Re: V-included categories

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

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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