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). 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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