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?

  • $\begingroup$ Is $(a \boxtimes b)\oplus (a’\boxtimes b’)$ in the essential image? $\endgroup$ – Piotr Achinger Nov 7 '19 at 15:20
  • $\begingroup$ 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 $1)$ $\endgroup$ – leibnewtz Nov 7 '19 at 16:47
  • $\begingroup$ 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. $\endgroup$ – 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 $\bigoplus_i a_i \boxtimes b_i \to \bigoplus_j c_j \boxtimes d_j$.

Unfortunately I cannot say anything about 2).

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 $(A \boxtimes_l B) = (A^{op} \boxtimes_r B^{op})^{op}$.

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  • $\begingroup$ 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? $\endgroup$ – wkf Apr 13 at 9:56

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