CoHochschild homology of chain coalgebras

Abstract

Communicated by C. Kassel MSC:Primary: 16E40 19D55 secondary: 18G60 55M20 55U10 81T30 a b s t r a c t Generalizing the work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra C is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex H (C) admits a natural comultiplicative structure. In particular, if K is a reduced simplicial set and C * K is its normalized chain complex, then H (C * K ) is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on H (C * K ) when K is a simplicial suspension.The coHochschild complex construction is topologically relevant. Given two simplicial maps g, h : K → L, where K and L are reduced, the homology of the coHochschild complex of C * L with coefficients in C * K is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of g and h, and this isomorphism respects comultiplicative structure. In particular, there is an isomorphism, respecting comultiplicative structure, from the homology of H (C * K ) to H * L|K |, the homology of the free loops on the geometric realization of K .

The Hochschild complex of a twisting cochain

“…is exactly the coHochschild complex on C with coefficients in N , as defined in [13]. In particular,…”

“…Theorem 2.36 also provides us with a new, more conceptual proof of Theorem B from [17], which was reformulated as Theorem 4.3 in [13] essentially as follows. Recall the DCSH retraction ρ C : BΩC → C from Example 2.33 and the chain map η C ⊗ Id ΩC :…”

“…Along similar lines, in [13], the authors specified conditions on a coalgebra C under which H (C) admits a comultiplication, a special case of the following result. …”

“…For any twisting cochain t : C → A, we define a chain complex H (t) that is a twisted tensor extension of A by C. The complex H (t) generalizes both the well-known Hochschild complex H (A) of a chain algebra A and the coHochschild complex H (C) of a chain coalgebra C [13].…”

“…of which both the well-known Hochschild complex of an augmented, associative algebra and the coHochschild complex of a coaugmented, coassociative coalgebra [13] are special cases. We therefore call H (t) the Hochschild complex of the twisting cochain t. We explore the extent of the naturality of the Hochschild complex construction and apply the results of this exploration to determining conditions under which H (t) admits multiplicative or comultiplicative structure.…”