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Stream: theory: category theory

Topic: ✔ pullback stability of adjunctions

Jonathan Beardsley (Jul 14 2023 at 22:14):

Suppose I've got an adjunction between categories C and D over a third category as below (the whole thing commutes). If I pick an object e∈E via a functor e:∗→E and pullback the whole thing along e, do I get another adjunction?
image.png

Jonathan Beardsley (Jul 14 2023 at 22:31):

oh i just found this answer https://math.stackexchange.com/questions/1735771/is-the-pullback-of-an-adjunction-along-any-functor-an-adjunction from @Kevin Arlin

Jonathan Beardsley (Jul 14 2023 at 22:32):

I happen to be working with ∞-categories..... so does that help with the homotopical problems??

Mike Shulman (Jul 14 2023 at 22:33):

If the whole adjunction lies in the slice 2-category ${\rm Cat}/\mathcal{E}$, which means that both functors make triangles commute and the unit and counit lie over the identity, then it's true because pullback is a 2-functor ${\rm Cat}/\mathcal{E} \to \rm Cat$ and hence preserves adjunctions.

Mike Shulman (Jul 14 2023 at 22:34):

If I read it correctly, the MSE question is only assuming that one of the functors lies in ${\rm Cat}/\mathcal{E}$.

Jonathan Beardsley (Jul 14 2023 at 22:34):

Oh I see.

Jonathan Beardsley (Jul 14 2023 at 22:34):

Okay nice... I think I need to check on the (co)unit condition

Jonathan Beardsley (Jul 14 2023 at 22:36):

okay so my unit is the identity, so that's nice, haha.

Jonathan Beardsley (Jul 14 2023 at 22:37):

ok yesssssss it all works

Jonathan Beardsley (Jul 14 2023 at 22:38):

:pray: :pray: :pray: :pray: :pray: :pray:

Jonathan Beardsley (Jul 14 2023 at 22:38):

that's nice though, i bet i can assemble that really precisely for ∞-categories from Riehl-Verity

Kevin Arlin (Jul 14 2023 at 23:04):

Yeah, you'd say something like "pullback is a cosmological functor between slice cosmoi so gives a 2-functor on homotopy 2-categories." The only little thing is that slice cosmoi's objects are isofibrations, not all functors, but that's rarely a problem since every functor is equivalent to an isofibration.

Jonathan Beardsley (Jul 14 2023 at 23:04):

ah crap but now i'm guess i'm stuck in a situation where i've got an adjunction in $Cat_\infty$ and i want to "lift" it to an adjunction in the slice

Jonathan Beardsley (Jul 14 2023 at 23:04):

wait how did Mike do latex

Jonathan Beardsley (Jul 14 2023 at 23:05):

$like this?$

Kevin Arlin (Jul 14 2023 at 23:05):

double dollar signs, yeah :(

Jonathan Beardsley (Jul 14 2023 at 23:10):

so in my case i'm looking at a really silly adjunction. I've got the diagram category $D=a\to b\to c$ and the subcategory $I=a\to b$, and so I'm left Kan extending along $I\subset D$ and then precomposing back again. Now I want to fix the image of $b$. And it's sort of obvious that the functors $R\colon C^{D}\to C^{I}$ and $Lan\colon C^{I}\to C^{D}$ both admit a map "evaluation at $b$" down to $C$. So fix some $z\in C$ and pullback along that. That should give me another adjunction between the two categories but with one vertex fixed at a chosen object, right?

Jonathan Beardsley (Jul 14 2023 at 23:11):

but the difficulty, it seems, in the ∞-categorical setting, is getting that original adjunction (in $Cat_\infty$) to lift to an adjunction in $Cat_\infty/C$.

Jonathan Beardsley (Jul 14 2023 at 23:15):

But maybe I can reduce the amount of "checking" I need to do by working in the homotopy 2-category or something?

Jonathan Beardsley (Jul 14 2023 at 23:29):

okay okay maybe this isn't so hard actually

Jonathan Beardsley (Jul 14 2023 at 23:29):

haha

Jonathan Beardsley (Jul 17 2023 at 19:58):

For the record I think the proof in the ∞-categorical setup (specifically Riehl-Verity) looks something like this:
image.png

Jonathan Beardsley (Jul 17 2023 at 19:59):

oops that has an arrow backwards

Notification Bot (Jul 17 2023 at 20:00):

Jonathan Beardsley has marked this topic as resolved.

Ralph Sarkis (Aug 03 2023 at 10:53):

I am not a moderator @Morgan Rogers (he/him) .