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Cyclic homology of categorical coalgebras and the free loop space

Manuel Rivera, Daniel Tolosa

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.

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 is naturally quasi-isomorphic to the -equivariant chains of the free loop space of . This statement does not require any hypotheses on 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.
Paper Structure (21 sections, 15 theorems, 189 equations, 4 figures)

This paper contains 21 sections, 15 theorems, 189 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Polytopes
  3. Freehedra
  4. Goodwillie polytopes
  5. Categorical coalgebras and dg categories
  6. Categorical coalgebras
  7. Differential graded categories
  8. Cobar functor
  9. The Hochschild complex of a dg category
  10. The coHochschild complex of a categorical coalgebra
  11. A natural chain contraction between the Hochschild and coHochschild complexes
  12. Chain models for path spaces
  13. Chains
  14. Models for path spaces
  15. A geometric model for the map $\pi$
  16. ...and 6 more sections

Key Result

Theorem 1

For any topological space $X$, there is a zig-zag of natural quasi-isomorphisms of mixed complexes As a consequence, the positive cyclic homology of the categorical coalgebra $\mathcal{C}(X)$ is naturally isomorphic to $H^{S^1}_\bullet(\mathcal{L}X;R)$, the $S^1$-equivariant homology of $\mathcal{L}X$.

Figures (4)

  • Figure 2.1: Two labelings of the freehedron $F_2$.
  • Figure 2.2: (left) $G_3^{0}$, (center) $G_3^{\epsilon}$, where $0< \epsilon < \frac{1}{2}$, and (right) $G_3^{\frac{1 }{ 2}}$.
  • Figure 2.3: Freehedra and Goodwillie polytopes in low dimensions.
  • Figure 2.4: Example of the decomposition of $G_n^{0<\epsilon< \frac{1}{2}}$ for $n=3$.

Theorems & Definitions (41)

  • Theorem 1: \ref{['maintheorem']}
  • Definition 2.1.1
  • Proposition 2.2.1
  • proof
  • Corollary 2.2.2
  • proof
  • Definition 3.1.1
  • Example 3.1.2
  • Example 3.1.3
  • Example 3.2.1
  • ...and 31 more