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Let $R$ be a commutative ring. A categorification of the concept of an $R$-module might be a $R$-linear cocomplete category (there are some reasons for this). For $R$-modules, we can define their tensor product as a classifying object of the $R$-bilinear maps on the product and it exists. My question is now if we can do the same thing for $R$-linear cocomplete categories.

Question. Let $C,D,E$ be $R$-linear cocomplete categories. A functor $C \times D \to E$ is called $R$-bilinear if it is cocontinuous and $R$-linear in each variable. We get a $2$-functor $\text{Bilin}(C \times D,-)$ from $R$-linear cocomplete categories to categories. Is it $2$-representable?

I've already tried to imitate the usual construction of the tensor product of $R$-modules, but the details do not work out. I'm pretty sure that it does work somehow and I would be glad if someone could explain this in detail.

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The answers to… are relevant (i.e., one makes a tensor product of module categories for R-mod). – David Ben-Zvi Jul 7 '11 at 1:00
I disagree that "$R$-linear cocomplete category" is the correct notion, because it allows for things that are much too large --- it would be like allowing $R$-modules to be non-sets. My favorite is "$R$-linear (locally) presentable category", and there the tensor product is relatively easy to describe --- it is given as an exercise in the book by Adamek and Rosicky (although I think they make a small error); we also describe some details in . – Theo Johnson-Freyd Jul 7 '11 at 1:19
@David: Interesting, but Deligne's construction uses specificly the abelian (and much more) structure. – Martin Brandenburg Jul 7 '11 at 2:43
@Theo: What a fantastic and interesting paper! Alternatively, feel free to impose cardinality bounds like the one by Lurie here… – Martin Brandenburg Jul 7 '11 at 8:13
Assuming what you mean by 2-representable is what I know by the name bi-representalbe, these questions are answered at the end of Max Kelly's book (although the pseudonaturality is not investigated there). Re Deligne's product, if you read past the definition you realise he proves its existence for a special class of abelian categories. – Nacho Lopez Jul 7 '11 at 9:52

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