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From: Vaughan Pratt
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Subject: Re: Terminology for point-free topology?
Date: Sun, 22 Jan 2023 13:32:59 -0800
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Hi Steve,
"Classically, it is not unreasonable to view lack of global points as a
pathology in the locale Y; and then the constructive tendency to lack
global points appears as pathology in the logic." (Your reply to me here
of Jan. 17)
Thanks for that and your accompanying remarks , Steve.
Space is both extroverted (Euclid's relatively clear Postulate 2 that a
finite straight line can be produced) and introverted (Euclid's vaguer
Definition 2 that a line (segment) is breadthless length).
>From a Topological Systems/Chuish perspective, I wonder if the extroverted
nature of space is best appreciated through points and its introverted
nature through states.
After all, we have Hoelder's 1901 notion of a linearly ordered group for
the former (and the free such on one generator will be the integers and
hence both abelian and Archimedean), while we have the
Pavlovich-P-Freyd-Leinster notion of the continuum as a final coalgebra,
which can be as small as the unit interval if you stick to midpoint
algebras (rather than continued fractions as Dusko and I did in 1999) and
as such ideal for filling in the gaps between consecutive integers.
That Euclid's Definition 2 is vaguer than his Postulate 2 is consistent
with the applicability of free algebras to the extroverted nature of space
appearing much earlier than that of final coalgebras to its introverted
nature.
These thoughts came to me after spending a few weeks mulling over a
conversation I had with my classmate (1962-5) Ross Street about our common
but independently arrived at interest, decades ago, in what Ross calls
"efficient" constructions of the reals.
And along a different line of thought, is Chu(Set,2) the right category for
topological systems, or might there be some advantage to Chu(E,k) where E
is the appropriate topos for the application at hand, or perhaps just the
free topos, and k its subobject classifier?
Best,
Vaughan
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