Tensor products of finitely cocomplete and abelian categories

TL;DR

This paper investigates the conditions under which Deligne's tensor product of abelian categories exists, showing it coincides with the tensor product of finitely cocomplete categories when both are abelian, and provides a counterexample.

Contribution

It establishes a precise criterion for the existence of Deligne's tensor product in terms of finitely cocomplete tensor products being abelian, and compares these constructions.

Findings

Deligne's tensor product exists iff the tensor product of finitely cocomplete categories is abelian.

Both tensor products coincide when they exist.

An example where Deligne's tensor product does not exist is provided.

Abstract

The purpose of this article is to study the existence of Deligne's tensor product of abelian categories by comparing it with the well-known ten- sor product of finitely cocomplete categories. The main result states that the former exists precisely when the latter is an abelian category, and moreover in this case both tensor products coincide. An example of two abelian categories whose Deligne tensor product does not exist is given.