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The Classification of Fusion 2-Categories

Thibault D. Décoppet, Peter Huston, Theo Johnson-Freyd, Dmitri Nikshych, David Penneys, Julia Plavnik, David Reutter, Matthew Yu

Abstract

We classify (multi)fusion 2-categories in terms of braided fusion categories and group cohomological data. This classification is homotopy coherent -- we provide an equivalence between the 3-groupoid of (multi)fusion 2-categories up to monoidal equivalences and a certain 3-groupoid of commuting squares of $\mathrm{B}\mathbb{Z}/2$-equivariant spaces. Rank finiteness and Ocneanu rigidity for fusion 2-categories are immediate corollaries of our classification.

The Classification of Fusion 2-Categories

Abstract

We classify (multi)fusion 2-categories in terms of braided fusion categories and group cohomological data. This classification is homotopy coherent -- we provide an equivalence between the 3-groupoid of (multi)fusion 2-categories up to monoidal equivalences and a certain 3-groupoid of commuting squares of -equivariant spaces. Rank finiteness and Ocneanu rigidity for fusion 2-categories are immediate corollaries of our classification.

Paper Structure

This paper contains 29 sections, 63 theorems, 74 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Classification of Fusion 2-Categories
  3. Profits of the Classification
  4. The Main Theorem: Relating Fusion 2-Categories and Delphic Squares
  5. From Witt squares to Delphic squares
  6. Outline
  7. Preliminaries
  8. Glossary
  9. Categorical Background
  10. Flagged Categories
  11. Fibrations
  12. Multifusion 2-Categories
  13. The Morita Category of Multifusion 2-Categories
  14. The Morita Category of Braided Multifusion 1-Categories
  15. Adjectives for 1-Morphisms
  16. ...and 14 more sections

Key Result

Theorem 1

Bosonic fusion $2$-categories are parameterised by the following data:

Figures (2)

  • Figure 1: This figure serves to display the different ways of presenting the data needed to define fusion 2-categories, and the fact that one can explicitly go between them.
  • Figure 2: Different models for $\mathsf{Spec}$ of the symmetric fusion 1-category $\mathcal{E}$

Theorems & Definitions (164)

  • Definition 1.1
  • Theorem 1: All bosons
  • Remark 1.2
  • Definition 1.3
  • Theorem 2: Emergent Fermions
  • Remark 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Corollary 3
  • ...and 154 more