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Stream: learning: questions

Topic: F-Alg as 2limit

Nico Beck (Sep 30 2023 at 08:38):

I want to write the category of $F$-algebras for an endofunctor $F:C\to C$ as a 2-limit. Is it correct that $F{alg}$ is the lax limit of $D:\mathcal J\to Cat$ where $\mathcal J$ is the 2-cat with a single 0-cell and a non-identity endo-1-cell, $D$ chooses $F$ and the weight is $\Delta 1$?

Mike Shulman (Sep 30 2023 at 15:03):

I think that's right.

John Baez (Sep 30 2023 at 15:10):

This is a nice counterpart of the more famous result expressing the Eilenberg-Moore category of a monad $T : C \to C$ as a lax limit, but with Nico's 2-category $\cal{J}$ replaced by the 'walking monad' one-object 2-category. That more famous result also gives the Kleisli category of $T$ as the lax colimit of the same data. What's the lax colimit of Nico's $D: \mathcal{J} \to \mathbf{Cat}$?

Mike Shulman (Sep 30 2023 at 18:15):

Nice question! I don't think I've ever thought about that before. I have a guess at the answer, but I'd only give it 50-50 odds of being right, so I won't say what it is. Does anyone know or feel like working it out carefully?

Tom Hirschowitz (Oct 02 2023 at 07:03):

It seems to be $C$, where you freely throw in $F$-algebra structure on every object and quotient out so that every morphism is an $F$-algebra morphism. Is this plausible?

Mike Shulman (Oct 02 2023 at 16:50):

But is there a more explicit description, like for the Kleisli category of a monad?

Tom Hirschowitz (Oct 03 2023 at 15:04):

Don't know... not assuming anything of $F$ or $C$? Maybe you could share your guess?

Tom Hirschowitz (Oct 03 2023 at 15:04):

(If only to demonstrate how ok it is to make mistakes on this zulip :innocent: )

Mike Shulman (Oct 03 2023 at 16:00):

Yeah, no one else seems to be taking a crack at it, so here's my guess: the objects of the lax colimit are the objects of $C$, and a morphism $X\to Y$ is a pair $(n,f)$ where $n\in \mathbb{N}$ and $f : X \to F^n Y$. The identity is $(0,\mathrm{id}_X)$ and the composite of $(n,f)$ and $(m,g)$ is $(n+m, F^n g \circ f)$. The inclusion from $C$ takes $f$ to $(0,f)$, and the $F$-algebra structure on the image of any object $X$ is $(1, \mathrm{id}_{FX})$.

Reid Barton (Oct 03 2023 at 16:16):

This is also my guess but I didn't put much thought into why it would be correct.

Mike Shulman (Oct 03 2023 at 20:48):

Hmm, actually there is a general reason for that: the lax colimit of any functor $G:D \to \mathrm{Cat}$ is its Grothendieck construction. In this case $D = \mathbf{B}\mathbb{N}$ is the delooping of the monoid of natural numbers. So there is only one fiber, but not all the morphisms lie in that fiber, and I think the above guess falls out of the definition of Grothendieck construction. I should have thought of that right away.

Tom Hirschowitz (Oct 04 2023 at 07:15):

Ooooh, nice!

You mean objects of $C$ of course, right?

Also, wouldn't the Grothendieck construction amount to formally adding coalgebra structures $c \to Fc$ for all $c$? This seems to agree with the nlab saying that the Grothendieck construction is an oplax colimit, so that in our case morphisms would rather go $f\colon F^n X \to Y$. This feels less right than your guess though, which nicely looks like a Kleisli over the free monad. Can someone set me straight please?

Mike Shulman (Oct 04 2023 at 07:19):

Lax, oplax, tomayto, tomahto...

Mike Shulman (Oct 04 2023 at 07:20):

Both the lax version and the oplax version are commonly called "Grothendieck construction". I meant the one that's a lax colimit. (-:O

Tom Hirschowitz (Oct 04 2023 at 07:27):

Oh, ok, thanks. So when reading "Grothendieck construction", one should mentally add "up to op". Note taken.

Mike Shulman (Oct 05 2023 at 17:36):

Mike Shulman said:

Lax, oplax, tomayto, tomahto...

In case anyone got the wrong idea from this, maybe I should emphasize that in nearly all situations there is a correct one of "lax" and "oplax" to use (although the choice between "oplax" and "colax" is up to you). The way I remember it is that lax monoidal functors preserve monoid objects, while colax monoidal functors preserve comonoid objects. Then you can reason by analogy or generalization/specialization to figure out the correct application of the terms to any other situation.

Nathanael Arkor (Nov 15 2023 at 20:05):

Mike Shulman said:

Yeah, no one else seems to be taking a crack at it, so here's my guess: the objects of the lax colimit are the objects of $C$, and a morphism $X\to Y$ is a pair $(n,f)$ where $n\in \mathbb{N}$ and $f : X \to F^n Y$. The identity is $(0,\mathrm{id}_X)$ and the composite of $(n,f)$ and $(m,g)$ is $(n+m, F^n g \circ f)$. The inclusion from $C$ takes $f$ to $(0,f)$, and the $F$-algebra structure on the image of any object $X$ is $(1, \mathrm{id}_{FX})$.

(I've just spotted that this description appears as Proposition 2.28 of Distributivity for endofunctors, pointed and co-pointed endofunctors, monads and comonads, if anyone is looking for a reference in the future.)