From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6788 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: RE: stacks (was: size_question_encore) Date: Thu, 14 Jul 2011 23:51:46 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1310722972 5501 80.91.229.12 (15 Jul 2011 09:42:52 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 15 Jul 2011 09:42:52 +0000 (UTC) Cc: categories@mta.ca To: marta.bunge@mcgill.ca Original-X-From: majordomo@mlist.mta.ca Fri Jul 15 11:42:44 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Qheuy-0001Bp-3F for gsmc-categories@m.gmane.org; Fri, 15 Jul 2011 11:42:44 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45217) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1QherK-0000et-LQ; Fri, 15 Jul 2011 06:38:58 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QherJ-0006Xa-Or for categories-list@mlist.mta.ca; Fri, 15 Jul 2011 06:38:57 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6788 Archived-At: On Wed, Jul 13, 2011 at 2:16 AM, Marta Bunge wrote= : > Bob Pare and I wanted to posit something like the existence of a "represe= ntative (or universal) cover" as an axiom (RC), such that GT =3D> RC =3D> A= SC, and also (ET + AC) =3D> RC. We hoped to show that RC had good propertie= s, and more importantly, that Eff satisfied it but this did not work. That's a very interesting question! There seem to be a lot of axioms of this flavor, which say in various different ways that AC fails "in only a small way". Some that I am aware of include: * Small Violations of Choice (Blass): http://nlab.mathforge.org/nlab/show/small+violations+of+choice * Small Cardinality Selection (Makkai): http://nlab.mathforge.org/nlab/show/small+cardinality+selection+axiom * Axiom of Multiple Choice (Moerdijk & Palmgren): http://nlab.mathforge.org/nlab/show/axiom+of+multiple+choice * Weakly Initial Sets of Covers (Roberts): http://nlab.mathforge.org/nlab/show/WISC * The ex/lex completion of Set (or the topos in question) is well-powered * The ex/lex completion of Set is a topos, i.e. Set has a generic proof Some of these hold in any Grothendieck topos, but others apparently need not, and I have no idea which of them might hold in Eff. It would be interesting to know if any of them imply ASC (or Andr=E9's proposed strengthening thereof). Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]