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Stream: theory: category theory

Topic: universality of the fundamental opindexed cat

Matteo Capucci (he/him) (Sep 12 2022 at 15:38):

Is there any sense in which the (op)indexed category

$$B/- : B \to \bf Cat$$

is universal?
It sends every object to the slice over it and every morphism acts by postcomposition (not pb)

Matteo Capucci (he/him) (Sep 12 2022 at 15:38):

I remember it was but I can't recall how.. It's not initial nor terminal, of course...

Matteo Capucci (he/him) (Sep 12 2022 at 15:56):

Mmh I recalled this thread:

https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/'Slice.20Yoneda'

and I think what I was going at with my answer is what Streicher (Benabou?) calls 'fibred Yoneda lemma':
image.png

Matteo Capucci (he/him) (Sep 12 2022 at 15:58):

Here $Sp(P)$ is defined as

$$Sp(P)(I) = Fib(B)(\underline I, P)$$

where in turn $\underline I = B/I \overset{cod}\to B$

Matteo Capucci (he/him) (Sep 12 2022 at 15:59):

Hence the statement reads

$$Fib(B)(U(S), P) \cong Sp(B)(S, Fib(B)(\underline -, P))$$

Matteo Capucci (he/him) (Sep 12 2022 at 15:59):

which is almost Yoneda-looking?

Matteo Capucci (he/him) (Sep 12 2022 at 16:02):

I mean it almost says

$$P \cong Fib(B)(\underline -, P)$$

Matteo Capucci (he/him) (Sep 12 2022 at 16:03):

Compare with:

$$P \cong Psh(B)(\underline -, P)$$

where now $P$ is a presheaf over $B$ and $\underline -$ is Yoneda embedding

Matteo Capucci (he/him) (Sep 12 2022 at 16:04):

So maybe my question is ill-framed, since I should really consider $B/-$ to land in $Fib(B)$, and then the answer becomes 'think of this as the Yoneda embedding for indexed categories'

Matteo Capucci (he/him) (Sep 12 2022 at 16:05):

In fact I'd bet a beer this is the Yoneda embedding for categories internal to $Set^{B^{op}}$... but this I can only eyeball

Matteo Capucci (he/him) (Sep 12 2022 at 16:06):

Matteo Capucci (he/him) said:

I mean it almost says

$$P \cong Fib(B)(\underline -, P)$$

Compare with:

$$P \cong Psh(B)(\underline -, P)$$

where now $P$ is a presheaf over $B$ and $\underline -$ is Yoneda embedding

:face_palm: these are the same exact statement, aren't they? Just on different sides of the Grothendieck construction

Matteo Capucci (he/him) (Sep 12 2022 at 16:07):

Matteo Capucci (he/him) said:

Hence the statement reads

$$Fib(B)(U(S), P) \cong Sp(B)(S, Fib(B)(\underline -, P))$$

But then I don't understand where splittings become relevant in all of this

Tobias Schmude (Sep 13 2022 at 04:36):

The fibered YL extends the "classical" one from discrete fibrations to arbitrary fibrations. The role of splittings vanishes in the classical case since every discrete fibration has a unique one.

Tobias Schmude (Sep 13 2022 at 04:39):

The equivalence $P \cong Fib(B)(\underline{-}, P)$ corresponds to the statement that the counit of the adjunction is an equivalence.

Matteo Capucci (he/him) (Sep 13 2022 at 09:32):

Thanks!

Matteo Capucci (he/him) (Sep 13 2022 at 09:32):

Do you know if it's still true that $Fib(B)$ is somehow generated by the fibrations $B/- \to B$, like presheaves are always colimits of representables?

Tobias Schmude (Sep 13 2022 at 13:29):

Presheaf categories are complete and cocomplete. Analogously, neither limits nor colimits should get you anything but discrete fibrations.
I don't know about other notions of "generated by", e.g. what's called a [[separator]] on the nlab.