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Stream: theory: category theory

Topic: a global view of Kan extensions

Christian Williams (Dec 08 2020 at 19:10):

The universal property of Kan extension along $p:A\to B$ is normally given in terms of an adjunction between hom-categories $A,C\simeq B,C$. This requires fixing a codomain $C$ -- is there a way to describe the "global" view, just in terms of the coslice 2-categories $A/\mathrm{Cat}$ and $B/\mathrm{Cat}$?

Christian Williams (Dec 08 2020 at 19:25):

Precomposition by $p$ is a 2-functor $-\circ p: B/\mathrm{Cat}\to A/\mathrm{Cat}$. The question is whether left Kan extension along $p$ defines a left 2-adjoint.

Christian Williams (Dec 08 2020 at 19:56):

First we at least need functoriality. Suppose we have a morphism
$(f,\varphi):(Y, h:A\to Y)\to (Z,k:A\to Z)$ in $A/\mathrm{Cat}$.

What is $lan_p(f,\varphi): lan_p(h)\to lan_p(k)$ in $B/\mathrm{Cat}$?

Christian Williams (Dec 08 2020 at 20:02):

Maybe it's better seen from an indexed perspective...
For any 2-category we have a hom $K(-,-):K^{op}\times K\to \mathrm{Cat}$.

Each $K(p,-):K(b,-)\to K(a,-)::K\to \mathrm{Cat}$ should have a left adjoint, which should induce an adjunction $\int K(a,-)\simeq a/K \leftrightharpoons b/K \simeq \int K(b,-)$. Hopefully.

Christian Williams (Dec 08 2020 at 20:59):

I think it's right. I'll try to find some time to write it up and post it here.

Christian Williams (Dec 08 2020 at 21:35):

The short story is that if $K$ is co/complete, the embedding $y:K^{op}\to [K,Cat]$ gives triple adjunctions and hence is of the form $K^{op}\to Adj(Adj([K,Cat]))$. We can then compose that with the Grothendieck construction $\int: [K,Cat]\to Cat$, to get

$K^{op}\xrightarrow{y} Adj(Adj([K,Cat])) \xrightarrow{A(A(\int))} Adj(Adj(Cat))$

which sends each $p:a\to b$ to a triple adjunction $a/K \xrightarrow{\Sigma_p \dashv \Delta_p \dashv \Pi_p} b/K$. We can glue these together to get a nice fibration over $K$, which might be something like an "internal language" of $K$.

Christian Williams (Dec 08 2020 at 21:43):

One thing I was confused about is the difference between strict slice categories and lax ones... I haven't thought a whole lot about the 2-Yoneda embedding, but it seems like if you want to keep all the information, you need to consider $K^{op}\to [K,Cat]_{lax}$, where each morphism in $K^{op}$ induces a lax 2-natural transformation rather than a strict one. That should give everything enough room to breathe.

Christian Williams (Dec 08 2020 at 21:43):

If anyone knows good references for any of this, I'd appreciate it.

Christian Williams (Dec 08 2020 at 21:50):

Christian Williams said:

you need to consider $K^{op}\to [K,Cat]_{lax}$, where each morphism in $K^{op}$ induces a lax 2-natural transformation rather than a strict one.

Hm, sorry I don't think this makes sense. I'm just looking for a way to describe how $\int K(a,-)\simeq a//K$ (the lax slice 2-category) in the same way that $\int C(c,-)\simeq c/C$ for 1-categories. But $K(a,-)$ goes to $\mathrm{Cat}$, not $2\mathrm{Cat}$. So we need some extra information about the fact that $K$ is more than just an indexed category, because it has 2-cells.

Christian Williams (Dec 08 2020 at 22:01):

Here @Mike Shulman considers some subtleties - https://ncatlab.org/michaelshulman/show/slice+2-category - and takes the pseudo definition to be more fundamental than the lax one. Do you have thoughts about how to form either one as (the fibers of) a variant of the Grothendieck construction? If you're busy, no problem. Thanks.

Christian Williams (Dec 08 2020 at 22:21):

(I guess that's a silly question, because we can just define the indexed 2-category $K^{op}\to 2\mathrm{Cat}$ which sends $a$ to $a//K$. What I really mean to ask is how/whether $a//K$ can be constructed from the representable $K(a,-)$.)

Mike Shulman (Dec 09 2020 at 03:00):

Are you talking about slice categories or co-slice categories? Usually the notation $a/\!/ K$ means a co-slice under $a$ rather than a slice over $a$.

Christian Williams (Dec 09 2020 at 03:10):

Sorry, yes I mean co-slice everywhere.

Mike Shulman (Dec 09 2020 at 03:28):

Well, the ordinary representable of a 1-category lands in Set, not Cat, but we can still do a Grothendieck construction by regarding sets as discrete categories. Have you tried the analogous thing with the embedding of Cat in 2-Cat?

Christian Williams (Dec 09 2020 at 03:31):

Ah! I see, yes I think that should work.

Christian Williams (Dec 09 2020 at 03:31):

Thanks.

Christian Williams (Dec 09 2020 at 03:33):

Then we get a 2-$\ast$-bifibration over any co/complete 2-category $K$, with the fibers being lax coslices and the adjoints given by extensions. Do you know if anyone has thought about this?

