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```* Terminology for point-free topology?
@ 2023-01-16 11:50 Steven Vickers
` (4 more replies)
0 siblings, 5 replies; 19+ messages in thread
From: Steven Vickers @ 2023-01-16 11:50 UTC (permalink / raw)
To: categories

I'm wondering if there's any consensus usage to found for "point-free" topology and related terms.

I've posted a detailed discussion on https://arxiv.org/abs/2206.01113, but I can summarize the question more succinctly.

It's not unusual to distinguish between two synonymous pairs:
point-set/pointwise = ordinary semantics of general topology,
point-free/pointless = reformed semantics of, e.g., locales or formal topology.

However, that is misleading, as locale theory can be validly done using points. See, e.g., Ng-Vickers on real exp and log, https://lmcs.episciences.org/9879. The trick is to restrict to geometric constructions and to apply them to *generalized* points, to be found in arbitrary Grothendieck toposes and not just Set (or your chosen base S).

Thus there are two distinctions to be made -

1 Ordinary semantics v. reformed
2 Use points v. avoid them

Some terms naturally fall into place.

Point-set = ordinary topology, points taken from a given set.

Pointwise = use points. Point-set is a subclass of pointwise, but strict,  as shown by the above example.

What about pointless and point-free? I'm piloting -

Pointless = avoid points (e.g. construct locale maps concretely as frame homomorphisms). There's some value judgement in my choice there, as very often the pointwise reasoning is simpler and more transparent, so there seems  to be no good reason for arguing pointlessly.

Point-free = reformed topology. I try to think of this as meaning that the points are liberated from their confinement to Set or S.

Does anyone have comments on these, or suggestions for other phrases for the concepts?

Happy New Year!

Steve Vickers.

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• `[parent not found: <a23cbfc0-7433-eec1-115e-af2041d4cfd1@uu.nl>]`
• ```* Re: Terminology for point-free topology?
@ 2023-01-18 12:12   ` Steven Vickers
2023-01-20  3:06     ` David Yetter
` (4 more replies)
0 siblings, 5 replies; 19+ messages in thread
From: Steven Vickers @ 2023-01-18 12:12 UTC (permalink / raw)
To: I.Moerdijk; +Cc: categories list

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 classifying topos of a continuous groupoid I" (1988), where you said -

"... in presenting many arguments concerning generalized, "pointless" spaces, 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 arguments are perfectly suitable for dealing with pointless spaces (at least as long 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 into hiding. In fact I explicitly wrote "Locales and toposes as spaces" as a guide 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 before I knew those papers of yours, but I felt right from the start that I was merely unveiling techniques already known to the experts - though I hope  you'll agree I've been more explicit about them and particularly the nature 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 McLarty to the effect that (modulo misrepresentation by me) Grothendieck was  aware of two different lines of reasoning with toposes: by manipulating sites concretely, or by using colimits and finite limits under the rules corresponding to Giraud's theorem. I imagine that as being something like the distinction between pointless and pointwise.

Best wishes,

Steve.

________________________________

Hi Steve,

A very early illustration of the strategy of using points in pointless topology 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 from general topos theory, which enable one to follow classical arguments about points of separable metric spaces rather closely. Although both approaches are equivalent, we will follow the second one, because it shows more clearly the interplay between general topos theory and arguments (somewhat similar 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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• ```* Re: Terminology for point-free topology?
2023-01-18 12:12   ` Steven Vickers
@ 2023-01-20  3:06     ` David Yetter
` (3 subsequent siblings)
4 siblings, 0 replies; 19+ messages in thread
From: David Yetter @ 2023-01-20  3:06 UTC (permalink / raw)
To: categories list

I seem to recall from back in my days as a grad student or new PhD that Peter Johnstone wrote a paper entitled "The Point of Pointless Topology".  Just in honor of that I've always favored "pointless topology" as the term for  the theory of locales and sheaves on locales.

Best Thoughts,
David Y.

________________________________
From: Steven Vickers <s.j.vickers.1@bham.ac.uk>
Sent: Wednesday, January 18, 2023 6:12 AM
To: I.Moerdijk@uu.nl <I.Moerdijk@uu.nl>
Cc: categories list <categories@mta.ca>
Subject: categories: Re: Terminology for point-free topology?

This email originated from outside of K-State.

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 classifying topos of a continuous groupoid I" (1988), where you said -

"... in presenting many arguments concerning generalized, "pointless" spaces, 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 arguments are perfectly suitable for dealing with pointless spaces (at least as long 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 into hiding. In fact I explicitly wrote "Locales and toposes as spaces" as a guide 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 before I knew those papers of yours, but I felt right from the start that I was merely unveiling techniques already known to the experts - though I hope  you'll agree I've been more explicit about them and particularly the nature 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 McLarty to the effect that (modulo misrepresentation by me) Grothendieck was   aware of two different lines of reasoning with toposes: by manipulating sites concretely, or by using colimits and finite limits under the rules corresponding to Giraud's theorem. I imagine that as being something like the distinction between pointless and pointwise.

Best wishes,

Steve.

