Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: cartesian morphisms as “fibred terminal”

Amar Hadzihasanovic (Oct 12 2022 at 06:46):

I was trying to better understand Grothendieck fibrations and I came up with this characterisation of cartesian morphisms which I thought was nice; I'm wondering if it's part of some more general story that I'm unaware of.

Let $p: \mathcal{C} \to \mathcal{D}$ be a functor. Say that an object $x$ of $\mathcal{C}$ is terminal relative to $p$ if the following are true:

• for all objects $y$ of $\mathcal{C}$ and morphisms $f: py \to px$ in $\mathcal{D}$, there exists a lift $\tilde{f}: y \to x$ of $f$ to $\mathcal{C}$;
• for all objects $y$ and pairs of morphisms $g, h: y \to x$ in $\mathcal{C}$, if $pg = ph$ in $\mathcal{D}$, then $g = h$.

This is a “fibred” version of the definition of terminal object, in the sense that an object $x$ of $\mathcal{C}$ is terminal if and only if it is terminal relative to the unique functor $\mathcal{C} \to 1$.

Then
Claim. A morphism $f: x \to y$ is $p$-cartesian if and only if it a terminal object relative to $p_*: \mathcal{C}/y \to \mathcal{D}/py$.

So $p$ is a Grothendieck fibration iff for all $y$, all objects of $\mathcal{D}/py$ admit a $p_*$-terminal lift to $\mathcal{C}/y$.

As it is this is just a “cute” rephrasing, but it made me wonder -- is there a general story about “fibred limits” of which this is an instance? The nLab didn't help.

Zhen Lin Low (Oct 12 2022 at 09:47):

It's probably related to the characterisation of fibrations in terms of adjoints and comma categories, and of adjoints in terms of initial/terminal objects...

Christian Williams (Oct 12 2022 at 17:21):

By inverse image, a functor $p$ is equivalent to a (normal) lax functor $p^\ast: D\to \mathrm{Prof}$. It is a fibration when each $p^\ast(d):p^\ast(d_0)\to p^\ast(d_1)$ is representable as $p^\ast(d_1)(1,d^\ast)$ cart-rep.png

Christian Williams (Oct 12 2022 at 17:25):

So I think this pertains to the equivalence of representation and terminal objects, which then comes up in the concept of fibrations.

fosco (Oct 12 2022 at 18:14):

I was about to write something similar: a functor $P : X\to Set$ is representable if and only if its category of elements has an initial/terminal object (depending on the variance of $P$)

Amar Hadzihasanovic (Oct 12 2022 at 18:43):

This stackexchange question by a certain @fosco seems to suggest that representability of a profunctor is not equivalent to a “bare” initiality/terminality condition on the category of elements.
At this point I would conjecture that it could, instead, be equivalent to initiality/terminality relative to one end of the two-sided discrete fibration, as in my definition?

Amar Hadzihasanovic (Oct 12 2022 at 18:48):

In any case, my question is not so much about fibrations, as it is about this “universality relative to a functor” definition, which happens to have cartesian morphisms as an instance.

It seems to me that there is a clear generalisation to “fibred limits” of each shape (just replace “objects” with “cocones” and “morphisms” with “morphisms of cocones”). I would like to know if the theory of these things has been worked out somewhere.

Amar Hadzihasanovic (Oct 12 2022 at 19:08):

Amar Hadzihasanovic said:

At this point I would conjecture that it could, instead, be equivalent to initiality/terminality relative to one end of the two-sided discrete fibration, as in my definition?

To make this more precise: consider a profunctor $H: \mathcal{D}^\mathrm{op} \times \mathcal{C} \to \mathbf{Set}$. To this we can associate a two-sided fibration $p: \mathcal{E} \to \mathcal{C}$, $q: \mathcal{E} \to \mathcal{D}$, where $p$ is a discrete opfibration and $q$ a discrete fibration.
Then:
Conjecture. $H$ is representable as $\mathcal{D}(-, F-)$ if and only if $\mathcal{E}$ has a terminal object relative to $p$, and is representable as $\mathcal{C}(G-, -)$ if and only if $\mathcal{E}$ has an initial object relative to $q$.
Update: no, this seems all wrong.

Amar Hadzihasanovic (Oct 12 2022 at 19:10):

I may have gotten some variances wrong, but I think this reduces to the usual condition for representability of (co)presheaves when either $\mathcal{C}$ or $\mathcal{D}$ is trivial.

fosco (Oct 12 2022 at 19:32):

Amar Hadzihasanovic said:

This stackexchange question by a certain fosco seems to suggest that representability of a profunctor is not equivalent to a “bare” initiality/terminality condition on the category of elements.
At this point I would conjecture that it could, instead, be equivalent to initiality/terminality relative to one end of the two-sided discrete fibration, as in my definition?

