The Classification of Fusion 2-Categories

TL;DR

This paper provides a classification of fusion 2-categories using braided fusion categories and group cohomology, establishing a homotopy coherent equivalence and deriving finiteness and rigidity results.

Contribution

It introduces a homotopy coherent classification of fusion 2-categories via braided fusion categories and cohomological data, connecting higher category theory with homotopy theory.

Findings

Classification of fusion 2-categories in terms of braided fusion categories and cohomology

Homotopy equivalence between 2-categories and spaces with group actions

Finiteness and rigidity results for 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 $\mathrm{B}\mathbb{Z}/2$-equivariant spaces. Rank finiteness and Ocneanu rigidity for fusion 2-categories are immediate corollaries of our classification.