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Fully-Dualizable and Invertible $\mathcal{E}_n$-Algebras

Pablo Bustillo Vazquez

Abstract

We prove a conjecture of Brochier, Jordan, Safronov, and Snyder [BJSS21], first formulated by Lurie [Lur09b], characterizing fully-dualizable and invertible $\mathcal{E}_n$-algebras viewed as objects in the higher Morita categories $\mathbf{Mor}_n(\mathcal{V})$ [Lur09b, Sch14, Hau17b, Hau23]. In other words, we characterize those $\mathcal{E}_n$-algebras which give rise to $(n + 1)$-dimensional topological quantum field theories (TQFT), and those which give rise to invertible theories.

Fully-Dualizable and Invertible $\mathcal{E}_n$-Algebras

Abstract

We prove a conjecture of Brochier, Jordan, Safronov, and Snyder [BJSS21], first formulated by Lurie [Lur09b], characterizing fully-dualizable and invertible -algebras viewed as objects in the higher Morita categories [Lur09b, Sch14, Hau17b, Hau23]. In other words, we characterize those -algebras which give rise to -dimensional topological quantum field theories (TQFT), and those which give rise to invertible theories.
Paper Structure (19 sections, 37 theorems, 27 equations, 3 figures)

This paper contains 19 sections, 37 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Section 2
  4. Section 3
  5. Section 4
  6. Acknowledgments
  7. Adjoints in (∞, n)-categories
  8. Colimits of (∞, n)-categories with adjoints have adjoints
  9. Freely Adding Adjoints and Subcategories with Adjoints
  10. Lifting Once-Fully-Adjoint Morphisms
  11. Invertibility
  12. Dualizability Data as Factorization Homology
  13. Homological Commutative Diagrams
  14. Handlebodies as Duality Datum
  15. Dualizability of Iterated Bimodules
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $\mathcal{C}\colon \mathbf{Cat}_{(\infty,2)}$. Precomposition by $L,R \colon c_k \rightarrow \mathbf{Adj}$ gives monomorphisms \begin{tikzcd} {[\Adj,\mathcal{C}]} && {[c_1, \mathcal{C}]} \arrow["{L^*}", curve={height=-6pt}, hook, from=1-1, to=1-3] \arrow["{R^*}"', curve={

Figures (3)

  • Figure 1: Pinching $\mathbb{R}^{3}_{\overline{1},\overline{\epsilon,\eta}} \mathrel{ \mkern2.5mu \hbox{${\mkern-2.5mu\relbar}$} \mathord{\mathop{ {$π_1$} { \hbox{\cleaders$\m@th\mkern-2.5muȀ\mkern-2.5mu$} } \clipbox{1em 0pt 1em 0pt}{} }\limits^{\pi_1}} \mkern-12mu\mathord{\twoheadrightarrow} } \mathbb{R}^{3}_{\overline{1},\overline{\eta}}$. The small arrows indicate the collapsing that occurs "inside" the handle.
  • Figure 2: Visualization of the diagram \ref{['eq:XY']}.
  • Figure 3: Visualization of the action on the left-hand side of \ref{['eq:action']} for $i = 2$. The background space is $\mathbb{R}^2 \times \mathbb{R}_{\geq 0}$ with $A$ supported on the $3$-dimensional strata (hence seen as an $\mathcal{E}_3$-algebra) and $M$ supported on the $2$-dimensional strata (hence seen as an $\mathcal{E}_3$-algebra acted on in an $\mathcal{E}_1$-one-sided-way by $A$). The $\mathcal{E}_1$-algebra of interest is then the factorization homology of an upper-half dome, with the $\mathcal{E}_1$-structure given by the radial symmetry, and it acts on $M$ pictured at the center of the bottom plane. This factorization homology can be interpreted as a relative form of Hochschild Homology.

Theorems & Definitions (95)

  • Definition 2.1: Adjoint
  • Theorem 2.1: Homotopy Uniqueness of Adjunctions RV16
  • Corollary 1: Homotopy Uniqueness of (higher) Adjunctions RV16
  • proof
  • Definition 2.2: $n$-Fold Simplicial Space with Adjoints
  • Lemma 1
  • proof
  • Corollary 2: Composition of Adjoints
  • proof
  • Lemma 2: Segalification Preserves Adjoints
  • ...and 85 more