Group actions on 2-categories

Abstract

We study actions of discrete groups on 2-categories. The motivating examples are actions on the 2-category of representations of finite tensor categories and their relation with the extension theory of tensor categories by groups. Associated to a group action on a 2-category, we construct the 2-category of equivariant objects. We also introduce the G-equivariant notions of pseudofunctor, pseudonatural transformation and modification. Our first main result is a coherence theorem for 2-categories with an action of a group. For a 2-category B with an action of a group G, we construct a braided G-crossed monoidal category Z_G(B) with trivial component the Drinfeld center of B. We prove that, in the case of a G-action on the 2-category of representation of a tensor category C, the 2-category of equivariant objects is biequivalent to the module categories over an associated G-extension of C. Finally, we prove that the center of the equivariant 2-category is monoidally equivalent to the equivariantization of a relative center, generalizing results obtained in [S. Gelaki, D. Naidu and D. Nikshych, Centers of graded fusion categories, Algebra Number Theory 3, No. 8 (2009), 959--990.]