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Stream: theory: category theory

Topic: Doctrinal adjunction

Brendan Murphy (Mar 03 2024 at 23:37):

Half of the statement of doctrinal adjunction is really a general property about conjoints/companions in double categories while the other half can be understood as saying that the forgetful functor from the double category of $T$-algebras to the double category of quintets creates conjunctions, and this second bit is specific to algebras over monads. Is there a conceptual/high level proof of this fact from the double category pov? It rhymes with the fact that ordinary monadic functors create limits, although conjunctions aren't a type of limit. I would like to say the functor $T\mathsf{-Alg} \to Q(K)$ is some sort of "doubly-monadic" functor, but there's some weirdness coming from the fact that we start with a monad on a globular category and end up with a functor between double categories... Does anyone have any thoughts on this? Is the notion of "monadic" I'm looking for known?

Mike Shulman (Mar 04 2024 at 00:59):

Here's the closest thing I know. Consider the 2-category of triple categories (strict is good enough), with 2-cells being "transversal transformations". Let $Q_3(K)$ be the "triple category of quintets" in $K$: its three underlying double categories are all the ordinary double category of quintets in $K$, and its cubes are commutative cubes in $K$. Then a 2-monad $T$ on $K$ induces a triple monad on $Q_3(K)$, and for its EM-object in the 2-category of triple categories we have:

• Its objects are $T$-algebras.
• Its transversal morphisms are strict $T$-morphisms.
• Its horizontal/vertical double category is the usual double category of lax and colax $T$-morphisms.
• Its other two underlying double categories are sub-double-categories of this in which one or the other kind of morphism is strict.

I don't know of any abstract explanation of the lifting property for doctrinal adjunctions in terms of this monadicity property, but I haven't thought about it much.

Mike Shulman (Mar 04 2024 at 01:02):

I don't think the lifting of a 2-monad to a triple monad is too weird; if anything it suggests that whatever result there is may be true more generally about triple monads. That's similar to what happened when Steve Lack and I studied lifting of limits to 2-categories of $T$-algebras using $\mathcal{F}$-categories: we started by making a 2-monad into an $\mathcal{F}$-monad, and ended up with a more general statement about arbitrary $\mathcal{F}$-monads.

Mike Shulman (Mar 04 2024 at 01:04):

(It would be really cool if someone could figure out a way to generalize those results about lifting of limits to the double-categorical case. There are some "limits" that exist involving lax and colax morphisms together, such as the comma object of a colax morphism over a lax one, but even though the resulting "comma square" is a square in the double category $T \mathrm{Alg}$ I've never been able to figure out even how to give it a "universal property" in that double category (or triple category), let alone how to characterize a class of such universal properties that lift along monadic functors.)

Brendan Murphy (Mar 04 2024 at 01:37):

So is the reason we need to go to triple categories is that a 2-monad on K doesn't induce a double monad on Q(K)? That's sort of surprising to me

Brendan Murphy (Mar 04 2024 at 01:37):

Or is it that the double category of lax/colax stuff (without strict information) isn't an eilenberg moore object?

Mike Shulman (Mar 04 2024 at 04:01):

Well, I would say part of the problem is that when you say "double monad" you're implicitly referring to a notion of "double transformation" to be its unit and counit, and if you want a double monad on $Q(K)$, which direction should those transformations go in?

Mike Shulman (Mar 04 2024 at 04:03):

Either direction is arbitrary. You could take them to be companion pairs, but there's more to worry about, because to get lax and colax morphisms as the two kinds of morphisms in the EM-object, you'll want the transformations to be lax on one direction and colax on the other direction, but for those transformations to form a 2-category (or even a 3-category) you need them to be strict (or at least pseudo) in one direction.

Mike Shulman (Mar 04 2024 at 04:04):

You could try to see whether it works with companion pairs, whether you could cook up some kind of higher structure with double categories, double functors, and companion-pair-transformations, and maybe some kind of 3-cell between them, such that $T \mathrm{Alg}$ is an EM object therein. I suppose it might.

Mike Shulman (Mar 04 2024 at 04:05):

I would be more surprised if you could get strict $T$-algebras as the objects that way, though. And the advantage of the triple-category approach is that you also get the strict $T$-morphisms as part of the induced structure, whereas the double category $T\mathrm{Alg}$ only knows about the lax, colax, and pseudo (= companion-pair) morphisms.

Brendan Murphy (Mar 04 2024 at 05:16):

Oh, I'm not really interested in the strict algebras

Brendan Murphy (Mar 04 2024 at 05:17):

I guess TAlg is pretty ambiguous notation

Mike Shulman (Mar 04 2024 at 06:09):

Why not?

Mike Shulman (Mar 04 2024 at 06:10):

Pseudo $T$-algebras are the same as strict $T'$-algebras, where $T'$ is the cofibrant replacement of $T$. So strict algebras are actually the more general notion. And standard objects like non-strict (biased) monoidal categories are strict algebras, but not pseudo algebras.

Mike Shulman (Mar 04 2024 at 06:11):

I guess another advantage of the triple category approach is that the 2-category of triple categories you use is the "most obvious" thing you could write down from the notion of triple category; you don't have to cook up anything special to make it work.

Brendan Murphy (Mar 04 2024 at 06:24):

That's fair enough, you've convinced me to be interested in them

Brendan Murphy (Mar 04 2024 at 06:24):

I didn't know that fact about cofibrant replacement

Mike Shulman (Mar 04 2024 at 06:27):

It's pretty cool! And it's actually a special case of another cool fact, that for any 2-monad $T$ on a nice 2-category and any $T$-algebra $A$, there is a cofibrant replacement $A'$ (or $QA$) such that strict $T$-morphisms $A'\to B$ are in bijection with pseudo $T$-morphisms $A\to B$.

Mike Shulman (Mar 04 2024 at 06:29):

The fact about pseudo algebras follows by showing that for a nice 2-category $K$ there is a 2-monad $M$ whose algebras are (finitary, or more generally $\kappa$-accessible) 2-monads on $K$, and for any object $A\in K$ there is an $M$-algebra $\langle A,A\rangle$ (an "endomorphism monad") such that strict (resp. pseudo) $T$-algebra structures on $A$ are in bijection with strict (resp. pseudo) $M$-morphisms $T\to \langle A,A\rangle$.