Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories

Abstract

We introduce group-theoretical fusion 2-categories, a strong categorification of the notion of a group-theoretical fusion 1-category. Physically speaking, such fusion 2-categories arise by gauging subgroups of a global symmetry. We show that group-theoretical fusion 2-categories are completely characterized by the property that the braided fusion 1-category of endomorphisms of the monoidal unit is Tannakian. Then, we describe the underlying finite semisimple 2-category of group-theoretical fusion 2-categories, and, more generally, of certain 2-categories of bimodules. We also partially describe the fusion rules of group-theoretical fusion 2-categories, and investigate the group gradings of such fusion 2-categories. Using our previous results, we classify fusion 2-categories admitting a fiber 2-functor. Next, we study fusion 2-categories with a Tambara-Yamagami defect, that is $\mathbb{Z}/2$-graded fusion 2-categories whose non-trivially graded factor is $\mathbf{2Vect}$. We classify these fusion 2-categories, and examine more closely the more restrictive notion of Tambara-Yamagami fusion 2-categories. Throughout, we give many examples to illustrate our various results.