________________________________

Hi Steve,

A very early illustration of the strategy of using points in pointless topology 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 from general topos theory, which enable one to follow classical arguments about points of separable metric spaces rather closely. Although both approaches are equivalent, we will follow the second one, because it shows more clearly the interplay between general topos theory and arguments (somewhat similar 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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• `[parent not found: <SN6PR05MB5213EBE225CB83D101EA0F57A2C59@SN6PR05MB5213.namprd05.prod.outlook.com>]`
• ```* Re: Terminology for point-free topology?
@ 2023-01-20 11:50       ` Steven Vickers
2023-01-21 19:42         ` ptj
0 siblings, 1 reply; 19+ messages in thread
From: Steven Vickers @ 2023-01-20 11:50 UTC (permalink / raw)
To: David Yetter, I.Moerdijk; +Cc: categories list

Dear David,

Yes, and it's an excellent paper with a witty title for which only "pointless" would do.

I particularly like what Peter said when explaining the significant difference in the absence of choice (such as in toposes of sheaves), and that "usually it is locales, not spaces, which provide the right context in which to  do topology".

He went on to say,

"This is the point which ... Andre Joyal began to hammer home in the early 1970s; I can well remember how, at the time, his insistence that locales were the real stuff of topology, and spaces were merely figments of the classical mathematician's imagination, seemed (to me, and I suspect to others) like unmotivated fanaticism. I have learned better since then."

This is all part of the argument for using a reformed topology, but there is nothing particular there about the pointwise style of reasoning for it. Hence we are still left with the question of how to reference the two concepts, the reformed topology and the reasoning without points.

Would you call Ng's paper with me pointless? Points are everywhere in it. (Of course, there's the separate issue of whether it was pointless in the sense of not worth the trouble. But an important feature of the style is that  it forces you to be careful to distinguish between Dedekind reals and 1-sided (lower or upper) reals, and in Ng's thesis this uncovered unexpected roles of 1-sided reals in the account of Ostrowski's Theorem and the Berkovich spectrum. So there is a bit of payoff.)

Best wishes,

Steve.

________________________________
From: David Yetter <dyetter@ksu.edu>
Sent: Friday, January 20, 2023 3:06 AM
To: I.Moerdijk@uu.nl <I.Moerdijk@uu.nl>; Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>
Cc: categories list <categories@mta.ca>
Subject: Re: categories: Re: Terminology for point-free topology?

I seem to recall from back in my days as a grad student or new PhD that Peter Johnstone wrote a paper entitled "The Point of Pointless Topology".  Just in honor of that I've always favored "pointless topology" as the term for  the theory of locales and sheaves on locales.

Best Thoughts,
David Y.

________________________________
From: Steven Vickers <s.j.vickers.1@bham.ac.uk>
Sent: Wednesday, January 18, 2023 6:12 AM
To: I.Moerdijk@uu.nl <I.Moerdijk@uu.nl>
Cc: categories list <categories@mta.ca>
Subject: categories: Re: Terminology for point-free topology?

This email originated from outside of K-State.

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 classifying topos of a continuous groupoid I" (1988), where you said -

"... in presenting many arguments concerning generalized, "pointless" spaces, 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 arguments are perfectly suitable for dealing with pointless spaces (at least as long 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 into hiding. In fact I explicitly wrote "Locales and toposes as spaces" as a guide 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 before I knew those papers of yours, but I felt right from the start that I was merely unveiling techniques already known to the experts - though I hope  you'll agree I've been more explicit about them and particularly the nature 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 McLarty to the effect that (modulo misrepresentation by me) Grothendieck was   aware of two different lines of reasoning with toposes: by manipulating sites concretely, or by using colimits and finite limits under the rules corresponding to Giraud's theorem. I imagine that as being something like the distinction between pointless and pointwise.

Best wishes,

Steve.

________________________________

Hi Steve,

A very early illustration of the strategy of using points in pointless topology 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 from general topos theory, which enable one to follow classical arguments about points of separable metric spaces rather closely. Although both approaches are equivalent, we will follow the second one, because it shows more clearly the interplay between general topos theory and arguments (somewhat similar 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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• ```* Re: Terminology for point-free topology?
2023-01-20 11:50       ` Steven Vickers
@ 2023-01-21 19:42         ` ptj
2023-01-23 11:44           ` Pedro Resende
0 siblings, 1 reply; 19+ messages in thread
From: ptj @ 2023-01-21 19:42 UTC (permalink / raw)
To: Steven Vickers; +Cc: categories list

I was wondering how long it would be before someone in this thread
referred to my `point of pointless topology' paper! Perhaps not so many
people know that the title was a conscious echo of an earlier paper
by Mike Barr called `The point of the empty set', which began with the
words (I quote from memory) `The point is, there isn't any point there;
that's exactly the point'.

As Steve says, to fit that title I had to use the word `pointless', but
on the whole I prefer `pointfree'; it carries the implication that you
are free to work without points or to use them (in a generalized sense),
as you prefer.

