On the stable equivalences between finite tensor categories

TL;DR

This paper investigates conditions under which stable equivalences between finite tensor categories can be lifted to tensor equivalences, advancing Morita theory in tensor triangulated categories.

Contribution

It establishes that stable equivalences induced by exact monoidal functors can be lifted to tensor equivalences under specific conditions for categories without projective simple objects.

Findings

Stable equivalences can be lifted to tensor equivalences.

Conditions for lifting are identified for categories with no projective simple objects.

Advances Morita theory in tensor triangulated categories.

Abstract

We aim to study Morita theory for tensor triangulated categories. For two finite tensor categories having no projective simple objects, we prove that their stable equivalence induced by an exact $\Bbbk$-linear monoidal functor can be lifted to a tensor equivalence under some certain conditions.