On 04/01/18 08:39, Thomas Streicher wrote:
I think what Grothendieck and Verdier could have meant is that Fonc(C,U) doesn't satisfy (C1) and (C2) if C is just a U-category.
For this Ob(C) has to be an element of U as well I guess that's what they call U-petit which is more restrictive than just a U-category.
Thomas
Already on this. I made some further checking. Kashiwara-Schapira take exactly SGA4 definition of universe U, and as in SGA4 define "X is a U-set if X belongs to U", and "X is U-small if it is bijective to a U-set", and define U-category as a category such that hom(x, y) is small (definitions 1.1.2 and 1.2.1) But it seems they do not read more than the very first definitions in SGA4, since under the pretense of working rigorously they soon become happily naive in 1.4.2 where they use that for a U-category, hom(x, y) is a U-set (in order to to have the representable functors). Mac Lane on the other hand differs from SGA4, he defines X to be U-small if it belongs to U (thus, for him U-set and U-small are the same). He emphasizes that very small sets are not U-small giving the example X = {U}, which is not U-small in his sense (although being a singleton) but it is clearly U-small in the sense of SGA4 (or Kashiwara-Schapira). The question is: Why SGA4 takes the trouble of carefully distinguish the two notions, U-set and U-small, while Mac Lane does not and seems get away with this. Obviously MacLane does as people working in naive category theory with universes, namely, the working mathematicians :=) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]