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: pullback in functor 2-category

Jonathan Beardsley (Dec 20 2023 at 22:03):

Say I've got a 1-category C so that I have a 2-category of functors Fun(C,Cat). In particular, given F, G:C→Cat, I have a category of natural transformations Nat(F,G). Now suppose I've got another functor E and a natural transformation E→F. This give a functor Nat(F,G)→Nat(E,G). I would like to say that this is left adjoint to a sort of "C-indexed" left Kan extension. Is this true? I'm happy to assume that C is complete and cocomplete and that F and G (but not necessarily E) factor through complete and cocomplete categories.

Jonathan Beardsley (Dec 20 2023 at 22:11):

Assuming that all works, is it then true that this recovers the normal restriction-extension adjunction if I restrict to one particular object x∈C and look at pulling back along the functor E(x)→F(x)?

fosco (Dec 20 2023 at 22:20):

you might want to reach out to @Luca Mesiti as he might have something to say here, given the talk he gave.. last monday!

fosco (Dec 20 2023 at 22:21):

(the way in which you compute Kan extensions is related to the GConstruction. But in any case, I believe he might have thought about something relevant to your problem! Good luck!)

Jonathan Beardsley (Dec 20 2023 at 22:27):

@fosco that's a good point. maybe it's easier to understand as something fibered over C

Jonathan Beardsley (Dec 20 2023 at 22:34):

It's funny. I think I more or less know where this is worked out for quasicategories in Lurie's work.

Jonathan Beardsley (Dec 20 2023 at 22:36):

I suppose in this case however I could just say that the "left adjoint" is the levelwise left Kan extension and then try to check that it's the right thing globally.

Jonathan Beardsley (Dec 20 2023 at 22:40):

Certainly feels like it should be "formal."

Jonathan Beardsley (Dec 20 2023 at 23:03):

I think Mac Lane calls these things "adjunctions with parameter" or "conjugate natural transformations."

fosco (Dec 20 2023 at 23:08):

Jonathan Beardsley said:

I think Mac Lane calls these things "adjunctions with parameter" or "conjugate natural transformations."

really, are they the same thing?

Jonathan Beardsley (Dec 20 2023 at 23:08):

fosco said:

Jonathan Beardsley said:

I think Mac Lane calls these things "adjunctions with parameter" or "conjugate natural transformations."

really, are they the same thing?

No, I take that back. They're related I think, but not the same thing. I was misunderstanding. I'm interested in the former.

Jonathan Beardsley (Dec 20 2023 at 23:09):

I think this turns out to be pretty straightforward since all I really want, I think, is a 2-functor Adj×C→Cat.

fosco (Dec 20 2023 at 23:09):

anyway, Street's "Conspectus of variable categories" deals, among other things, with equipping pseudofunctors C -> Cat with a yoneda structure, which then gives a formula for pointwise Kan extensions. You might want to look into that if you haven't already

fosco (Dec 20 2023 at 23:11):

(I recall having a hard time finding pseudo-Lan's in the category of prederivators, and I recall having a hard time reading that paper :stuck_out_tongue:)

Jonathan Beardsley (Dec 20 2023 at 23:11):

Haha yeah looks like it might be tough going.

Jonathan Beardsley (Dec 20 2023 at 23:12):

For the moment I sort of am trusting my intuition above that in this special case this is really very straightforward. But I'll guess I'll see if I get stuck.

Mike Shulman (Dec 20 2023 at 23:37):

I would like to say that this is left adjoint to a sort of "C-indexed" left Kan extension

(I assume you mean either that it is right adjoint to a left Kan extension, or that it is left adjoint to a right Kan extension.) What happens is, as you suggest, analogous to ordinary categories, but one has to be careful to make everything indexed. In the caseto of a left Kan extension, there is an indexed left Kan extension as long as $G$ is "indexed cocomplete" and $E$ is "indexed small".

There are some variations in what those phrases can mean, but one version is that $G$ is indexed-cocomplete if it lands in cocomplete categories and cocontinuous functors, plus each of its transition functors have left adjoints satisfying the Beck-Chevalley condition; and $E$ is "indexed small" if it is generated by a small diagram of a certain kind, of internal categories and profunctors in $C$. Another version requires $G$ to land only in finitely cocomplete categories and finitely cocontinuous functors (but still to have left adjoints to transition functors), while $E$ must be induced by a single internal category in $C$.

