Dear Ross, I agree with you on all points: Pseudo-fibrations is much better than weak fibrations, it describes more precisely the notion defined. I hope it will be adopted by the category-commmunity. I am also convinced that these pseudo fibrations have a mathematical role. The example I gave has two interests: (i) To show that calling them fibrations would violate all our intuitions about the idea of fibration. (ii) to provide a new, important, and perhaps not known example of such pseudo fibrations. Of course what makes things work in that example is that, if G is a groupoid, for cospans with codomain G comma objects and pseudo pullbacks coincide. This is also true in any 2-Category C, if you define a groupoid of C to be an object G such that for each object X of C, the category C(X,G) is a groupoid. Best wishes, Jean Le 24 juil. 2014 à 00:03, Ross Street a écrit :
On 23 Jul 2014, at 3:42 pm, Jean Bénabou <jean.benabou@wanadoo.fr> wrote:
In view of this example I suggest that the name of fibrations should be used exclusively for Grothendieck fibrations, the usual ones or their internalizations along the lines I described, and another name, e.g. weak fibrations, be given to the notion defined by Street. Woud you agree with this, Ross ?
Dear Jean
Yes, I do agree that ``weak fibration'' or ``pseudo fibration’’ would be good terminology. The ``pseudo-fibres” rather than the strict fibres are the relevant concept.
There is no doubt that Grothendieck fibrations are very important. But the pseudo-fibrations do have a mathematical role, as outlined is a message sent to ``categories’’ (by a group of us) a year or two ago.
Best wishes, Ross
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