On 07/05/13 13:11, Jean B?nabou wrote:

For more than forty years now I have struggled to convince the category community that fibered categories are much better than indexed ones, and that the latter ought to be briefly described for the sake of completeness, but immediately discarded.

Pare and Scumacher (SLN 661) among other considerations to justify their choice of indexed categories (= cloven fibrations) over fibrations say: "We have tried to make our theory conform as closely as possible to actual mathematical practice" Grothendiek (SGA 1, SLN 224, http://arxiv.org/abs/math/0206203) among other considerations to justify his choice of fibrations over cloven fibrations (= indexed categories) say: "Il est d?ailleurs probable que, contrairement a l?usage encore preponderant maintenant, lie a d?anciennes habitudes de pensee, il finira par s?averer plus commode dans les problemes universels, de ne pas mettre l?accent sur une solution supposee choisie une fois pour toutes, mais de mettre toutes les solutions sur un pied d?egalite" "actual mathematical practice" = "l?usage encore preponderant maintenant" Grothendieck ads "lie a d'anciennes habitudes de pensee" Interesting enough, categorical thinking is hard to swallow. e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]