From: David Yetter <email@example.com>
To: categories list <firstname.lastname@example.org>
Subject: Re: Terminology for point-free topology?
Date: Fri, 20 Jan 2023 03:06:03 +0000 [thread overview]
Message-ID: <E1pJ2WOemail@example.com> (raw)
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 <firstname.lastname@example.org>
Sent: Wednesday, January 18, 2023 6:12 AM
To: I.Moerdijk@uu.nl <I.Moerdijk@uu.nl>
Cc: categories list <email@example.com>
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.
A very early illustration of the strategy of using points in pointless topology 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 from general topos theory, which enable one to follow classical arguments about points of separable metric spaces rather closely. Although both approaches are equivalent, we will follow the second one, because it shows more clearly the interplay between general topos theory and arguments (somewhat similar 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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next prev parent reply other threads:[~2023-01-20 3:06 UTC|newest]
Thread overview: 19+ messages / expand[flat|nested] mbox.gz Atom feed top
2023-01-16 11:50 Steven Vickers
[not found] ` <firstname.lastname@example.org>
2023-01-18 12:12 ` Steven Vickers
2023-01-20 3:06 ` David Yetter [this message]
[not found] ` <SN6PR05MB5213EBE225CB83D101EA0F57A2C59@SN6PR05MB5213.namprd05.prod.outlook.com>
2023-01-20 11:50 ` Steven Vickers
2023-01-21 19:42 ` ptj
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
You may reply publicly to this message via plain-text email
using any one of the following methods:
* Save the following mbox file, import it into your mail client,
and reply-to-all from there: mbox
Avoid top-posting and favor interleaved quoting:
* Reply using the --to, --cc, and --in-reply-to
switches of git-send-email(1):
git send-email \
* If your mail client supports setting the In-Reply-To header
via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line
before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox