ( f k )+ n − X k =1 ( − n − k + m ∂ (0 ... ˆ k...n ) ( f n − ) . (3)For f ∈ Hom m (( X, a ) , ( Y, b )) and g ∈ Hom l (( Y, b ) , ( Z, c )), their composi-tion composition is given by(4) ( g ◦ f ) n = n X k =0 ( − m ( n − k ) ∂ ( k...n ) ( g n − i ) ◦ ∂ (0 ...k ) ( f k ) . (cid:3) We now apply these formulas to the cosimplicial system (1). Note thatwhile each category in (1) has a single object, this would not hold for thehomotopy limit, where the data of an object includes morphisms. Denoteholim A • =: A . Theorem 3.2.
An object a in A is an infinite sequence { a i } i ≥ with a i ∈ ( A ⊗ i ) − i and a homotopy invertible, subject to relations(5) d ( a ) = 0 d ( a ) = a ⊗ a − ∆( a ) . . .d ( a n ) = − n − X k =1 ( − n − k a n − k ⊗ a k + n − X k =1 ( − n − k (id ⊗ k − ⊗ ∆ ⊗ id ⊗ n − k − )( a n − ) . . . A morphism f : a → b of degree m is also an infinite sequence { f n } n ≥ with f n ∈ ( A ⊗ n ) − n , with differential given by d ( f ) n = d ( f n ) + n − X k =1 ( − n − k a k ⊗ f n − k − n − X k =1 ( − m ( n − k +1) f i ⊗ b n − k + n − X k =1 ( − n − k + m (id ⊗ k − ⊗ ∆ ⊗ id ⊗ n − k − )( f n − )(6)and composition given by(7) ( g ◦ f ) n = n X k =0 ( − m ( n − k ) g n ⊗ f n − k OMOTOPY CHARACTERS AS A HOMOTOPY LIMIT 7
Proof.
This is a straightforward application of Theorem 3.1. As in the the-orem, denote an object of the homotopy limit by (
X, a ). In our cosimplicialsystem (1), A = k has only one object, so X = ∗ . Then the identities (2)translate to (5), the formula for the differential (3) corresponds to (6), andthe formula for the composition corresponds to (7). (cid:3) Below we present several interpretations of this data.4.
Maurer-Cartan elements in Cobar
We interpret the homotopy limit category A in terms of Cobar construc-tion for the DG-coalgebra A . Proposition 4.1.
The objects of A are exactly the Maurer-Cartan elementsof Cobar( A ), with one extra condition that their first component is homo-topy invertible. Proof.
The Maurer-Cartan equation dx + [ x, x ] = 0 translates preciselyinto the formulas (5). (cid:3) In any DG algebra A a Maurer-Cartan element c allows to twist thedifferential: d c ( a ) = d ( a ) + [ c, a ]Denote the new algebra by c C c . For two Maurer-Cartan elements c and c , denote by c C c a complex obtained by considering A with the newdifferential(8) d ( c ,c ) ( a ) = d ( a ) + c a − ( − | a | ac . This will not be a DG-algebra anymore (for the lack of multiplication satis-fying the Leibniz rule), but it will be a c C c - c C c DG-bimodule.
Proposition 4.2.
In the DG-category A , the complex of morphisms A ( a, b ) = a Cobar( A ) b . Proof.
The formula (8) for the twisted differential corresponds precisely tothe formula (6). (cid:3)
So as a graded vector space, every A ( a, b ) is always equal to Cobar( A ). Proposition 4.3.
Under this assignment, the composition A ( a, b ) ⊗ A ( b, c ) → A ( a, c ) corresponds to the multiplication in Cobar( A ). Proof.
This is the formula (7). (cid:3)
In Cobar( A ), there is a distinguished nontrivial Maurer-Cartan element,namely, 1 A ∈ A . Denote the corresponding object of A by I . Its endomor-phisms are A Cobar( A ) A ≃ Cobar coaug ( A ). As explain in Appendix A, thisis a model for the homotopy limit of our cosimplicial system but taken inthe category DGAlg ( k ).Recall the notion of gauge equivalence for Maurer-Cartan elements. SERGEY ARKHIPOV AND DARIA POLIAKOVA
Definition 4.4.
