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Stream: theory: category theory

Topic: Pseudo cartesian natural transformations and fibrations

Beppe Metere (Jun 05 2025 at 07:34):

Hi guys, here is a question I recently published on MO.

Is anybody out there who can help, please? Thanks!

Beppe.

Given the notion of cartesian natural transformation, i.e. one such that all naturality squares are pullbacks, one can define a pseudo version of it: I would say that a pseudonatural transformation between pseudofunctors is pseudocartesian if all naturality squares are pseudopullbacks.

Now, my question is: if we take two pseudofunctors 

$$F,G\colon \mathsf{C}^{op}\to \mathsf{CAT}$$

and a pseudonatural transformation 

$$\alpha\colon F\Rightarrow G$$

what kind on notion does $\alpha$ translate into, if we regard $F,G$ as fibrations over $\mathsf{C}$, and $\alpha$ as a morphism between them?

A simplified version is asking the same question for two presheaves over  $\mathsf{C}$. In this case,  $\alpha$ would be cartesian, rather than pseudocartesian.

Matteo Capucci (he/him) (Jun 05 2025 at 15:52):

After thinking about this for a minute, it seems that $\alpha$ gets weak cartesian lifts, though (1) I ignored the pseudoness and (2) it might very well have strong cartesian lifts, I just didn't check

Mike Shulman (Jun 05 2025 at 16:23):

I have a guess that in the case of presheaves, the condition is that the functor $\int F \to \int G$ is also an opfibration. But even if this is true, I don't think it generalizes to pseudofunctors, so I didn't bother working out the details and posting it as an answer on MO.

Beppe Metere (Jun 05 2025 at 16:41):

Thanks @Matteo Capucci (he/him) and @Mike Shulman !

Mike, in fact, my first guess was that the functor $\int F\to \int G$ (call it $A$) was an opfibration, and indeed you can lift arrows of $\int G$ with their domains coming from $\int F$, but such liftings do not seem to be cocartesian. In other words, it seems to me in this case that $A$ is star surjective on the domains, and indeed things get worst when you switch to the pseudoversion.

Matteo, indeed I am not very comfortable with the notion of weak cartesian lifting. If you mean Benabou's precartesian maps... well I have to check this.

Matteo Capucci (he/him) (Jun 05 2025 at 16:49):

I now realize I started thinking about covariant indexed categories, so weakly cocartesian checks out

Matteo Capucci (he/him) (Jun 05 2025 at 16:52):

@Beppe Metere maybe? I don't remember that definition. A functor admitting weakly Cartesian lifts is a prefibration though.
Concretely it means being terminal as lifts of a certain morphism with a given codomain.

Mike Shulman (Jun 06 2025 at 01:33):

Beppe Metere said:

indeed you can lift arrows of $\int G$ with their domains coming from $\int F$, but such liftings do not seem to be cocartesian.

I thought that the full universal property of the pullback would tell you that covariant lifts don't just exist but are unique, and in the discrete case existence and uniqueness of lifts is sufficient to make them opcartesian.

Beppe Metere (Jun 06 2025 at 06:15):

Mike Shulman said:

Beppe Metere said:

indeed you can lift arrows of $\int G$ with their domains coming from $\int F$, but such liftings do not seem to be cocartesian.

I thought that the full universal property of the pullback would tell you that covariant lifts don't just exist but are unique, and in the discrete case existence and uniqueness of lifts is sufficient to make them opcartesian.

@Mike Shulman, your argument is correct, universal property of pullbacks makes the covariant lifts cocartesian. Thanks! I wonder how this interacts with the cartesian structure.

I tried to do the same with the pseudo-version, but it does not work, I fear.

Mike Shulman (Jun 06 2025 at 06:21):

Yes, I suspect it would work for pseudofunctors to Gpd, but not for pseudofunctors to Cat.

Beppe Metere (Jun 06 2025 at 06:25):

@Matteo Capucci (he/him) here is what I mean.

Let $P \colon \mathsf X \to \mathsf B$ be a functor. A map $k \colon Y \to X$ is precartesian iff for every map $f \colon Z \to X$ with $P(f) = P(k)$ there exists a unique vertical map $v \colon Z \to Y$ with $f = kv$.

So, possibly, for a strict cartesian transformation between pseudofunctors, the corresponding morphism of fibrations is a precofibration, right? aha!

However, if the transformation is pseudocartesian, things get confused... sob!

Matteo Capucci (he/him) (Jun 06 2025 at 06:54):

Well if the maps are strong cocartesian (as it seems to be the case) then they are also weak cocartesian. So maybe this just gives something new for pseudofunctors into Cat rather than Set?

Mike Shulman (Jun 06 2025 at 06:55):

In the pseudocartesian case you probably just want to switch from ordinary (op)fibrations to [[Street fibrations]].

Mike Shulman (Jun 06 2025 at 06:56):

But it's not clear to me why you even get preopcartesian liftings; where do the factorizations come from?

Mike Shulman (Jun 06 2025 at 06:57):

BTW, this is one case where I think "op" is preferable to "co", particularly on the word "opfibration" -- a "cofibration" is generally something with an extension property (a "fibration in the opposite category") whereas an opfibration still has a lifting property.

Beppe Metere (Jun 06 2025 at 08:53):

Mike Shulman said:

BTW, this is one case where I think "op" is preferable to "co", particularly on the word "opfibration" -- a "cofibration" is generally something with an extension property (a "fibration in the opposite category") whereas an opfibration still has a lifting property.

Do you think this should apply also to liftings? I mean, opcartesian instead of cocartesian?

Beppe Metere (Jun 06 2025 at 09:30):

Mike Shulman said:

In the pseudocartesian case you probably just want to switch from ordinary (op)fibrations to [[Street fibrations]].

Yes, this seems to be like a natural direction to follow when you consider the fibrational side of the problem.

However, what puzzles me is that from the pseudofunctorial point of view, the fact that a pseudonatural transformation is pseudocartesian should translate into a consistent notion in $\mathsf{Fib}(\mathsf{C})$, without bringing Street-fibrations into the game.

E.g. one feature of pseudocartesian transformation between fibrations should be that, whenever $\mathsf C$ has initial object, the transformation $\alpha$ is determined by its $0$-component, i.e. the corresponding morphisms $A$ is determined by its restriction to $0$-fibres...

Mike Shulman (Jun 06 2025 at 14:47):

Beppe Metere said:

Do you think this should apply also to liftings? I mean, opcartesian instead of cocartesian?

For consistency, yes.

Mike Shulman (Jun 06 2025 at 14:47):

Beppe Metere said:

the fact that a pseudonatural transformation is pseudocartesian should translate into a consistent notion in $\mathsf{Fib}(\mathsf{C})$, without bringing Street-fibrations into the game.

I don't see any reason to expect that.

Mike Shulman (Jun 06 2025 at 14:49):

Oh, maybe I wasn't clear. I'm still thinking you would be in the 2-category Fib(C) of ordinary fibrations. I was just saying that the map between such fibrations could only be expected to be a Street opfibration.

Beppe Metere (Jun 06 2025 at 15:01):

Mike Shulman said:

Oh, maybe I wasn't clear. I'm still thinking you would be in the 2-category Fib(C) of ordinary fibrations. I was just saying that the map between such fibrations could only be expected to be a Street opfibration.

Ah... ok! Now I got it, thanks! This would be reasonable, and interesting.