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Stream: learning: questions

Topic: Algebra-preserving reflectors

Bernd Losert (Aug 28 2024 at 07:43):

I was wondering if there are any results about algebra-preserving reflectors. Let's say you have a category $\mathbf{C}$ and an object $(A, a)$ of the category of Eilenberg-Moore algebras $\mathbf{C}^T$ (for some monad $T$). Given a reflexive subcategory $\mathbf{B}$ with reflector $R : \mathbf{C} \to \mathbf{B}$, under what conditions is $(RA, Ra)$ also an Eilenberg-Moore algebra.

One obvious requirement that I see is that $R \circ T = T \circ R$. Of course this is not enough to guarantee that $Ra$ is algebra.

I am hoping for a result along the lines of "if $R$ is polynomial functor, then $R$ preserves algebras".

Bernd Losert (Aug 28 2024 at 07:54):

Also, it would be useful to know, for a fixed reflector $R$, what monads $T$ would allow $R$ to preserve algebras.

Nathan Corbyn (Aug 28 2024 at 10:19):

This is a special case of the question of when two monads can be composed to give a new monad whose algebras support both structures in a compatible way. This is because reflective subcategories are equivalently EM categories for idempotent monads. This question is well understood and a good starting point is the nLab page: https://ncatlab.org/nlab/show/distributive+law#monad_distributing_over_monad

John Baez (Aug 28 2024 at 12:07):

Are there some special theorems about monads that distribute over idempotent monads? I feel there should be. Or maybe over idempotent commutative monads.

Nathan Corbyn (Aug 28 2024 at 14:48):

I'm not aware of any, but I'm definitely not an expert

Bernd Losert (Aug 28 2024 at 16:56):

Thanks @Nathan Corbyn. That's a good entry point.

John Baez (Aug 29 2024 at 15:09):

Now I remember what I was thinking about. All this may help @Bernd Losert very little, but I'll say it anyway.

Bernd's question made me think about ways to lift a monad $T$ on a category $C$ to a monad on some reflective subcategory of $C$. The reflector $R : C \to C$ is an idempotent monad, and the reflective subcategory of $C$ is the category of $R$-algebras. So Bernd made me think about this:

• Given a monad $T$ and an idempotent monad $R$ on a category $C$, what are the ways to lift $T$ to the category of $R$-algebras? (These are called distributive laws.)

However, Bernd wasn't really asking this question: he was taking a particular $T$-algebra $A$ and trying to make $R A$ into a $T$-algebra.

Then I remembered this:

In his paper Distributive laws via admissibility, Charles Walker proved an interesting theorem about distributive laws between a monad and idempotent monad - but he actually proved such a theorem in a 2-categorical context, so he was talking about pseudomonads and lax-idempotent pseudomonads (which he calls "KZ pseudomonads"). If I understand correctly, he showed that there's essentially at most one way to lift a lax idempotent pseudomonad on a 2-category $C$ to the 2-category of pseudoalgebras of a pseudomonad on $C$

If we strip off all the 2-category theory and guess what this means for ordinary monads, it seems to say

• Given a monad $T$ and an idempotent monad $R$ on a category $C$, there's at most one way to lift $R$ to the category of $T$-algebras.

I'd like to know if this is true! But this is answering a different question than the bulleted question earlier in my comment. One question is about ways for $R$ to distribute over $T$; the other is about ways for $T$ to distribute over $R$. (Don't ask me which is which.)

Bernd Losert (Aug 31 2024 at 10:45):

@John Baez Thanks for the reference to the Walker paper. I will have a look.