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: learning: questions

Topic: End/coend formulas for Kan lifts

James Deikun (Oct 15 2024 at 04:48):

The nLab has articles on both Kan extensions and Kan lifts. The [[Kan extensions]] article is well-developed, and has formulas for pointwise Lan and Ran, however the [[Kan lifts]] article is sparse and the only formulas it has are not for Cat. I kind of feel like it should be possible to figure them out but I have no confidence I would get all the ops and cos in the right place. Are these formulas already a known thing and my googling is just weak?

Mike Shulman (Oct 15 2024 at 05:28):

I don't think Kan lifts exist in general in Cat.

Nathanael Arkor (Oct 15 2024 at 07:01):

Kan lifts are essentially the same as relative adjoints (more precisely, relative adjoints are the appropriate notion of "pointwise" Kan lifts), so you can compute them in some situations using (relative) adjoint functor theorems, but in general there's not a straightforward way to compute them like there is with Kan extensions.

Mike Shulman (Oct 15 2024 at 07:07):

And in general, just as not every functor has an adjoint, not every Kan lift exists, right?

David Corfield (Oct 15 2024 at 07:58):

James Deikun said:

however the [[Kan lifts]] article is sparse and the only formulas it has are not for Cat

Did we gather all that was usefully said on this nCafe post, Kan Lifts?

James Deikun (Oct 15 2024 at 09:02):

Ah. So Kan extensions may not exist in general either, but Kan lifts can somehow be absent in a deeper, more essential sense...

James Deikun (Oct 15 2024 at 09:41):

Hm, I kind of buy it but not totally. There is a nice characterization of when adjoints are "essentially" vs "incidentally" absent ((co)continuity), and a formula (in terms of potentially large limits, though, but the ends for Kan extensions can be large in some circumstances too!) for left/right adjoint, where these limits might fail to exist but iff they exist the adjoint exists, and they're even pointwise.

James Deikun (Oct 15 2024 at 09:54):

(Hm, the fact that right relative adjoints can fail to be unique is more troubling though. Under which criteria would a formula pick one?)

James Deikun (Oct 15 2024 at 09:57):

(Oh, but the right relative adjoint is not a lift!)

Nathanael Arkor (Oct 15 2024 at 12:04):

James Deikun said:

Ah. So Kan extensions may not exist in general either, but Kan lifts can somehow be absent in a deeper, more essential sense...

I suppose in the sense that, when a category is cocomplete enough, you have all Kan extensions, but there is not a corresponding property that implies that all Kan lifts exist.

Nathanael Arkor (Oct 15 2024 at 12:06):

(Non-relative) adjoints are special because they can be identified either with Kan extensions or Kan lifts. However, there is a bifurcation with relative adjoints: left relative adjoints are lifts, whereas right relative adjoints are extensions.

Nathanael Arkor (Oct 15 2024 at 12:07):

(And dually for relative coadjunctions.)

James Deikun (Oct 15 2024 at 12:17):

It seems like a pointwise left Kan lift $\mathrm{Lift}_p F$ must exist when $p$ preserves limits and the domain of $p$ is complete enough. When $F$ is dense this is also necessary, and perhaps the density comonad of $F$ somehow measures how much $p$ can fail to preserve limits.

James Deikun (Oct 15 2024 at 12:41):

Since the nice conditions on $p$ for all left Kan lifts to exist also imply a(n absolute) left adjoint exists, perhaps this is one reason Kan lifts are less prominent than Kan extensions.

Nathanael Arkor (Oct 15 2024 at 12:42):

When $F$ is dense, you can use the representable functor theorem to find when a lift/left relative adjoint exists, which will involve continuity of $p$ and the solution set condition.

James Deikun (Oct 15 2024 at 12:44):

Are you sure that's not continuity of $p$? (And it was fixed, I see.)

Nathanael Arkor (Oct 15 2024 at 12:51):

Oops, that's right.

James Deikun (Oct 15 2024 at 13:45):

So it seems like a formula for $\mathrm{Lift}_p F$ where $p : D' \to D$ would be:

$$\mathrm{Lift}_p F(c) = \underset{x \in Fc \downarrow p}{\mathrm{lim}} \small{\Pi}_2x$$

with the following caveats:

• the formula will be evaluated in $D'$ but its effectiveness will be checked in $D$ so it only works to the extent $p$ preserves the limit
• the limit in practice will probably be large so you'll need to get clever to ensure it exists and evaluate it.

James Deikun (Oct 15 2024 at 13:50):

I haven't proved this formula though and I'm worried it might be a bit too pointwise to be real since it doesn't seem to take into account the properties of $F$ much.