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

Understand

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.