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Aaron David Fairbanks (Dec 08 2024 at 16:13):Anybody know where I might cite for the following fact? Or something similar stated in lesser or greater generality?
Let V be a monoidal category. In the virtual double category of monoids and bimodules in V, monads on an object m are monoids in V under m, monad maps are monoid maps in V under m, and bimodules are just bimodules in V.
Mike Shulman (Dec 08 2024 at 17:53):Well, in the non-virtual case I stated basically this in Examples 11.3 and 11.6 of Framed bicategories and monoidal fibrations. I'm sure it was known long before that.
Mike Shulman (Dec 08 2024 at 17:54):Some greater generality is that a monad in Prof(V) on an object C is a V-category with an identity-on-objects functor from C.
Nathanael Arkor (Dec 08 2024 at 17:58):Again in the non-virtual setting, Theorem 2.3.18 of Spivak–Schultz–Rupel's String diagrams for traced and compact categories are oriented 1-cobordisms gives a general characterisation of for an exact equipment with local reflexive coequalisers.