Peter Johnstone

On Jan 21 2023, Steven Vickers wrote:

>Dear David,
>
> Yes, and it's an excellent paper with a witty title for which only
> "pointless" would do.
>
> I particularly like what Peter said when explaining the significant
> difference in the absence of choice (such as in toposes of sheaves), and
> that "usually it is locales, not spaces, which provide the right context
> in which to do topology".
>
>He went on to say,
>
> "This is the point which ... Andre Joyal began to hammer home in the
> early 1970s; I can well remember how, at the time, his insistence that
> locales were the real stuff of topology, and spaces were merely figments
> of the classical mathematician's imagination, seemed (to me, and I
> suspect to others) like unmotivated fanaticism. I have learned better
> since then."
>
> This is all part of the argument for using a reformed topology, but there
> is nothing particular there about the pointwise style of reasoning for
> it. Hence we are still left with the question of how to reference the two
> concepts, the reformed topology and the reasoning without points.
>
> Would you call Ng's paper with me pointless? Points are everywhere in it.
> (Of course, there's the separate issue of whether it was pointless in the
> sense of not worth the trouble. But an important feature of the style is
> that it forces you to be careful to distinguish between Dedekind reals
> and 1-sided (lower or upper) reals, and in Ng's thesis this uncovered
> unexpected roles of 1-sided reals in the account of Ostrowski's Theorem
> and the Berkovich spectrum. So there is a bit of payoff.)
>
>Best wishes,
>
>Steve.
>
> ________________________________ From: David Yetter <dyetter@ksu.edu>
> Sent: Friday, January 20, 2023 3:06 AM To: I.Moerdijk@uu.nl
> <I.Moerdijk@uu.nl>; Steven Vickers (Computer Science)
> <s.j.vickers.1@bham.ac.uk> Cc: categories list <categories@mta.ca>
> Subject: Re: categories: Re: Terminology for point-free topology?
>
> I seem to recall from back in my days as a grad student or new PhD that
> Peter Johnstone wrote a paper entitled "The Point of Pointless Topology".
> Just in honor of that I've always favored "pointless topology" as the
> term for the theory of locales and sheaves on locales.
>
>Best Thoughts,
>David Y.
>
>________________________________
>From: Steven Vickers <s.j.vickers.1@bham.ac.uk>
>Sent: Wednesday, January 18, 2023 6:12 AM
>To: I.Moerdijk@uu.nl <I.Moerdijk@uu.nl>
>Cc: categories list <categories@mta.ca>
>Subject: categories: Re: Terminology for point-free topology?
>
>This email originated from outside of K-State.
>
>
>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
> classifying topos of a continuous groupoid I" (1988), where you said -
>
> "... in presenting many arguments concerning generalized, "pointless"
> spaces, 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 arguments are perfectly suitable for
> dealing with pointless spaces (at least as long 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
> into hiding. In fact I explicitly wrote "Locales and toposes as spaces"
> as a guide 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 before I knew those papers of yours, but I felt right from the start
> that I was merely unveiling techniques already known to the experts -
> though I hope you'll agree I've been more explicit about them and
> particularly the nature 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
> McLarty to the effect that (modulo misrepresentation by me) Grothendieck
> was aware of two different lines of reasoning with toposes: by
> manipulating sites concretely, or by using colimits and finite limits
> under the rules corresponding to Giraud's theorem. I imagine that as
> being something like the distinction between pointless and pointwise.
>
>Best wishes,
>
>Steve.

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• ```* Re: Terminology for point-free topology?
2023-01-21 19:42         ` ptj
@ 2023-01-23 11:44           ` Pedro Resende
2023-01-30 21:59             ` categories: " Wesley Phoa
0 siblings, 1 reply; 19+ messages in thread
From: Pedro Resende @ 2023-01-23 11:44 UTC (permalink / raw)
To: categories list

In addition to all the deeper reasons, `pointless’ can be taken to be derogatory, so preferably it should be used only when in tongue-in-cheek mode. At least that’s what I tell my students — just as I ask them not to say `abstract nonsense’ too enthusiastically… :)

Pedro

> On Jan 21, 2023, at 7:42 PM, ptj@maths.cam.ac.uk wrote:
>
> I was wondering how long it would be before someone in this thread
> referred to my `point of pointless topology' paper! Perhaps not so many
> people know that the title was a conscious echo of an earlier paper
> by Mike Barr called `The point of the empty set', which began with the
> words (I quote from memory) `The point is, there isn't any point there;
> that's exactly the point'.
>
> As Steve says, to fit that title I had to use the word `pointless', but
> on the whole I prefer `pointfree'; it carries the implication that you
> are free to work without points or to use them (in a generalized sense),
> as you prefer.
>
> Peter Johnstone
>
> On Jan 21 2023, Steven Vickers wrote:
>
>> Dear David,
>>
>> Yes, and it's an excellent paper with a witty title for which only
>> "pointless" would do.
>>
>> I particularly like what Peter said when explaining the significant
>> difference in the absence of choice (such as in toposes of sheaves), and
>> that "usually it is locales, not spaces, which provide the right context
>> in which to do topology".
>>
>> He went on to say,
>>
>> "This is the point which ... Andre Joyal began to hammer home in the
>> early 1970s; I can well remember how, at the time, his insistence that
>> locales were the real stuff of topology, and spaces were merely figments
>> of the classical mathematician's imagination, seemed (to me, and I
>> suspect to others) like unmotivated fanaticism. I have learned better
>> since then."
>>
>> This is all part of the argument for using a reformed topology, but there
>> is nothing particular there about the pointwise style of reasoning for
>> it. Hence we are still left with the question of how to reference the two
>> concepts, the reformed topology and the reasoning without points.
>>
>> Would you call Ng's paper with me pointless? Points are everywhere in it.
>> (Of course, there's the separate issue of whether it was pointless in the
>> sense of not worth the trouble. But an important feature of the style is
>> that it forces you to be careful to distinguish between Dedekind reals
>> and 1-sided (lower or upper) reals, and in Ng's thesis this uncovered
>> unexpected roles of 1-sided reals in the account of Ostrowski's Theorem
>> and the Berkovich spectrum. So there is a bit of payoff.)
>>
>> Best wishes,
>>
>> Steve.
>>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• ```* categories: Re: Terminology for point-free topology?
2023-01-23 11:44           ` Pedro Resende
@ 2023-01-30 21:59             ` Wesley Phoa
2023-02-01  9:41               ` Martin Hyland
0 siblings, 1 reply; 19+ messages in thread
From: Wesley Phoa @ 2023-01-30 21:59 UTC (permalink / raw)
To: Pedro Resende; +Cc: categories list

I've been out of mathematics for three decades, so I feel qualified to
represent the lay audience in this discussion.

