To: Steven Vickers <firstname.lastname@example.org>
Cc: categories list <email@example.com>
Subject: Re: Terminology for point-free topology?
Date: 21 Jan 2023 19:42:15 +0000 [thread overview]
Message-ID: <E1pJhLM-0002Kh-DU@rr.mta.ca> (raw)
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.
On Jan 21 2023, Steven Vickers wrote:
> 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.)
> ________________________________ From: David Yetter <firstname.lastname@example.org>
> Sent: Friday, January 20, 2023 3:06 AM To: I.Moerdijk@uu.nl
> <I.Moerdijk@uu.nl>; Steven Vickers (Computer Science)
> <email@example.com> Cc: categories list <firstname.lastname@example.org>
> 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.
>From: Steven Vickers <email@example.com>
>Sent: Wednesday, January 18, 2023 6:12 AM
>To: I.Moerdijk@uu.nl <I.Moerdijk@uu.nl>
>Cc: categories list <firstname.lastname@example.org>
>Subject: categories: Re: Terminology for point-free topology?
>This email originated from outside of K-State.
> 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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next prev parent reply other threads:[~2023-01-21 19:42 UTC|newest]
Thread overview: 19+ messages / expand[flat|nested] mbox.gz Atom feed top
2023-01-16 11:50 Steven Vickers
[not found] ` <email@example.com>
2023-01-18 12:12 ` Steven Vickers
2023-01-20 3:06 ` David Yetter
[not found] ` <SN6PR05MB5213EBE225CB83D101EA0F57A2C59@SN6PR05MB5213.namprd05.prod.outlook.com>
2023-01-20 11:50 ` Steven Vickers
2023-01-21 19:42 ` ptj [this message]
2023-01-23 11:44 ` Pedro Resende
2023-01-30 21:59 ` categories: " Wesley Phoa
2023-02-01 9:41 ` Martin Hyland
[not found] ` <18E1AA5F-0054-4CA3-B231-BD9B799B03A2@tecnico.ulisboa.pt>
2023-01-23 13:47 ` Steven Vickers
[not found] ` <YQXPR01MB26464DF33EAE7481847A4F82E5C99@YQXPR01MB2646.CANPRD01.PROD.OUTLOOK.COM>
2023-01-24 12:20 ` categories: " Robert Pare
2023-01-27 17:55 ` Pedro Resende
2023-01-28 5:43 ` Patrik Eklund
2023-01-29 23:16 ` dawson
2023-01-28 10:48 ` categories: complete Galois groups Clemens Berger
2023-01-30 17:34 ` categories: " Eduardo J. Dubuc
[not found] ` <LNXP265MB1049E00AEC9CE5BE1233CCEF95C69@LNXP265MB1049.GBRP265.PROD.OUTLOOK.COM>
2023-01-22 21:32 ` Terminology for point-free topology? Vaughan Pratt
[not found] ` <CAL7kZqCz081nyFQuo_QFfaGzGe+UBORJHOZWsO9VgNzpDJ9_Gw@mail.gmail.com>
2023-01-23 13:25 ` Steven Vickers
[not found] ` <LNXP265MB104912A7940157738582CE2595C89@LNXP265MB1049.GBRP265.PROD.OUTLOOK.COM>
2023-01-23 23:17 ` categories: " Vaughan Pratt
[not found] ` <CAL7kZqAPzmzf=wt=qKNBjjeb_dGtG4eDu7tv5Eku-AVZD7wWtw@mail.gmail.com>
2023-01-24 11:45 ` Steven Vickers
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