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```From: Steven Vickers <s.j.vickers.1@bham.ac.uk>
To: Vaughan Pratt <pratt@cs.stanford.edu>
Cc: "categories@mta.ca" <categories@mta.ca>
Subject: Re:  Terminology for point-free topology?
Date: Mon, 23 Jan 2023 13:25:04 +0000	[thread overview]
Message-ID: <E1pK54j-00041C-5R@rr.mta.ca> (raw)

Dear Vaughan,

I've worked point-free on midpoint algebras (my paper that [-1,1] is an Escardo-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 whether 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 = R^2). It may be that a good way to do that is (1) (introvert?) define a midpoint homomorphism from [0, 1] to a region of S^1 close to 1, with 0 mapping to 1, and then (2) (extrovert?) use the homomorphism property to extend to the whole of R.

I'll turn now to the Chu spaces. If I understand properly what you're suggesting, it's to replace (Set, 2) by some (E, k). That's not going to solve the issue with lack of points that I was talking about. If anything, it makes 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 chosen base topos. However, insisting on a set on the points side smashes too much topological structure to work well in general. As my example with Sierpinski showed, it in effect forces you to approximate bundles with local homeomorphisms, and that can leave you with nothing useful that is available for the points side of the Chu space.

When you switch to generalized points, there are now enough. In fact the generic point in the topos of sheaves, on its own, is enough for most purposes. 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 <pratt@cs.stanford.edu>
Sent: Sunday, January 22, 2023 9:32 PM
To: Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>
Cc: categories@mta.ca <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 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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```

```next prev parent reply	other threads:[~2023-01-23 13:25 UTC|newest]

Thread overview: 19+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2023-01-16 11:50 Steven Vickers
2023-01-18 12:12   ` Steven Vickers
2023-01-20  3:06     ` David Yetter
2023-01-20 11:50       ` Steven Vickers
2023-01-21 19:42         ` ptj
2023-01-23 11:44           ` Pedro Resende
2023-01-30 21:59             ` categories: " Wesley Phoa
2023-02-01  9:41               ` Martin Hyland
2023-01-23 13:47       ` Steven Vickers
2023-01-24 12:20       ` categories: " Robert Pare
2023-01-27 17:55     ` Pedro Resende
2023-01-28  5:43       ` Patrik Eklund
2023-01-29 23:16         ` dawson
2023-01-28 10:48       ` categories: complete Galois groups Clemens Berger
2023-01-30 17:34         ` categories: " Eduardo J. Dubuc
2023-01-22 21:32   ` Terminology for point-free topology? Vaughan Pratt
2023-01-23 13:25   ` Steven Vickers [this message]
2023-01-23 23:17   ` categories: " Vaughan Pratt
2023-01-24 11:45   ` Steven Vickers
```

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