Re: V-included categories

[Paul Blain Levy <P.B.Levy@cs.bham.ac.uk>]

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




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