Re: Composition of Fibrations

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Dear Jean,

Thank you for your detailed comments.

Something I should say straight away is that the duality argument I had in mind, dualizing 2-cells, might be OK to deal with left adjoints to reindexing  but was completely wrong for right adjoints. Already, Richard Garner and Claudio Ermida (thanks to both of them) have shown me that it doesn't do the job.

I also want to stress that at no point did I intend to set up my own definition of fibration. I was following Street's "Fibrations and Yoneda's lemma in  a 2-category", which defines fibrations as those 1-cells that carry pseudoalgebra structure for a certain 2-monad, and then proves (Proposition 9) that  this is equivalent to what Street refers to as the Chevalley condition. If "Vickers' definition" is not equivalent to that then I have made a mistake somewhere. 

Have you found a discrepancy between the "Vickers definition" and Street? At  one point you write "Of course, I don't refer here to Street's notion which  describes a totally different kind of fibration, stable by equivalences."

I agree that the concept I have been using includes cleavage (and, for a bifibration, cocleavage). I cannot assume AC in what I do, and I rather imagined that structure something like the Chevalley criterion was needed in order to deal with its absence. However, I admit I am not so familiar with the fully general notion of fibration. For me the Chevalley condition seemed enough to do what I needed in the 2-category Loc of locales and my remarks were based on that experience.

Best wishes,

Steve.



> On 20 Jul 2014, at 17:18, Jean Bénabou <jean.benabou@wanadoo.fr> wrote:
> 
> A few weeks ago there has been a discussion about stability by composition  of fibrations, bifibrations, and similar notions. Obviously the results depend on how such notions are defined.  I would like to make a few comments, in particular about Steve Vickers' mail, since all the other participants to the discussion seemed to accept his approach.
> 
> (I )    VICKERS' DEFINITION OF FIBRATION
> if C is a 2-category with comma objects and 2-pullbacks, a one cell p: B -> A is a fibration iff it satisfies the Chevalley condition.
> 
> Let us test this definition in special cases.
> If  S is a category with finite limits the 2-category Cat(S) of internal categories in S satisfies Vickers' conditions hence we know when an internal functor is a fibration. 
> Let Set be the category of sets, except WE DON'T ASSUME THE AXIOM OF CHOICE (AC). Then Cat(Set), abbreviated by Cat, is the 2-Category of small categories.
> An easy verification shows that a functor p: B -> A  satisfies the Chevalley condition iff it is a fibration which admits a cleavage. Thus Vickers' argument, in that case, gives as result: fibrations WHICH ADMIT A CLEAVAGE are  stable by composition. 
> On the other hand, it is easy to show that: Every small fibration has a cleavage is equivalent to AC. This well known fact can be very much strengthened by the following example:
> 
> If AC does not hold in Set, one can construct in Cat a bifibration  p: B -> A  with internal products and coproducts where A and B are pre-ordered sets, with pullbacks preserved by p, every map of B is both cartesian and cocartesian, and add each of the following conditions:
> (i)  p has neither a cleavage nor a cocleavage. 
> (ii) A bit surprisingly:   p is a split fibration but has no cocleavage.
> (iii) Dual of (ii):   p is a cosplit cofibration but has no cleavage.
> 
> And of course we don't need AC to show that arbitrary fibrations in Cat are stable by composition.
> 

...


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]