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 ${\mathcal B}$ with an action of a group G, we construct a braided G-crossed monoidal category $\mathcal {Z}_G({\mathcal B})$ with trivial component the Drinfeld center of ${\mathcal B}$. We prove that, in the case of a G-action on the 2-category of representation of a tensor category ${\mathcal C}$, the 2-category of equivariant objects is biequivalent to the module categories over an associated G-extension of ${\mathcal C}$. 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 Gelaki et al. (Algebra Number Theory 3(8):959–990, 2009).