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Stream: learning: questions

Topic: cotensors in a Kleisli category

fosco (Apr 01 2022 at 10:55):

I am looking for sufficient conditions ensuring that the Kleisli category of a monad $T$ admits some co/limits, and in particular, working $V$-enriched, conditions on $T$ such that $Kl(T)$ has co/tensors.

I am aware that coproducts always exist in $Kl(T)$, when the domain $C$ of $T$ has them; I think a similar argument can prove that there are $V$-tensors in $Kl(T)$ if $C$ has them.

This is basically where my knowledge stops. Nondiscrete colimits, products, and nondiscrete limits, in general tend not to exist. $Kl(T)$ tends to miss infinite products, but what about finite ones? What about $Set$-cotensors (so, slightly less than all products, because you only need products $\prod_i A_i$ to exists when all $A_i$ are equal)?

One result that I know, regarding completeness, is that when $C$ is complete and the canonical functor $J : Kl(T)\to EM(T)$ has a right adjoint. Then $Kl(T)$ is also complete. This is Theorem 3.1 here A highly unsatisfying condition because

• the paper has exactly zero examples: how uncommon is this condition in nature, given that limits in Kleisli categories tend to not exist?
• the proof goes as follows: note that a cone for a diagram $D : X \to Kl(T)$ is precisely a cone for $U_T D : X \to C$. consider the limit $\lim U_TD$, say $(L, q_c : L \to U_TDc)$. $L$ is a $T$-algebra, in general (of course) not free. But $(L, h : TL\to L)$ can be "coreflected" (I know it's not exactly a coreflection, yes) to obtain a "best-approximating" free algebra. This is the limit of $D$ in $Kl(T)$.

Very good: but let's say, now, that I don't want all limits, just cotensors, finite limits, finite weighted limits... Can I get away without asking that $J$ has a right adjoint, or the only thing that I can relax is the limits that have to exist in $C$?

Morgan Rogers (he/him) (Apr 01 2022 at 12:12):

I spent some time thinking about limits in the Kleisli category a long time ago. As soon as C has limits of a given shape and T preserves them, that's enough to guarantee that the functor C -> Kl(T) preserves them too, so we get limits of diagrams coming from C. More generally, that functor is a left adjoint so preserves all colimits. Thus if T preserves products of a given size, then Kl(T) inherits those from C, and inherits and coproducts which exist in C. Limits and colimits over more general diagrams require much more fiddly conditions, however.

If I recall correctly, the existence of the adjoint should be easier to show than you might expect: we just require certain split coequalizers to exist in the Kleisli category in order to construct the adjoint. However, this is rather circular when one is trying to establish what colimits etc are available!

Without that adjoint it will be pretty tricky to demonstrate universal properties; you will need some extra hypotheses to guarantee their existence, I expect.