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Stream: theory: category theory

Topic: Category of elements vs Grothendieck construction

Tomáš Jakl (May 29 2023 at 13:49):

As stated on nCatLab, the Grothendieck construction $[\mathcal C^{op}, \mathbf{Cat}] \to \mathbf{Cat} \downarrow \mathcal C$ (where the codomain is the usual comma category) has a left adjoint which sends a functor $p : \mathcal A \to \mathcal C$. to the functor, which sends an object $c \in \mathcal C$ to the comma category $c \downarrow p$.

We can view the construction of the category of elements as a special type of Grothendieck construction of type $[\mathcal C^{op}, \mathbf{Set}] \to \mathbf{Cat} \downarrow \mathcal C$. I'm working in a setting where for my example $\mathcal C$ the construction of the category of elements has a left adjoint.

Do you know if this known in larger generality or if there are any other known examples of this?

Mike Shulman (May 29 2023 at 15:22):

The inclusion $\rm Set \hookrightarrow Cat$ has a left adjoint, so you should be able to get an adjoint for all $C$ by applying this pointwise and composing with the first adjunction.

Tomáš Jakl (May 29 2023 at 15:51):

Very nice, thank you! I had no idea about the left adjoint to $\mathrm{Set} \hookrightarrow \mathrm{Cat}$, thanks!

Patrick Nicodemus (May 29 2023 at 17:04):

Mac Lane has said in Sheaves in Geometry and Logic that he was working on the special case of the category of elements before Grothendieck gave that construction. Just a historical note

Tomáš Jakl (Jun 07 2023 at 08:46):

It turns out that the fact that the discrete Grothendieck construction $[\mathcal C^{op}, \mathrm{Set}] \to \mathrm{Cat} \downarrow \mathcal C$ has a left adjoint was already mentioned in Kelly's Basic concepts of enriched category theory. He even gives an explicit formula in (4.77).

Tomáš Jakl (Jun 07 2023 at 08:48):

Kelly also mentions that the discrete Grothendieck construction is fully faithful. Do you know if this holds in general, for the (full) Grothendieck construction $[\mathcal C^{op}, \mathrm{Cat}] \to \mathrm{Cat} \downarrow \mathcal C$?

Tomáš Jakl (Jun 07 2023 at 08:49):

Also, is there a good reference for the fact that the (discrete) Grothendieck construction is fully faithful? Kelly does not provide any justification. I'm not sure if it was folklore back and if a proof of it appears anywehere in the literature.

Nathanael Arkor (Jun 07 2023 at 10:20):

The $\Theta$ in Kelly is fully faithful because it's the right adjoint of a reflective subcategory, which Kelly mentions just above (4.78).

Tomáš Jakl (Jun 07 2023 at 12:33):

Yes, I know, he mentions that but he does not give a proof of it (unless I missed something). It is not clear to me from Kelly's writing if this was folklore back then or if he just claimed it and the reader was expected to figure out the details by himself, and now this is attributed to Kelly.

Tomáš Jakl (Jun 07 2023 at 14:51):

Now when I think about this, the question is basically who proved first that this adjunction restricts to the equivalence with discrete opfibrations...

Mike Shulman (Jun 07 2023 at 16:13):

I don't know the original reference. The fact that discrete opfibrations are a reflective subcategory of ${\rm Cat}/C$ is a special case of the [[comprehensive factorization system]].

The general Grothendieck construction is not fully faithful to ${\rm Cat}/C$; its image is the category of opfibrations and morphisms of opfibrations (preserving opcartesian arrows).

Tomáš Jakl (Jun 07 2023 at 19:06):

Interesting, thanks!