From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9641 Path: news.gmane.org!.POSTED!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Point-free affine real line? Date: Fri, 01 Jun 2018 11:11:56 +0100 Message-ID: References: Reply-To: Steve Vickers NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: blaine.gmane.org 1528037028 31230 195.159.176.226 (3 Jun 2018 14:43:48 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 3 Jun 2018 14:43:48 +0000 (UTC) Cc: Categories To: Johannes.Huebschmann@math.univ-lille1.fr Original-X-From: majordomo@mlist.mta.ca Sun Jun 03 16:43:44 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 1fPUEO-0007xY-Ou for gsmc-categories@m.gmane.org; Sun, 03 Jun 2018 16:43:41 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52837) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1fPUG3-0007b0-F0; Sun, 03 Jun 2018 11:45:23 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1fPUFT-0003Ma-LX for categories-list@mlist.mta.ca; Sun, 03 Jun 2018 11:44:47 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9641 Archived-At: Dear Johannes, "Point-free" is as opposed to "point-set" - there is no assumption that the real numbers form a set. Instead, they are taken to be the models of a (geometric) theory of Dedekind sections of the rationals (which do form a set). More generally, a point-free space is one where the points are defined as models of a propositional geometric theory. The topology is then defined intrinsically, opens being the geometric propositions. My motivation for this comes from topos theory. In any elementary topos, each point-free space does in fact have a set (object) of points. Topologically, it amounts to approximating bundles by local homeomorphisms. However, that construction is not geometric - preserved by inverse image functors, and by pullback of bundles -, so the point-set version is not robust under change of base. There are advantages to staying within the geometric fragment of elementary topos logic, and I am exploring how far that can be taken. I also now have other semantics using arithmetic universes, where the point-set versions simply don't exist. If it helps, the point-free real line in an elementary topos E is the localic geometric morphism p: F -> E got from F as topos of sheaves of the internal locale of formal reals in E. If E is an S-topos, then you can do this generically over S to get R as the S-classifier for Dedekind sections, and then p is just the bipullback E x_S R. A point-set R would be some local homeomorphism over E (then equipped separately with a topology), but in general it is not got as a bipullback of a local homeomorphism over S. 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. To put it another way, am I following in Grothendieck's footsteps in the way I think of toposes, or am I mishandling his ideas in ways that have no bearing on what he was trying to do? All the best, Steve. On 01/06/2018 10:22, huebschm@math.univ-lille1.fr wrote: > Dear Steve > > I am not sure whether I understand. > What precisely do you mean by "point-free"? > > > > On Thu, 31 May 2018, Steve Vickers wrote: > >> Algebraic geometry defines the affine line over a field k as an >> affine scheme, > the spectrum of k[X]. It includes a copy of k, each element a being > present as the irreducible polynomial X-a, > with local ring the ring of fractions got by inverting polynomials > f(X) such that f(a) is non-zero. >> >> You can carry this out for the real line R, but it is very much R as >> a set, and the copy of R in the underlying space of the spectrum has >> the discrete topology. > > > > > The standard approach leads to the Zariski topology. > > > >> Does algebraic geometry provide an analogous construction that could >> lead to the point-free R? Can the locally ringed space be >> topologized (point-free) so that the copy of R has its usual topology? >> >> I've run into various problems. >> >> 1. It is not obvious to me that R[X] exists point-free. By that I >> mean that, without presupposing a set R[X], or using non-geometric >> constructions, I can't see how to define a geometric theory whose >> models are the polynomials. The problem comes with trying to pin down >> the requirement that all but finitely many of the coefficients of a >> polynomial must be zero. You cannot continuously define the degree of >> a polynomial, because the function R -> N, a |-> degree(aX + 1), is >> not continuous. >> >> That suggests the construction as Spec(R[X]) might have to be >> adjusted. Is there still some locally ringed space that does the trick? >> >> 2. The "structure sheaf" cannot be a sheaf. We hope its fibres are >> point-free local rings, but, whatever they are, they must be >> R-algebras and so cannot have the discrete topology. The space is >> locally ringed by some bundle other than a sheaf (local homeomorphism). >> >> 3. The usual local rings, got as rings of fractions as described >> above, may be problematic point-free in the same way as R[X] is. >> I don't know what would do instead. The power series ring R[[X]]? > > > > The power series ring recovers a single point ("formal geometry" in > the sense of Grothendieck, Gelfand-Kazhdan, Kontsevich, etc.). > > > Best Johannes > > > > >> (At least as fibre over 0.) >> It does have the property of inverting those polynomials f for which >> f(0) > is non-zero. And it can be defined point-free, as R^N. (However, the > finitely presented approximations R[X|X^n = 0] happily exist point-free.) >> >> Thanks for any references you can provide, >> >> Steve. >> >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]