On the Dualizability of Fusion 2-Categories

TL;DR

This paper proves that fusion 2-categories over characteristic zero fields are fully dualizable within a Morita 4-category framework, enabling a better understanding of their algebraic and topological properties.

Contribution

It establishes the existence of the relative 2-Deligne tensor product for separable module 2-categories and proves their full dualizability in the Morita 4-category.

Findings

Separable module 2-categories have a well-defined tensor product.

Fusion 2-categories are fully dualizable over characteristic zero fields.

Results extend to any field of characteristic zero.

Abstract

Over an arbitrary field, we prove that the relative 2-Deligne tensor product of two separable module 2-categories over a compact semisimple tensor 2-category exists. This allows us to consider the Morita 4-category of compact semisimple tensor 2-categories, separable bimodule 2-categories, and their morphisms. Categorifying a result of arXiv:1312.7188, we prove that separable compact semisimple tensor 2-categories are fully dualizable objects therein. In particular, it then follows from the main theorem of arXiv:2211.04917 that, over an algebraically closed field of characteristic zero, every fusion 2-category is a fully dualizable object of the above Morita 4-category. We explain how this can be extended to any field of characteristic zero. Finally, we discuss the field theoretic interpretation of our results.