From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7813 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories,gmane.spam.detected Subject: Reply to Eduardo Dubuc Date: Fri, 26 Jul 2013 04:55:44 -0400 Message-ID: References: Reply-To: Marta Bunge NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1374861993 5315 80.91.229.3 (26 Jul 2013 18:06:33 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 26 Jul 2013 18:06:33 +0000 (UTC) To: , E D Original-X-From: majordomo@mlist.mta.ca Fri Jul 26 20:06:35 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 1V2mPR-0004P0-9z for gsmc-categories@m.gmane.org; Fri, 26 Jul 2013 20:06:33 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54322) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1V2mOQ-0003tx-Sd; Fri, 26 Jul 2013 15:05:30 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1V2mOQ-0007lV-Km for categories-list@mlist.mta.ca; Fri, 26 Jul 2013 15:05:30 -0300 Precedence: bulk X-Spam-Report: 7.3 points; * 1.9 DATE_IN_PAST_06_12 Date: is 6 to 12 hours before Received: date * 2.5 LOCALPART_IN_SUBJECT Local part of To: address appears in Subject * 1.1 FORGED_HOTMAIL_RCVD2 hotmail.com 'From' address, but no 'Received:' * 1.8 MIME_QP_LONG_LINE RAW: Quoted-printable line longer than 76 chars Xref: news.gmane.org gmane.science.mathematics.categories:7813 gmane.spam.detected:5073978 Archived-At: > Dear Eduardo,=20 >=20 > Thank you for keeping me au courant. Difficult as it is to write in my iPa= d, I want to tell you my position and what I think of your question. >=20 > My position (as that of any topos theorist) is that >=20 > 1) If you work in an elementary topos SS, then it is true that you can do s= o "as in Sets", provided you do not use excluded middle ( as the internal lo= gic is intuitionistic) or use Choice (as the set theory intrinsic to SS is c= onstructive). >=20 > 2) If on the other hand you work in a topos EE that is bounded over SS, th= en you cannot work "as in Sets" concerning constructions in EE that involve S= S. Here it is necessary to resort to either the "teoria delle categorie sopr= a un topos di base" (indexed categories) or to the theory of fibrations.=20 >=20 > Concerning your question, let me point out that, although you pretend to w= ork entirely inside SS, you do not, as your construction involves EE. It cer= tainly makes sense, since EE is locally small, and CC is small ( meaning int= ernal to SS). What I want to warn you about is how you proceed from there wi= thout resorting to EE as a bounded SS- topos.=20 > I apologize for previous (private) irrelevant remarks as I misunderstood y= our question - until now, that is. >=20 > Concerning Diaconescu's theorem, AC implies Booleaness can be done "as in S= ets" but the proof is not trival. If you mean the theorem which gives a char= acterization of bounded SS-toposes as a classifying topos, it requires fibra= tions over SS.=20 >=20 > Regards, > Marta >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]