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In the bicategory $\mathsf{Bimod}$ of rings, bimodules and bimodule morphisms there is also a direct sum making every category $\mathsf{Bimod}(\mathsf{R},\mathsf{S})$ of $(\mathsf{R},\mathsf{S})$-bimodules an abelian category. This direct sum is compatible with the tensor product in the sense that we have a distributive law.

Is there a notion of "distributive bicategory" (or something else) generalizing this structure?

I could only find this on nLab about distributive monoidal categories, but there seems to be no mention of a 2-/bi-category version.

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  • $\begingroup$ There certainly is. In fact, most categorification involves $2$-categories with additive structure. Not sure what name to give them though (I am mostly familiar with the more specialized versions with finitary $2$-categories). $\endgroup$ Mar 27, 2018 at 11:06

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I have heard this called an abelian 2-category, although I don't like the terminology. It's just a particular kind of enriched bicategory, see the nLab.

Namely, a bicategory $B$ enriched in the monoidal 2-category of additive categories has additive hom-categories $B(x,y)$ together with additive composition functors-in particular, composition with a fixed morphism is additive, so preserves biproducts. I would probably call such a thing a locally additive bicategory. Now the bicategory of bimodules happens to be locally abelian, but the composition functors are not exact, rather, cocontinuous-this is a common assumption on the composition in bicategories that are locally cocomplete, see here: https://ncatlab.org/nlab/show/local+colimit

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