In a DG-algebra A , the gauge action of a degree 0 invertibleelement f on a Maurer-Cartan element a is given by f.a = f af − + f d ( f − ) . One checks that this is again a Maurer-Cartan element. Two Maurer-Cartan elements are called gauge equivalent if they belong to the same orbitof gauge action.
Proposition 4.5.
Gauge equivalent Maurer-Cartan elements of Cobar( A )are strictly isomorphic as objects of A . Proof.
The very same invertible element provides the closed isomorphismwhen viewed as an element of the Hom-complex. Upon explicitly check-ing closedness, the rest follows from composition being reinterpreted as themultiplication in Cobar( A ). (cid:3) CoMorita equivalences
For any DG algebra A and Maurer-Cartan elements a , b it holds that a A b ⊗ b A b b A a = a A a , so on the nose a A b and b A a are inverse bimodules. This gives an expecta-tion for a Morita equivalence between A and Cobar( A ). However, sometimesthese bimodules may be acyclic, and derived tensoring by an acyclic bimod-ule cannot induce an equivalence of derived categories. To make thingswork one needs to consider not derived categories but instead Positselski’scoderived categories, where the class of acyclic objects is replaced by asmaller class of coacyclic objects. For detailed exposition see [Pos]. Definition 5.1.
For a DG algebra A , the subcategory CoAcycl ⊂ Ho ( A )is the smallest triangulated subcategory containing totalizations of exacttriples of modules and closed with respect to infinite direct sums. Definition 5.2.
The coderived category D co ( A ) is defined as the Verdierquotient of the homotopy category Ho ( A ) by the full subcategory CoAcycl .For the proof of the next lemma, recall the notion of CDG-algebras andtheir morphisms.
Definition 5.3.
A curved DG-algebra (for brevity, a CDG-algebra) is agraded algebra A equipped with a degree 1 derivation d and a closed curva-ture element h ∈ A , satisfying d ( x ) = [ h, x ]A morphism of CDG-algebras A → B is a pair ( f, b ) where f : B → C is amultiplicative map and c ∈ B is the change of curvature, i.e. they satisfy(9) f ( d A ( x )) = d B ( f ( x )) + [ a, x ](10) d ( h A ) = h B + d B ( b ) + b The composition of CDG-morphisms is( g, c ) ◦ ( f, b ) = ( g ◦ f, c + g ( b ))A DG-algebra can be viewed as a CDG-algebra with zero curvature, butthe inclusion DGAlg ( k ) ֒ → CDGAlg ( k ) is not full. Lemma 5.4.
For any DG algebra A there is an equivalence of coderivedcategories D co ( a A a ) ≃ D co ( b A b ) Proof. a A a and b A b are isomorphic as CDG-algebras (with zero curvature).The CDG-isomorphism a A a → b A b is given by ( id, − a ), where (9) corre-sponds to the formula for twisting the differential, and (10) corresponds toMaurer-Cartan equation for a . Coderived categories are preserved underCDG-isomorphisms. (cid:3) Remark 5.5.
Compare the calculation above of the explicit representativefor the homotopy limit of the DG-algebras considered as DG-categories withthe following.(1) In the paper [AØ2] the authors solve a similar problem for the ho-motopy limit of the derived categories of DG-modules over the DG-algebras in the cosimplicial system. The answer can be interpretedas the derived category of DG-modules over the reduced Cobar con-struction for the original DG-Hopf algebra (Theorem 4.1.1).(2) Conjecturally the statement remains true also for the homotopy limitof the corresponding enhanced coderived categories: one obtains thecoderived category of DG-modules over the Cobar construction forthe original DG-Hopf algebra.Now take the category of DG-modules over the DG-category of Maurer-Cartan elements A . While its derived category obviously differs from the de-rived category that appears in (1), its coderived category is quasi-equivalentto the answer in (2).We will now make this precise. Let B be an arbitrary DG-algebra. Definition 5.6.
