From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6606 Path: news.gmane.org!not-for-mail From: Andrej Bauer Newsgroups: gmane.science.mathematics.categories Subject: Re: Constitutive Structures Date: Fri, 8 Apr 2011 12:18:16 +0200 Message-ID: References: Reply-To: Andrej Bauer NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1302302197 29166 80.91.229.12 (8 Apr 2011 22:36:37 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 8 Apr 2011 22:36:37 +0000 (UTC) To: "Ellis D. Cooper" , categories Original-X-From: majordomo@mlist.mta.ca Sat Apr 09 00:36:33 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 1Q8KI4-0000T5-CI for gsmc-categories@m.gmane.org; Sat, 09 Apr 2011 00:36:32 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:53459) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Q8KI0-0007vC-7a; Fri, 08 Apr 2011 19:36:28 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Q8KHx-0004xh-PD for categories-list@mlist.mta.ca; Fri, 08 Apr 2011 19:36:25 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6606 Archived-At: You may wish to look at Davorin Le=C5=A1nik's Ph.D. thesis, where he studies real numbers in a constructive setting (without choice). He identifies suitable categories inside of which the real numbers exist as an object with a universal property that determines the reals up to isomorphism. The various categories correspond to the various substructure of the reals (order, additive group, ring, etc.) An interesting question is where to find his Ph.D. thesis. I will make him publish it somewhere on the web and will come back to you with a link. With kind regards, Andrej On Thu, Apr 7, 2011 at 2:50 PM, Ellis D. Cooper wro= te: > 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 s= o > 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 ma= kes > 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/ ]