Re: Terminology for point-free topology?

[ptj@maths.cam.ac.uk]

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 <dyetter@ksu.edu>
> Sent: Friday, January 20, 2023 3:06 AM To: I.Moerdijk@uu.nl
> <I.Moerdijk@uu.nl>; Steven Vickers (Computer Science)
> <s.j.vickers.1@bham.ac.uk> Cc: categories list <categories@mta.ca>
> 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 <s.j.vickers.1@bham.ac.uk>
>Sent: Wednesday, January 18, 2023 6:12 AM
>To: I.Moerdijk@uu.nl <I.Moerdijk@uu.nl>
>Cc: categories list <categories@mta.ca>
>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.

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