Maurer-Cartan DG-category MC ( B ) has Maurer-Cartanelements of B as morphisms, and Hom-complexes are given byHom MC ( B ) ( a, b ) = a B b . The definitions 5.1 and 5.2 can be directly generalized from DG-algebrasto DG-categories, so for a DG-category C one can consider a category D co ( C ). Proposition 5.7.
For any DG-algebra B and a Maurer-Cartan element b ∈ B there is an equivalence of categories D co ( MC ( B )) ≃ D co ( b B b ) . Proof.
This is a statement of the type “modules over a connected groupoidare the same as modules over endomorphisms of an object in this groupoid”, with a similar proof.Let F : DGMod ( MC ( B )) → DGMod ( b B b )be given by restricting to b , F ( M ) = M ( b ) . Define G : DGMod ( b B b ) → DGMod ( MC ( B ))by setting, for a ∈ MC ( b ), G ( N )( a ) = a B b ⊗ b B b N and for f ∈ MC ( B )( a , a ) = a B a let the corresponding map G ( f ) : a B b ⊗ b B b N → a B b ⊗ b B b N be simply multiplication by f on the left. We would like to check thatthese functors induce an equivalence on coderived categories. First we checkthat they give an equivalence at the level of DG-categories. It is clear that F G = Id DGMod ( b B b )) . For M ∈ DGMod ( MC ( B )) and a ∈ MC ( B ), we have GF ( M )( a ) = a B b ⊗ b B b M ( b ) . Then the isomorphism GF ( M ) → M is given at a by f ⊗ m M ( f )( m )and its inverse is m ⊗ M (1)( m )where 1 ∈ a B b is viewed as a map a → b .We are left to verify that F and G preserve coacyclic objects. To do so,they need to preserve exact triples, and commute with totalizations, conesand infinite direct sums. For DG-modules over a DG-category, exactness ischecked objectwise, and totalizations, cones and direct sums are also formedobjectwise. Thus for F the statements hold trivially. For G , the statementsabout totalizations, cones and sums hold trivially, and the statement that G respects exact triples follows from flatness of b B b -modules a B b . They areindeed flat, because their underlying graded modules are just free of rank 1,and flatness does not depend on the differential. (cid:3) Note that in particular this proposition establishes a coMorita equivalencebetween MC ( B ) and B itself, as B can be seen as endomorphism algebra of0 ∈ MC ( B ). Also note that Lemma 5.4 follows from this proposition, butwe keep its proof via CDG-isomorphism because it is conceptually correct. Corollary 5.8.
There is an equivalence of coderived categories D co ( A ) ≃ D co (Cobar coaug ( A )) . (cid:3) OMOTOPY CHARACTERS AS A HOMOTOPY LIMIT 11
Here we are considering reduced Cobar construction for the sake of com-paring with the result in [AØ2] and with the computation in
DGAlg ( k ).Reduced and non-reduced Cobar constructions are coMorita equivalent byProposition 5.7 (though not Morita equivalent).6. Homotopy characters
Recall the notion of an A ∞ -comodule over a DG-coalgebra ( A ∞ -comodulescan be considered over any A ∞ -coalgebra, but this generality will not beneeded). For detailed exposition see [AØ2] or, on the dual side, [Kel]. Definition 6.1.
The A ∞ -comodule structure on a graded vector space M over a DG-coalgebra C is a DG-module structure on M ⊗ Cobar( C ) overCobar( C ). Explicitly, it is given by a sequence of coaction maps, for all n ≥ µ n : M → C ⊗ n − ⊗ M with µ n of degree 1 − n and all the collection of maps together satisfyingthe A ∞ -identities for each n ≥ − n − n X i =0 (id ⊗ i ⊗ d ⊗ id ⊗ n − i − ) µ n + µ n d + n − X i =1 ( − i (id ⊗ i ⊗ µ n − i ) µ i + n − X i =0 ( − i (id ⊗ i ⊗ ∆ ⊗ id ⊗ n − i − ) µ n − = 0(11) Definition 6.2.
