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Stream: learning: questions

Topic: ✔ What is this left adjoint ?

Robin Jourde (Apr 16 2025 at 13:16):

Hi everyone, I am reading a paper with the following claim and I don't see how to prove it. I guess it should be simple as it is not justified by anything.
Let $F : \mathcal A \to \mathcal B$ be an identity-on-object $\mathcal V$-functor between small $\mathcal V$-categories, then the functor $[F, 1] : [\mathcal B, \mathcal V] \to [\mathcal A, \mathcal V]$ has a left adjoint and strictly creates colimits.
I understand $[F, 1]$ as doing the following on objects : $X \mapsto X \circ F$, I hope this is the right thing.
Does someone knows how this works ? Thank you in advance !

Morgan Rogers (he/him) (Apr 16 2025 at 13:17):

The left adjoint should be "left Kan extension along $F$ ", so you need some conditions on $\mathcal{V}$ for that to exist

Robin Jourde (Apr 16 2025 at 13:27):

They work with $\mathcal V$ that are locally presentable, would that be enough ? I don't think they have any other assumptions
Is 4.1 in https://ncatlab.org/nlab/show/Kan+extension what you are refering to ?

Sam Speight (Apr 16 2025 at 13:46):

Yes, because locally presentable categories are cocomplete, which is enough.

Morgan Rogers (he/him) (Apr 16 2025 at 14:01):

I think you also need a smallness condition that is also verified by lfp categories, otherwise I agree.

Nathanael Arkor (Apr 16 2025 at 14:31):

You only need the size condition on $F$ (for which $\mathcal A$ being small is enough). Local presentability is not necessary.

Kevin Carlson (Apr 16 2025 at 14:54):

(But cocompleteness is, which they’re getting from local presentability.)

Robin Jourde (Apr 16 2025 at 14:57):

Thank you everyone for your answers. Indeed, corollary X.3.2 in Mac Lane says (if I reverse it to left extensions) that I need $\mathcal A$ to be small and $\mathcal V$ to be cocomplete

Notification Bot (Apr 16 2025 at 14:58):

Robin Jourde has marked this topic as resolved.