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Stream: learning: questions

Topic: left adjoint functors from the category of simplicial sets

Leopold Schlicht (Oct 22 2021 at 16:59):

Is every left adjoint functor from the category of simplicial sets to a cocomplete category $\mathcal C$ the left Kan extension of some functor $\Delta\to\mathcal C$ along the Yoneda embedding? I found this claim in the title (but not in the actual text) of section 4 in this document. The title also seems to claim that there is a unified presentation of all left adjoint functors from the category of simplicial sets to any (not necessarily cocomplete) category $\mathcal C$ (for which I can find no indication whatsoever).

Mike Shulman (Oct 22 2021 at 17:02):

Yes, because left adjoints are cocontinuous, and presheaf categories are [[free cocompletions]].

Leopold Schlicht (Oct 22 2021 at 17:58):

Thanks, so one just has to precompose with the Yoneda embedding to obtain the desired functor $\Delta\to \mathcal C$. I think it's a bit surprising that each cocontinuous functor $\mathbf{sSet}\to\mathcal C$ is left adjoint.

Mike Shulman (Oct 22 2021 at 18:07):

Yes, the adjoint functor theorem is a bit magic.

John Baez (Oct 22 2021 at 18:10):

But also, cocontinous functors should be left adjoints, so we're just saying here that nothing horrible happens. Which is of course a bit surprising, but only like

DISASTER DID NOT STRIKE TODAY!!!

John Baez (Oct 22 2021 at 18:12):

(The headline you never see in the paper, because it hasn't happened yet.)

Mike Shulman (Oct 22 2021 at 19:17):

Why "should" cocontinuous functors be left adjoints unless you believe in the adjoint functor theorem?

Fawzi Hreiki (Oct 22 2021 at 20:35):

Also, this is true for any [[total category]].

John Baez (Oct 23 2021 at 03:49):

I believe in the adjoint functor theorem.

John Baez (Oct 23 2021 at 03:58):

I just meant to tell Leopold, who may just be starting to learn about this stuff, that when you see a cocontinuous functor you should expect it to be a left adjoint, because in most situations you come across, it will be.