I know the guy, you shouldn't trust him :stuck_out_tongue:

This is why I didn't write anything in the end, I didn't know how to relate the two things precisely, given that result :frown: incidentally it is related to the annoying fact that the collage of two categorie along a profunctor is not the cat of elements of the profunctor

fosco (Oct 12 2022 at 19:32):

for a profunctor regarded as a presheaf over $A^{op}\times B$ representability is a splitting into two Yoneda's

fosco (Oct 12 2022 at 19:33):

Instead what you want is a profunctor $A^{op}\times B\to Set$ whose transposed $B\to[A^{op},Set]$ is objectwise representable

fosco (Oct 12 2022 at 19:33):

(also, I will gladly talk about this tomorrow in person, now I really am too tired to say something meaningful!)

fosco (Oct 12 2022 at 19:34):

but hmu with developments, this is interesting and I thought a lot about how to see basic facts of fibered categories for a generic displayed category $A\to Prof$

fosco (Oct 12 2022 at 20:03):

ah, you left me with a nice toy before bed!

If $A,B$ are discrete categories a profunctor $p : A\times B\to Set$ is a matrix of sets, and $p$ is representable if and only if the matrix is made only of 0s and 1s (or rather, $\varnothing$'s and $\{*\}$'s) and each row has exactly one 1 (I hope I didn't transpose the matrix I'm considering...)

fosco (Oct 12 2022 at 20:04):

this means that there is a function $f : A\to B$ such that $p(a,b)=B(fa,b)$

fosco (Oct 12 2022 at 20:04):

and $p(a,b)$ is 1 if and only if $fa=b$, 0 otherwise

fosco (Oct 12 2022 at 20:07):

can you relate this to your conjecture?

Amar Hadzihasanovic (Oct 12 2022 at 20:08):

We can try to figure it out tomorrow or Friday, I'm off to bed now :D

fosco (Oct 12 2022 at 20:10):

same! :grinning:

Amar Hadzihasanovic (Oct 12 2022 at 20:48):

(Not in bed obviously) on second thought I don't think my conjecture makes sense, in this case you need fibrewise terminal/initial objects which is different from what “fibred” terminal is

Amar Hadzihasanovic (Oct 12 2022 at 20:51):

I.e. I think for profunctors the condition for e.g. representability as $\mathcal{D}(-, F-)$ is that, for $c$ in $\mathcal{C}$, each fiber $p^* c$ has a terminal object, which is very different from $\mathcal{E}$ having a terminal object rel $p$...

Taichi Uemura (Oct 12 2022 at 21:29):

Section 4.3.1 of Higher Topos Theory on relative colimits might be related.

Amar Hadzihasanovic (Oct 13 2022 at 05:08):

Amar Hadzihasanovic said:

I.e. I think for profunctors the condition for e.g. representability as $\mathcal{D}(-, F-)$ is that, for $c$ in $\mathcal{C}$, each fiber $p^* c$ has a terminal object, which is very different from $\mathcal{E}$ having a terminal object rel $p$...

Ah, perhaps this works:

Revised conjecture. $H$ is representable as $\mathcal{D}(-, F-)$ if and only if every $c$ in $\mathcal{C}$ lifts to a terminal object in $\mathcal{E}$ relative to $p$, and is representable as $\mathcal{C}(G-, -)$ if and only if every $d$ in $\mathcal{D}$ lifts to an initial object in $\mathcal{E}$ relative to $q$.

Amar Hadzihasanovic (Oct 13 2022 at 05:09):

Taichi Uemura said:

Section 4.3.1 of Higher Topos Theory on relative colimits might be related.

I will take a look, thank you!

Amar Hadzihasanovic (Oct 13 2022 at 09:59):

Ok so after in-person discussion with Fosco I think we have figured out the following.

Given $p: \mathcal{C} \to \mathcal{D}$, the property that “every object of $\mathcal{D}$ has a lift to $\mathcal{C}$ that is terminal relative to $p$” is equivalent to “$p$ has a fully faithful right adjoint”, so this recovers an equivalent definition of fibrations (according to Fosco, due to Chevalley): $p$ is a Grothendieck fibration iff for all $c$, the functor $p_*: \mathcal{C}/c \to \mathcal{D}/pc$ has a fully faithful right adjoint.

