Cyclic homology of categorical coalgebras and the free loop space

Abstract

We prove that the cyclic chain complex of the categorical coalgebra of singular chains on an arbitrary topological space $X$ is naturally quasi-isomorphic to the $S^1$-equivariant chains of the free loop space of $X$. This statement does not require any hypotheses on $X$ or on the commutative ring of coefficients. Along the way, we introduce a family of polytopes, coined as Goodwillie polytopes, that controls the combinatorics behind the relationship of the coHochschild complex of a categorical coalgebra and the Hochschild complex of its associated differential graded category.

Revised typos and added a necessary reparametrization step into Adams' map to make families of constant Moore loops to be parametrized by the zero length interval. Added references and acknowledgements. Recommended for acceptance in Advances in Mathematics