On 03/01/18 00:20, edubuc@dm.uba.ar wrote:
On 1/1/18 19:46, Paul Blain Levy wrote:
Dear Eduardo,
On 01/01/18 21:14, edubuc@dm.uba.ar wrote:
Have you considered the following at the very begining of SGA4:
Remarque 1.1.2. ??? Soit D une cat??gorie poss??dant les propri??t??s suivantes : (C1) L???ensemble ob(D) est contenu dans l???univers U . (C2) Pour tout couple (x, y) d???objets de D, l???ensemble HomD(x, y) est un ??l??ment de U .
(Les cat??gories usuelles construites ?? partir d???un univers U poss??dent ces deux propri??t??s: U -Ens, U -Ab,. . .).
Soit C une cat??gorie appartenant ?? U . Alors la cat??gorie Fonct(C, D) ne poss??de pas en g??n??ral les propri??t??s (C1) et (C2). Par exemple la cat??gorie Fonct(C,U-Ens) ne poss??de aucune des propri??t??s (C1) et (C2). Thanks for your reply but I am mystified by this statement in SGA4.?? It appears to me Fonct(C,U-Ens) does satisfy both (C1) and (C2), so I must be missing something.
It is clear that U-Ens^C satisfy (C1) and (C2) (see the practice of category theory by any mathematician). OK, I mistakenly assumed you were endorsing the statement you quoted.?? Sorry for boring you with this obvious proof. Now, it is necessary to see what exactly means "Fonct(C,U-Ens)" and/or?? "appartenant ?? U" in SGA4. :-)???? Alternatively: the authors just made a mistake.?? And evidently, had they not made this mistake, they would have defined "U-category" by (C1)--(C2), since they regard these conditions as a priori natural.?? That's good to see.
Paul [For admin and other information see: http://www.mta.ca/~cat-dist/ ]