On the dualizability of fusion 2-categories

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 Douglas, Schommer-Pries, Snyder [Mem. Amer. Math. Soc. 268 (2021)], we prove that separable compact semisimple tensor 2-categories are fully dualizable objects therein. In particular, it then follows from the main theorem of Décoppet [Comp. Math.] 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.