From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10909 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Steven Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology for point-free topology? Date: Mon, 23 Jan 2023 13:25:04 +0000 Message-ID: References: Reply-To: Steven Vickers Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="19955"; mail-complaints-to="usenet@ciao.gmane.io" Cc: "categories@mta.ca" To: Vaughan Pratt Original-X-From: majordomo@rr.mta.ca Mon Jan 23 23:15:14 2023 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.75]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1pK55u-0004r1-Db for gsmc-categories@m.gmane-mx.org; Mon, 23 Jan 2023 23:15:14 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:39542) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1pK55O-0007sq-0v; Mon, 23 Jan 2023 18:14:42 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1pK54j-00041C-5R for categories-list@rr.mta.ca; Mon, 23 Jan 2023 18:14:01 -0400 In-Reply-To: Accept-Language: en-US Content-Language: en-US Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10909 Archived-At: Dear Vaughan, I've worked point-free on midpoint algebras (my paper that [-1,1] is an Esc= ardo-Simpson interval object), and I think I understand what you're getting= at there. Here's a particular mathematical problem I've been looking at, to check whe= ther my thinking on extrovert/introvert is in line with yours. Now we have satisfactory point-free accounts of exp and log, can we do the = same with trigonometry? That boils down to defining group homomorphisms from R to the circle group = S^1 (viewed as a sublocale of C =3D R^2). It may be that a good way to do t= hat is (1) (introvert?) define a midpoint homomorphism from [0, 1] to a reg= ion of S^1 close to 1, with 0 mapping to 1, and then (2) (extrovert?) use t= he homomorphism property to extend to the whole of R. I'll turn now to the Chu spaces. If I understand properly what you're sugge= sting, it's to replace (Set, 2) by some (E, k). That's not going to solve t= he issue with lack of points that I was talking about. If anything, it make= s it worse, because without choice it's harder to find points. The problem lies, rather, in the fact that the Chu space relies on pairing = two *sets*. I'm perfectly happy to allow "set" to mean object in some chose= n base topos. However, insisting on a set on the points side smashes too mu= ch topological structure to work well in general. As my example with Sierpi= nski showed, it in effect forces you to approximate bundles with local home= omorphisms, and that can leave you with nothing useful that is available fo= r the points side of the Chu space. When you switch to generalized points, there are now enough. In fact the ge= neric point in the topos of sheaves, on its own, is enough for most purpose= s. But then, if you wanted to adapt the Chu spaces somehow to allow that in= , you might as well take it on its own. Hope that helps, Steve. ________________________________ From: Vaughan Pratt Sent: Sunday, January 22, 2023 9:32 PM To: Steven Vickers (Computer Science) Cc: categories@mta.ca Subject: Re: categories: Terminology for point-free topology? Hi Steve, "Classically, it is not unreasonable to view lack of global points as a pat= hology in the locale Y; and then the constructive tendency to lack global p= oints appears as pathology in the logic." (Your reply to me here of Jan. 1= 7) Thanks for that and your accompanying remarks , Steve. Space is both extroverted (Euclid's relatively clear Postulate 2 that a fin= ite straight line can be produced) and introverted (Euclid's vaguer Definit= ion 2 that a line (segment) is breadthless length). >From a Topological Systems/Chuish perspective, I wonder if the extroverted = nature of space is best appreciated through points and its introverted natu= re through states. After all, we have Hoelder's 1901 notion of a linearly ordered group for th= e former (and the free such on one generator will be the integers and hence= both abelian and Archimedean), while we have the Pavlovich-P-Freyd-Leinste= r notion of the continuum as a final coalgebra, which can be as small as th= e unit interval if you stick to midpoint algebras (rather than continued fr= actions as Dusko and I did in 1999) and as such ideal for filling in the ga= ps between consecutive integers. That Euclid's Definition 2 is vaguer than his Postulate 2 is consistent wit= h the applicability of free algebras to the extroverted nature of space app= earing much earlier than that of final coalgebras to its introverted nature= . These thoughts came to me after spending a few weeks mulling over a convers= ation I had with my classmate (1962-5) Ross Street about our common but ind= ependently arrived at interest, decades ago, in what Ross calls "efficient"= constructions of the reals. And along a different line of thought, is Chu(Set,2) the right category for= topological systems, or might there be some advantage to Chu(E,k) where E= is the appropriate topos for the application at hand, or perhaps just the = free topos, and k its subobject classifier? Best, Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]