From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6789 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: RE: stacks (was: size_question_encore) Date: Fri, 15 Jul 2011 06:27:43 -0400 Message-ID: Reply-To: Marta Bunge 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 1310824735 12943 80.91.229.12 (16 Jul 2011 13:58:55 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 16 Jul 2011 13:58:55 +0000 (UTC) Cc: To: Mike Shulman , Marta Bunge Original-X-From: majordomo@mlist.mta.ca Sat Jul 16 15:58:49 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 1Qi5OK-0002yu-Br for gsmc-categories@m.gmane.org; Sat, 16 Jul 2011 15:58:48 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54189) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1Qi5Lm-0002A5-RW; Sat, 16 Jul 2011 10:56:10 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Qi5Lm-00075V-5R for categories-list@mlist.mta.ca; Sat, 16 Jul 2011 10:56:10 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6789 Archived-At: Dear Mike=2C Thank you for the information. It would take me a while to check them all. = "Weakly initial covers" sounds a bit like "RC" or like one of its variatio= ns.=20 All the best=2C Marta > Date: Thu=2C 14 Jul 2011 23:51:46 -0700 > Subject: Re: categories: RE: stacks (was: size_question_encore) > From: mshulman@ucsd.edu > To: marta.bunge@mcgill.ca > CC: categories@mta.ca >=20 > On Wed=2C Jul 13=2C 2011 at 2:16 AM=2C Marta Bunge wrote: >> Bob Pare and I wanted to posit something like the existence of a "repre= sentative (or universal) cover" as an axiom (RC)=2C such that GT =3D> RC = =3D> ASC=2C and also (ET + AC) =3D> RC. We hoped to show that RC had good p= roperties=2C and more importantly=2C that Eff satisfied it but this did not= work. >=20 > That's a very interesting question! There seem to be a lot of axioms > of this flavor=2C which say in various different ways that AC fails "in > only a small way". Some that I am aware of include: >=20 > * 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=2C i.e. Set has a generic proof >=20 > Some of these hold in any Grothendieck topos=2C but others apparently > need not=2C 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). >=20 > Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]