From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1913 Path: news.gmane.org!not-for-mail From: Martin Escardo Newsgroups: gmane.science.mathematics.categories Subject: A universal characterization of the unit interval Date: Tue, 10 Apr 2001 15:28:00 +0100 (BST) Message-ID: <15059.6003.377874.445195@henry.cs.bham.ac.uk> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018200 1177 80.91.229.2 (29 Apr 2009 15:16:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:16:40 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Apr 12 16:43:43 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3CIpus11314 for categories-list; Thu, 12 Apr 2001 15:51:56 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: VM 6.43 under 20.4 "Emerald" XEmacs Lucid Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 11 Original-Lines: 32 Xref: news.gmane.org gmane.science.mathematics.categories:1913 Archived-At: The following extended abstract is now available on the Web. A universal characterization of the closed Euclidean interval ABSTRACT. We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the category of topological spaces, the interval objects are closed and bounded intervals with the Euclidean topology. We also prove that an interval object exists in any elementary topos with natural numbers object. http://www.cs.bham.ac.uk/~mhe/papers/lics2001-revised.ps Best wishes, Martin Escardo & Alex Simpson -- P.s. A draft full version with proofs is available on-line at http://www.dcs.ed.ac.uk/home/als/Research/interval.ps