Amar Hadzihasanovic (Oct 13 2022 at 10:08):

And now, maybe we should be able to also answer my “revised conjecture”: we have that, given the graph $p: \mathcal{E} \to \mathcal{C}, g: \mathcal{E} \to \mathcal{D}$ of a profunctor $H: \mathcal{D}^\mathrm{op} \times \mathcal{C} \to \mathbf{Set}$, the condition that every $c$ lifts to a $p_*$-terminal object of $\mathcal{E}$ is equivalent to $p$ having a fully faithful right adjoint, i.e. exhibiting $\mathcal{C}$ as a reflective subcategory of $\mathcal{E}$

Amar Hadzihasanovic (Oct 13 2022 at 10:09):

Which sounds very much like the answer to the SE question mentioned above, if we accept that maybe the collage of a profunctor was mixed-up with the graph of a profunctor

fosco (Oct 13 2022 at 10:10):

(The result is quoted in Gray's "Fibered and cofibered categories", where Gray attributes it to Chevalley, and the fibrations you get are only the cloven ones; see also the Monoidal toplogy book)

Amar Hadzihasanovic (Oct 13 2022 at 10:10):

(Unless maybe there's also a dual characterisation in terms of reflective subcategories of the collage, which may well be)

Tobias Schmude (Oct 13 2022 at 10:15):

Christian Williams said:

By inverse image, a functor $p$ is equivalent to a (normal) lax functor $p^\ast: D\to \mathrm{Prof}$. It is a fibration when each $p^\ast(d):p^\ast(d_0)\to p^\ast(d_1)$ is representable as $p^\ast(d_1)(1,d^\ast)$

It's just a prefibration then though, right? For a fibration it needs to be a pseudofunctor as well.
(Sorry if this has already been mentioned, I've only skimmed the rest of the discussion so far)

Beppe Metere (Oct 13 2022 at 15:23):

Amar Hadzihasanovic said:

Given $p: \mathcal{C} \to \mathcal{D}$, the property that “every object of $\mathcal{D}$ has a lift to $\mathcal{C}$ that is terminal relative to $p$” is equivalent to “$p$ has a fully faithful right adjoint”, so this recovers an equivalent definition of fibrations (according to Fosco, due to Chevalley): $p$ is a Grothendieck fibration iff for all $c$, the functor $p_*: \mathcal{C}/c \to \mathcal{D}/pc$ has a fully faithful right adjoint.

Well, I am not sure this story is exactly as described above. Let me say that Theorem 2.10 on Gray's 1966 paper states (among other things, and adapting the notation) that your functor $p$ is a fibration iff for each $c$, $p_*: \mathcal{C}/c \to \mathcal{D}/pc$ has a rari, i.e. a right adjoint right inverse. Now, if I am not wrong, this condition is slightly stronger than having a fully faithful right adjoint, because in the first case the counit is an identity, while in the second one it is just an iso. However, the picture from the Monoidal Topology book provided by @fosco is correct, since it asks for the counit to be an identity.
Concerning Chevalley, this is another bit of the same story.
It is true that Gray mentions some handwritten notes of a seminar given by Chevalley at Berkley in 1962 - i.e. some three/four years before his paper. What he says, however, is that such notes "treated these questions from a slightly different point of view". Then, he points the reader to Proposition 3.11, where he claims to discuss the "Chevalley condition". Therefore, I do not see any reason to credit Theorem 2.10 to Chevalley. As far as I understand, such result is of Gray himself.
On there other hand, there exists this Chevalley criterion, which is stated in Proposition 3.11. It says that a functor $p$ as above is a fibration iff the canonical comparison functor $\mathcal{C}/\mathcal{C}\to\mathcal{D}/p$ has a rari. Notice that such a condition is stated globally, while that of Theorem 2.10 is a collection of local conditions.
More details on Chevalley, with some care to the internal (formal) aspects can be found on a paper of mine (and coauthors):

• A. S. Cigoli, S. Mantovani, G. Metere, Discrete and Conservative Factorizations in Fib(B), APCS 29 (2021).

This is indeed a final cut.
You can find the arxived version v1 (https://arxiv.org/pdf/1909.10822v1.pdf), which, together with some mistakes corrected in the final version, contains (in the appendix) the explicit proof of the internal version of the Chevalley criterion.

Amar Hadzihasanovic (Oct 13 2022 at 15:31):

Thanks, yes, I was being a bit careless with the rari / reflective subcategory / fully faithful right adjoint distinction!

Beppe Metere (Oct 13 2022 at 15:38):

Don't worry! It took me ages to get this point!

Morgan Rogers (he/him) (Oct 14 2022 at 05:52):

If it helps, slice-wise ff right adjoints give Street fibrations.