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From: Steven Vickers
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Subject: Re: Terminology for point-free topology?
Date: Mon, 23 Jan 2023 13:25:04 +0000
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To: Vaughan Pratt
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Dear Vaughan,
I've worked point-free on midpoint algebras (my paper that [-1,1] is an Esc=
ardo-Simpson interval object), and I think I understand what you're getting=
at there.
Here's a particular mathematical problem I've been looking at, to check whe=
ther my thinking on extrovert/introvert is in line with yours.
Now we have satisfactory point-free accounts of exp and log, can we do the =
same with trigonometry?
That boils down to defining group homomorphisms from R to the circle group =
S^1 (viewed as a sublocale of C =3D R^2). It may be that a good way to do t=
hat is (1) (introvert?) define a midpoint homomorphism from [0, 1] to a reg=
ion of S^1 close to 1, with 0 mapping to 1, and then (2) (extrovert?) use t=
he homomorphism property to extend to the whole of R.
I'll turn now to the Chu spaces. If I understand properly what you're sugge=
sting, it's to replace (Set, 2) by some (E, k). That's not going to solve t=
he issue with lack of points that I was talking about. If anything, it make=
s it worse, because without choice it's harder to find points.
The problem lies, rather, in the fact that the Chu space relies on pairing =
two *sets*. I'm perfectly happy to allow "set" to mean object in some chose=
n base topos. However, insisting on a set on the points side smashes too mu=
ch topological structure to work well in general. As my example with Sierpi=
nski showed, it in effect forces you to approximate bundles with local home=
omorphisms, and that can leave you with nothing useful that is available fo=
r the points side of the Chu space.
When you switch to generalized points, there are now enough. In fact the ge=
neric point in the topos of sheaves, on its own, is enough for most purpose=
s. But then, if you wanted to adapt the Chu spaces somehow to allow that in=
, you might as well take it on its own.
Hope that helps,
Steve.
________________________________
From: Vaughan Pratt
Sent: Sunday, January 22, 2023 9:32 PM
To: Steven Vickers (Computer Science)
Cc: categories@mta.ca
Subject: Re: categories: Terminology for point-free topology?
Hi Steve,
"Classically, it is not unreasonable to view lack of global points as a pat=
hology in the locale Y; and then the constructive tendency to lack global p=
oints appears as pathology in the logic." (Your reply to me here of Jan. 1=
7)
Thanks for that and your accompanying remarks , Steve.
Space is both extroverted (Euclid's relatively clear Postulate 2 that a fin=
ite straight line can be produced) and introverted (Euclid's vaguer Definit=
ion 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 natu=
re through states.
After all, we have Hoelder's 1901 notion of a linearly ordered group for th=
e 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-Leinste=
r notion of the continuum as a final coalgebra, which can be as small as th=
e unit interval if you stick to midpoint algebras (rather than continued fr=
actions as Dusko and I did in 1999) and as such ideal for filling in the ga=
ps between consecutive integers.
That Euclid's Definition 2 is vaguer than his Postulate 2 is consistent wit=
h the applicability of free algebras to the extroverted nature of space app=
earing much earlier than that of final coalgebras to its introverted nature=
.
These thoughts came to me after spending a few weeks mulling over a convers=
ation I had with my classmate (1962-5) Ross Street about our common but ind=
ependently 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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