Mathematicians use the word "space" to refer to three concepts which, to a
lay person, seem completely unrelated:

1. a space of parameters: e.g. a space of moduli, a configuration space,
parameter space for a neural network
2. a thing with a shape: e.g. a doughnut, a coffee cup, a Klein bottle, a
tesseract
3. empty space: e.g. Euclidean space, curved spacetimes, the higher
dimensional spaces in string theory

These concepts have quite different (lay) intuitions associated with them:*

1. this kind of space obviously has points, but it's tricky to grasp what
cohesion means
2. this kind of space is obviously cohesive, but it's a leap to think of it
as made up of points
3. it doesn't obviously/naively make sense to talk about either points or
cohesion when there's nothing there

The fact that there are formalisms in which #1 and #2 are "the same thing"
is surprising, amazing and powerful. And the fact that there are several
formalisms, even more so!

So you wouldn't expect there to be a single language that feels natural to
everyone, in all three settings.* Any more than you would expect to find a
single "best" formalism.

Wesley

*Further confusion ensues as some of these concepts ramify further, e.g.
"cohesion" into continuity, smoothness etc., "point" as a bare point, a
point with symmetries, a point with an extent... We've gotten used to
regarding these as living inside different subject areas within
mathematics, but that wasn't obvious* ex ante*.

On Mon, Jan 23, 2023 at 2:18 PM Pedro Resende <
pedro.m.a.resende@tecnico.ulisboa.pt> wrote:

> In addition to all the deeper reasons, `pointless’ can be taken to be
> derogatory, so preferably it should be used only when in tongue-in-cheek
> mode. At least that’s what I tell my students — just as I  ask them not to
> say `abstract nonsense’ too enthusiastically… :)
>
> Pedro
>
>> On Jan 21, 2023, at 7:42 PM, ptj@maths.cam.ac.uk wrote:
>>
>> I was wondering how long it would be before someone in this thread
>> referred to my `point of pointless topology' paper! Perhaps not so many
>> people know that the title was a conscious echo of an earlier paper
>> by Mike Barr called `The point of the empty set', which began with the
>> words (I quote from memory) `The point is, there isn't any point there;
>> that's exactly the point'.
>>
>> As Steve says, to fit that title I had to use the word `pointless', but
>> on the whole I prefer `pointfree'; it carries the implication that you
>> are free to work without points or to use them (in a generalized sense),
>> as you prefer.
>>
>> Peter Johnstone

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• ```* categories: Re: Terminology for point-free topology?
2023-01-30 21:59             ` categories: " Wesley Phoa
@ 2023-02-01  9:41               ` Martin Hyland
0 siblings, 0 replies; 19+ messages in thread
From: Martin Hyland @ 2023-02-01  9:41 UTC (permalink / raw)
To: Wesley Phoa; +Cc: Pedro Resende, categories list

[-- Warning: decoded text below may be mangled, UTF-8 assumed --]
[-- Attachment #1: Type: text/plain, Size: 4289 bytes --]

The lay audience is very sensible. Further in the direction of not expecting
any best formulation I add some remarks.

1. From maybe mid last century we came to see many structures where there
is a notion of point but where it is important that there are not enough points.
(There is more than one sense of that … .) It is a question in the history of
thought whether the idea of a space as made up of points predates set
theory. Bill Lawvere liked to stress that in Greek geometry there were
other figures - lines, triangles whatever.
2. Thinking of Bill suggests taking as a *modern* starting point the idea
that a space is an object in a category of spaces. That is parallel to
the idea that a vector is an element in a vector space. But of course that
idea has limitations as e.g. in the theory of forces on a rigid body.
Similarly a category of spaces may only get one so far. Wesley
mention points with symmetries as e.g. in the space of triangles.
We have yet to develop a background theory there?

None of that helps re nomenclature which we can influence though
hardly control. But I do not know what any of us can do beyond
stressing the value of abstract mathematics. Not easy in a
scornful world … .