For two A ∞ -comodules over a DG-algebra A , Hom-complexbetween them is defined byHom m ( M, N ) = ∞ Y i =0 Hom m − ik ( M, C ⊗ i ⊗ N )with differential d ( f ) n = n − X k =1 ( − n − k (id ⊗ n − k − ⊗ ∆ ⊗ id ⊗ k ) f n − + X i =0 ( − i (id ⊗ i ⊗ µ n − i ) f i +1 + n X p =1 ( − p | f | (id ⊗ p − ⊗ f n − p +1 ) µ p (12)The composition is given by(13) ( g ◦ f ) n = n X l =1 ( − | g | ( l − (id ⊗ l − ⊗ g n − l +1 ) f l Proposition 6.3.
The DG-category A is isomorphic to the subcategory of1-dimensional (non-counital) A ∞ -comodules over A . Proof.
For M = k a structure map µ n : k → A ⊗ n ⊗ k is indeed given by anelement a n ∈ A ⊗ n . The A ∞ -relations (11) correspond to the formulas (5). The formula for the differential (12) corresponds to (6), and the formula forthe composition (13) corresponds to (7). (cid:3)
Note that if A was the coalgebra of functions on some group, then comod-ules over this coalgebra would correspond to representations of the group.This leads us to the following interpretation of our data. A ∞ -comodulesover a Hopf DG-algebra can be viewed as representations up to homotopy of the corresponding derived group. Within this category, one-dimensionalcomodules correspond to homotopy characters . Group representations up tohomotopy have been defined and studied (for non-derived Lie groupoids) byAbad-Crainic in [AC].In the case when A is a Hopf algebra of functions on a group (concentratedin degree 0), our category has honest characters as objects, and the Homcomplexes compute Exts between them. Example 6.4.
Let G be the group of invertible upper triangular 2 × C . Consider the following functions: x (cid:18) a c b (cid:19) = a ; y (cid:18) a c b (cid:19) = b ; z (cid:18) a c b (cid:19) = c. The Hopf algebra of regular functions on G is C [ x ± , y ± , z ], with comul-tiplication ∆( x ± ) = x ± ⊗ x ± ;∆( y ± ) = y ± ⊗ y ± ;∆( z ) = x ⊗ z + z ⊗ y. xy − are two characters of G . We have Ext (1 , xy − ) = C . In ourHolim category, the Hom complex between 1 and xy − is C −→ C [ x ± , y ± , z ] −→ C [ x ± , y ± , z ] ⊗ −→ . . . where the first differential is multiplication by 1 − xy − , and the seconddifferential is given by d ( f ) = f ⊗ xy − ⊗ f + ∆( f ). The kernel of it isgenerated by 1 − xy − and y − z , the latter being a representative for thenontrivial first Ext.7. Tensor products and multibraces
One can see that the data of multiplication in A does not come up in theanswer so far. This however suggests that A is equipped with additionalstructure. We notice that a commutative DG-algebra is a monoidal
DG-category with one object, and while the passage to homotopy limit mightnot preserve this structure, at least something can be expected to survive.Indeed, in [ACD] the authors construct the monoidal structure on the ho-motopy category of all representations up to homotopy, which in particularrestricts to the subcategory of characters. We obtain a similar answer for
OMOTOPY CHARACTERS AS A HOMOTOPY LIMIT 13 noncommutative DG-Hopf algebras as well.Let a = { a i } and b = { b i } be two homotopy characters. Then a and b are homotopy invertible and homotopy grouplike, and so is a b . Indeed, if a ⊗ a − ∆( a ) = d ( a ) and b ⊗ b − ∆( b ) = d ( b ), then a b ⊗ a b − ∆( a b )= ( a ⊗ a )( b ⊗ b ) − ∆( a )∆( b )= (∆( a ) + d ( a ))(∆( b ) + d ( b )) − ∆( a )∆( b )= (∆( a ) + d ( a )) d ( b ) + d ( a )∆( b )= ( a ⊗ a ) d ( b ) + d ( a )∆( b )= d (( a ⊗ a ) b + a ∆( b )) . We notice that ( a b , ( a ⊗ a ) b + a ∆( b ) , . . . ) starts looking like thebeginning of another homotopy character. There is an asymmetry between a and b , but there is a certain freedom to modify the formulas above, so wecould have also obtained ( a b , a ( b ⊗ b ) + ∆( a ) b , . . . ). Theorem 7.1.
