On the Classification of Pointed Fusion Categories up to weak Morita Equivalence

TL;DR

This paper characterizes when two pointed fusion categories are weakly Morita equivalent using cohomological conditions, aiding classification of these categories based on their global dimension.

Contribution

It provides necessary and sufficient cohomological criteria for weak Morita equivalence of pointed fusion categories, extending classification methods.

Findings

Cohomological conditions for weak Morita equivalence derived

Use of Lyndon-Hochschild-Serre spectral sequence in classification

Potential to classify pointed fusion categories by global dimension

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 $C$ and $D$ are weakly Morita equivalent if there exists an indecomposable right module category $M$ over $C$ such that $Fun_C(M,M)$ and $D$ 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 may permit to classify the equivalence classes of pointed fusion categories of any given global dimension.