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). Now, it is necessary to see what exactly means "Fonct(C,U-Ens)" and/or "appartenant ?? U" in SGA4.
Here is my proof; please would you point outwhere I'm going wrong?
I imagine your carefully proof must be valid, anyway it is clear that under the interpretation you give to "Fonct(C,U-Ens)" and "appartenant ?? U" U-Ens^C satisfy (C1) and (C2) best Eduardo.
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
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