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By the Eilenberg-Watts theorem there is a monoidal equivalence between the category of $R^e$-bimodules and the category of cocontinuous, additive endofunctors on the category of $R$-bimodules. $R$ is a ring and $R^e=R\otimes \overline{R}$ its enveloping ring.

The category of cocontinuous, additive functors carries (apart from composition) another monoidal structure: Day convolution. This is defined like this: Take two concontinuous, additive endofunctors $F$ and $G$ on the category of $R$-bimodules (with the tensor product of $R$-bimodules as the monoidal product $\otimes$). Their Day convolution is defined as the left Kan extension of the external tensor product $\otimes \circ (F \times G)$ along $\otimes$.

Does this Kan extension exist? While the category $\mathcal{C}$ of $R$-bimodules is cocomplete, it is neither small nor essentially small. Also I am not sure that the left Kan extension is cocontinuous and additive. I think it amounts to the existence of this coend for all $Z\in \mathcal{C}$: $$\int^{X,Y\in \mathcal{C}}\mathcal{C}(X\otimes_R Y,Z)\otimes_{\mathbb{Z}}(F(X)\otimes_R G(Y)).$$ Is that true? And does this coend exist?

What does the Day convolution tensor product correspond to in the category of $R^e$-bimodules under the Eilenberg-Watts equivalence?

Margaret
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  • It doesn't answer your question, but for the existence you can use that a category of modules $\mathcal{C} = R-Mod$ is locally finitely presentable, meaning that the subcategory of finiteley presentable objects $\mathcal{C}^{fp}$ generates the whole category under filtered colimits. – MPos Mar 20 '24 at 10:53
  • This subcategory is essentially small, so one can extend if you restrict your source to this subcategory. Co-continuity of the Left Kan extension is not obvious to me either at first glance. But here is a discussion on it: https://mathoverflow.net/questions/112137/when-do-kan-extensions-preserve-limits-colimits Not sure if that helps though. Your coend formula seems correct! – MPos Mar 20 '24 at 10:56

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