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From: Steven Vickers
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Subject: Re: Terminology for point-free topology?
Date: Fri, 20 Jan 2023 11:50:33 +0000
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To: David Yetter , "I.Moerdijk@uu.nl"
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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
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