Let a = ( a , a , . . . ) and b = ( b , b , . . . ) be homotopy char-acters. Then there exists a homotopy character a ⊗ b , given by the formulas(14) ( a ⊗ b ) n = X i + ... + i k = n ( a i ⊗ . . . ⊗ a i k )(∆ i − ⊗ . . . ⊗ ∆ i k − )( b n ) . There also exists a homotopy character given by the formulas(15) ( a ⊗ b ) n = X i + ... + i k = n (∆ i − ⊗ . . . ⊗ ∆ i k − )( a n )( b i ⊗ . . . ⊗ b i k )Both tensor products of objects are strictly associative. Proof.
It can be explicitly checked that Maurer-Cartan equation holds inboth cases. Strict associativity of these tensor products is obtained by adirect computation. (cid:3)
The formulas above are the same as in Corollary 5.10 in [ACD] – in theirnotation, these are ω and ω . Theorem 5.6 in [ACD] states that the twodifferent tensor products are actually homotopy equivalent.The formulas (14) and (15) have an interpretation in terms of Kadeishvili’smultibraces that exist on the Cobar-construction of a bialgebra and assem-ble into homotopy Gerstenhaber algebra structure. Recall the followingdefinitions. Definition 7.2.
For a DG-algebra B with multiplication µ , its Bar-constructionis, as a graded vector space,Bar( B ) = T ( B [1]) = ∞ M i =0 B [1] ⊗ i . The comultiplication is that of a tensor coalgebra. The differential is givenby d = d B + µ into the cogenerators and extends to the rest of the coalgebraby coLeinbiz rule. Definition 7.3.
A DG-algebra B is a homotopy Gerstenhaber algebra(hGa) if it is equipped with a family of operations (multibraces) E , k : B ⊗ B ⊗ k → B that induce a associative multiplication on Bar( B ) consistent with its tensorcomultiplication. Remark 7.4.
A multiplication on Bar( B ) is a coalgebra map E : Bar( B ) ⊗ Bar( B ) → Bar( B ) . As a coalgebra map, it is uniquely determined by its part that lands into thecogenerators, B . Denote its component B ⊗ l ⊗ B ⊗ k → B by E l,k . A familyof E l,k that gives rise to an associative multiplication is known as Hirschalgebra structure on B . In Definition 7.3 we restrict ourselves to familieswhere E l,k vanish when l = 1.For elements b and b , . . . , b k we write E ,k ( b ; b , . . . , b k ) = b { b , . . . b k } (thus the term multibraces). We can naturally modify the definitions aboveto also obtain operations E k, , for which we will write E ,k ( b , . . . , b k ; b ) = { b , . . . b k } b . Let us call operations E ,k left multibraces, and operations E k, right multibraces.In Section 5 of [Ka] the author constructs (left) hGa structure on B =Cobar( A ) for a bialgebra A . For tensors x = x (1) ⊗ . . . ⊗ x ( n ) ∈ B and y , y , . . . , y k ∈ B , the left multibrace E ,k is given by E ,k ( x ; y , . . . , y k ) = X ≤ i <...
The results of [ACD] on tensoring morphisms also work in ourgenerality of non-commutative DG-Hopf algebra. However, extracted fromits natural (operadic) framework, the formula looks totally unenlightening:( f ⊗ g ) n == X i + ... + i k = n ≤ m ≤ k g ( a i ⊗ . . . ⊗ a i m − ⊗ f i m ⊗ x i m +1 ⊗ . . . ⊗ x i k )(∆ i − ⊗ . . . ⊗ ∆ i k − ) b k + X i + ... + i k = n f ( x i ⊗ . . . ⊗ x i k )(∆ i − ⊗ . . . ⊗ ∆ i k − ) g k + X i + j = ni + ... + i k = ij + ...j l = j ≤ m ≤ k ( a i ⊗ . . . ⊗ a i m − ⊗ f i m ⊗ x i m +1 ⊗ . . . ⊗ x i k ⊗ x j ⊗ . . . ⊗ x j l )(∆ i − ⊗ . . . ⊗ ∆ i k − ) b k ⊗ (∆ j − ⊗ . . . ⊗ ∆ j l − ) g l . We do not spell out the signs here, since the formula is already sufficientlyintimidating in their absence. The tensor product of morphisms given bythis formula is associative up to homotopy, and respects compositions up tohomotopy. Packaging the data of all these higher homotopies is the goal ofour ongoing project.
