### A remark about Street fibrations

Jean Bénabou <jean.benabou <at> wanadoo.fr>

2014-07-23 05:42:41 GMT

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In order to stay simple I shall consider only the case where the 2-category with comma objects and
2-pullbacks is Cat, and I shall even accept the axiom of universes and AC.
In that case, but only in that case, the Grothendieck construction makes sense and the theorem, proved by
Grothendieck himself, stating that indexed categories and fibered ones are 2-equivalent is, of course, correct.
Nevertheless, as Dubuc points out, Grothendieck discarded indexed categories for fibrations, and
neither he nor his school ever used indexed categories.
For ages, I have tried to convince people, in particular on this mailing list, that indexed categories
ought to be abandoned, not only because of the authority of Grothendieck, but for mathematical reasons
which I explained, and I wrote the paper in the Journal of Symblic Logic (JSL) to explain my views.
This had no effect, and sometimes got me a lot of abuse. In particular the JSL paper was qualified as a
pamphlet, and I still consider it as the deepest paper I ever wrote. That is why I dedicated it to Grothendieck.
I'm glad to see that in the discussion, only fibrations were considered, and such notions as fibrations
with internal sums and products, which I introduced, were used.
So far, I have had no reaction from Bill Lawvere who introduced indexed categories, and whose influence
determined the wrong choice of indexed versus fibered made by many people.
More surprisingly, no reaction either from Peter Johnstone who made in the Elephant a complete mess of the
whole question. I hope he will give his opinion.
I would like to thank Ross Street for his answer, and make a few comments on his notion of non evil fibration to
answer a question asked by Steve Vickers which I quote:
But that seems to claim that in Cat it doesn't matter whether you use iso or equality in the Chevalley
condition. Does that accord with your understanding?

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