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The name pretty much says it all: there's a well-defined operation on categories equivalent to modules over some ring: if $\mathcal{C}_1=A\operatorname{-mod}$ and $\mathcal{C}_2=B\operatorname{-mod}$, then $\mathcal{C}_1\boxtimes\mathcal{C}_2 =A\otimes B\operatorname{-mod}$ is independent of the choice of $A$ and $B$ (since Morita equivalence of one of the factors will induce Morita equivalence of the tensor product).

Is there some clever name for this operation?

Apologies if I should know this; it's not such an easy thing to Google for.

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    $\begingroup$ If you use dualizable modules then the Deligne tensor product should work. I think there's a version for some kind of presentable categories but I'm not so familiar with the non-derived version; if you use derived categories then the tensor product of presentable infty-categories does this. $\endgroup$
    – Dylan Wilson
    Commented Oct 13, 2018 at 2:55
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    $\begingroup$ I believe abstractly this is $\operatorname{Adj}({\mathcal C}_1^{\mathrm{op}},{\mathcal C}_2)$, the category of contravariant adjoint pairs between these categories (an $A\otimes B$-module $M$ corresponds to the pair $\left\langle\operatorname{Hom}_A(-,M),\operatorname{Hom}_B(-,M)\right\rangle$, and any such pair $\left\langle L,R\right\rangle$ is isomorphic to one such, with $M\cong L(A)\cong R(B)$). I would call this the category of Galois connections but I never met either this or any other name for it. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 13, 2018 at 4:34
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    $\begingroup$ If I understand correctly, you're asking not just for a name, but for a categorical construction (and its possible name, if it has been done somewhere) which when specified to $A$-mod and $B$-mod yields $A\otimes B$-mod (I guess $A,B$ are meant to be commutative algebras over a common commutative ring). $\endgroup$
    – YCor
    Commented Oct 13, 2018 at 7:45
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    $\begingroup$ Very related to what @მამუკაჯიბლაძე was saying, I might call this the "cocomplete tensor product". This works in general for enriched category over a complete cocomplete symmetric monoidal category $V$, but taking $V = Ab$ as here, for $C, D$ two ($V$-)cocomplete categories there is a cocomplete $C \boxtimes D$ and a functor $C \times D \to C \boxtimes D$ that is separately cocontinuous in its two arguments, that is the universal such: any separately cocontinuous $F: C \times D \to E$ to cocomplete $E$ extends uniquely (up to isomorphism) to a cocontinuous $\hat{F}: C \boxtimes D \to E$. $\endgroup$
    – Todd Trimble
    Commented Oct 13, 2018 at 11:40

2 Answers 2

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I don't know of any "clever" name, but there are names that appear in the literature, such as the tensor product of Grothendieck categories, or the category of (additive) locally presentable categories. Here are two references:

  • Martin Brandenburg, Alexandru Chirvasitu, and Theo Johnson-Freyd, Reflexivity and dualizability in categorified linear algebra, Theory and Applications of Categories, Vol. 30, 2015, No. 23, pp 808-835. [link] See Lemma 2.7.

  • Julia Ramos González, On the tensor product of large categories, Thesis, University of Antwerp 2017. [link] See section 2.4.

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    $\begingroup$ BCJ-F is really a delight to read and I second the recommendation! $\endgroup$
    – Noah Snyder
    Commented Oct 13, 2018 at 20:43
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I say “Deligne-Kelly tensor product” for this. Technically Deligne tensor product is for certain abelian categories and right exact functors, and Kelly is for finitely cocomplete categories and right exact functors, while here you want locally presentable categories and cocontinuous functors. But they’re all similar enough, see Section 3.1-3.2 of Ben-Zvi-Brochier-Jordan for a quick summary and some references.

(Aside: If you want understand the technical differences between Deligne and Kelly's versions the must-read paper is Lopez Franco.)

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