Tensor of cocomplete categories
Let $C$, $D$ and $E$ be cocomplete categories.
Is there a construction $C \otimes D$ such that there is a correspondence between functors $C \otimes D \to E$ preserving colimits and functors $C \times D \to E$ preserving colimits in each of its compontents ("bilinear")? How can I construct it? Why not?
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$\begingroup$For simplicity, I will discuss categories with colimits of $\kappa$-small diagrams, where $\kappa$ is a regular cardinal. Specifically, consider the following category $\mathbf{K}$:
- The objects are small categories equipped with chosen $\kappa$-ary coproducts and coequalisers of parallel pairs.
- The morphisms are functors that strictly preserve the chosen colimits.
By standard arguments, $\mathbf{K}$ is a locally $\kappa$-presentable category. Given objects $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ in $\mathbf{K}$, define the following set: $$\mathbf{K} (\mathcal{A}, \mathcal{B} ; \mathcal{C}) = \{ F \in \mathrm{Fun} (\mathcal{A} \times \mathcal{B}, \mathcal{C}) : F \text{ strictly preserves colimits in each variable} \}$$ It is easy to see that $\mathbf{K} (\mathcal{A}, \mathcal{B} ; -) : \mathbf{K} \to \mathbf{Set}$ preserves all limits and also $\kappa$-filtered colimits. Thus, by the accessible adjoint functor theorem, it is represented by some object $\mathcal{A} \otimes \mathcal{B}$ in $\mathbf{K}$. Unfortunately, everything here is strict, so this doesn't quite do what we want.
The core issue is this: given objects $\mathcal{C}$ and $\mathcal{D}$ in $\mathbf{K}$, there may be functors $\mathcal{C} \to \mathcal{D}$ that preserve (up to isomorphism) colimits of $\kappa$-small diagrams that are not isomorphic to any functor $\mathcal{C} \to \mathcal{D}$ that strictly preserve the chosen colimits. If I understand the general theory correctly, this can be fixed: there exist an object $\tilde{\mathcal{C}}$ in $\mathbf{K}$ and a morphism $p : \tilde{\mathcal{C}} \to \mathcal{C}$ in $\mathbf{K}$ that is fully faithful and essentially surjective on objects such that there are enough morphisms $\tilde{\mathcal{C}} \to \mathcal{D}$.
Thus, given $\mathcal{A}$ and $\mathcal{B}$ in $\mathbf{K}$, the desired tensor product is not $\mathcal{A} \otimes \mathcal{B}$ itself but rather $\tilde{\mathcal{A}} \otimes \tilde{\mathcal{B}}$. I'm afraid I do not have a more explicit description of this category, but all of this is basically a souped-up version of the usual construction of tensor products by generators and relations.
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