E-infinity coalgebra structure on chain complexes with coefficients in Z
aa r X i v : . [ m a t h . A T ] A p r E-INFINITY COALGEBRA STRUCTURE ON CHAINCOMPLEXES WITH COEFFICIENTS IN Z JES ´US S ´ANCHEZ-GUEVARA
Abstract.
The aim of this paper is to construct an E ∞ -operad R and provethat this operad induces an E ∞ -coalgebra structure on chain complexes withcoefficients in Z . The operad R is an alternative to the description of the E ∞ -coalgebra structure on chain complexes by the Barrat-Eccles operad. Introduction
In [10], Smith describes an E ∞ -coalgebra structure on the chain complex of asimplicial set when the coefficients ring is Z . In order to do this, he uses an E ∞ -operad, denoted S , with components RΣ n , the Σ n -free bar resolution of Z . Themorphisms f n : RΣ n ⊗ C ∗ ( X ) → C ∗ ( X ) ⊗ n determined by the E ∞ -coalgebra struc-ture contains a family of higher diagonals on C ∗ ( X ), starting with an homotopicversion of the iterated Alexander-Whitney diagonal (given by x f n ([ ] n ⊗ x )). Theconstruction made by Smith can be seen as a version of the Barratt-Eccles operad(see [1]). Moreover, Berger and Fresse (see [2]) construct a explicit coaction overthe normalized chain complex associated to a simplicial set by the Barrat-Ecclesoperad that extend the structure given by the Alexander-Whitney diagonal.In this article, it is constructed an E ∞ -operad R which is used to give an alter-native description of the E ∞ -structure on the chain complex of an simplicial set.The method used to construct R gives an simply way to produce E ∞ -operads.The operad R presents similarities with the bar-cobar resolution of Ginzburg-Kapranov (see [6]). Berger and Moerdij (see [3]) show that this resolution canidentified with the W -construction of Boardman and Vogt (see [4]), given as a resultthat applied to the Barratt-Eccles operad, the W -construction gives a cofibrantresolution of it. Then, the construction of R can be seen as a middle point betweenthe Barratt-Eccles operad and its W -construction.The results in this article are based in the Phd thesis of the author [9], where theconstruction of E ∞ -operads is needed to study homotopy properties, described byAlain Prout in [7] and [8], of structures associated to chain complexes determinatedby the Eilenberg-Mac lane transformation.2. Preliminaries
Differential graded modules. A Z -module M is graded if there is a collec-tion { M i } i ∈ Z of submodules of M such that M = L i ∈ Z M i . A differential gradedmodule with augmentation and coefficients in Z , or DGA -module for short, is agraded module M together with an application ∂ : M → M of degree − ∂ = 0, an applications ǫ : M → Z , η : Z → M of degree 0, called the augmentation The author was supported by Universidad de Costa Rica. and coaugmentation of M , respectively, such that ǫ ◦ η = 1 Z . The category of DGA -modules is denoted
DGA -Mod.2.2.
Operads.
An operad P on the monoidal category DGA -Mod is a collections of
DGA -modules { P ( n ) } n ≥ together with right actions of the symmetric group Σ n oneach component P ( n ), and morphisms of the form γ : P ( r ) ⊗ P ( i ) ⊗ P ( i r ) → P ( i + · · · + i r ), which satisfies the usual conditions of existence of an unit, asociativity andequivariance. The morphisms γ will be called composition morphisms or simply thecomposition of the operad. A morphism between operads f : P → Q , is a collectionof DGA -morphisms f n : P ( n ) → Q ( n ) of degree 0, respecting the units, compositionand equivariance. The category of operads is denoted OP If we forget the composition morphism of an operad P , the collections with theright actions by the symmetrics groups are called S -modules. They form a categorydenoted S -Mod. The forgetful functor U : OP → S -Mod has a right adjoint denoted F : S -Mod → OP , called the free operad functor. Definition 2.1.
Let P be an operad on the category of DGA- Z -modules, withcomposition γ . A sub S -module I of U ( P ) is called an operadic ideal of P ifit satisfies γ ( x ⊗ y ⊗ · · · ⊗ y k ) ∈ I , whenever some of the elements x, y , . . . , y k belongs to I . Definition 2.2.
