Re: cleavages and choice

["Jean Bénabou" <jean.benabou@wanadoo.fr>]

Dear Thomas,

I am a bit surprised that you, of all people, should defend cleavages, i.e. indexed categories. As far as I remember, there not many of them in the notes you wrote on Fibered categories a la Benabou.
I remind you that these notes were written not only after lectures I gave, but after long conversations we had, many times, in my flat, and also a few days you spent in my house in the south of France, where, for at least 10 hours everyday I explained in detail to you my work on fibered categories and corrected many mistakes you made in first drafts  of that paper.
Nevertheless, for the sake of the people on the category list,to whom this message is also addressed, I shall answer your questions an remarks.

Let  p: X --> S be a surjective group homomorphism. It is a fibration, and a cleavage is a section of p (in Set). This was explicitly noted by Grothendieck more than 50 years ago! 
Does such a  p come equipped with a cleavage? Take for p the morphim :  R --> R/Z  of the reals on the circle. Suppose I were to teach periodic functions, i.e functions with domain R which factor through R/Z. Wouldn't it be ridiculous to use a section of p ? Which one by the way?

Take the theorem : The composite of two fibrations is a fibration. Does it need cleavages, i.e. AC for classes, to be proved? Of course, if you are cleavage fan, as you seem to be, you can add that, given cleavages of p and q one gets an associated cleavage of  pq.

Let's look at an important example,namely categories S with pull backs, not with choice of pull backs mind you. This is a first order notion, saying that : for every cospan of S there exists a universal span making the obvious square commutative.
Then, without AC, you can prove that the functor  Codom: S^2 --> S  is a fibration. And again, if you are cleavage happy, add that a cleavage of this fibration, if it exists, (I'm not assuming AC)  is a choice of pullbacks in S.

I could multiply the examples. But let's look at an important question. Suppose you prove an intrinsic result about fibrations, using cleavages, in principle you'd have to see what happens when you change cleavages. And don't wave your hands and tell me that, for formal 2-categorical reasons, the result is obvious. I'll believe you only when you write a precise metatheorem which covers ALL the cases.
You are convinced, and I am convinced, and everybody is convinced that such a metatheorem is not necessary. But that is NOT A PROOF !  And why are you convinced? Because, even if you say the contrary, deep in your mind you KNOW that intrinsic properties of fibrations AND cartesian functors should not refer to cleavages. Let me insist on the fact that the mythic metatheorem should also cover cartesian functors  F: X --> X' where you change the cleavages of both X and X'

Of course the theorem I mentioned in my mail on pre foliations, applies to fibrations and gives new results in that case. But this theorem is true for  F: X --> X'  where X is a prefoliation hence, even with AC, has no cleavage, and X' is an arbitrary category over S, i.e. has even less cleavages than X.

In order not to make this mail too long I have not, but I should have, mentioned internalization where cleavages are even more problematic.

Best to all,
Jean


Le 30 juil. 2014 à 17:06, Thomas Streicher a écrit :

> Dear Jean,
> 
> of course, you are right when emphasizing that one need choice for
> classes to endow an "anonymous" fibration with a cleavage.
> But that applies also to catgeories with say binary products. One
> needs choice for classes in order to choose a product cone for every
> pair of objects.
> In many instances, however, categories come together with a choice of
> products and fibrations come together with a choice of a cleavage. 
> 
> For example Set comes with a choice of a cleavage. Fibrations arising
> from internal categories are even split. Many constructions on fibrations 
> allow one to choose a cleavage given cleavages for the arguments.
> Do you know of any construction on fibrations which is not "cleavage
> preserving" in this sense?
> 
> Of course, one should not require cartesian functors to preserve
> cleavages just as one should not require functors to preserve chosen
> products.
> 
> Thomas



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