CoHochschild homology of chain coalgebras

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 CC is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex ℋ̂(C) admits a natural comultiplicative structure. In particular, if KK is a reduced simplicial set and C∗KC∗K is its normalized chain complex, then ℋ̂(C∗K) is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on ℋ̂(C∗K) when KK is a simplicial suspension.The coHochschild complex construction is topologically relevant. Given two simplicial maps g,h:K→Lg,h:K→L, where KK and LL are reduced, the homology of the coHochschild complex of C∗LC∗L with coefficients in C∗KC∗K is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of gg and hh, and this isomorphism respects comultiplicative structure. In particular, there is an isomorphism, respecting comultiplicative structure, from the homology of ℋ̂(C∗K) to H∗L|K|H∗L|K|, the homology of the free loops on the geometric realization of KK.