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Stream: learning: questions

Topic: Algebras as presheaves on the Kleisli category

Paolo Perrone (Oct 26 2023 at 08:31):

Let $T$ be a monad on a category $C$. Denote its Kleisli category by $C_T$ and it's Eilenberg-Moore category by $C^T$.
It is a well known fact of category theory that $C^T$ is equivalent to the full subcategory of $[C_T^{op}, \mathrm{Set}]$ of those presheaves whose restriction to $C$ along the left-adjoint is representable.

Suppose now that $C$ is a monoidal category, $T$ is a monoidal (= strong commutative) monad, and that $C^T$ has enough coequalizers so that the tensor product of $T$-algebras is defined.
Is the equivalence of categories mentioned above a monoidal equivalence? (As tensor product of presheaves we take the Day convolution.)
Does anyone know if this has been worked out anywhere?

Matteo Capucci (he/him) (Oct 26 2023 at 08:48):

The nLab says it works on every 2-category equipped with a Yoneda structure (citing this). I'm pretty sure $\cal C \mapsto {\bf Psh}(C)$ with Day convolution is a Yoneda structure on $\bf MonCat$ (though maybe this not very helpful -- I converted a pretty plausible statement to a different plausible statement :laughing: )

Paolo Perrone (Oct 26 2023 at 09:26):

Oh, I think I've just proven it directly.
Still, I'm interested in whether there is a 2-categorical abstract proof, for example something along the lines of Matteo's idea.

Matteo Capucci (he/him) (Oct 26 2023 at 09:40):

Probably the easiest way to prove my claim is to dig out from the Yoneda structures literature a result that says 'the algebras of a nice pseudomonad/2-monad on a 2-category with Yoneda structure inherit the Yoneda structure', which seems likely to have been proven in someday in Australia. Maybe @Ivan Di Liberti @fosco @Nathanael Arkor know where to find it.

fosco (Oct 26 2023 at 09:48):

Matteo Capucci (he/him) said:

The nLab says it works on every 2-category equipped with a Yoneda structure (citing this). I'm pretty sure $\cal C \mapsto {\bf Psh}(C)$ with Day convolution is a Yoneda structure on $\bf MonCat$ (though maybe this not very helpful -- I converted a pretty plausible statement to a different plausible statement :laughing: )

It's not that simple, the kan extensions are not monoidal by default iirc. I made this mistake already once, Ivan knows.

fosco (Oct 26 2023 at 09:50):

Point is, it's very likely that a kz monad (and you have one, the proof is in ImgKelly) goes a long way proving facts about the yoneda embedding without the full power of a yoneda structure

Nathanael Arkor (Oct 26 2023 at 09:58):

It is standard that the small-presheaf construction lifts to monoidal categories and lax monoidal functors, e.g. in Im–Kelly or Walker's Distributive laws via admissibility. Every lax-idempotent pseudomonad induces a Yoneda structure (cf. Walker's Yoneda structures and KZ doctrines). I think it is then likely that Proposition 22 of Street–Walters could be applied, but there are details to check (e.g. that the monoidal structure on the category of algebras makes the category into an algebra object for the monad). (In particular, I suspect that, in this instance, a direct proof would be shorter and clearer.)