From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7809 Path: news.gmane.org!not-for-mail From: "Olivia Caramello" Newsgroups: gmane.science.mathematics.categories Subject: R: disjoint_coproducts_? Date: Wed, 24 Jul 2013 13:04:41 +0200 Message-ID: References: Reply-To: "Olivia Caramello" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1374750507 7950 80.91.229.3 (25 Jul 2013 11:08:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 25 Jul 2013 11:08:27 +0000 (UTC) To: "'Eduardo J. Dubuc'" , "'Categories list'" Original-X-From: majordomo@mlist.mta.ca Thu Jul 25 13:08:29 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1V2JPJ-0001mn-EL for gsmc-categories@m.gmane.org; Thu, 25 Jul 2013 13:08:29 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:53652) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1V2JNO-0005Y6-J3; Thu, 25 Jul 2013 08:06:30 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1V2JNN-0002fR-3o for categories-list@mlist.mta.ca; Thu, 25 Jul 2013 08:06:29 -0300 In-Reply-To: Content-Language: it Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7809 Archived-At: Dear Eduardo, Your statement is valid internally in SS, that is once formalized in the internal language of the topos SS; this can be done in geometric logic, = by considering a (possibly infinitary) disjunction over all the arrows f: C ---> X for C in CC (interpreted by the arrows Ff in SS) and existential quantifications. If you want a statement valid 'externally', you should instead use generalized elements in SS and epimorphic families involving their domains. I hope this helps. Best regards, Olivia =20 > -----Messaggio originale----- > Da: Eduardo J. Dubuc [mailto:edubuc@dm.uba.ar] > Inviato: mercoled=EC 24 luglio 2013 00:45 > A: Categories list > Oggetto: categories: disjoint_coproducts_? >=20 > Hello, I have the following question: >=20 > Assume a topos SS as the base topos, and work in this topos as in = naive set > theory (without choice or excluded middle). Take a Grothendieck topos = EE --- >> SS with a site of definition CC. As usual in the literature (Joyal-Tierney, > Moerdijk, Bunge, and many more) consider that CC has objects, and that > these objects are objects of EE which are generators in the sense = that given > any X in EE, the family of all > f: C ---> X, all C in CC, is epimorphic. Consider F: CC ---> SS to be = the inverse > image of a point. Then the family Ff: FC ---> FX is epimorphic in SS. >=20 > My question is: >=20 > Can I do the following ? (meaning, is it correct the following = arguing, certainly > valid if SS is the topos of sets): >=20 > Given a in FX, take f:C ---> X and c in FC such that a =3D Ff(c). >=20 > We can break this question in two: >=20 > 1) Does it make sense to take >=20 > E =3D COPRODUCT_{all f: C ---> X, all C in CC} FC ? >=20 > We have g: E ---> FX an epimorphism, so we can take c in E such that a = =3D g(c). >=20 > Then we would need the validity of: >=20 > 2) Given x in COPRODUCT_{i in I} S_i , then x in S_i for some i in I. >=20 > greetings e.d. >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]