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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 01/01/18 13:10, Paul Blain Levy wrote:
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. Another advantage: the category of V-included categories is W-small.
Paul
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
But this is the usual definition of a locally small category in V - objects form a class (more precisely, are equipotent to a class) and hom-collections are sets. Jiri Rosicky Dne 2018-01-01 14:10, Paul Blain Levy napsal:
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I believe that in his paper "Notions of topos" Ross Street used the name "V-moderate category" for this or a closely related notion. There the point was that V-moderate categories have another advantage over locally V-small ones, namely that their objects can (assuming the axiom of choice) be well-ordered with all initial segments being V-small. On Mon, Jan 1, 2018 at 5:10 AM, Paul Blain Levy <P.B.Levy@cs.bham.ac.uk> wrote:
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
-
Eduardo Julio Dubuc -
Michael Shulman -
Paul Blain Levy -
rosicky