From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2259 Path: news.gmane.org!not-for-mail From: F W Lawvere Newsgroups: gmane.science.mathematics.categories Subject: Re: equivalent varieties Date: Sun, 27 Apr 2003 22:14:35 -0400 (EDT) Message-ID: Reply-To: wlawvere@acsu.buffalo.edu NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018532 3291 80.91.229.2 (29 Apr 2009 15:22:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:22:12 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Apr 28 09:06:53 2003 -0300 X-Keywords: Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 28 Apr 2003 09:06:53 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19A7Nx-0006r9-00 for categories-list@mta.ca; Mon, 28 Apr 2003 09:05:29 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 37 Xref: news.gmane.org gmane.science.mathematics.categories:2259 Archived-At: An invariant way to see the particular equivalence discussed is to note that the topos of presheaves on the category of finite sets is the classifying topos for p-algebras for any given p > 1. That is because any non-empty set is a retract of a finite power of p and because left-exactness is equivalent to preserving finite products in this particular case. This representation suggests a different interpretation from the usual "truth of properties" point of view concerning the essential content of Boolean algebra. Namely, it concerns finite partitions of a hypothetical whole and shuffling of these induced by arbitrary maps between the index sets for the partitions, nothing more. Coordinatizing the above shuffling of partitions using p = 3 has some advantages over p = 2, namely, the unary operations of the theory suffice to characterize ultrafilters, i.e. to insure that perceived points of a finite set are actually there; more formally, the contravariant functor represented by 3 from finite sets to M-sets is full where M is the 27-element monoid of these unary operations. ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************