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What’s the definition, or at least, when in the video does Bénabou define them?
I have no time these days to inspect the video, will do as soon as I finish a big overdue task I need to finish these days; I am very eager to study the video in detail. I was told those definitions by Igor, but I may be quite wrong in my memory. It is about factorization systems vertical vs. horizontal. There is a weaker notion of prefoliated category. In think, there is a remark about foliated categories in the lectures a la Benabou on fibered categories written by Thomas Streicher.
That’s ok. I just wondered if you knew.
Has anyone with an understanding of spoken French listened to the section 9:53-14:00, where I think foliated categories are defined. Certainly Bénabou defines prefoliated categories there, but I get lost when he starts describing the equivalence of two conditions he calls $(C_0)$ and $(C_1)$. These might just be something that holds for prefoliated categories. I’m going to continue to watch the video, and then transcribe at least the definition at foliated category. EDIT: no, he definitely hasn’t defined foliated categories at that point.
And for future reference, Streicher mentions the term foliated category, but never defines it, and the example very briefly mentioned is not really sufficient to figure out what’s going on.
Ok, I think the definition of a foliated category comes in at about 46:27, beginning with the definition of a ’horizontal arrow’. There’s also something to do with the closure of cartesian arrows under composition.
PS Someone coughs right at the wrong moment, and I think the obscured word is either ’horizontal’ or ’cartesian’. Perhaps it can be reconstructed from context.
PPS I think the definitions go like this, if someone confirms, I can add it to the nLab page. The first one is ok, the second one is conjectural, based on my incomplete understanding of French.
Definition Given a functor $P:X\to S$, the category $X$ is prefoliated if every arrow of $X$ admits a factorisation into a vertical arrow followed by a cartesian arrow.
“Definition” A prefoliated category is foliated if it is equipped with a class of arrows called horizontal such that whenever $f:x\to y$ is cartesian, and $h:y\to z$ is horizontal, then $h\circ f:x\to z$ is cartesian.
I’m not 100% sure whether a foliated category is equipped with horizontal arrows, as extra structure, or if they arise some other way.
What’s the motivation?
For Bénabou or me? If you want to know the former, then I suppose it’s the usual search for general axioms that give nice theorems in pure category theory. For me, well, I’m just curious.
What is one of the nice theorems that he’s after? What is a motivating example? Why do we consider this definition?
I heard of a number of naturally occuring examples, but I have no exact memory of them (sorry, I have quite a bad memory). He spent few years working on it, and I was told it is quite a lot behind it. From a point of view, the theory is roughly and partly about a connection between factorization systems (horizontal vs. vertical) and the theory of generalized fibrations.In 2011 (cf. link) Benabou gave a talk
Title : A NEW APPROACH TO THE NOTION OF “BEING CARTESIAN”
Abstract:
Cartesian functors and maps were introduced by Grothendieck in 1960-61 for prefibrations and fibrations. Since then they have been studied essentially only in the case of fibrations. I shall present general definitions covering many other situations, which reduce to the classical ones in the previous cases, and give important examples which do not fit in the classical setting, and shall prove many results which were not known even in the case of fibrations. Some of these results need assumptions, much weaker than fibrations or even prefibrations, namely prefoliations and foliations, which I shall introduce and study briefly, with a focus on examples and counter examples.
6: about horizontal vs. cartesian see some sutble points in a comment of Thomas Streicher at category list archive.
Thanks, Zoran. It seems that the notion of fibration is singled out by the Grothendieck-construction-equivalence. What happens to that as we generalize the notion of fibration? Is there something natural on the dual side that generalizes 2-functors to Cat?
I think not. It is not about generalization of pseudofunctor, but about good categories over categories. There are so many categorical constructions in which one has somewhat richer category in some construction being equipped with a natural projection which has this or that property for liftings, uniqueness, factorizations, special horizontal morphisms and so on so that extensions of the theory are very interesting.
Of course, the Grothendieck construction can be extended to lax functors, there do not need to be pseudofunctors, and for the cofibered case to oplax functors.
