On the classification of pointed fusion categories up to weak Morita equivalence

Bernardo Uribe

Vol. 290 (2017), No. 2, 437–466

DOI: 10.2140/pjm.2017.290.437

Abstract

A pointed fusion category is a rigid tensor category with finitely many isomorphism classes of simple objects which moreover are invertible. Two tensor categories $\mathcal{C}$ and $\mathcal{D}$ are weakly Morita equivalent if there exists an indecomposable right module category $\mathcal{ℳ}$ over $\mathcal{C}$ such that ${Fun}_{\mathcal{C}}\left(\mathcal{ℳ},\mathcal{ℳ}\right)$ and $\mathcal{D}$ are tensor equivalent. We use the Lyndon–Hochschild–Serre spectral sequence associated to abelian group extensions to give necessary and sufficient conditions in terms of cohomology classes for two pointed fusion categories to be weakly Morita equivalent. This result allows one to classify the equivalence classes of pointed fusion categories of any given global dimension.

Keywords

tensor category, pointed tensor category, weak Morita equivalence, fusion category

Mathematical Subject Classification 2010

Primary: 18D10

Secondary: 20J06

Milestones

Received: 20 July 2016

Revised: 7 February 2017

Accepted: 10 March 2017

Published: 25 July 2017