# Questions about Deligne's tensor product of abelian categories

Let $$A$$ and $$B$$ be abelian categories, then $$A \times B$$ is an abelian category. Also denote $$Ab$$ to be the category of abelian groups or any abelian category.

If want to study a bi-additive (also called bilinear) functor $$F(-,-): A\times B \rightarrow Ab$$ , which is right exact in each variable, then Deligne's tensor product $$A\boxtimes B$$ which is an abelian category allows us to study bi-additive functor F on $$A\times B$$ by studying an additive right exact functor $$\overline{F}:A\boxtimes B \rightarrow Ab$$.

More precisely, we have a functor $$\boxtimes: A \times B \rightarrow A\boxtimes B$$ such that for any $$F$$ described above, then there exists a unique additive right exact functor $$\overline{F}: A\boxtimes B \rightarrow Ab$$ such that $$F= \overline{F} \circ \boxtimes$$.

I have a few questions regarding this construction.

1) Is $$\boxtimes$$ essentially surjective? (put any conditions if you want)

2) For $$A$$= the oppositive category of finite typed commutative group schemes over a field $$k$$ and $$B$$= the category of finite typed commutative group schemes over $$k$$. Does $$A \boxtimes B$$ exist? If yes, can we describe its objects if $$\boxtimes$$ is not essentially surjective?

3) Deligne's tensor product allows us to deal with bi-additive functor that is right exact in each variable as mentioned. Does there exist an alternative constuction allowing us to deal with a bi-additive functor that is left exact in each variable?

• Is $(a \boxtimes b)\oplus (a’\boxtimes b’)$ in the essential image? – Piotr Achinger Nov 7 '19 at 15:20
• You could probably use that the analogous map from a product of abelian groups to its tensor product is not surjective to construct a counter example for $1)$ – leibnewtz Nov 7 '19 at 16:47
• You guys are right. I was being silly. I should've asked whether the essential image generates (with direct sum) all the objects in $A \boxtimes B$. Tbh I have almost no idea how to construct the box tensor category here. But I really want to work with an abelian category that has objects closely related to $Ob(A) \times Ob(B)$. Also studying left exactness is more important for me. For example, $Hom$ functor is considered as a bi-additive functor $Ab^{op} \times Ab \rightarrow Ab$ which is left-exact in each variable. – wkf Nov 7 '19 at 17:35

1) In general, no. But $$A \boxtimes B$$ is the closure of $$\{a \boxtimes b : a \in A,\, b \in B\}$$ under finite colimits. Specifically, every object can be written as the cokernel of some map $$\bigoplus_i a_i \boxtimes b_i \to \bigoplus_j c_j \boxtimes d_j$$.
3) If $$A \boxtimes_r B$$ classifies functors which are right exact in both variables, and $$A \boxtimes_l B$$ classifies functors which are left exact in both variables, then $$(A \boxtimes_l B) = (A^{op} \boxtimes_r B^{op})^{op}$$.
• Thank you very much for the answers. Regarding 2) or in general the existence of Deligne's tensor product, I only see some relevant results concerning categories enriched in k-vector spaces (k: fields) namely k-linear categories e.g. the product exists when both $A$ and $B$ are length artinian categories. Is there any relevant results concerning ordinary abelian categories? Or maybe in the simplest case (or not), does the product exist when we take both $A$ and $B$ as the category of abelian groups? – wkf Apr 13 at 9:56