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, Volume 159, Issue 1–2, pp 81–115 | Cite as

Group actions on 2-categories

  • Eugenia Bernaschini
  • César Galindo
  • Martín MombelliEmail author
Article
  • 22 Downloads

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).

Mathematics Subject Classification

18D05 18D10 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Eugenia Bernaschini
    • 1
  • César Galindo
    • 2
  • Martín Mombelli
    • 1
    Email author
  1. 1.Facultad de Matemática, Astronomía y FísicaUniversidad Nacional de CórdobaCórdobaArgentina
  2. 2.Departamento de MatemáticasUniversidad de los AndesBogotáColombia

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