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Stream: learning: questions

Topic: Monadic adjunctions for indexed categories

Marco Paviotti (Aug 22 2025 at 07:32):

Hi there,
let $\mathbb{N}$ be the discrete natural numbers category and $\Delta : \mathcal{C} \to \mathcal{C}^\mathbb{N}$ be the constant functor $\Delta X (n) = X$. Is the adjunction $\Delta \dashv \Pi$ monadic? somehow it feels it shouldn't because that would imply that $\mathcal{C}^\mathbb{N}$ is equivalent to the Eilenberg-Moore category for the countable power monad $T$

$$\mathcal{C}^\mathbb{N} \simeq \mathcal{C}^{T}$$

where $T X = \Pi \Delta X \cong X^\mathbb{N}$. However, I cannot find a Π-split coequaliser that is not created by Π, this is because this Π-split coequaliser should be computed point wise, but it seems hard to find a counter example.

Is there any other monad $T$ for which the above can be an equivalence?

Thanks!

fosco (Aug 22 2025 at 12:07):

Your category has to have countable products for the adjunction to exist. Let's see if something instructive comes out from doing it on ${\cal C}=\bf Set$. You're asking to characterize the algebras for the "reader" or "environment" monad, a notoriously not-easy problem ( https://math.stackexchange.com/questions/868262/algebras-of-the-environment-monad/ ). In particular, the category of algebras is not ${\bf Set}^{\mathbb N} = \prod_n {\bf Set}$, but it's much more complicated.

fosco (Aug 22 2025 at 12:12):

well, on second thought I can't make the description of algebras more explicit than the MSE post. nLab mentions that in case the set to which your monad powers has 2 elements, algebras for the reader monad are idempotent semigroups https://ncatlab.org/nlab/show/rectangular+band

Kevin Carlson (Aug 22 2025 at 21:39):

Actually, as far as I can tell, $\Pi_{\mathbb N}$ does create $\Pi$-split coequalizers. Certainly this is true for products over finite sets instead of $\mathbb N$. But even in the finite case, $\Delta \dashv \Pi$ is not monadic! How can that be? Well, $\Pi$ is not conservative!

Kevin Carlson (Aug 22 2025 at 22:26):

Ah, hold on, $\Pi_X$ preserves but doesn't create $\Pi_X$-split coequalizers, at least if $X$ is finite, for similar reasons to the failure of conservativity: Any cofork in $\mathbf{Set}^X$, all of whose objects has at least one value empty, is sent to the split coequalizer $0\rightrightarrows 0\to 0$ in $\mathbf{Set}.$

Mike Shulman (Aug 22 2025 at 23:21):

Note that the [[monadicity theorem]] can be phrased either as (having a left adjoint and)

• creating coequalizers of $\Pi$-split pairs, or
• being conservative and having and preserving coequalizers of $\Pi$-split pairs.

So if it has and preserves such coequalizers, then it creates such coequalizers if and only if it is conservative.

Marco Paviotti (Aug 23 2025 at 05:01):

Mike Shulman said:

Note that the [[monadicity theorem]] can be phrased either as (having a left adjoint and)

• creating coequalizers of $\Pi$-split pairs, or

On this note, what I find fascinating about this theorem is that is not just saying "it creates coequalisers", but it creates coequalisers only of those pairs which have already a split coequaliser under $\Pi$. Is there any intuition for why this condition is necessary?

Mike Shulman (Aug 23 2025 at 05:13):

Well, there's a general fact that the category of algebras for a monad $T$ inherits all colimits that are preserved by $T$ and $T^2$, and split coequalizers are preserved by all functors. Is that the sort of thing you mean?

Marco Paviotti (Aug 23 2025 at 05:43):

Mike Shulman said:

and split coequalizers are preserved by all functors

interesting, I've never thought of that. but yeah, I guess in the end this theorem is really saying, a functor is monadic if it behaves the in same way as the forgetful functor for EM-algebras.

Mike Shulman (Aug 23 2025 at 05:55):

There's another version of the theorem that refers to [[absolute coequalizers]] rather than split ones.