Let P be an operad and I an operadic ideal of P . We definethe quotient operad P / I as the operad with components given by ( P / I )( k ) = P ( k ) /I ( k ) for every k ≥
1, and composition induced by the composition of P . Remark . Clearly the operad structure P / I is well defined by the properties ofthe ideal, which allows the pass to the quotient of the composition in P .2.3. The Bar Resolution. Σ n will denote the symmetric group on of the set [ n ] = { , . . . , n } . The chain complex with coefficients in Z given by the Σ n -free bar reso-lution of Z is denoted R Σ n . Recall that degree m elements of R Σ n are Z -linear com-binations of elements of the form σ [ σ / · · · /σ m ], where σ, σ , . . . , σ m ∈ Σ n and theirborder is determinated by the equations ∂ = P mi =0 ( − i ∂ i , where ∂ [ σ / · · · /σ m ] = σ [ σ / · · · /σ m ], for 0 < i < m ∂ i [ σ / · · · /σ m ] = [ σ / · · · /σ i σ i +1 / · · · /σ m ], and ∂ m [ σ / · · · /σ m ] = [ σ / · · · /σ m − ]. In degree zero, the Z [Σ n ]-module is generatedby the element writed [ ].The contracting chain homotopy for the chain complex R Σ n is the application ψ n : R Σ n → R Σ n of degree 1 defined by the relations ψ n [ σ / · · · /σ m ] = 0 and ψ n σ [ σ / · · · /σ m ] = [ σ/σ / · · · /σ m ].2.4. E ∞ -Operads.Definition 2.4. An operad P on the category DGA -Mod is called E ∞ -operad ifeach component P ( n ) is a Σ n -free resolution of Z . Definition 2.5.
We call E ∞ -algebra any P -algebra with P an E ∞ -operad. Andin the same way, an E ∞ -coalgebra is an P -coalgebra where the operad P is an E ∞ -operad.We introduce a notion of morphism between E ∞ -coalgebras which is well suitedfor our purpose. Definition 2.6.
Let P be an E ∞ -operad on the category DGA -Mod, and let
A, B P -coalgebras. A morphism f : A → B of P -coalgebras is a morphism of DGA -Mod -INFINITY COALGEBRA STRUCTURE ON CHAIN COMPLEXES WITH COEFFICIENTS IN Z which preserves the P -coalgebra structure up to homotopy, that is, the followingdiagram(2.1) P ( n ) ⊗ A ϕ An / / ⊗ f (cid:15) (cid:15) A ⊗ nf ⊗ n (cid:15) (cid:15) P ( n ) ⊗ B ϕ Bn / / B ⊗ n is commutative up to homotopy for every n >
0, where ϕ An and ϕ Bn are theassociated morphisms of the P -coalgebra structure of A and B , respectively. Thecategory of coalgebras on the operad P is denoted P -CoAlg.3. The Operad R In this section, it is constructed an E ∞ -operad R which is used to describe C ∗ ( X ) as a E ∞ -coalgebra. Roughly speaking, to construct the operad R , first takethe S -module with components the Z [Σ n ]-free bar resolutions of Z , and then makethe quotient of the free operad on this S -module by a suitable operad ideal I (see[6] § Definition 3.1.
Let S be the be the S -module on the category DGA -Mod, withcomponents S ( n ) = RΣ n , the Z [Σ n ]-free bar resolution of Z . Define the operad R as the quotient operad F ( S ) / J , where J is the operadic ideal of the free operad F ( S ) generating by the elements of degree zero of F ( S ) of the form x − y , where x and y are not null. Theorem 3.2.
