Equivariant Morita theory for graded tensor categories

TL;DR

This paper extends Morita theory to graded tensor categories, establishing that graded Morita equivalence corresponds to equivalence of their equivariant Drinfeld centers as braided $G$-crossed tensor categories.

Contribution

It introduces a framework for graded Morita equivalence in finite tensor categories and links it to the structure of equivariant Drinfeld centers.

Findings

Graded Morita equivalence characterized by equivariant Drinfeld centers

Equivalence of centers implies graded Morita equivalence

Framework applies to finite tensor categories graded by finite groups

Abstract

We extend categorical Morita equivalence to finite tensor categories graded by a finite group $G$. We show that two such categories are graded Morita equivalent if and only if their equivariant Drinfeld centers are equivalent as braided $G$-crossed tensor categories.