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).
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Bernaschini, E., Galindo, C. & Mombelli, M. Group actions on 2-categories. manuscripta math. 159, 81–115 (2019). https://doi.org/10.1007/s00229-018-1031-2
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DOI: https://doi.org/10.1007/s00229-018-1031-2