From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10905 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: Fri, 20 Jan 2023 11:50:33 +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="17222"; mail-complaints-to="usenet@ciao.gmane.io" Cc: categories list To: David Yetter , "I.Moerdijk@uu.nl" Original-X-From: majordomo@rr.mta.ca Sat Jan 21 02:20:47 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 1pJ2Yo-0004Fl-Lt for gsmc-categories@m.gmane-mx.org; Sat, 21 Jan 2023 02:20:46 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:39330) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1pJ2YT-0001or-4a; Fri, 20 Jan 2023 21:20:25 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1pJ2Xz-0007s2-14 for categories-list@rr.mta.ca; Fri, 20 Jan 2023 21:19:55 -0400 In-Reply-To: Accept-Language: en-US Content-Language: en-US Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10905 Archived-At: Dear David, Yes, and it's an excellent paper with a witty title for which only "pointle= ss" would do. I particularly like what Peter said when explaining the significant differe= nce in the absence of choice (such as in toposes of sheaves), and that "usu= ally it is locales, not spaces, which provide the right context in which to= do topology". He went on to say, "This is the point which ... Andre Joyal began to hammer home in the early = 1970s; I can well remember how, at the time, his insistence that locales we= re the real stuff of topology, and spaces were merely figments of the class= ical mathematician's imagination, seemed (to me, and I suspect to others) l= ike unmotivated fanaticism. I have learned better since then." This is all part of the argument for using a reformed topology, but there i= s nothing particular there about the pointwise style of reasoning for it. H= ence we are still left with the question of how to reference the two concep= ts, the reformed topology and the reasoning without points. Would you call Ng's paper with me pointless? Points are everywhere in it. (= Of course, there's the separate issue of whether it was pointless in the se= nse of not worth the trouble. But an important feature of the style is that= it forces you to be careful to distinguish between Dedekind reals and 1-si= ded (lower or upper) reals, and in Ng's thesis this uncovered unexpected ro= les of 1-sided reals in the account of Ostrowski's Theorem and the Berkovic= h spectrum. So there is a bit of payoff.) Best wishes, Steve. ________________________________ From: David Yetter Sent: Friday, January 20, 2023 3:06 AM To: I.Moerdijk@uu.nl ; Steven Vickers (Computer Science) = Cc: categories list Subject: Re: categories: Re: Terminology for point-free topology? I seem to recall from back in my days as a grad student or new PhD that Pet= er Johnstone wrote a paper entitled "The Point of Pointless Topology". Jus= t in honor of that I've always favored "pointless topology" as the term for= the theory of locales and sheaves on locales. Best Thoughts, David Y. ________________________________ From: Steven Vickers Sent: Wednesday, January 18, 2023 6:12 AM To: I.Moerdijk@uu.nl Cc: categories list Subject: categories: Re: Terminology for point-free topology? This email originated from outside of K-State. Dear Ieke, Thanks for mentioning that. It's a beautiful paper, both in its results and= in its presentation, and one I still return to. Another place where I think you were even more explicit was in "The classif= ying topos of a continuous groupoid I" (1988), where you said - "... in presenting many arguments concerning generalized, "pointless" space= s, I have tried to convey the idea that by using change-of-base-techniques = and exploiting the internal logic of a Grothendieck topos, point-set argume= nts are perfectly suitable for dealing with pointless spaces (at least as l= ong as one stays within the 'stable' part of the theory)." (Would you still say that "pointless" and "point-set" are the right phrases= there? I'm proposing "point-free" and "pointwise".) On the other hand, in your book with Mac Lane, those ideas seemed to go int= o hiding. In fact I explicitly wrote "Locales and toposes as spaces" as a g= uide to reading the points back into the book. My first understanding of these pointwise techniques came in the 1990's, as= I developed the exposition of "Topical categories of domains". That was b= efore I knew those papers of yours, but I felt right from the start that I = was merely unveiling techniques already known to the experts - though I hop= e you'll agree I've been more explicit about them and particularly the nat= ure and role of geometricity. I still don't know as much as I would like about the origin and history of = those techniques. It would certainly improve my arXiv notes if I could say = more. Might they even have roots in Grothendieck? I once saw a comment by Colin M= cLarty to the effect that (modulo misrepresentation by me) Grothendieck was= aware of two different lines of reasoning with toposes: by manipulating s= ites concretely, or by using colimits and finite limits under the rules cor= responding to Giraud's theorem. I imagine that as being something like the = distinction between pointless and pointwise. Best wishes, Steve. ________________________________ Hi Steve, A very early illustration of the strategy of using points in pointless topo= logy is in my paper with Wraith (published 1986). I just looked at it again= , and the strategy is explicitly stated in the introduction : "the strategy is to use adequate extensions of the base topos available fro= m general topos theory, which enable one to follow classical arguments abou= t points of separable metric spaces rather closely. Although both approache= s are equivalent, we will follow the second one, because it shows more clea= rly the interplay between general topos theory and arguments (somewhat simi= lar to those) from topology" We used it to prove an actual theorem. Of course I used this strategy much = more often, e.g. in my two 1990 papers with Joyal. Ieke [For admin and other information see: http://www.mta.ca/~cat-dist/ ]