Appendix A. Homotopy limit in DG-algebras
For any combinatorial model category C and a diagram X of the shape∆, one can use Bousfeld-Kan formula to find the homotopy limit as the fat totalization, see Example 6.4 in [AØ1]:holim ∆ X = Z ∆ + R ( X n ) n where R is some functor C → C ∆ opp which sends an object c ∈ C to its sim-plicial resolution, i.e. a Reedy-fibrant replacement of the constant simplicial diagram with value c .We first present functorial simplicial resolutions for C ≃ DGVect ( k ), andthen extend the construction to C ≃ DGAlg ( k ). We then apply the fattotalization formula to compute the homotopy limit of a cosimplicial systemassociated with a DG-bialgebra.A.1. Simplicial resolutions in
DGVect ( k ) . Let us present functorial sim-plicial resolutions for
DGVect ( k ).Recall a simplicial vector space X • is under Dold-Kan correspondencesent to its Moore complex N ( X ) • , given by N ( X ) − n = X n /D n , where D n is the degenerate part of X n . The differential is the alternating sum of faces.For n ≥
0, let k ∆[ n ] be the linearization of standard simplex, and set L n = N ( k ∆[ n ]). Explicitly, this complex is spanned by elements f i <...
0, with i ≥ i k ≤ n – these are the nondegenerate simplicesof ∆[ n ] that correspond to faces with vertices i , . . . , i k . The differential inthis basis is d ( f i <...
For X ∈ DGVect ( k ), the simplicial system X [ − ] gives asimplicial resolution of X , i.e. it is Reedy-fibrant, and there exists a mapconst( X ) → X [ − ] that is a levelwise quasiisomorphism. Proof.
The map r : X → X [ n ] is is given by x r ( x ) where r ( x )( f i ) = x for all i , and r ( x )( f i <...
0. This respects differentials: wehave r ( d X ( x ))( f i ) = d X ( x ) = d X ( r ( x )( f i )) − r ( x )( d L n ( f i )) = d X [ n ] ( r ( x ))( f i )and ( d X ( x ))( f i We check that r is a quasiisomorphism. We first check that it is injectiveon cohomology. Let x ∈ X be a closed element such that its image vanishesin cohomology, r ( x ) = d X [ n ] ( s ) for some s : L n → X . Then x = r ( x )( f ) = d X [ n ] ( s )( f ) = d X ( s ( f )) − s ( d L ( f ))so x = d X ( s ( f )), i.e. it vanishes in cohomology.We now check r is surjective on cohomology. Let s : L n → X be a closedmorphism. Then r ( s ( f )) − s = d X [ n ] ( t ), where t ( f ) = 0 t ( f i ) = s ( f t ( f i <... 00 if i = 0For different n , these maps r ( n ) are consistent with cosimplicial structure:for φ : [ m ] → [ n ] we have r ( m ) ( x )( f i <... φ ∗ ( r ( n ) ( x ))( f i <... DGVect ( k ), i.e. surjections. By definition, the n th matching object M n is M n = lim δ ([ n ] ↓ (∆ op ) − ) X [ − ] = lim [ m ] ֒ → [ n ] X [ m ] . These are morphisms from a subcomplex of L n ⊂ L n that is spanned byeverything except f <... Simplicial resolutions in DGAlg ( k ) . We now enhance our construc-tion of simplicial resolutions from DGVect ( k ) to DGAlg ( k ). The result ismotivated by Holstein resolutions in DGCat ( k ) (see [Hol], [AP]) but simpler. Proposition A.2. The cosimplicial system of complexes L • can be up-graded to a cosimplicial system of DG-coalgebras, by introducing the fol-lowing comultiplication:∆( f i <...
