It seems the notion of tensor product of abelian categories exists naturally. Does someone know the reference of the construction?

## 3 Answers

Deligne's article, 'Categories Tannakiennes,' section 5 would be a good place to look. It was published in the Grothendieck Festschrift, vol. 2.

A generalization of Deligne's construction may be found here: http://arxiv.org/abs/0911.4979. To recoved Deligne's construction one simply takes the perspective that any abelian category is a module category over $Vec$. Here we define $V\otimes N$, for any finite dimensional vector space $V$ and object $N$ to be the unique object representing the functor $N\mapsto Hom(V, Hom(N, N))$ (really internal hom).

In general let $\mathcal{M}, \mathcal{N}$ be right, left $\mathcal{C}$-module categories for $\mathcal{C}$ any tensor category. Then the $relative$ $tensor$ $product $ $\mathcal{M}\boxtimes_{\mathcal{C}}\mathcal{N}$ is defined as the unique (up to a unique equivalence) universal object for right exact in each variable $\mathcal{C}$-balanced bifunctors from the cartesian product $\mathcal{M}\times\mathcal{N}$. As such it follows that $\mathcal{M}\boxtimes_{Vec}\mathcal{N}=\mathcal{M}\boxtimes\mathcal{N}$ where $\mathcal{M}\boxtimes\mathcal{N}$ denotes the product of abelian categories defined in ``Categories Tannakiennes".

Yes Julian you are exactly right. In the paper you cite the Deligne product is extended to general (bi-)module categories in a way that reduces to the product of abelian categories in the case that the monoidal category involved is $Vec$. There it is also shown that certain classical formulas (adjunction of hom and tensor, for example) hold on the level of module categories.

In an upcoming article the tensor product of module categories is studied from the perspective that it is a special case of a generalization of the center of a module category as introduced by Drinfeld and studied to great effect by Müger et. al., the so called *relative center*.