Re: cleavages and choice

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

Dear Eduardo,

Of course I fully agree with what you said. Let me just add a few remarks.
 
When Grothendieck defined fibrations it was for precise purposes, namely to axiomatize the notion of inverse image, and to use this axiomatization for descent (the titles of his talks are quite clear about that)

My view about fibrations is to try to get rid of sets, as much as possible, in category theory. This is what I tried to explain in my paper at the JSL.
Thus if some fibrations arising from set theory come equipped with more or less artificial cleavages, it does not impress me at all. 

The same process of elimination of sets, makes very sensitive, for fibrations but also for other domains, to the possibility of internalization.

Let me give an example of a totally different nature. In the paper where I introduced bicategories, their definition is in section 1.1 and takes 3 pages. But immediately after, it took me 10 pages to show in detail that they could be internalized, in any category with pullbacks. Let me point out that this paper was written in 1966 and published in 1967.

I think of course that ZFC has been a huge progress in the history of mathematics, but category theory has given us the possibility to explore new and fascinating countries. This goes for many domains of mathematics, and in particular.... for fibered categories.

Best to all,
Jean


Le 1 août 2014 à 18:47, Eduardo J. Dubuc a écrit :

> 
> Dear all,
> 
> I prefer fibrations over fibrations furnished with a cleavage (indexed categories) many times for reasons purely pragmatical mixed with an aesthetic philosophy.
> 
> Suppose you are dealing with fibrations where a canonical cleavage is present, suppose even that these cleavages come first and that the fibration is just a  conceptual context around them. Even in this case, faced to the need to produce a proof, if you succeed to find one without utilizing the cleavages, you will have something much nicer than the cleavage arguing. It will also give you a deeper understanding and a truthful light on the situation.
> 
> Suppose you do not care about foundations, axiom of choice, or things of that sort. You should still prefer fibrations. They are simpler, more to the point, and CERTAINLY AHEAD IN THE PROGRESS OF MATHEMATICS.
> 
> 
> NOTE: I wonder why so many people are so happy working with pull-backs and pull-back preserving functors (*) without even thinking in introducing a choice of pullbacks, and when it comes to fibrations, feel the need to introduce and work with cleavages.
> 
> (*) for example even when dealing with the category of sets (or categories whose objects have an underlying set), which are plenty of choices of pull-backs, for example, inverse image of a subset, the standard construction as a subset of the set of pairs, etc. We precisely teach in category theory courses that you should not work with any particular choice between the choices.
> 
> We all agree that it is neither necessary not good to choose a choice between all possible choices. This is precisely the progress that represents category theory thinking over set theory thinking. See for example, in the dawn of category theory, the considerations of Mac Lane concerning the fact that a quotient of a quotient of a group is not a quotient (as it is still now taught in algebra courses, category theory thinking has not arrived there yet).
> 
> You may say that the choice of a cleavage is at a different level than all this, but, essentially, deep down, for me it is the same. There is an old way of thinking (as Grothendieck said, SLN224, page 193) that hesitates in face of fibrations and prefer to work with a chosen cleavage.
> 
> 
> 

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