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Stream: learning: questions

Topic: Eilenberg-Moore as a "colimit completion" of Kleisli

Aaron David Fairbanks (Jul 10 2025 at 04:01):

The nLab page [[Eilenberg-Moore category]] describes the Eilenberg-Moore category of a monad as a "colimit completion" of the Kleisli category. Is there any sense in which this is a free cocompletion under some class of colimit?

Bryce Clarke (Jul 10 2025 at 04:19):

https://m.youtube.com/watch?v=-RbDGHTE8OQ

Bryce Clarke (Jul 10 2025 at 04:20):

John Bourke, Bicategorical enrichment in algebra

Bryce Clarke (Jul 10 2025 at 04:20):

Abstract: In category theory, sometimes one does not wish to work with categories per se but instead categories over a fixed base. Such concrete categories can be viewed as categories enriched in a quantoloid, a certain bicategory. Garner showed this perspective is illuminating, using it to characterise topological categories as bicategory-enriched categories which are total.

In this talk, I will explain how the same enrichment is useful in algebra, where we also sometimes work over a fixed base. We will use the bicategorically-enriched perspective to show that Eilenberg-Moore categories of monads are free cocompletions of their Kleisli categories, which is false from the traditional point of view, and use this to give a nice proof of Beck's monadicity theorem. This is a report on ongoing work with Soichiro Fujii.

Bryce Clarke (Jul 10 2025 at 04:23):

@Aaron David Fairbanks I remember watching this talk but I don’t know if there are slides or a paper yet. You could certainly contact John (maybe after CT next week) for more info if you’re interested.

Aaron David Fairbanks (Jul 10 2025 at 04:23):

Thanks, I'll take a look!

Nathanael Arkor (Jul 10 2025 at 06:32):

There are slides from Soichiro's PSSL talk, but no paper yet.

Aaron David Fairbanks (Jul 10 2025 at 19:09):

That's very interesting and relevant, thanks. So if $T$ is a monad on a category $C$, the forgetful functor from the Eilenberg-Moore category $C^T \to C$ is in a certain sense a free cocompletion of the forgetful functor from the Kleisli category $C_T \to C$. But from John's talk abstract it sounds like it's not true that the Eilenberg-Moore category itself is a cocompletion of the Kleisli category itself under a class of colimits. Can we see this?

I wonder, is there an example of two monads $T_1$ and $T_2$ on a category $C$ and a functor $F \colon C_{T_1} \to C^{T_2}$ (from the Kleisli category $C_{T_1}$ of $T_1$ to the Eilenberg-Moore category $C^{T_2}$ of $T_2$) such that the induced extension $\mathrm{Psh}(C_{T_1}) \to \mathrm{Psh}(C_{T_2})$ does not restrict to a functor $C^{T_1} \to C^{T_2}$?