From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6774 Path: news.gmane.org!not-for-mail From: =?ISO-8859-1?Q?Andr=E9_Joyal?= Newsgroups: gmane.science.mathematics.categories Subject: Re: RE: stacks (was: size_question_encore) Date: Tue, 12 Jul 2011 11:04:39 -0400 Message-ID: Reply-To: =?ISO-8859-1?Q?Andr=E9_Joyal?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=ISO-8859-1; format=flowed; delsp=yes Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1310495783 21825 80.91.229.12 (12 Jul 2011 18:36:23 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 12 Jul 2011 18:36:23 +0000 (UTC) Cc: martabunge@hotmail.com, categories , david.roberts@adelaide.edu.au To: mshulman@ucsd.edu Original-X-From: majordomo@mlist.mta.ca Tue Jul 12 20:36:19 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 1Qghof-0005p7-GD for gsmc-categories@m.gmane.org; Tue, 12 Jul 2011 20:36:17 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:35630) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1Qghm5-0006Mk-9P; Tue, 12 Jul 2011 15:33:37 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Qghm4-0001Xr-E4 for categories-list@mlist.mta.ca; Tue, 12 Jul 2011 15:33:36 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6774 Archived-At: Dear Michael, You wrote: > Are there known examples of elementary toposes which violate the =20 axiomof stack completions? Here is my favorite example. Let C(2) be the cyclic group of order 2. It suffices to construct a topos E for which the cardinality of set of isomorphism classes of C(2)-torsor is larger than the cardinality of the set of global sections of any object of E. Let G=3DC(2)^I be the product of I copies of C(2), where I is an =20 infinite set. The group G is compact totally disconnected. Let me denote the topos of continuous G-sets by BG. There is then a canonical bijection between the following three sets 1) the set of isomorphism classes of C(2)-torsors in BG 2) the set of isomorphism classes of geometric morphisms BC(2)--->BG 3) the set of continuous homomomorphisms G-->C(2). Each projection G-->C(2) is a continuous homomomorphism. Hence the cardinality of set of isomorphism classes of C(2)-torsors in =20= BG must be as large as the cardinality of I. The topos E=3DBG is thus an example when I is a proper class. For those who dont like proper classes, we may and take for E the topos of continuous G-sets in a Grothendieck universe and I to be a set larger than this universe. Best, Andre -------- Message d'origine-------- De: viritrilbia@gmail.com de la part de Michael Shulman Date: lun. 11/07/2011 21:20 =C0: Marta Bunge Cc: david.roberts@adelaide.edu.au; Joyal, Andr=E9; categories@mta.ca Objet : Re: categories: RE: stacks (was: size_question_encore) Is the "axiom of stack completions" related to the "axiom of small cardinality selection" used by Makkai to prove that the bicategory of anafunctors is cartesian closed? I think I recall a remark in Makkai's paper to the effect that the stack completion of a category C is at least morally the same as the category Ana(1,C) of "ana-objects" of C. Are there known examples of elementary toposes which violate the axiom of stack completions? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]