Cyclic homology of categorical coalgebras and the free loop space

Manuel Rivera; Daniel Tolosa

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arXiv · 2403.08116

Cyclic homology of categorical coalgebras and the free loop space

Manuel Rivera·Daniel Tolosa

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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.

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Manuel Rivera, Daniel Tolosa. 2024-03-12. Cyclic homology of categorical coalgebras and the free loop space. https://arxiv.org/abs/2403.08116

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