From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9640 Path: news.gmane.org!.POSTED!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Point-free affine real line? Date: Thu, 31 May 2018 18:45:47 -0700 Message-ID: Reply-To: Vaughan Pratt NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" X-Trace: blaine.gmane.org 1528036997 29331 195.159.176.226 (3 Jun 2018 14:43:17 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 3 Jun 2018 14:43:17 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sun Jun 03 16:43:12 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 1fPUDw-0007W3-KW for gsmc-categories@m.gmane.org; Sun, 03 Jun 2018 16:43:12 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52821) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1fPUEw-0007WP-Bk; Sun, 03 Jun 2018 11:44:14 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1fPUEH-0003Lm-9H for categories-list@mlist.mta.ca; Sun, 03 Jun 2018 11:43:33 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9640 Archived-At: (FMCS and BLAST attendees will have seen this before.) Apropos of affine spaces over a field (the affine real line being a one-dimensional example), here are some axioms for a language with one binary operation ab, with (ab)c abbreviated to abc (left-associative convention as for combinatory algebra etc.), satisfying A1. aa = a A2. abb = a A3. abc = ac(bc) A4. abcd = adcb (Side remark: Axiom Ai requires i variables.) The intended model of ab is 2b - a over an arbitrary abelian group, e.g. the integers. (A4 forces abelian. Obviously the free algebra on n generators for n < 2 has n elements. Identify the free algebra F2 on {a,b} with the integers ..., -3, -2, -1, 0, 1, 2, 3, 4, ... as follows. ... baba, aba, ba, a, b, ab, bab, abab, ... (Exercise: This are the only elements of F2.) As usual the elements of F2 can be understood as binary operations on F2. Abbreviate the binary operation identified with n as a[n]b; thus a[0]b = a, a[1]b = b, a[2]b = ab, a[-1]b = ba, etc. Interpreting ab as 2b - a over the integers, a[n]b = a + n(b - a), and 0[n]1 = n, making F2 yet another binary notation for the integers, albeit exponentially longer. Now expand this one-operation language with a family c1, c2, c3, c4, ... of operations of respective finite arities 1, 2, 3, 4, ... satisfying c1(a) = a cm(a1, ..., am)[n]cn(a1, ..., an) = an for all n > 1 where m = n-1 cn(a1, ..., am, cm(a1, ..., am))[n]b = b ditto The intended model of cn(a1, ..., an) is the centroid or mean of a1, ..., an over an arbitrary field, i.e. (a1 + ... + an)/n. Claim. The variety defined by this equational theory, including its homomorphisms as standardly defined, is equivalent to the category of affine spaces over the rationals. (Affine transformations over the rationals are linear combinations whose coefficients sum to 1, e.g. centroid. The idea is that other models such as the real line can be pulled apart by homomorphisms into uncountably many copies of the rational line because the operations of the theory can only make rational connections between points of the line.) I mention this in case it has any bearing on Steve's question about a point-free version of the real affine line. Not that I see one myself. Vaughan Pratt On Thu, May 31, 2018 at 2:40 AM, 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. > > 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]]? (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/ ]