From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6608 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Constitutive Structures Date: Sat, 09 Apr 2011 15:53:31 -0700 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: dough.gmane.org 1302465445 22875 80.91.229.12 (10 Apr 2011 19:57:25 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 10 Apr 2011 19:57:25 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Sun Apr 10 21:57:21 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Q90l6-0006VX-25 for gsmc-categories@m.gmane.org; Sun, 10 Apr 2011 21:57:20 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52226) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Q90kk-0005WU-C8; Sun, 10 Apr 2011 16:56:58 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Q90ki-000258-9D for categories-list@mlist.mta.ca; Sun, 10 Apr 2011 16:56:56 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6608 Archived-At: One answer to this question is that the continuum is the final F-coalgebra in a suitable category for a suitable F. The first result along those lines was Pavlovic & P, "The continuum as a final coalgebra", TCS 280(1-2):105-122, May 2002, originally presented at CMCS'99 in Amsterdam. It made explicit the double coinduction implicit in the various continued-fraction representations of the reals. The category was Posets and only the topological and order structure was represented. This was subsequently extended in papers by Peter Freyd and by Tom Leinster to express as well the algebraic structure, and also to reduce the double coinduction to a single coinduction in exchange for giving up uniqueness of representation of reals (the continued fractions are in bijection with the nonnegative reals). Vaughan Pratt On 4/7/2011 5:50 AM, Ellis D. Cooper wrote: > What might be the proper categorical framework to discuss, for > example, the fact that the Real Numbers have constitutive structures > such as additive abelian group, multiplicative abelian group, > topology generated by open intervals, totally ordered infinite set, and > so on? > At first one might think of forgetful functors, but then what would > be the category in which Real Numbers is one object among many? > Or, one might say take a category with exactly one object and a > functor to each of the categories of the constitutive structures. This > makes > the Real Numbers look like an "element" of the "intersection" of > diverse categories. Then the Complex Numbers or the Hyperreal Numbers > which contain > the Real Numbers as sub-objects in certain ways are "elements" of > other "intersections" of categories. What am I talking about? > > Ellis D. Cooper > > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]