From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6791 Path: news.gmane.org!not-for-mail From: Eduardo Dubuc Newsgroups: gmane.science.mathematics.categories Subject: RE: stacks (was: size_question_encore) Date: Fri, 15 Jul 2011 16:01:24 -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: 7bit X-Trace: dough.gmane.org 1310824823 13488 80.91.229.12 (16 Jul 2011 14:00:23 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 16 Jul 2011 14:00:23 +0000 (UTC) Cc: categories To: unlisted-recipients:; (no To-header on input) Original-X-From: majordomo@mlist.mta.ca Sat Jul 16 16:00:16 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 1Qi5Pg-0003dB-UL for gsmc-categories@m.gmane.org; Sat, 16 Jul 2011 16:00:13 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58073) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1Qi5Nk-0002MD-8K; Sat, 16 Jul 2011 10:58:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Qi5Nj-0007AM-HG for categories-list@mlist.mta.ca; Sat, 16 Jul 2011 10:58:11 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6791 Archived-At: Is interesting to compare Joyal and Johnstone examples of Giraud faux topos, which are very similar and very different at the same time. Take a proper class (large set) I, and let A = N (natural numbers) or A = Z/2Z (cyclic group of order 2) Let M = A^(I) = {f | f(i) = 0 except for finitely many i, A = N }. or M = A^I = { all f, A = Z/2Z }. For any (small) subset K c I, let M_K = A^(K) or A^K respectively. We have a continuous surjective morphism M ---> M_K. M_K is a (small) set with a continuous action of M. Let E = \beta M. E is a Giraud topos. M_K \in E. (A = N is Johnstone example, A = Z/2Z is Joyal's) In Johnstone case M_K has at least K different subobjects, used to disprove that E is an elementary topos (can not have a subobject classifier). In Joyal case, all these subobjects dissappear, M_K is connected, and does not have any non trivial subobjects. Joyal knows that in this case (i have not tried to prove it) E has a subobject classifier, and it is an elementary topos. However we still have all the M ---> M_K, in particular one for each singleton {i} c I, used to disprove the axiom of stalk completion. **************** For any K, M_K is a monoid (group in Joyal case), and for K c J, a continuous surjective morphism M_J ---> M_K, thus a large strict promonoid (or progroup), and we have a large pro-object of Grothendieck topoi E_K = \beta M_K and connected transition morphisms. It is known (Kennison, Tierney, Moerdijk) that in this case \beta commutes with the inverse limits, denoted lM, lE respectively, \beta lM = lE (provided the KTM result holds for large limits). We have a cones M ---> M_K and E ---> E_K, which determine arrows M ---> lM, E ---> lE. In Joyal case both arrows are isomorphisms (seems easy for M, then by KTM it follows for E), which may be behind the fact that E = lE is an elementary topos. The category lE may not even have small hom sets in Johnstone case. **************** Let now MA, EA be Joyal's case, and MP, EP be Johnstone case. We have dense continous morphisms MP ---> MA and MP_K ---> MA_K which determine arrows: EP ---> EA (as categories EA Full ---> EP) EP_K ---> EA_K (as categories EA_K Full ---> EP_K) lEP ---> lEA (as categories lEA Full ---> lEP) The indexed by K are Kosher, the others are False, some more false than others. *************** [For admin and other information see: http://www.mta.ca/~cat-dist/ ]