From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9652 Path: news.gmane.org!.POSTED!not-for-mail From: Peter Johnstone Newsgroups: gmane.science.mathematics.categories Subject: Re: Point-free affine real line? Date: Sun, 10 Jun 2018 14:54:56 +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 1528737449 17545 195.159.176.226 (11 Jun 2018 17:17:29 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Mon, 11 Jun 2018 17:17:29 +0000 (UTC) Cc: categories@mta.ca To: Vaughan Pratt Original-X-From: majordomo@mlist.mta.ca Mon Jun 11 19:17:25 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 1fSQRW-0004Lo-4u for gsmc-categories@m.gmane.org; Mon, 11 Jun 2018 19:17:22 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54738) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1fSQSS-0005qU-Sd; Mon, 11 Jun 2018 14:18:20 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1fSQRv-0001Lm-Ok for categories-list@mlist.mta.ca; Mon, 11 Jun 2018 14:17:47 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9652 Archived-At: Sorry, what I wrote was a bit sloppy. Vaughan is right that the problem doesn't arise with the passage from considering Q as a linear space to considering it as an affine space, since it already has order- reversing linear automorphisms. For the order to be definable from the algebraic structure, you need to consider Q as a field, which is what the usual Dedekind-section construction does. Peter Johnstone On Thu, 7 Jun 2018, Vaughan Pratt wrote: > > 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. > > Morphism-wise, since the affine transformations are just the composition of > a linear transformation with a translation, and translation of the rational > line preserves order, affinity can't be the problem here. > > Structure-wise, one can equip the rational line with either its linear > combinations or its linear order, or both. Using both eliminates the > order-reversing linear transformations. "Affine" only makes sense in the > context of having the linear combinations, as "affine" limits the linear > combinations to those whose coefficients sum to one. If it is ok for the > linear combinations and the linear order to coexist, it must be even more > ok for the affine combinations and the linear order to coexist. > > So whether one considers the morphisms or the structure they preserve, > affinity (affineness?) must be a red herring here: any problem for the > rational line as an affine space is surely also a problem for it as a > vector space. > > Vaughan Pratt > > > > On Tue, Jun 5, 2018 at 3:55 AM, Peter Johnstone wrote: > >> 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/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]