Martin

> On 30 Jan 2023, at 21:59, Wesley Phoa <doctorwes@gmail.com> wrote:
>
> I've been out of mathematics for three decades, so I feel qualified to
> represent the lay audience in this discussion.
>
> Mathematicians use the word "space" to refer to three concepts which, to a
> lay person, seem completely unrelated:
>
> 1. a space of parameters: e.g. a space of moduli, a configuration space,
> parameter space for a neural network
> 2. a thing with a shape: e.g. a doughnut, a coffee cup, a Klein bottle, a
> tesseract
> 3. empty space: e.g. Euclidean space, curved spacetimes, the higher
> dimensional spaces in string theory
>
> These concepts have quite different (lay) intuitions associated with them:*
>
> 1. this kind of space obviously has points, but it's tricky to grasp what
> cohesion means
> 2. this kind of space is obviously cohesive, but it's a leap to think of it
> as made up of points
> 3. it doesn't obviously/naively make sense to talk about either points or
> cohesion when there's nothing there
>
> The fact that there are formalisms in which #1 and #2 are "the same thing"
> is surprising, amazing and powerful. And the fact that there are several
> formalisms, even more so!
>
> So you wouldn't expect there to be a single language that feels natural to
> everyone, in all three settings.* Any more than you would expect to find a
> single "best" formalism.
>
> Wesley
>
> *Further confusion ensues as some of these concepts ramify further, e.g.
> "cohesion" into continuity, smoothness etc., "point" as a bare point, a
> point with symmetries, a point with an extent... We've gotten used to
> regarding these as living inside different subject areas within
> mathematics, but that wasn't obvious* ex ante*.
>
> On Mon, Jan 23, 2023 at 2:18 PM Pedro Resende <
> pedro.m.a.resende@tecnico.ulisboa.pt> wrote:
>
>> In addition to all the deeper reasons, `pointless’ can be taken to be
>> derogatory, so preferably it should be used only when in tongue-in-cheek
>> mode. At least that’s what I tell my students — just as I  ask them not to
>> say `abstract nonsense’ too enthusiastically… :)
>>
>> Pedro
>>
>>> On Jan 21, 2023, at 7:42 PM, ptj@maths.cam.ac.uk wrote:
>>>
>>> I was wondering how long it would be before someone in this thread
>>> referred to my `point of pointless topology' paper! Perhaps not so many
>>> people know that the title was a conscious echo of an earlier paper
>>> by Mike Barr called `The point of the empty set', which began with the
>>> words (I quote from memory) `The point is, there isn't any point there;
>>> that's exactly the point'.
>>>
>>> As Steve says, to fit that title I had to use the word `pointless', but
>>> on the whole I prefer `pointfree'; it carries the implication that you
>>> are free to work without points or to use them (in a generalized sense),
>>> as you prefer.
>>>
>>> Peter Johnstone
>

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• `[parent not found: <18E1AA5F-0054-4CA3-B231-BD9B799B03A2@tecnico.ulisboa.pt>]`
• ```* Re: Terminology for point-free topology?
@ 2023-01-23 13:47       ` Steven Vickers
0 siblings, 0 replies; 19+ messages in thread
From: Steven Vickers @ 2023-01-23 13:47 UTC (permalink / raw)
To: categories list

Dear Pedro,

Of course, that's the very reason why I wanted to transfer it to the style of working without points.

That's slightly unfair, in that in many cases of reasoning algebraically, without points, it's not at all clear how to do it pointwise.

You and I have certainly experienced that in our work on quantales, which are much more purely algebraic gadgets. Our approach via localic suplattices  (algebras for the lower hyperspace monad) gives a more point-free approach  to the subject, but it takes effort - I think you'll agree - to work with the hyperspaces in a pointwise manner.

Do you think there's a less derogatory term for the style of reasoning without points?

All the best,

Steve.

________________________________
From: pedro.m.a.resende@tecnico.ulisboa.pt <pedro.m.a.resende@tecnico.ulisboa.pt>
Sent: Monday, January 23, 2023 11:44 AM
To: ptj@maths.cam.ac.uk <ptj@maths.cam.ac.uk>
Cc: Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>; categories list <categories@mta.ca>
Subject: Re: categories: Re: Terminology for point-free topology?

In addition to all the deeper reasons, `pointless’ can be taken to be derogatory, so preferably it should be used only when in tongue-in-cheek mode.  At least that’s what I tell my students — just as I ask them not to say `abstract nonsense’ too enthusiastically… :)

Pedro

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• `[parent not found: <YQXPR01MB26464DF33EAE7481847A4F82E5C99@YQXPR01MB2646.CANPRD01.PROD.OUTLOOK.COM>]`
• ```* categories: Re: Terminology for point-free topology?
@ 2023-01-24 12:20       ` Robert Pare
0 siblings, 0 replies; 19+ messages in thread
From: Robert Pare @ 2023-01-24 12:20 UTC (permalink / raw)
To: categories

Dear Steve,

I've been following this thread with interest, though I've never worked
on the subject. In trying to understand what it's actually all about, I've
come to the conclusion that it's "copoint topology". Is that Crazy?

Cheers,

Bob
________________________________
From: Steven Vickers <s.j.vickers.1@bham.ac.uk>
Sent: January 23, 2023 9:47 AM
To: categories list <categories@mta.ca>
Subject: categories: Re: Terminology for point-free topology?

Dear Pedro,

Of course, that's the very reason why I wanted to transfer it to the style of working without points.

That's slightly unfair, in that in many cases of reasoning algebraically, without points, it's not at all clear how to do it pointwise.

You and I have certainly experienced that in our work on quantales, which are much more purely algebraic gadgets. Our approach via localic suplattices   (algebras for the lower hyperspace monad) gives a more point-free approach  to the subject, but it takes effort - I think you'll agree - to work with the hyperspaces in a pointwise manner.

Do you think there's a less derogatory term for the style of reasoning without points?

All the best,

Steve.

