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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hello,
The authors are Verdier and Grothendieck, I doubt they made mistakes, and specially in the very basic definitions of the whole theory.
This made me smile, but I agree!
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.
I think the explanation is rather simple. The first point is that the foundations here are Bourbaki set theory: everything is a set, including a category. When they say that C belongs to U, they literally mean that C, when rigorously formalised as a set, is an element of U. The same goes for a functor. The second point is that U-smallness of a set X is defined to be a set which is isomorphic to an element of U, not necessarily an actual element of U. The word 'isomorphic' is underlined. Let us consider (C2) first. The axioms of a universe only allow one to construct sets in the universe from other sets in the universe. Since U is not in U, there is no way that any definition/construction involving it can produce a set in U. Certainly one needs to involve U to be able to define the Hom sets of Func(C, U-Ens). But the Hom sets of Func(C, U-Ens) are isomorphic to elements of U, i.e. are U-small. This is because U-Ens is a U-category, i.e. the Hom sets are U-small. One can make the same point about the set of objects of Func(C, U-Ens). One needs U to define it, so it is certainly not an element of U. It is not even isomorphic to an element of U, because this would imply that the cardinality of U is strictly less than U. In (C1), it is not exactly this that is asked, but rather that the set of objects of Func(C, U-Ens) is a subset of U. This is impossible for the same kind of reasons: one will need to consider for instance a subset of the set C x U to define it, and there is no way that show that this is a subset of U (it is perfectly possible if instead of U we have some element u of U, because then the product of C and u belongs to U). Hence Func(C, U-Ens) is, as SGA claims, the prototypical example to illustrate why the definition of a U-category is exactly the way it is. Best wishes, Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
-
Eduardo Julio Dubuc -
Richard Williamson