From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9648 Path: news.gmane.org!.POSTED!not-for-mail From: Peter Johnstone Newsgroups: gmane.science.mathematics.categories Subject: Re: Point-free affine real line? Date: Tue, 5 Jun 2018 11:55:14 +0100 (BST) Message-ID: References: Reply-To: Peter Johnstone NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII; format=flowed X-Trace: blaine.gmane.org 1528205667 15803 195.159.176.226 (5 Jun 2018 13:34:27 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 5 Jun 2018 13:34:27 +0000 (UTC) Cc: Steve Vickers , Johannes.Huebschmann@math.univ-lille1.fr, Categories To: Vaughan Pratt Original-X-From: majordomo@mlist.mta.ca Tue Jun 05 15:34:23 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1fQC6Q-00040c-JH for gsmc-categories@m.gmane.org; Tue, 05 Jun 2018 15:34:22 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:53315) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1fQC7r-0000VI-U9; Tue, 05 Jun 2018 10:35:51 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1fQC7I-00043V-Pi for categories-list@mlist.mta.ca; Tue, 05 Jun 2018 10:35:16 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9648 Archived-At: On Sun, 3 Jun 2018, Vaughan Pratt wrote: > > The question about the affine real line represents a challenge to this >> geometric approach, and I'd like to form a better idea of whether it is >> simply a difficult problem, or a fundamental limitation to my approach. > > An affine space over any given field differs *k* from a vector space over > *k* only in its algebraic structure, not its topological structure. > Whereas the algebraic operations of a vector space over *k* consist of all > finitary linear combinations with coefficients drawn from *k*, those of an > affine space consist of the subset of those combinations whose coefficients > sum to unity, the barycentric combinations. Since the former includes the > constant 0 as a linear combination while the latter does not, a consequence > is that 0 is a fixpoint of linear transformations but not of affine > transformations, whence the latter can include the translations. > > This is equally true whether *k* is the rationals or the reals. So > whatever method you use to obtain the real line from the rational line > should also produce the affine real line from the affine rational line. > Not quite: the affine rational line doesn't have a definable total order, since it has order-reversing automorphisms, so any definition using Dedekind sections is problematic. However, it does have a ternary `betweenness' relation, and it should be possible to rewrite the geometric theory of Dedekind sections of Q, as presented on p. 1015 of `Sketches of an Elephant', in terms of this relation (but note that sections will have to be unordered rather than ordered pairs of subobjects of Q). Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]