________________________________
From: pedro.m.a.resende@tecnico.ulisboa.pt <pedro.m.a.resende@tecnico.ulisboa.pt>
Sent: Monday, January 23, 2023 11:44 AM
To: ptj@maths.cam.ac.uk <ptj@maths.cam.ac.uk>
Cc: Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>; categories list <categories@mta.ca>
Subject: Re: categories: Re: Terminology for point-free topology?

In addition to all the deeper reasons, `pointless’ can be taken to be derogatory, so preferably it should be used only when in tongue-in-cheek mode.   At least that’s what I tell my students — just as I ask them not to say `abstract nonsense’ too enthusiastically… :)

Pedro

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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<br>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>From:</b> Robert Pare &lt;R.Pare@Dal.Ca&gt;<br>
<b>Sent:</b> January 24, 2023 8:19 AM<br>
<b>To:</b> Steven Vickers &lt;s.j.vickers.1@bham.ac.uk&gt;<br>
<b>Cc:</b> categories@mta &lt;categories@mta&gt;<br>
<b>Subject:</b> Re: categories: Re: Terminology for point-free topology?</font>
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Dear Steve,</div>
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I've been following this thread with interest, though I've never worked</div>
<div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)">
on the subject. In trying to understand what it's actually all about, I've</div>
<div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)">
come to the conclusion that it's &quot;copoint topology&quot;. Is that Crazy?</div>
<div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)">
<br>
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Cheers,</div>
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<br>
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Bob<br>
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<div id="x_divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" color="#000000" style="font-size:11pt"><b>From:</b> Steven Vickers &lt;s.j.vickers.1@bham.ac.uk&gt;<br>
<b>Sent:</b> January 23, 2023 9:47 AM<br>
<b>To:</b> categories list &lt;categories@mta.ca&gt;<br>
<b>Subject:</b> categories: Re: Terminology for point-free topology?</font>
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<br>
Dear Pedro,<br>
<br>
Of course, that's the very reason why I wanted to transfer it to the style of working without points.<br>
<br>
That's slightly unfair, in that in many cases of reasoning algebraically, without points, it's not at all clear how to do it pointwise.<br>
<br>
You and I have certainly experienced that in our work on quantales, which are much more purely algebraic gadgets. Our approach via localic suplattices&nbsp; (algebras for the lower hyperspace monad) gives a more point-free approach&nbsp; to the subject, but it takes
effort - I think you'll agree - to work with the hyperspaces in a pointwise manner.<br>
<br>
Do you think there's a less derogatory term for the style of reasoning without points?<br>
<br>
All the best,<br>
<br>
Steve.<br>
<br>
________________________________<br>
From: pedro.m.a.resende@tecnico.ulisboa.pt &lt;pedro.m.a.resende@tecnico.ulisboa.pt&gt;<br>
Sent: Monday, January 23, 2023 11:44 AM<br>
To: ptj@maths.cam.ac.uk &lt;ptj@maths.cam.ac.uk&gt;<br>
Cc: Steven Vickers (Computer Science) &lt;s.j.vickers.1@bham.ac.uk&gt;; categories list &lt;categories@mta.ca&gt;<br>
Subject: Re: categories: Re: Terminology for point-free topology?<br>
<br>
In addition to all the deeper reasons, `pointless’ can be taken to be derogatory, so preferably it should be used only when in tongue-in-cheek mode.&nbsp; At least that’s what I tell my students — just as I ask them not  to say `abstract nonsense’ too enthusiastically…
:)<br>
<br>
Pedro<br>
<br>
<br>
[For admin and other information see: <a href="http://www.mta.ca/~cat-dist/">http://www.mta.ca/~cat-dist/</a> ]<br>
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• ```* categories: Re: Terminology for point-free topology?
2023-01-18 12:12   ` Steven Vickers
` (3 preceding siblings ...)
@ 2023-01-27 17:55     ` Pedro Resende
2023-01-28  5:43       ` Patrik Eklund
2023-01-28 10:48       ` categories: complete Galois groups Clemens Berger
4 siblings, 2 replies; 19+ messages in thread
From: Pedro Resende @ 2023-01-27 17:55 UTC (permalink / raw)
To: Steven Vickers; +Cc: categories list

Hi Steve,

Sorry for the radio silence, it’s been a hectic week.

Concerning your question about a less derogatory expression… I think I like `algebraic reasoning’ versus `point-based reasoning’ (which to me sounds better than `pointwise', I don’t know why).

This is analogous to commutative algebra versus algebraic geometry.

In any case, am I right that it seems to be somewhat consensual (in this thread) that `pointfree topology’ is the appropriate terminology for the kind of topology that *can* (but not necessarily has to) be studied without reasoning in terms of points?

Incidentally, in my mind the `pointfree' terminology should also apply to more general notions, such as quantales, or at least some classes of them. For instance, inverse quantal frames are `the same' as localic etale groupoids, and they have associated etendues.

