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Stream: learning: questions

Topic: A bimonoid in [C,C]

fosco (Dec 20 2023 at 09:48):

On the nLab, at the [[bimonoid]] page, I read: "In the completely noncommutative situation of the monoidal category of endofunctors, one can look at various compatibilities between monads and comonads or monads and tensor products, for example involving distributive laws."

which is hinting at the fact that (perhaps) a distributive law fixes the lack of symmetry of the monoidal structure and allows to define, and find examples, of functors that are both a monad and a comonad, and compatibly so. I think there is a general, well-known theorem here, but googling didn't return much information.

So: what is a nontrivial example of a bimonoid in $[C,C]$?

Matteo Capucci (he/him) (Dec 20 2023 at 09:57):

If $C$ is cartesian and $M$ a monoid therein then $M \times -$ should be an example..?

Tobias Fritz (Dec 20 2023 at 09:57):

This sounds like the notion of bimonad considered here on the nlab. That page gives a pointer to Notes on bimonads and Hopf monads for more details. If I understand their Example 2.2 correctly, it's saying in particular that if $M$ is a bimonoid in a symmetric monoidal category $C$, then the functor $M \otimes -$ is a bimonad on $C$. They're actually doing this more generally though.

fosco (Dec 20 2023 at 10:02):

ah, of course, I should have remembered Hopf monads (and M x _...)