From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5270 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: topos and magic Date: Fri, 13 Nov 2009 11:34:44 -0800 Message-ID: References: Reply-To: Vaughan Pratt 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: ger.gmane.org 1258208675 30488 80.91.229.12 (14 Nov 2009 14:24:35 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 14 Nov 2009 14:24:35 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Sat Nov 14 15:24:28 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1N9JXn-0006h7-J3 for gsmc-categories@m.gmane.org; Sat, 14 Nov 2009 15:24:03 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N9J1T-0003zA-0M for categories-list@mta.ca; Sat, 14 Nov 2009 09:50:39 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5270 Archived-At: Andre Joyal wrote: > I find the notion of elementary topos absolutly extraordinary, almost magical. > Like every important mathematical discovery, it stands out as a colorful gem on a bed of grey stones. > A classical example of gem is the field of complex numbers. Historically these two gems emerged as entirely independent developments. However they are arguably facets of a single gem, the abelian-topos categories Peter Freyd wrote about on this list in November 1997, archived at http://blog.gmane.org/gmane.science.mathematics.categories/day=19971031 or on Karel Stokkerman's topic-indexed archive at http://www.mta.ca/~cat-dist/catlist/1999/atcat and http://www.mta.ca/~cat-dist/catlist/1999/prattsli The complex numbers live within the ring M(2,R) of 2x2 real matrices as a subring of M(2,R) that happens to form a field. (The general linear group GL(2,R) is a larger *skew* field in M(2,R), but is there a larger *field* than the complex numbers therein?) M(2,R) in turn forms a one-object full subcategory of the abelian category Vct_R, with matrix multiplication (hence complex number multiplication) realized as composition. So the complex numbers form a subcategory of an abelian category. As Peter's treatment makes clear, abelian categories are a quite minor variant on toposes. An abelian category (resp. topos) is an abelian-topos category all of whose objects X are of type A (resp. T), meaning that the first (resp. second) projection of Xx0, namely from Xx0 to X (resp. 0), is an iso. This difference is expressed as a very simple elementary (first-order) predicate whose intuitive meaning is clear: simply multiply X by zero and see whether it remains X or collapses to zero. So within the relatively small universe of abelian-topos categories (by comparison with *all* categories) we find lurking therein both the field of complex numbers and the toposes (and hence in particular the effective topos, yet another gem Andre mentioned). Over on the Foundations of Mathematics mailing list, FOMers would presumably connect complex numbers to set theory by observing that the complex numbers form a set which lives within the universe of sets axiomatized by the Zermelo-Fraenkel axioms. To me the path from complex numbers to toposes via matrices and abelian categories seems somehow more intimate. Simply calling the complex numbers a set seems dry as dust (literally). (Peter J., are abelian-topos categories in the Elephant? They seem an obvious candidate yet I couldn't find them in either the table of contents or the index.) Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]