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Stream: theory: category theory

Topic: ✔ pullback in functor 2-category

Jonathan Beardsley (Dec 21 2023 at 22:10):

So @Mike Shulman here's, I think, an example of what I'm thinking of. Say I've got three symmetric monoidal categories $A$, $B$ and $C$, and a symmetric monoidal functor $\alpha\colon A\to B$ and I realize them as "Segal" functors, say $A',B',C'\colon\Gamma^{op}\to Cat$ and a natural transformation $\alpha'\colon A'\Rightarrow B'$. Let's assume $A$, $B$ and $C$ are all complete and cocomplete and small. Now there's a pullback functor $\alpha^\ast\colon SMC(B,C)\to SMC(A,C)$ and I guess I'm asking if this functor has a right adjoint. Is this not generally true? And then that would correspond to a functor $Nat(A',C')\to Nat(B',C')$?

Mike Shulman (Dec 21 2023 at 22:12):

So in this case your proposal to take the levelwise right Kan extension corresponds to taking the ordinary right Kan extension along the underlying ordinary functor $\alpha:A\to B$, and my question of whether you've tried to show that what you get is a natural transformation corresponds to asking whether you've tried to check that this is still a symmetric monoidal functor.

Jonathan Beardsley (Dec 21 2023 at 22:12):

Right, yeah, okay.

Jonathan Beardsley (Dec 21 2023 at 22:14):

Alright. Fair enough. I think my reasoning was a little bit like: you can consider on underlying categories the functor $\alpha_0\colon A\to B$ to get a pullback $Fun(B,C)\to Fun(A,C)$ which has a right adjoint. Now note that both of those functor categories have symmetric monoidal structures coming from Day convolution and the pullback functor is strong monoidal w/r/t that monoidal structure (but maybe I'm wrong about it being strong monoidal?) and so has lax monoidal right adjoint.

Jonathan Beardsley (Dec 21 2023 at 22:16):

And so that right adjoint gives a functor from lax monoidal functors $A\to C$ to lax monoidal functors $B\to C$.

Jonathan Beardsley (Dec 21 2023 at 22:17):

And I'm fine with assuming my functor $\alpha\colon A\to B$ is strong monoidal.

Jonathan Beardsley (Dec 21 2023 at 22:22):

There's an analogous result for Cartesian fibrations of $\infty$-categories $p\colon E\to\Gamma^{op}$ and $q\colon F\to \Gamma^{op}$ which says that if you give me a functor $\Theta\colon E\to F$ over $\Gamma^{op}$ such that $\Theta_n$ has a right adjoint $\Omega_n\colon F_n\to E_n$ over each $\langle n \rangle\in \Gamma^{op}$ (assuming the skeletal version of $\Gamma^{op}$ of course) AND the functor $\Theta$ preserves Cartesian morphisms then $\Theta$ admits a right adjoint $\Omega$ over $\Gamma^{op}$.

Mike Shulman (Dec 21 2023 at 22:23):

I don't think that result is analogous, cf my comment above about the indexed AFT.

Jonathan Beardsley (Dec 21 2023 at 22:23):

Guh, right, that's the same mistake I keep making. Sorry.

Jonathan Beardsley (Dec 21 2023 at 22:24):

I can't keep it straight for some reason.

Mike Shulman (Dec 21 2023 at 22:24):

Jonathan Beardsley said:

and the pullback functor is strong monoidal w/r/t that monoidal structure (but maybe I'm wrong about it being strong monoidal?)

This is also where I instinctively smell a rat. Have you tried to prove that it's strong monoidal?

Jonathan Beardsley (Dec 21 2023 at 22:24):

Mike Shulman said:

Jonathan Beardsley said:

and the pullback functor is strong monoidal w/r/t that monoidal structure (but maybe I'm wrong about it being strong monoidal?)

This is also where I instinctively smell a rat. Have you tried to prove that it's strong monoidal?

No. That's where I'll start I guess.

Jonathan Beardsley (Dec 21 2023 at 22:25):

Oh, well, ughhhhhh, Day convolution, haha. I guess I'll try to see if I can do it.

Jonathan Beardsley (Dec 21 2023 at 22:26):

Well, or alternatively: https://math.stackexchange.com/questions/2257437/pullback-functor-preserves-day-convolution

Jonathan Beardsley (Dec 21 2023 at 22:26):

Darn. Okay.

Jonathan Beardsley (Dec 21 2023 at 22:26):

:smiling_face_with_tear:

Mike Shulman (Dec 21 2023 at 22:33):

Well, there you go I guess.

Jonathan Beardsley (Dec 21 2023 at 22:34):

Notification Bot (Dec 21 2023 at 22:47):

Jonathan Beardsley has marked this topic as resolved.