Braided Picard groups and graded extensions of braided tensor categories

Davydov, Alexei; Nikshych, Dmitri

doi:10.1007/s00029-021-00670-1

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Braided Picard groups and graded extensions of braided tensor categories

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Davydov, Alexei
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s00029-021-00670-1
(Publisher version)

https://doi.org/10.48550/arXiv.2006.08022
(Preprint)

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Citation

Davydov, A., & Nikshych, D. (2021). Braided Picard groups and graded extensions of braided tensor categories. Selecta Mathematica, 27(4): 65. doi:10.1007/s00029-021-00670-1.

Cite as: https://hdl.handle.net/21.11116/0000-000B-02C6-3

Abstract

We classify various types of graded extensions of a finite braided tensor
category $\cal B$ in terms of its $2$-categorical Picard groups. In particular,
we prove that braided extensions of $\cal B$ by a finite group $A$ correspond
to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard
group of $\cal B$ (consisting of invertible central $\cal B$-module
categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane
cohomology. We describe in detail braided $2$-categorical Picard groups of
symmetric fusion categories and of pointed braided fusion categories.