nLab
Deligne tensor product of abelian categories

Context

Additive and abelian categories

Monoidal categories

Contents

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 AA and BB two abelian categories, their Deligne tensor product ABA \boxtimes B is the abelian category such that for any other abelian category CC right exact functors of the form ABCA \boxtimes B \to C are equivalent to functors A×BCA \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 kk, there is a finite-dimensional algebra AA over kk and a kk-linear equivalence of categories

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

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

Proposition

For A,BAlg kA, B \in Alg_k two finite-dimensional associative algebras over a field kk, 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 kBA \otimes_k B:

AMod fdBMod fd(A kB)Mod fd. 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).

References

The construction was introduced in

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

A survey is in

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

Here the author points 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.

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

In the set of lecture notes

the Deligne tensor product is discussed in lecture 9.

Revised on July 24, 2014 22:56:30 by John Baez (138.23.235.90)