From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6547 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: RE: Subobject Classifier Algorithm Date: Thu, 24 Feb 2011 12:11:29 -0500 Message-ID: References: Reply-To: "F. William Lawvere" 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 1298606620 16571 80.91.229.12 (25 Feb 2011 04:03:40 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 25 Feb 2011 04:03:40 +0000 (UTC) To: , categories Original-X-From: majordomo@mlist.mta.ca Fri Feb 25 05:03:36 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Psoty-0007ah-CO for gsmc-categories@m.gmane.org; Fri, 25 Feb 2011 05:03:34 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:37841) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Psoti-0001Pq-AF; Fri, 25 Feb 2011 00:03:18 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Psotd-00035b-Fx for categories-list@mlist.mta.ca; Fri, 25 Feb 2011 00:03:13 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6547 Archived-At: For sheaves on a finite site (which by a theorem of AGV in SGA4 vol 2 is th= e same as all presheaves on some smaller finite category C)=2C take the cat= egory of all functors from C^op to bold 2. It is a finite poset=2C in fact= =2C a Heyting algebra (indeed even bi-Heyting) belying the old misconcepti= on that one deviates from Boole only for infinite sets. If for each given A= in C we do the same for C/A=2C we get the figures of shape A in the Omega = of the topos. The adjoints to maps induced by A'->A give a concrete model o= f tense logic. By the same AGV (not only these C/A' ->C/A but) any functor between finite = categories induces a geometric morphism that is even "essential". Actually=2C taking sheaves valued in the topos of finite sets would be inte= resting=2C providing a more objective version of number theorythan the abst= ract exponential rig traditionally called "natural". Topos theory is bristling with potential examples that we "generalists" hav= e been slow to take up. Anyway=2C the above construction of Omega is manifestly exponential=2C henc= e an effort to find computable sub cases is clearly needed=2C as you sugges= t Ellis. Bill > Date: Wed=2C 23 Feb 2011 10:16:56 -0500 > To: categories@mta.ca > From: xtalv1@netropolis.net > Subject: categories: Subobject Classifier Algorithm >=20 > What are the general rules for calculating the sub-object classifier > of a topos? Or=2C for what class of toposes is there an algorithm for > calculating the sub-object classifier of its members? >=20 > Ellis D. Cooper >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]