On 1/3/18 03:44, Paul Blain Levy wrote:
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.
you should do this
:-) Alternatively: the authors just made a mistake.
The authors are Verdier and Grothendieck, I doubt they made mistakes, and specially in the very basic definitions of the whole theory. There is something odd here but I am not inclined to take time to clarify it, I will sit and wait to see if some in the list come out with an explanation. best Eduardo.
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
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