This is worked out in even greater generality in my paper on enriched indexed categories, although unfortunately the extra generality means it is probably no easier to read than Street's.

Jonathan Beardsley (Dec 20 2023 at 23:59):

Mike Shulman said:

I would like to say that this is left adjoint to a sort of "C-indexed" left Kan extension

(I assume you mean either that it is right adjoint to a left Kan extension, or that it is left adjoint to a right Kan extension.) What happens is, as you suggest, analogous to ordinary categories, but one has to be careful to make everything indexed. In the caseto of a left Kan extension, there is an indexed left Kan extension as long as $G$ is "indexed cocomplete" and $E$ is "indexed small".

There are some variations in what those phrases can mean, but one version is that $G$ is indexed-cocomplete if it lands in cocomplete categories and cocontinuous functors, plus each of its transition functors have left adjoints satisfying the Beck-Chevalley condition; and $E$ is "indexed small" if it is generated by a small diagram of a certain kind, of internal categories and profunctors in $C$. Another version requires $G$ to land only in finitely cocomplete categories and finitely cocontinuous functors (but still to have left adjoints to transition functors), while $E$ must be induced by a single internal category in $C$.

This is worked out in even greater generality in my paper on enriched indexed categories, although unfortunately the extra generality means it is probably no easier to read than Street's.

Thanks very much Mike. What do you mean by transition functors here? Are those the functors in the image of G?

Jonathan Beardsley (Dec 21 2023 at 00:01):

I'll have to think about this situation of indexed cocompleteness. For me C is quite simple, it's $Fin_*$, and G is the functor describing the smash product symmetric monoidal structure on $Set_*$, all pointed sets.

Jonathan Beardsley (Dec 21 2023 at 00:03):

Ah, and actually, right, I want my "pullback" functor to be a _left adjoint_ and therefore have a right Kan extension in the other direction. So presumably I need indexed completeness.

Jonathan Beardsley (Dec 21 2023 at 00:06):

But I think this is a problem because I don't think the smash product will preserve limits!

Jonathan Beardsley (Dec 21 2023 at 00:07):

And that's one of the functors in the image of G.

Jonathan Beardsley (Dec 21 2023 at 00:14):

(the reason my case isn't precisely asking for a symmetric monoidal adjunction is that not all of my Segal maps associated to E and F are necessarily equivalences)

Jonathan Beardsley (Dec 21 2023 at 00:16):

But it's hard for me to see what goes wrong, in this simple case, if one simply takes the levelwise right Kan extension. You really need every functor in the image of G to preserve all small limits??

Jonathan Beardsley (Dec 21 2023 at 00:28):

Hm, maybe this is related https://link.springer.com/chapter/10.1007/3-540-57867-6_14

Jonathan Beardsley (Dec 21 2023 at 00:32):

Yeesh, there are several papers out there on fibered and indexed adjunctions, and none of them are available to me, haha.

Jonathan Beardsley (Dec 21 2023 at 00:41):

It seems like there's a pretty tractable indexed adjoint functor theorem. https://ncatlab.org/nlab/show/indexed+adjoint+functor+theorem

Mike Shulman (Dec 21 2023 at 01:51):

Yes, by transition functors I mean the functors in the image of $G$.

Mike Shulman (Dec 21 2023 at 02:05):

The indexed AFT uses the same kinds of indexed limits and colimits that I was talking about. I'm not sure that it's helpful for you, though: it's about defining indexed adjoints to indexed functors between indexed categories, a.k.a. adjoint (pseudo)natural transformations between Cat-valued functors. You're looking instead for an adjoint to a functor between ordinary categories whose objects happen to be indexed functors.

Mike Shulman (Dec 21 2023 at 02:07):

Jonathan Beardsley said:

But it's hard for me to see what goes wrong, in this simple case, if one simply takes the levelwise right Kan extension.

Have you tried to show that what you get is a natural transformation?

Jonathan Beardsley (Dec 21 2023 at 03:49):

Right. Blergh. Ok. I'll have to try to think through this tomorrow. I'm getting stuck.