Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: taking algebras is the algebra of the taking monad monad

Matteo Capucci (he/him) (Jan 18 2023 at 11:15):

Here's a series of wild conjectures (springing from @Oscar Cunningham question on this Mastodon thread):

1. On the 3-category of 2-categories, there is a pseudomonad $\bf Mnd$ which takes a 2-category $\cal K$ to its 2-category of [[monads]]. The unit $\eta_{\cal K} : \cal K \to \bf Mnd(\cal K)$ sends an object $a$ to its identity $(a, 1_a)$, while multiplication $\mu_{\cal K} : \bf Mnd(Mnd(\cal K)) \to \bf Mnd(\cal K)$ uses the equivalence between monads in monads and [[distributive laws]] (I don't really know how this construction works, does anyone have reference?) to produce a new monad as the composite monad arising from such distributive law
2. This pseudomonad, when restricted to 2-categories with enough [[lax limits]], becomes a co-KZ doctrine, since taking [[Eilenberg-Moore objects]] is right adjoint to the unit

$${\cal K}(x, {\bf EM}(t)) \cong {\bf Mnd(\cal K)}(\eta(x), t)$$

This equivalence is based on the (unproven but plausible) fact ${\bf Mnd(\cal K)}(\eta(x), t) \cong t{\bf Mod}(x)$.

3. Similarly, when we restrict to 2-categories with enough [[lax colimits]], $\bf Mnd$ becomes a KZ doctrine, since taking [[Kleisli objects]] is left adjoint to the unit (now we use the unproven but plausible dual fact that ${\bf Mnd(\cal K)}(t, \eta(x)) \cong {\bf Mod}t(x)$)

The last two points are especially speculative since I don't know whether I can conclude $\bf Mnd(\cal K)$ is lax co/complete from the fact $\cal K$ is. Does anyone know whether this construction is known?

Nathanael Arkor (Jan 18 2023 at 11:45):

I would recommend reading Street's The formal theory of monads. Street states that Mnd forms a 3-monad (strict, not weak) on Cat, though he only defines the 2-monad structure. I'm not quite sure what your "Mod" notation means, but Mnd is neither KZ or coKZ (existence of limits/colimits are irrelevant). You also seem to be conflating Mnd (relevant for the EM construction) and Mnd°(-°) (relevant for the Kleisli construction). However, I think reading Street's paper will clarify many of your questions.

Nathanael Arkor (Jan 18 2023 at 11:46):

Lack–Street's The formal theory of monads II also clarifies the difference between Mnd and the completion under Kleisli or EM objects, which is relevant to your questions about KZ/coKZ doctrines.

Matteo Capucci (he/him) (Jan 18 2023 at 12:50):

Thanks Nathanael!

Matteo Capucci (he/him) (Jan 18 2023 at 12:51):

to clarify, $t {\bf Mod}(x)$ are algebras of $t:a \to a$ with domain $x$

Matteo Capucci (he/him) (Jan 18 2023 at 12:54):

Three pages in I already see I got something wrong: image.png

Matteo Capucci (he/him) (Jan 18 2023 at 12:54):

The left 2-adjoint to $\eta$ is $U:\bf Mnd(\cal K) \to \cal K$ which forgets the monad and only keeps the object. Very obvious in hindsight. Moreover, when EM exists, it forms a right adjoint to $\eta$. So in that case we have a string of adjoints

$$U \dashv \eta \dashv \bf EM$$

Matteo Capucci (he/him) (Jan 18 2023 at 12:56):

So $\bf Mnd$ is not co-KZ because while this algebra is left adjoint to the unit, other algebras aren't?

Nathanael Arkor (Jan 18 2023 at 15:14):

Yes, exactly, there can be Mnd-algebras for which the algebra structure is not right adjoint to the unit (though I will admit I can't think of any good examples off the top of my head).

Matteo Capucci (he/him) (Jan 19 2023 at 10:59):

Matteo Capucci (he/him) said:

3. Similarly, when we restrict to 2-categories with enough [[lax colimits]], $\bf Mnd$ becomes a KZ doctrine, since taking [[Kleisli objects]] is left adjoint to the unit (now we use the unproven but plausible dual fact that ${\bf Mnd(\cal K)}(t, \eta(x)) \cong {\bf Mod}t(x)$)

So coming back to this point...

We do have indeed that $\cal K(a_t, x) \cong {\bf Mod}t(x)$ as its definition, and ${\bf Mod}t(x) \cong {\bf Mnd}((a, t), (x, 1))$ by inspection.
So this would support the conclusion that ${\bf Kl} \dashv \eta$, but we know $U$ is instead,

The point here is that a right $t$-module really isn't a monad morphism $(a,t) \to (x,1)$. The problem is the laws have to be dualized, so this would work in $\cal K^{op}$. You'd also have to reverse the monad morphisms, and this explains @Nathanael Arkor's remark about using ${\bf Mnd}^{op}(\cal K^{op})$ to talk about Kleisli
In that case it seems that you can dualize the case of algebras and say $\cal K$ admits the Kleisli construction iff $\eta$ (for ${\bf Mnd}^{op}(\cal K^{op})$) admits a right 2-adjoint.