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Stream: learning: questions

Topic: Isomorphisms in Kleisli categories

Nathan Corbyn (Mar 15 2024 at 15:24):

Monads $T$ on $\mathrm{Set}$ often have the useful property that the induced map $\mathrm{Set} \to \mathrm{Set}_T$ is conservative (reflects isomorphisms). Are there general conditions we can put on a monad over an arbitrary category that ensure that the map into the Kleisli category is conservative?

Matt Earnshaw (Mar 15 2024 at 15:49):

To get the ball rolling, the inclusion is faithful iff $\eta$ is pointwise monic, and this would imply conservativity in the case that $\mathbb{C}$ is balanced.

Nathan Corbyn (Mar 15 2024 at 16:01):

Fantastic—thank you

Nathan Corbyn (Mar 15 2024 at 16:04):

I think in my case C isn’t balanced but this is nice to know!

Ralph Sarkis (Mar 15 2024 at 16:06):

Matt Earnshaw said:

To get the ball rolling, the inclusion is faithful iff $\eta$ is pointwise monic, and this would imply conservativity in the case that $\mathbb{C}$ is balanced.

I was also thinking that $\eta_A$ being split mono would help (and it is also often the case), but I was only able to prove (I think) conservativity when $\eta$ is split mono, i.e. the left inverses of $\eta_A$ are also natural in $A$ (with no assumption on $\mathbf{C}$).

Morgan Rogers (he/him) (Mar 15 2024 at 16:09):

I deleted the original comment, but the functor to the Kleisli category does in fact reflect monos. (It's just more complicated to check than I thought).

Morgan Rogers (he/him) (Mar 15 2024 at 16:15):

So explicitly, a morphism $x:X \to Y$ is mapped to an isomorphism in $\mathbb{C}_T$ iff there exists $y:Y \to TX$ such that $x;y = \eta_X$ and $y;Tx = \eta_Y$.

Morgan Rogers (he/him) (Mar 15 2024 at 16:16):

(assuming I haven't cancelled too much stuff in my diagram chasing)

Morgan Rogers (he/him) (Mar 15 2024 at 16:20):

@Ralph Sarkis I don't quite see how you get the inverse from the split monos?

Ralph Sarkis (Mar 15 2024 at 16:26):

Let $r: T \Rightarrow \mathrm{id}$ be the left inverse of $\eta$. Since $x;y = \eta_X$, $x;y;r_X = \mathrm{id}_X$, then we also have $y;Tx;r_Y = \eta_Y;r_Y = \mathrm{id}_Y$, so by naturality of $r$, we have $y;r_X;x = \mathrm{id}_Y$, and we conclude $y;r_X$ is the inverse of $x$.

Morgan Rogers (he/him) (Mar 15 2024 at 16:50):

Matt Earnshaw said:

To get the ball rolling, the inclusion is faithful iff $\eta$ is pointwise monic, and this would imply conservativity in the case that $\mathbb{C}$ is balanced.

Most of the examples I can think of on Set follow this pattern; @Nathan Corbyn are there any properties of the category you have in mind that might help? Ralph's criterion is nice but pretty strong!

Nathan Corbyn (Mar 15 2024 at 16:51):

It’s LFP but there’s not much else you can say

Nathan Corbyn (Mar 15 2024 at 16:54):

I would expect the unit to be monic though

Morgan Rogers (he/him) (Mar 15 2024 at 16:56):

It's sufficient that $T$ is conservative, but that's probably unhelpful.

Ralph Sarkis (Mar 15 2024 at 16:58):

It is even enough that $T$ reflects monos to get that the unit is monic because one of the triangle defining monads says $T\eta_A$ is a split mono. The other triangle says $\eta_{TA}$ is a split mono.

Ralph Sarkis (Mar 15 2024 at 16:59):

I said "even enough" but I am not sure if we can compare being conservative and reflecting monos, is one stronger than the other ?

Morgan Rogers (he/him) (Mar 15 2024 at 17:01):

Not without further structure on the category, no :sweat_smile:

Mike Shulman (Mar 15 2024 at 18:15):

But if $\mathbb{C}$ has pullbacks, and $T$ preserves pullbacks and is conservative, then it reflects monos, since $m:A\to B$ is mono iff $\Delta_m : A \to A\times_B A$ is iso.

Paolo Perrone (Mar 15 2024 at 19:39):

I don't know if it can help, but we know from categorical probability that for a strongly affine monad on a cartesian monoidal category, the Kleisli inclusion reflects isomorphisms.

Graham Manuell (Mar 16 2024 at 12:27):

Ralph Sarkis said:

Matt Earnshaw said:

To get the ball rolling, the inclusion is faithful iff $\eta$ is pointwise monic, and this would imply conservativity in the case that $\mathbb{C}$ is balanced.

I was also thinking that $\eta_A$ being split mono would help (and it is also often the case), but I was only able to prove (I think) conservativity when $\eta$ is split mono, i.e. the left inverses of $\eta_A$ are also natural in $A$ (with no assumption on $\mathbf{C}$).

Isn't $\eta$ being pointwise extremal monic enough? I think this is just the dual of the result that a right adjoint is faithful and conservative if the counit is a pointwise extremal epi.