  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.
