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Stream: learning: questions

Topic: compatible algebras and opalgebras

Nathanael Arkor (Nov 23 2021 at 14:31):

Let $t$ be a monad on an object $a$ in a 2-category. We can consider algebras and opalgebras for $t$ (also called "left-modules" and "right-modules", see [[module over a monad]]). An algebra is a 1-cell $x : b \to a$ and a 2-cell $tx \Rightarrow x$ satisfying some conditions and an opalgebra is a 1-cell $y : a \to b$ and a 2-cell $yt \Rightarrow y$ satisfying some conditions. It seems natural then to consider a pair of an algebra and an opalgebra such that the two 2-cells $ytx \Rightarrow yx$ are equal. Do these pairs have a name (alternatively: can anything interesting be said about them), or is there some observation that makes them not worth considering? (These are not the same as endo-bimodules, but they do appear when defining the composition of bimodules.)

Nathanael Arkor (Nov 23 2021 at 14:37):

An additional potential condition one could impose is that $xy \cong t$, which is satisfied for the Kleisli or EM object, or for (op)algebras induced by adjunctions generating $t$.

Todd Trimble (Dec 26 2021 at 20:58):

Nathanael Arkor said:

Let $t$ be a monad on an object $a$ in a 2-category. We can consider algebras and opalgebras for $t$ (also called "left-modules" and "right-modules", see [[module over a monad]]). An algebra is a 1-cell $x : b \to a$ and a 2-cell $tx \Rightarrow x$ satisfying some conditions and an opalgebra is a 1-cell $y : a \to b$ and a 2-cell $yt \Rightarrow y$ satisfying some conditions. It seems natural then to consider a pair of an algebra and an opalgebra such that the two 2-cells $ytx \Rightarrow yx$ are equal. Do these pairs have a name (alternatively: can anything interesting be said about them), or is there some observation that makes them not worth considering? (These are not the same as endo-bimodules, but they do appear when defining the composition of bimodules.)

Do you have some examples that arise in nature? Do me, it seems like it would hold pretty rarely.

Nathanael Arkor (Dec 26 2021 at 21:00):

The motivating examples I had in mind were purely formal: i.e. Kleisli/EM objects and (op)algebras induced by adjunctions.

Nathanael Arkor (Dec 26 2021 at 21:01):

But the definition, and these examples, seem very natural.

Todd Trimble (Dec 26 2021 at 21:14):

Sorry, I don't think I'm following yet, so let me say what I'm getting from your post, and you can tell me what I'm misunderstanding. We can take the EM case: say $U: C \to D$ is monadic with left adjoint $F$ and counit $\varepsilon: FU \to 1_D$, and put $T = UF$. We have a left action $U\varepsilon: TU \to U$ and a right action $\varepsilon F: FT \to F$. But the two maps you allude to are $FU\varepsilon, \varepsilon FU: FUFU \to FU$. Except in specialized cases (such as the unit being an isomorphism), these are usually unequal.

Nathanael Arkor (Oct 12 2022 at 13:05):

@Todd Trimble: I just realised I never responded to your message. I think I had planned to try to reconcile my confusion with your message, and never got around to doing so. I think I must have just made a silly mistake somewhere, because I don't see now why I thought these maps would coincide in general. Thank you for pointing out my mistake, and apologies that it took me so long to get back to you!

Todd Trimble (Oct 12 2022 at 14:05):

Hi Nathanael. No worries at all! Thanks for getting back.