As Igor’s work in progress shows the question on variants of the notion of fibered category becomes much richer for fibered bicategories where there is a plethora of variants with different lifting properties. There is an interesting role of double bicategories and even double tricategories he discovered in classifying those.
I see, thanks. Would be interesting to know, though, which of these generalizations are dually equivalent to which kind of higher functors.
For instance looking at simplicial sets, there is a plethora of different kind of fibrations known Kan fibration, inner fibration, left fibration, Cartesian fibration etc. While they are all sort of useful, many of them do not have any intrinsic meaning in category theory, as they vialote the principle of equivalence. They are just tools for combinatorics. The only one with intrinsic meaning in the above list is that of Cartesian fibration.
Is the notion of foliated category invariant under equivalence of categories?
Urs, it seems so … IF I have the definition above correct, then we need to talk about equivalences of categories with extra structure (the horizontal arrows). This may not count as having intrinsic meaning to you, if you want equivalence under arbitrary equivalences of categories.
Bénabou spent at least half an hour talking about applications of prefoliated categories (which I skimmed through, not trying to pick up details), then introduced foliated categories, and then had only a few minutes left, and I didn’t try to figure out what the applications were. I’d email and ask, but he may well just accuse me of trying to steal his work. Streicher might tell me, but there is nothing publicly available, not even lecture notes, so he may be loath to release Bénabou’s work without his permission.
Other than that, I could ask on MO, and see if anyone bites. (The categories mailing list I imagine would also evoke a response from Bénabou, so I don’t feel like trying there).
Hi David,
don’t worry, I was just trying to see why you are talking about this definition. It wasn’t clear to me that it wasn’t clear to you. :-)
I suppose you are following common practice, much honored on MO, too, to assume that since Bénabou is famous, whatever he says is valuable even without explanation.
Probably you are right.
This reminds me of a joke that somebody once made at the hight of the excitement about string dualities in the 1990s. He said: “If Edward Witten went insane tomorrow but kept posting articles, would we notice it?”
Mostly I wanted to write down the definition on the nLab, because all that is there at the moment is a link to this video. And since the definition is nowhere to be found otherwise, I thought it might be useful to someone if it was written down. I don’t have any reason to use the thing myself, and I don’t really see how the notion is useful except in a kind of reverse mathematical way.
I suppose you are following common practice, much honored on MO, too, to assume that since Bénabou is famous, whatever he says is valuable even without explanation.
well, no. But I can see how it might appear that way. I actually find the sort of things that he focuses on not very interesting, and even orthogonal to my own interests.
Okay, sorry. I misunderstood what’s going on here.
Just a link so I can come back to this: slides from the talk discussed above (or a duplicate of it), and an annotated video. I added the link to the slides at foliated category.
Added the definition, both of foliated categories themselves, and their functors.
Added a video reference where Bénabou talks about this:
- Jean Bénabou, Du vieux et du neuf sur la construction de Grothendieck, March 2019, YouTube
I’m not sure I got the definition of functor between foliated categories correct, I’m trying to base it on the discussion starting at this timestamp - it’s in French and I’m mostly working from key words and the little Bénabou writes on the board.
Hmm, reading back over my attempted capture of the definition above in comment #6, the definition I added in #21 would be of a prefoliated category. So here is where my lack of knowledge of French is biting me. I don’t have the time to try to figure out the old video and square it with the new one I just added.
OK, I just checked the definition in the categories mail list email supplied in #11 and corrected in #20, and there’s a closure condition I missed. I will add it.
Also, possibly Bénabou uses a weaker notion of cartesian morphism …. :-S
It looks like foliated categories are almost those categories for which ($P$-vertical maps, $P$-cartesian maps) forms a strict factorisation system on $\mathcal{E}$, but without uniqueness of factorisation (though we do have that the “image of any factorisation under $P$” is unique). I don’t know whether there is a name for such structures (it is not a weak factorisation system, because we do not have orthogonality either).
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