Exact sequences of tensor categories with respect to a module category

TL;DR

This paper extends the concept of exact sequences in tensor categories to include module categories, establishing their properties, equivalences, and implications for semisimplicity, thereby generalizing known results for Hopf algebras.

Contribution

It introduces a new definition of exact sequences of tensor categories with respect to a module category and proves their fundamental properties and equivalences.

Findings

The Deligne tensor product forms an exact sequence in this new sense.

The dual of an exact sequence is also exact.

Semisimplicity of the middle term follows from the other terms being semisimple.

Abstract

We generalize the definition of an exact sequence of tensor categories due to Brugui\`eres and Natale, and introduce a new notion of an exact sequence of (finite) tensor categories with respect to a module category. We give three definitions of this notion and show their equivalence. In particular, the Deligne tensor product of tensor categories gives rise to an exact sequence in our sense. We also show that the dual to an exact sequence in our sense is again an exact sequence. This generalizes the corresponding statement for exact sequences of Hopf algebras. Finally, we show that the middle term of an exact sequence is semisimple if so are the other two terms.