Cohochschild Homology of Chain Coalgebras

Generalizing 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 c 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 c H (C∗K) is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on c 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 a isomorphism, respecting comultiplicative structure, from the homology of c H (C∗K) to H∗L|K|, the homology of the free loops on the geometric realization of K.