Finite Pre-Tensor Categories that are Morita Equivalent to Finite Tensor Categories

TL;DR

This paper characterizes finite pre-tensor categories that are Morita equivalent to finite tensor categories, linking this equivalence to the properties of their Drinfeld centers.

Contribution

It provides a complete characterization of when a finite pre-tensor category is Morita equivalent to a finite tensor category, based on the Drinfeld center.

Findings

A finite pre-tensor category is Morita equivalent to a finite tensor category iff its Drinfeld center is a finite tensor category.

The paper discusses higher algebraic consequences of this characterization.

It extends the notion of Morita equivalence to finite pre-tensor categories.

Abstract

A finite pre-tensor category is a finite abelian category equipped with a right exact tensor product for which every projective object has duals. Finite tensor categories, for which every object has duals, are notable examples. More generally, the category of bimodules over an algebra in a finite tensor category is a finite pre-tensor category. In particular, it is natural to extend the notion of Morita equivalence between finite tensor categories to finite pre-tensor categories. We characterize completely those finite pre-tensor categories that are Morita equivalent to finite tensor categories. More precisely, we show that a finite pre-tensor category $\mathcal{C}$ is Morita equivalent to a finite tensor category if and only if the Drinfeld center of $\mathcal{C}$ is a finite tensor category. We also discuss higher algebraic consequences of our characterization.