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James Deikun (May 01 2025 at 19:53):For monads one can define a notion of "right module" or "opalgebra". The obvious (and universal) example is the Kleisli category of the monad. The definition of these things is pretty abstract, though. What does the (op)algebraic structure on these things look like in Cat? For one thing, they're functors out of the monad's base category--this already makes them difficult to get a handle on, unlike the small objects that we usually deal with on the algebra side.
I've heard somewhere they could be identified with free algebras in Cat and that's why people don't talk about them, but I don't see how that works.
Nathanael Arkor (May 01 2025 at 20:02):They can be identified with functors out of the Kleisli category, which is why they aren't discussed so much in Cat. But they do appear in the literature: for instance, in Hirschowitz and Maggesi's Modules over monads and initial semantics.
Nathanael Arkor (May 01 2025 at 20:03):(That paper has several examples.)