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 $2Vect$. 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.

Substantial revision in response to referees' feedback. In particular, the statement of proposition 3.4.4 was corrected

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Cited by in corpus (6)

• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Fusion 3-Categories for Duality Defects
• Twisted gauging and topological sectors in (2+1)d abelian lattice gauge theories
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Non-invertible symmetries of two-dimensional Non-Linear Sigma Models
• Compact Semisimple Tensor 2-Categories are Morita Connected