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From: Vaughan Pratt
Date: Mon, 23 Jan 2023 15:17:09 -0800
Subject: categories: Re: Terminology for point-free topology?
To: Steven Vickers
CC: "categories@mta.ca"
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Reply-To: Vaughan Pratt
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" 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."
In my case the problem I'm having with Chu(E,k) is that I have no intuition
about E-enriched categories. A given Chu space (a,r,x) would have a and x
each be an object of E, with x a frame defined by inclusion. What I can't
picture is r: a x x =E2=86=92 k.
"In fact the generic point in the topos of sheaves, on its own, is enough
for most purposes."
Would that be the dense point?
"But then, if you wanted to adapt the Chu spaces somehow to allow that in,
you might as well take it on its own."
The idea of a dense point seems right. However maybe there's a simple way
to approach it in Chu(Set,2). My earlier question was, how many points are
needed to support representing a local as a Chu space over 2? It seems to
me that a dense set of points (suitably defined) would suffice, e.g. in a
flat n-dimensional space, D^n where D consists of the dyadic rationals.
The intersection of the line y =3D x with the unit circle in the upper righ=
t
quadrant would presumably be the dense point at (1,1)/=E2=88=9A2.
But Euclidean geometry is just real algebraic geometry limited to degree 2,
suggesting replacing D by the field of constructible numbers. You wouldn't
need the notion of a dense point because all Euclidean constructions would
produce a real point, but space would still be just as countable as D^n.
Space would still have atomless parts, namely the uncountably many points
created by Dedekind or Cantor (the latter using Cauchy sequences) that
aren't needed for Euclidean geometry.
For real algebraic geometry, just drop the limit of 2 on degree of
polynomials. And drop "real" for general algebraic geometry.
All this is to permit locales to be represented as Chu spaces over 2,
namely by requiring their set of states to be a locale where the order is
given by set inclusion.
But if "point-free" means replacing the concrete notion of the point
(1,1)/=E2=88=9A2 itself with the generic dense point located there, I no lo=
nger see
how to represent a locale as a Chu space of any kind.
If there's better (e.g. more pedagogically suitable) language than what I
used above, I'm all ears.
Vaughan
On Mon, Jan 23, 2023 at 5:25 AM Steven Vickers
wrote:
> 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 th=
e
> same with trigonometry?
>
> That boils down to defining group homomorphisms from R to the circle grou=
p
> S^1 (viewed as a sublocale of C =3D 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 pairin=
g
> 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 wit=
h
> 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 allo=
w
> 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
> 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 extroverte=
d
> 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 spac=
e
> 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 commo=
n
> 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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