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.?? Here is my proof; please would you point out where I'm going wrong?
Firstly: C is in U, so ob C and every object, every homset and every morphism of C are in U, by transitivity of U. For (C1), we must show that any functor F from C to U-Ens is in U.?? Any such F is an ordered pair (ob F, mor F).?? - ob F is a set of ordered pairs (c,x) where c is a C-object and x is in U. Such an ordered pair is in U.?? So ob F is a subset of U and its cardinality is that of ob C so ob F is in U (Proposition 7 in the Appendix of SGA4, p98). - mor F is a set of triples (c,d,p) where c and d are C-objects and p is a map from C(c,d) to Fc->Fd hence a subset of C(c,d) * (Fc -> Fd).?? And Fc and Fd are in U, so Fc -> Fd is too by the Corollary to Proposition 6 (on p98).?? So C(c,d) * (Fc -> Fd) is in U, so p is in U, so (c,d,p) is in U.?? So mor F is a subset of U, and its cardinality is that of (ob C)*(ob C) which is in U.?? By Proposition 7, mor F is in U. In conclusion F = (ob F, mor F) is in U. For (C2), let F and G be functors from C to U-Ens.?? The set of natural transformations F -> G is a subset of Prod_{c in ob C} (Fc --> Gc).?? For any c in C, we know that Fc and Gc are in U, so Fc -> Gc is in U.?? So by the Corollary to Proposition 6, Prod_{c in ob C} (Fc -> Gc) is in U, so the set of natural transformations F -> G is in U.?? Best regards, Paul
C???est ce qui justifie la d??finition adopt??e de U-cat??gorie, de pr??f??rence ?? la notion plus restrictive par les conditions (C1) et (C2) ci-dessus.
best?? e.d.
El 1/1/18 a las 10:10, Paul Blain Levy escribi??:
Hi,
Let V be a Grothendieck universe.?? A "V-set" is an element of V, and a "V-class" is a subset of V.
Say that a category C is "V-included" when it has the following two properties.
(1) ob C is a V-class.
(2) C(x,y) is a V-set for all x,y in ob C.
The advantage of V-inclusion over local V-smallness (i.e. condition (2) alone) is that V-included categories are W-small for every universe W greater than V, whereas locally V-small categories are not, in general.
Furthermore, all the standard categories constructed from V are V-included.?? (Except for the ones that are not even locally V-small, like the category of V-included categories.)
Is there a standard name for V-inclusion?
Paul
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