From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6557 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: RE: Subobject Classifier Algorithm Date: Tue, 1 Mar 2011 11:52:25 -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 1299027264 5342 80.91.229.12 (2 Mar 2011 00:54:24 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 2 Mar 2011 00:54:24 +0000 (UTC) To: , categories Original-X-From: majordomo@mlist.mta.ca Wed Mar 02 01:54:18 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 1PuaKT-0000Xh-1O for gsmc-categories@m.gmane.org; Wed, 02 Mar 2011 01:54:13 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:37197) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PuaKA-00028a-NG; Tue, 01 Mar 2011 20:53:54 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PuaK7-0005Ou-O4 for categories-list@mlist.mta.ca; Tue, 01 Mar 2011 20:53:51 -0400 Importance: Normal In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6557 Archived-At: Of course the initial responses given by Fred and me were notthemselves al= gorithmic=2C but why can they not be a prelude to such?Fred recalled the se= quence of UMPs that defines Omega and I recalled what results in the case o= f a finite site. The hope is that experts in Maple and the like will be abl= e to solve the sort of problem needed to generate displays. That is=2C give= n a finite presentation of a category C=2Cfind a presentation of the result= ing Heyting algebra with the action ofC on it. Of course as stated this inc= ludes the Burnside problem =2C the Post problem etc . Hence the need for re= cognizing solvable subproblems.Even for the class of graphic monoids=2C wh= ere the structure is finite=2C it growsexponentially =2C which has been use= d in the past by other computer scientists as an excuse not to consider it.= But if we are given a few fixed finite C=2C we can consider the class of c= ategories discretely fiberedover them (equivalent to objects of the toposes= ) and try to see whetherOmega is computable in the above sense for them. I said contravariant 2-valued functors and of course these factor through = the poset reflection. Calculating that poset reflection is a search problem= wrt=20 the underlying graph=2C but actually wrt composition as well since we have = todo it for the slice categories C/A.Bill> Date: Fri=2C 25 Feb 2011 11:20:3= 6 +0000 > From: pt11@PaulTaylor.EU > To: categories@mta.ca > Subject: categories: Subobject Classifier Algorithm >=20 > Ellis Cooper asked=2C >=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 > Thanks to Bill and Fred for describing the constructions. >=20 > I would suggest=2C however=2C that it is rather stretching the meaning > of the word "algorithm" to describe them as such. What kind of > machine might be able to perform these operations? >=20 > A propos of this question=2C it is well known both to new students > of category theory and to those who like to use the subject to > discuss Life=2C The Universe And Everything=2C that: >=20 > (1) a topos is a cartesian closed category with > (2) an internal Heyting algebra Omega=2C >=20 ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]