2007/11/07 by Kathryn Hess, Hess, Kathryn, Paul-Eugène Parent +3
Mathematics · #16E40 #18G60 #19D55 (Primary) #55M20 #55U10 #81T30 (Secondary) #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.0711.1023
openalex publication_date 2007/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \cohoch (C) admits a natural comultiplicative structure. In particular, if K is a reduced simplicial set and C*K is its normalized chain complex, then \cohoch (C*K) is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on \cohoch (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 \cohoch(C*K) to H*\op L|K|, the homology of the free loops on the geometric realization of K.