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Stream: learning: questions

Topic: Kleisli left adjoint preserving limits?

Paolo Perrone (Jul 27 2024 at 13:22):

Hi all.
Let $C$ be a category and $T$ a monad on $C$.
Is it true that the left-adjoint $C\to Kl(T)$ preserves all those limits which exist in $C$ and which are preserved by $T$?
(If so, is this written anywhere?)

Ivan Di Liberti (Jul 28 2024 at 08:53):

it is indeed true by direct inspection (and it is an iff) but I do not have a reference for it.

Paolo Perrone (Jul 28 2024 at 09:01):

Thank you.

Paolo Perrone (Jul 28 2024 at 09:23):

Do you see if one can prove it more abstractly, using the fact that monadic functors create limits? (And then using the fact that the resulting algebra must be free?)

Todd Trimble (Jul 28 2024 at 13:35):

Yes, like you say, the monadic functor $U_T$ preserves and reflects limits, and so if $D: J \to C$ is a diagram in $C$ with a limit preserved by $T$, then the lift $F: C \to EM(T)$ also preserves that limit. The only other thing needed is that $F$ factors through the full inclusion $Kl(T) \to EM(T)$, and full inclusions always reflect limits (they might not preserve them, but that's another story) -- something one easily checks by direct inspection.

Paolo Perrone (Jul 28 2024 at 13:40):

Oh, very nice, thanks.

Paolo Perrone (Jul 28 2024 at 14:10):

Okay, the result is now on the nLab.
(I hope I haven't made any mistakes.)

Todd Trimble (Jul 28 2024 at 14:15):

Looks good to me!