Mike Shulman (Dec 09 2020 at 04:00):

Well, I don't think that co/completeness of $K$ ensures that all left and right Kan extensions exist; for that you need ("internal") co/completeness of the codomain. I would also be surprised if Kan extensions are adjoints to composition in this sense; I would expect the left adjoint, in particular, to instead be some kind of pushout.

Christian Williams (Dec 09 2020 at 05:00):

Oh, yes I was being loose with the term co/completeness... is there a sufficient condition on $K$ for the existence of all extensions, or is that probably asking too much? And I'm curious about your second sentence. Is it not true that $lan_p$ can be understood (under certain conditions) as giving a left adjoint to $-\circ p: K(b,-)\to K(a,-)$?

Mike Shulman (Dec 09 2020 at 15:43):

Well, you can assume that "$K$ has all extensions". But I don't know offhand of any 2-categories that do.

Mike Shulman (Dec 09 2020 at 15:45):

Certainly $\mathrm{Lan}_p$ is a left adjoint to $-\circ p : K(b,c) \to K(a,c)$ for any fixed $c$ that admits all $p$-extensions. But these adjoints don't generally vary naturally with $c$.

Mike Shulman (Dec 09 2020 at 16:15):

There are two different indexed categories here that I think may be being confused. On the one hand there is the indexed 2-category $K^{\mathrm{op}}\to 2Cat$ that sends $a$ to $a/\!/K$. In that case, the left adjoint to precomposition should be some kind of pushout, and if all such exist it would be some kind of 2-bifibration (though I haven't thought about the laxness issues.)

Mike Shulman (Dec 09 2020 at 16:16):

On the other there is the co-indexed category $K\to Cat$ that sends $c$ to $K(a,c)$ for some fixed $a$. In that case precomposition is not a restriction operation within one indexed category, but a map from one indexed category to another (i.e. a natural transformation), and in this case its adjoints should indeed be some kind of Kan extensions (insofar as they exist), although I think in general they will only form lax or colax transformations.

Christian Williams (Dec 09 2020 at 18:39):

Mike Shulman said:

Certainly $\mathrm{Lan}_p$ is a left adjoint to $-\circ p : K(b,c) \to K(a,c)$ for any fixed $c$ that admits all $p$-extensions. But these adjoints don't generally vary naturally with $c$.

Okay, I see why now. But it's precisely the absolute extensions which do vary naturally in this way, right?

Christian Williams (Dec 09 2020 at 22:35):

Mike Shulman said:

Well, you can assume that "$K$ has all extensions". But I don't know offhand of any 2-categories that do.

Maybe the 2-category of co/complete categories and co/continuous functors? Or something similar.

Christian Williams (Dec 09 2020 at 22:37):

As for your latter point, I see I was mixing two levels; I need to think more about this. But it seems to me that under the right conditions there should be a 2-fibration with adjoints given by extensions.

Christian Williams (Dec 09 2020 at 22:43):

Well, I wasn't exactly mixed up. If you look at the composite above, I'm considering the embedding $a\mapsto K(a,-)$. If we restrict to absolute extensions, then each $-\circ p: K(b,-)\to K(a,-)$ has left and right adjoints, because they vary naturally with composition.

Then taking the 2-Grothendieck of those representables (after including $\mathrm{Cat\to 2Cat}$, as you say) we get $a//K$. So... somehow it seems like left extension is getting mapped to pushout? Something strange/interesting happens, unless I am indeed confused.

Mike Shulman (Dec 22 2020 at 23:56):

Yes, absolute extensions do vary naturally that way. But absolute extensions are pretty hard to come by; the usual pointwise Kan extensions constructed via limits and colimits are not in general absolute.

Mike Shulman (Dec 22 2020 at 23:58):

A cocomplete category as target admits all left Kan extensions along functors with small domain. But a cocomplete category itself is never small (unless it's a poset), so you can't get a 2-category admitting all Kan extensions that way unless you go down to something like complete lattices.

Christian Williams (Dec 23 2020 at 00:45):

Yes. I thought that absolute extensions were the same as relative adjunctions, but now I see that those are absolute liftings. (Considering that relative monads are becoming more popular and applied, I was thinking there might be an interesting construction here.)

Christian Williams (Dec 23 2020 at 00:47):

And yes, the conflict between completeness and size is constrictive.

Christian Williams (Dec 23 2020 at 00:48):

The reason I'm interested is that extensions generalize dependent sum and product. If there were a 2-categorical version of the codomain hyperdoctrine, it would expand the scope of dependent type theory substantially.

Christian Williams (Dec 23 2020 at 00:50):

I understand that there are significant conditions, obstacles, considerations; but I imagine there are ways it can work.

Christian Williams (Dec 23 2020 at 04:28):

(I'm returning to your nLab pages about 2-topos theory, because I realize you've thought a lot about the internal logic of 2-categories.)

Mike Shulman (Dec 23 2020 at 18:39):

I think I'd need to see written out in more detail exactly what constructions you're intending to perform, and this isn't really the format. If you do write something up, feel free to share it by email.