Cyclic homology of categorical coalgebras and the free loop space

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 S¹-equivariant chains of 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.