Best wishes,

Pedro

> On Jan 23, 2023, at 1:47 PM, Steven Vickers <s.j.vickers.1@bham.ac.uk> wrote:
>
> Dear Pedro,
>
> Of course, that's the very reason why I wanted to transfer it to the style of working without points.
>
> That's slightly unfair, in that in many cases of reasoning algebraically, without points, it's not at all clear how to do it pointwise.
>
> You and I have certainly experienced that in our work on quantales, which are much more purely algebraic gadgets. Our approach via localic suplattices (algebras for the lower hyperspace monad) gives a more point-free approach to the subject, but it takes effort - I think you'll agree - to work with the hyperspaces in a pointwise manner.
>
> Do you think there's a less derogatory term for the style of reasoning without points?
>
> All the best,
>
> Steve.
>
> From: pedro.m.a.resende@tecnico.ulisboa.pt > Sent: Monday, January 23, 2023 11:44 AM
> To: ptj@maths.cam.ac.uk <mailto:ptj@maths.cam.ac.uk> <ptj@maths.cam.ac.uk <mailto:ptj@maths.cam.ac.uk>>
> Cc: Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk <mailto:s.j.vickers.1@bham.ac.uk>>; categories list <categories@mta.ca <mailto:categories@mta.ca>>
> Subject: Re: categories: Re: Terminology for point-free topology?
>
> In addition to all the deeper reasons, `pointless’ can be taken to be derogatory, so preferably it should be used only when in tongue-in-cheek mode. At least that’s what I tell my students — just as I ask them not to say `abstract nonsense’ too enthusiastically… :)
>
> Pedro

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• ```* categories: Re: Terminology for point-free topology?
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
1 sibling, 1 reply; 19+ messages in thread
From: Patrik Eklund @ 2023-01-28  5:43 UTC (permalink / raw)
To: Pedro Resende; +Cc: Steven Vickers, categories list

Quantales are interesting examples also from the viewpoint how they may
be used for many-valued truth in logic. There are certain mechanisms
where "points can be recovered", but in my view this is not a sufficient
justification for working only algebrally in logic, or algebraically in
topology for that matter.

Propositional two-valued logic is a bit similar. The boolean values are
not pointfree but "term-free", if the analogy is allowed. We can do
something with propositional logic, but representation  of "condition"
or "state" requires "insideness, in the sense of inside points". Points
as such representing states to me makes no sense in practice. We can
surely create fancy examples, but we cannot define things like "asthma"
or logically differentiate between Alzheimer's and vascular dementia
using pointfree topology.

Pointfree or not, term-free or not, I think it is important to justify
freeness whenever the calculation machinery allows it, but at the same
time refrain from being overenthusiastic about pointfreeness in the
sense of "I can work totally without points". Such things I would call
not just pointfree but indeed pointless, in particular as such
pointlessness kind of intentionally shuts out any possibility for
real-world application.

Some parts of theoretical mathematics is about seductive tricks, and
some mathematicians fall for it. Potential practicality of even "deepest
theoretical theory" keeps feet on the ground, even if practicality is
not realizable or desirable. But my view is that we must keep
"real-world applicability" at least as a "general burden" in the sense
that all science must useful, in one way or another. Science should
never be just "aus liebe zur Kunst".

Patrik

On 2023-01-27 19:55, Pedro Resende wrote:
> Hi Steve,
>
> Sorry for the radio silence, it???s been a hectic week.
>
> Concerning your question about a less derogatory expression??? I think I
> like `algebraic reasoning??? versus `point-based reasoning??? (which to me
> sounds better than `pointwise', I don???t know why).
>
> This is analogous to commutative algebra versus algebraic geometry.
>
> In any case, am I right that it seems to be somewhat consensual (in
> this thread) that `pointfree topology??? is the appropriate terminology
> for the kind of topology that *can* (but not necessarily has to) be
> studied without reasoning in terms of points?
>
> Incidentally, in my mind the `pointfree' terminology should also apply
> to more general notions, such as quantales, or at least some classes
> of them. For instance, inverse quantal frames are `the same' as
> localic etale groupoids, and they have associated etendues.
>
> Best wishes,
>
> Pedro
>
>> On Jan 23, 2023, at 1:47 PM, Steven Vickers <s.j.vickers.1@bham.ac.uk>
>> wrote:
>>
>> Dear Pedro,
>>
>> Of course, that's the very reason why I wanted to transfer it to the
>> style of working without points.
>>
>> That's slightly unfair, in that in many cases of reasoning
>> algebraically, without points, it's not at all clear how to do it
>> pointwise.
>>
>> You and I have certainly experienced that in our work on quantales,
>> which are much more purely algebraic gadgets. Our approach via localic
>> suplattices (algebras for the lower hyperspace monad) gives a more
>> point-free approach to the subject, but it takes effort - I think
>> you'll agree - to work with the hyperspaces in a pointwise manner.
>>
>> Do you think there's a less derogatory term for the style of reasoning
>> without points?
>>
>> All the best,
>>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• ```* categories: Re: Terminology for point-free topology?
2023-01-28  5:43       ` Patrik Eklund
@ 2023-01-29 23:16         ` dawson
0 siblings, 0 replies; 19+ messages in thread
From: dawson @ 2023-01-29 23:16 UTC (permalink / raw)
To: categories list

Patrik Eklund wrote:

> But my view is that we must keep "real-world applicability"
> at least as a "general burden" in the sense
> that all science must useful, in one way or another.
> Science should never be just "aus liebe zur Kunst".

You should, of course, follow your own ethical guidance, but I
respectfully disagree that "we" must do so.

Most of us agree that a painting or a novel can be beautiful without
having an improving message, that fine wine is worthwhile even if its
health benefits are dubious, etc.  Why should science be required to
clear a utilitarian bar that other fields of human endeavour do not?
Surely the onus is on anybody making such a claim to prove it.

