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Factorization homology of enriched $\infty$-categories

David Ayala, John Francis, Aaron Mazel-Gee, Nick Rozenblyum

Abstract

For an arbitrary symmetric monoidal $\infty$-category $\mathcal{V}$, we define the factorization homology of $\mathcal{V}$-enriched $(\infty,1)$-categories over (possibly stratified) 1-manifolds and study some of its basic properties. In the case of spectral enrichment, we show that the value of factorization homology on a circle is topological Hochschild homology.

Factorization homology of enriched $\infty$-categories

Abstract

For an arbitrary symmetric monoidal -category , we define the factorization homology of -enriched -categories over (possibly stratified) 1-manifolds and study some of its basic properties. In the case of spectral enrichment, we show that the value of factorization homology on a circle is topological Hochschild homology.

Paper Structure

This paper contains 21 sections, 12 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Categorified integration in quantum field theory
  3. Hochschild homology, cyclic homology, and the cyclotomic trace
  4. Factorization homology of enriched $(\infty,1)$-categories
  5. Miscellaneous remarks
  6. Outline
  7. Notation and conventions
  8. Recollections of factorization homology in dimension 1
  9. The category ${\boldsymbol{\mathcal{D}}}$
  10. The category ${\boldsymbol{\mathcal{M}}}$
  11. Factorization homology
  12. Right-lax functoriality of factorization homology
  13. Flagged enriched $(\infty,1)$-categories
  14. Right-lax functoriality of factorization homology
  15. Enriched factorization homology
  16. ...and 6 more sections

Key Result

Theorem A

Let $(\mathcal{V},\boxtimes)$ be a $\otimes$-presentable symmetric monoidal $\infty$-category. For each $\mathcal{V}$-enriched $(\infty,1)$-category $\mathcal{C}$ factorization homology over the circle defines a $\mathbb T$-module in $\mathcal{V}$: In the case that $\mathcal{V} = {\mathcal{S} {\sf p}}$, and that $\mathcal{C}$ is presented by an ordinary category $\mathsf C$ enriched over the ordi

Theorems & Definitions (62)

  • Theorem A
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem: circle
  • Definition \oldthetheorem: circle
  • Proposition \oldthetheorem: circle
  • Theorem \oldthetheorem: circle
  • Definition \oldthetheorem
  • ...and 52 more