From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10907 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: ptj@maths.cam.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology for point-free topology? Date: 21 Jan 2023 19:42:15 +0000 Message-ID: References: Reply-To: ptj@maths.cam.ac.uk Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=ISO-8859-1 Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="22504"; mail-complaints-to="usenet@ciao.gmane.io" Cc: categories list To: Steven Vickers Original-X-From: majordomo@rr.mta.ca Sun Jan 22 21:54:54 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 1pJhMb-0005a3-Ua for gsmc-categories@m.gmane-mx.org; Sun, 22 Jan 2023 21:54:54 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:39438) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1pJhM9-0007OI-E8; Sun, 22 Jan 2023 16:54:25 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1pJhLM-0002Kh-DU for categories-list@rr.mta.ca; Sun, 22 Jan 2023 16:53:37 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10907 Archived-At: I was wondering how long it would be before someone in this thread referred to my `point of pointless topology' paper! Perhaps not so many people know that the title was a conscious echo of an earlier paper by Mike Barr called `The point of the empty set', which began with the words (I quote from memory) `The point is, there isn't any point there; that's exactly the point'. As Steve says, to fit that title I had to use the word `pointless', but on the whole I prefer `pointfree'; it carries the implication that you are free to work without points or to use them (in a generalized sense), as you prefer. Peter Johnstone On Jan 21 2023, Steven Vickers wrote: >Dear David, > > Yes, and it's an excellent paper with a witty title for which only > "pointless" would do. > > I particularly like what Peter said when explaining the significant > difference in the absence of choice (such as in toposes of sheaves), and > that "usually 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 were the real stuff of topology, and spaces were merely figments > of the classical mathematician's imagination, seemed (to me, and I > suspect to others) like unmotivated fanaticism. I have learned better > since then." > > This is all part of the argument for using a reformed topology, but there > is nothing particular there about the pointwise style of reasoning for > it. Hence we are still left with the question of how to reference the two > concepts, 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 > sense 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-sided (lower or upper) reals, and in Ng's thesis this uncovered > unexpected roles of 1-sided reals in the account of Ostrowski's Theorem > and the Berkovich 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 > Peter Johnstone wrote a paper entitled "The Point of Pointless Topology". > Just 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 > classifying topos of a continuous groupoid I" (1988), where you said - > > "... in presenting many arguments concerning generalized, "pointless" > spaces, 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 arguments are perfectly suitable for > dealing with pointless spaces (at least as long 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 > into hiding. In fact I explicitly wrote "Locales and toposes as spaces" > as a guide 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 before 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 hope you'll agree I've been more explicit about them and > particularly the nature 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 > McLarty to the effect that (modulo misrepresentation by me) Grothendieck > was aware of two different lines of reasoning with toposes: by > manipulating sites concretely, or by using colimits and finite limits > under the rules corresponding to Giraud's theorem. I imagine that as > being something like the distinction between pointless and pointwise. > >Best wishes, > >Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]