From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7808 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: disjoint_coproducts_? Date: Tue, 23 Jul 2013 19:45:02 -0300 Message-ID: Reply-To: "Eduardo J. Dubuc" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1374661334 27461 80.91.229.3 (24 Jul 2013 10:22:14 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 24 Jul 2013 10:22:14 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Wed Jul 24 12:22:16 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 1V1wD1-0005iN-0v for gsmc-categories@m.gmane.org; Wed, 24 Jul 2013 12:22:15 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:53180) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1V1wB4-0000sB-1w; Wed, 24 Jul 2013 07:20:14 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1V1wB2-0006lJ-Sa for categories-list@mlist.mta.ca; Wed, 24 Jul 2013 07:20:12 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7808 Archived-At: Hello, I have the following question: 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. My question is: Can I do the following ? (meaning, is it correct the following arguing, certainly valid if SS is the topos of sets): Given a in FX, take f:C ---> X and c in FC such that a = Ff(c). We can break this question in two: 1) Does it make sense to take E = COPRODUCT_{all f: C ---> X, all C in CC} FC ? We have g: E ---> FX an epimorphism, so we can take c in E such that a = g(c). Then we would need the validity of: 2) Given x in COPRODUCT_{i in I} S_i , then x in S_i for some i in I. greetings e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]