From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10908 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology for point-free topology? Date: Sun, 22 Jan 2023 13:32:59 -0800 Message-ID: References: Reply-To: Vaughan Pratt Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="9590"; mail-complaints-to="usenet@ciao.gmane.io" Cc: "categories@mta.ca" To: Steven Vickers Original-X-From: majordomo@rr.mta.ca Mon Jan 23 23:13:24 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 1pK543-00029u-JB for gsmc-categories@m.gmane-mx.org; Mon, 23 Jan 2023 23:13:19 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:39520) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1pK53G-0007bh-Rk; Mon, 23 Jan 2023 18:12:30 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1pK52m-0003zm-1g for categories-list@rr.mta.ca; Mon, 23 Jan 2023 18:12:00 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10908 Archived-At: Hi Steve, "Classically, it is not unreasonable to view lack of global points as a pathology in the locale Y; and then the constructive tendency to lack global points appears as pathology in the logic." (Your reply to me here of Jan. 17) Thanks for that and your accompanying remarks , Steve. Space is both extroverted (Euclid's relatively clear Postulate 2 that a finite straight line can be produced) and introverted (Euclid's vaguer Definition 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 nature through states. After all, we have Hoelder's 1901 notion of a linearly ordered group for the 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-Leinster notion of the continuum as a final coalgebra, which can be as small as the unit interval if you stick to midpoint algebras (rather than continued fractions as Dusko and I did in 1999) and as such ideal for filling in the gaps between consecutive integers. That Euclid's Definition 2 is vaguer than his Postulate 2 is consistent with the applicability of free algebras to the extroverted nature of space appearing much earlier than that of final coalgebras to its introverted nature. These thoughts came to me after spending a few weeks mulling over a conversation I had with my classmate (1962-5) Ross Street about our common but independently 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/ ]