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From: Steven Vickers
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Subject: Re: Terminology for point-free topology?
Date: Wed, 18 Jan 2023 12:12:13 +0000
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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 be=
fore I knew those papers of yours, but I felt right from the start that I w=
as merely unveiling techniques already known to the experts - though I hope=
you'll agree I've been more explicit about them and particularly the natur=
e 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 si=
tes concretely, or by using colimits and finite limits under the rules corr=
esponding to Giraud's theorem. I imagine that as being something like the d=
istinction 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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