Compatibility with differentials and and with cosimplicial structureis checked by an elementary explicit computation. (cid:3) Remark A.3. Conceptually this is the comultiplication in standard sim-plices that is responsible for the existence of cup-product in singular coho-mology.Now, for any monoidal DG-category C , if X is a coalgebra in C and Y is an algebra in C , then the complex C ( X, Y ) is a DG-algebra by means ofconvolution: C ( X, Y ) ⊗ C ( X, Y ) ≃ C ( X ⊗ X, Y ⊗ Y ) (∆ X ,µ Y ) −−−−−→ C ( X, Y )We are working in the case when C is the category of chain complexes, DGVect ( k ). Coalgebras in DGVect ( k ) are DG-coalgebras and algebras in DGVect ( k ) are DG-algebras. So for A a DG-algebra, the Hom-complexHom • ( L n , A ) has a DG-algebra structure. Denote this algebra by A [ n ] . Proposition A.4. For a DG-algebra A , the simplicial system A [ − ] gives asimplicial resolution of A , i.e. it is Reedy-fibrant, and there exists a mapconst( A ) → A [ − ] that is a levelwise quasiisomorphism. Proof. The map r : A → A [ n ] is exactly the same as in the case of DGVect ( k )- namely, a r ( a ) where r ( a )( f i ) = a for all i , and r ( a )( f i <... 0. We check that this map is compatible with multiplication:( r ( a ) ∗ r ( b ))( f i ) = µ A ( r ( a ) ⊗ r ( b ))( f i ⊗ f i ) = ab = r ( ab )( f i ) . and for k > r ( a ) ∗ r ( b ))( f i <...
Fat totalizations in DGVect ( k ) and DGAlg ( k ) . Let X • be the cosim-plicial complex in whose homotopy limit we are interested. Thenholim ∆ X • = Z ∆ + ( X n ) [ n ] = Eq Y n ≥ Hom • ( L n , X n ) ⇒ Y [ m ] ֒ → [ n ] Hom • ( L m , X n ) . This is the complex Nat ∆ + ( L • , X • ) of natural transformations between twofunctors ∆ + → DGVect ( k ). OMOTOPY CHARACTERS AS A HOMOTOPY LIMIT 19 Proposition A.5. As a graded vector space, the homotopy limit of a cosim-plicial vector space X • is given byholim ∆ X • = ∞ Y n =0 X n [ − n ] . For an element x = ( x , x , . . . ), its differential is given by(16) d ( x ) n = d X n ( x n ) − n X i =0 ∂ (0 ... b i...n ) ( x n − ) . Proof. A natural transformation φ : L • → X • consists of maps φ n : L n → X n for all n . For all indexing subsets I smaller than { < . . . < n } , thegenerator f I is in the image of i ∗ : L m → L n for some i : [ m ] ֒ → [ n ] ∈ ∆ + , m < n . Thus the only part of φ n that is not determined by φ m for m < n is its value φ n ( f <... The underlying complex of holim ∆ ( A • ) is as describedin Proposition A.5. For two elements a = ( a , a , . . . ) and b = ( b , b , . . . ),their product is given by(17) ( a · b ) n = n X i =0 ∂ (0 ...i ) ( a i ) · ∂ ( i...n ) ( b n − i ) Proof. The description of the underlying complex follows from the fact thatsimplicial resolutions in DGVect ( k ) are the underlying complexes of simpli-cial resolutions in DGAlg ( k ). We now recover the multiplication given byconvolution. Let φ and ψ be two natural transformations corresponding to a = ( a , a , . . . ) and b = ( b , b , . . . ). Then( φ ∗ ψ ) n ( f <... Application to the cosimplicial system of a DG-bialgebra. Let A be a DG-bialgebra, and let A • be its associated cosimplicial system of DG-algebras, as in (1). Let us use the above formulas to compute its homotopylimit. Proposition A.7. holim ∆ ( A • ) ≃ Cobar coaug ( A ) . Proof.