nLab Deligne tensor product of abelian categories

Idea

The natural tensor product operation on finite abelian categories is known as the Deligne tensor product or Deligne box product, introduced in (Deligne 90).

For $A$ and $B$ two abelian categories, their Deligne tensor product $A \boxtimes B$ is the abelian category such that for any other abelian category $C$ right exact functors of the form $A \boxtimes B \to C$ are equivalent to functors $A \times B \to C$ that are right exact in each argument separately.

This tensor product exists for finite abelian categories but not generally on all abelian categories. However a slight variant does: instead of abelian categories one can consider categories with finite colimits, see Franco 12 and (Chirvasitu&Johnson-Freyd 11, remark 2.2.8).

Properties

In terms of categories of modules and tensor product of algebras

Recall that for every finite abelian category over $k$, there is a finite-dimensional algebra $A$ over $k$ and a $k$-linear equivalence of categories

$$\mathcal{C} \simeq A Mod_{fd} \,$$

where the right side consists of $A$-modules which are finite-dimensional as vector spaces over $k$. $A$ is uniquely determined up to Morita equivalence.

Proposition

For $A, B \in Alg_k$ two finite-dimensional associative algebras over a field $k$, the Deligne tensor product of their categories of finite-dimensional modules is the category of finite-dimensional modules of the tensor product of algebras $A \otimes_k B$:

$$A Mod_{fd} \boxtimes B Mod_{fd} \simeq (A \otimes_k B) Mod_{fd} \,.$$

This appears for instance as (EGNO, prop. 1.46.2). Without the finiteness constraints and using the tensor product of categories with finite colimits, this appears as (Chirvasitu&Johnson-Freyd 11, remark 2.2.8).

Related concepts

• tensor product of presentable (infinity,1)-categories

References

The construction was introduced in

• Pierre Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II. Progr. Math. 87, 111–195. Birkhäuser Boston. 1990.

Ignacio López Franco has pointed out that while Deligne’s tensor product always exists for finite abelian categories, it does not always exist for general abelian categories. He argues that in this case it is better to use Kelly’s tensor product of finitely cocomplete categories, because it always exists, and it agrees with Deligne’s tensor product when the latter exists. See:

• Ignacio López Franco?, Tensor products of finitely cocomplete and abelian categories, Journal of Algebra 396, (2013) 207-219. (web)

• Ignacio López Franco?, Tensor products of finitely cococomplete and abelian categories (2012) (pdf of slides)

Kelly’s tensor product can be found in section 6.5 of

• Max Kelly, Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press 1982, 245 pp.. Reprint: TAC reprints 10, tac,pdf

Similar remarks (in the context of 2-rings/2-modules) are from corollary 2.2.5 on in

• Alexandru Chirvasitu, Theo Johnson-Freyd, The fundamental pro-groupoid of an affine 2-scheme, Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469–522. (DOI, arXiv:1105.3104)

In this set of lecture notes

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Topics in Lie theory and Tensor Categories, Lecture notes (Spring 2009) (web)

the Deligne tensor product is discussed in lecture 9.