Group actions on 2-categories


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 [S. Gelaki, D. Naidu and D. Nikshych, Centers of graded fusion categories, Algebra Number Theory 3, No. 8 (2009), 959–990.]

Manuscripta Math. 159 (2019), no. 1-2, 81–115