# Bicategory of bimodules over internal monoids

In a monoidal category $(C,\otimes)$ one can consider internal monoids and bimodules over these monoids (provided some additional requirements like existence of coequalizers hold). These monoids together with bimodules as 1-morphisms and bimodule morphisms as 2-morphisms should constitute a bicategory $\mathsf{Bimod}$.

Does anyone know a reference for this result? The only thing I could find so far is this page on nLab, but there is no proof given.

• This work by Haugseng: arxiv.org/abs/1412.8459 might be relevant for you Aug 6 '18 at 14:07
• Specifically, Section 2 of Haugseng does what you're looking for in the further generality where $C$ is an $(\infty,1)$-category, using Segal spaces as the formalism for $(\infty,1)$-categories and $(\infty,2)$-categories. Since you just want the discrete setting and can use the easier notion of bicategory, it should be easier. But I don't know of the top of my head somewhere that really does the details. I thought it might be in Müger or Yamagami, but both of them just seem to treat this as obvious. It really should just be a tedious check, is there a specific step that tripped you up? Aug 6 '18 at 15:00
• Thanks for the help so far! I agree that this proof should a "simple" check of a long list of small statements. Nevertheless, when I came to defining the associativity natural isomorphism I really lost the overview, which made me wondering if there is conceptually clearer way to show all of this. And besides that I wanted to know if this has been done, or if it is just folklore (quite convincing, but nevertheles...). Aug 6 '18 at 15:41
• It probably is not the earlier reference, but all is proved carefully in this work of Joyal and Gambino (chapter 4), if I recall correctly. Also I remembered doing this myself in the context of pseudo double categories once and it was not that long, so it might worth integrating the proofs you want directly into your work if you don't find precise references. Aug 11 '18 at 7:53