The operad R is an E ∞ -operad and induces an E ∞ -coalgebra es-tructure on C ∗ ( X ) .Proof. It suffices to exhibit in each arity an contracting chain homotopy. In arity n , the contracting chain homotopy Φ n : R ( n ) → R ( n ) is obtained by extending on R ( n ) the contracting chain homotopy ψ n from the component RΣ n of S as follows. R (2) is isomorphic to S (2), so the contracting chain homotopy remains the same.When n > R ( n ) has two types of elements: the elements from the injection S ( n ) → R ( n ) and the elements of the form γ ( x ; y , . . . , y r ), where x ∈ S ( r ) and y j ∈ R ( i j ). In the first case Φ n will behaves as the contracting chain homotopy in S ( n ), and for the second case, we define Φ n γ ( x ; y , . . . , y r ) = γ (Φ n ( x ); y , . . . , y r ).To check that ∂ Φ n + Φ n ∂ = 1, let x of the form [ σ | · · · | σ l ], with σ j ∈ Σ r . Now ∂ Φ n γ ( x ; y , . . . , y r ) = ∂γ (Φ n ( x ); y , . . . , y r ) = 0. On the other hand,Φ n ∂γ ( x ; y , . . . , y r )(3.1) =Φ n γ ( ∂x ; y , . . . , y r ) + (sign) X Φ n γ ( x ; y , . . . , ∂y j , . . . , y r )(3.2) = γ (Φ n ∂x ; y , . . . , y r ) + (sign) X γ (Φ n x ; y , . . . , ∂y j , . . . , y r )(3.3) = γ ( x − ∂ Φ n x ; y , . . . , y r )(3.4) = γ ( x ; y , . . . , y r )(3.5) JES ´US S ´ANCHEZ-GUEVARA
When x has the form σ [ σ | · · · | σ l ] the verification is similar, because the com-positions γ satisfy the following equivariance relation: γ ( σ [ σ | · · · | σ l ]; y , . . . , y r ) = γ ([ σ | · · · | σ l ]; y σ − (1) , . . . , y σ − ( l ) ).Now, the universal property of the coaugmentation ι of the adjunction F ⊢ U ,gives the commutative diagram:(3.6) S i ! ! ❈❈❈❈❈❈❈❈❈ ι / / F ( S ) p (cid:15) (cid:15) S Where the morphism i is the identity of S -modules. It is easy to see that p respect the ideal J because, when the free operad construction is interpreted byrooted trees, p is essentially the contraction of vertices of trees. Thus p pass tothe quotient and we obtain a morphism of operads p : R → S , which implies thatevery S -coalgebra is an R -coalgebra. (cid:3) Corollary 3.3.
The construction in theorem 3.2 is functorial.Proof.
The functoriality of the S -coalgebra structure is heredited by the R -coalgebraestructure by the operad morphism p : R → S in the proof of theorem 3.2, as showsthe following commutative diagramm for every morphism f : X → Y :(3.7) R p / / S / / & & ▲▲▲▲▲▲▲▲▲▲▲ CoEnd( C ∗ ( X )) f ∗ (cid:15) (cid:15) CoEnd( C ∗ ( Y )) (cid:3) We can understand the relation between the operad R and the operad S by thefollowing proposition. Corollary 3.4.
There is an operad ideal I such that S ∼ = R / I .Proof. This is because the underlying S -module of S is S , and a direct consequenceof the definition of compositions γ of S (see [10]), in the sense that, the operadicideal I is defined by the identification needed for γ . (cid:3) In [5] Vallette and Dehling describe an operad similar to R and they show thatthis operad can be used to explicitly state (by the use relations) the definition of E ∞ -algebras, as it is already possible for A ∞ -algebras. Corollary 3.5.
Let A be a DGA -module together with: (1)
For every integer m ≥ , n ≥ and σ, σ , . . . , σ n ∈ Σ m , morphisms ofdegree n : µ σ [ σ / ··· /σ n ] m : A → A ⊗ n . (2) For every integer m ≥ and σ ∈ Σ m , applications of degree : µ σ [ ] m : A → A ⊗ n . Suppose these morphisms satisfy the following relations: (1) µ σx = µ x σ , where σ is the right action on n factors. -INFINITY COALGEBRA STRUCTURE ON CHAIN COMPLEXES WITH COEFFICIENTS IN Z (2) µ x + y = µ x + µy and ∂µ x = µ ∂x . (3) ( µ [ ] m ⊗ · · · ⊗ µ [ ] mn ) µ [ ] n = µ [ ] m ··· + mn .Then, A is an R -coalgebra if and only if A has an structure of this type.Proof. This is directly implied by the operad morphism
R →
Coend( A ). (cid:3) References
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