From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6777 Path: news.gmane.org!not-for-mail From: Eduardo Dubuc Newsgroups: gmane.science.mathematics.categories Subject: RE: stacks (was: size_question_encore) Date: Tue, 12 Jul 2011 16:12:17 -0300 Message-ID: References: Reply-To: Eduardo Dubuc NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1310579830 2092 80.91.229.12 (13 Jul 2011 17:57:10 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 13 Jul 2011 17:57:10 +0000 (UTC) Cc: mshulman@ucsd.edu, martabunge@hotmail.com, categories , david.roberts@adelaide.edu.au To: =?ISO-8859-1?Q?Andr=E9_Joyal?= Original-X-From: majordomo@mlist.mta.ca Wed Jul 13 19:57:03 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 1Qh3gA-00066n-QS for gsmc-categories@m.gmane.org; Wed, 13 Jul 2011 19:56:59 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49431) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1Qh3dW-0002pN-DO; Wed, 13 Jul 2011 14:54:14 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Qh3dV-0005m8-MV for categories-list@mlist.mta.ca; Wed, 13 Jul 2011 14:54:13 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6777 Archived-At: Dear all: If we take the filtered poset of finite subsets of I, then taking the=20 finite products of C(2) gives an strict pro group, and this is the=20 example of a "Faux topos", SGA4 SLN 269 page 322. Now, this example is exhibited as what we some times call Giraud topos=20 (all exactness properties but without generators) My question is (answer probably well known to the experts): Are Giraud topoi elementary topoi (that is, do they have an Omega and=20 exponentials) ? greetings e.d. On 07/12/2011 12:04 PM, Andr=E9 Joyal wrote: > Dear Michael, > > You wrote: > > > Are there known examples of elementary toposes which violate the > 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 infini= te > 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 = 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]