# Abelian categories with tensor product

Is there a standard notion in the literature of abelian category with tensor product?

The definition ought to be wide enough to encompass all the usual examples of abelian categories with standard `tensor product'. I'd guess something like "symmetric monoidal bi-functor $\otimes \colon \mathcal{A} \times \mathcal{A} \rightarrow \mathcal{A}$ which preserves finite colimits" would do, but I wonder if there is a reference for this?

• See also here. – Zhen Lin Jan 17 '15 at 21:44

Let $(\mathcal{A},\otimes)$ be a monoidal category. It is called abelian monoidal if $\mathcal{A}$ is an abelian category and $\otimes$ is an additive bifunctor (see here, Section 1.5). An example given the linked paper is the category of bimodules over a ring (in fact any closed abelian monoidal category can be exactly embedded in a bimodule category).