Re: Composition of Fibrations

["Eduardo J. Dubuc" <edubuc@dm.uba.ar>]

I always have present that Grothendieck himself explicitly discarded his
notion of "cat?gorie cliv?e" (ie indexed category) introduced in the
S?minaire Bourbaki (1959) in favor of his notion of "cat?gorie fibr?e"
introduced in SGA1 (1961) (see SLN Vol 224 remark page 193).


On 20/07/14 13:18, Jean B?nabou 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.
>
> (II)     2-CTEGORICAL FIBRATIONS
> Other definitions of fibrations in an arbitrary 2-Category C have been proposed. The principal one, based on Yoneda, is:  A one cell  p: B -> A of  C  is a fibration iff for every object X of C the obvious functor  C(X,B) -> C(X,A) is a fibration in Cat, functorial in X.
> If C has comma objects and 2-pullbacks, it is easy to see that this is equivalent to Vickers' notion, and we have already seen how it can be inadequate.
> Of course, I don't refer here to Sreet's notion which describes a totally different kind of fibration, stable by equivalences.
>
> (III)    BIFIBRATIONS
> For bifibrations the situation is even more confusing: Ghani defines them, in Cat, by the existence of left adjoints to the reindexing functors, Except that without AC reindexing functors need not exist. Vickers uses two duals of the 2-category C where the fibration lives. However if C has comma objects and 2-pullbacks, there is no reason why these duals have the same properties. Moreover, even in Cat, Vickers' approach will work only for bifibrations which have both a cleavage and a co-cleavage.
> Thus the wide generalization asserted by Vickers imposes in the well known situations drastic and unnecessary restrictions. (Compare with the example at the end of (I))
>

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