Secondly, supposing [purely for the sake of argument!] that science
(oddly and uniquely) has no value except its utility. In many cases the
science that proves most useful was not obviously so when it was done.
Consider, for an overused but yet valid example, the number theory of
whose inutility Hardy wrote so proudly, and which now protects much of
the world's electronic commerce. Thus, we can't use this as a guide to
what we should study _now._

Best wishes,
Robert

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• ```* categories: complete Galois groups
2023-01-27 17:55     ` Pedro Resende
2023-01-28  5:43       ` Patrik Eklund
@ 2023-01-28 10:48       ` Clemens Berger
2023-01-30 17:34         ` categories: " Eduardo J. Dubuc
1 sibling, 1 reply; 19+ messages in thread
From: Clemens Berger @ 2023-01-28 10:48 UTC (permalink / raw)
To: Pedro Resende; +Cc: Steven Vickers, categories list

Dear all,

may I ask a question which isn't really at the heart of the discussion
but nonetheless touches upon one aspect of point-free topology (and is
related to the Morita-equivalence mentioned by Pedro) ?

It is well-known that connected, locally connected, Boolean
Grothendieck toposes are equivalent to classifying toposes of localic
groups. It is also well-known that the adjunction between locales and
spaces does not carry over to induce an adjunction between localic
groups and topological groups. Yet there are subclasses of groups where
such an adjunction on group objects exists and underpins a
Morita-equivalence of the corresponding classifying toposes, namely
profinite groups and prodiscrete groups (these two are treated more or
less explicitly in SGA1/3, resp. SGA4).

More recently (in work of Noohi, Bhatt-Scholze and Caramello) an even
larger class of topological groups has been outlined where such an
adjunction should exist and the corresponding ``Galois representation
theorem'' should hold, namely the class of ``complete Galois groups''.

I use the term ``Galois group'' for a topological group whose topology
is generated (in an obvious way) by its open subgroups. Since every open
subgroup is also closed, the underlying space of a Galois group is
zero-dimensional.

Galois groups form a reflective subcategory of the category of all
topological groups, and the classifying topos of a topological group is
equivalent to the classifying topos of its Galois group reflection.
Thus, from a Morita-equivalence perspective, it is enough to study
Galois groups.

A Galois group is said to be complete if it is complete for its
two-sided uniformity. Bhatt and Scholze show that a Galois group G is
complete iff the induced morphism from G to the automorphism group
Aut(p_G) of the canonical point p_G of BG is an isomorphism. They show
more precisely that for any Galois group G, the induced morphism
G->Aut(p_G) is completion w/r to the two-sided uniformity of G. They
finally show that a locally connected Boolean Grothendieck topos with a
conservative ``tame'' point p is equivalent to BAut(p).

My question is: what is the corresponding picture on the localic group
side ? How can we characterise ``intrinsically'' toposes that are of the
form BG for a complete Galois group G ? This would somehow distinguish
those cases where point set topology still gets his hands on, from those
where we are forced to use localic techniques.

Two concluding remarks:

(1) Banaschewski (following Kriz) showed that a topological group G is
complete for its two-sided uniformity iff the frame of opens of G is a
cogroup in frames. So, in particular, we get a zero-dimensional localic
group out of any complete Galois group. Does this induce a Morita
equivalence of the respective classifying toposes ?

(2) Perhaps the generic example of a complete Galois group, which is not
prodiscrete, is the group Sigma_N of permutations of the (discrete) set
of natural numbers, endowed with the compact-open topology. The
classifying topos BSigma_N is the celebrated Schanuel topos of nominal
sets. So, to some extent, all this is about how to apply Galois
theoretical ideas to toposes that behave like the topos of nominal sets.

All the best,
Clemens.

Le 2023-01-27 18:55, Pedro Resende a ??crit??:
> Hi Steve,
>
> Sorry for the radio silence, it???s been a hectic week.
>
> Concerning your question about a less derogatory expression??? I think I
> like `algebraic reasoning??? versus `point-based reasoning??? (which to me
> sounds better than `pointwise', I don???t know why).
>
> This is analogous to commutative algebra versus algebraic geometry.
>
> In any case, am I right that it seems to be somewhat consensual (in
> this thread) that `pointfree topology??? is the appropriate terminology
> for the kind of topology that *can* (but not necessarily has to) be
> studied without reasoning in terms of points?
>
> Incidentally, in my mind the `pointfree' terminology should also apply
> to more general notions, such as quantales, or at least some classes
> of them. For instance, inverse quantal frames are `the same' as
> localic etale groupoids, and they have associated etendues.
>
> Best wishes,
>
> Pedro
>
>> On Jan 23, 2023, at 1:47 PM, Steven Vickers <s.j.vickers.1@bham.ac.uk>
>> wrote:
>>
>> Dear Pedro,
>>
>> Of course, that's the very reason why I wanted to transfer it to the
>> style of working without points.
>>
>> That's slightly unfair, in that in many cases of reasoning
>> algebraically, without points, it's not at all clear how to do it
>> pointwise.
>>
>> You and I have certainly experienced that in our work on quantales,
>> which are much more purely algebraic gadgets. Our approach via localic
>> suplattices (algebras for the lower hyperspace monad) gives a more
>> point-free approach to the subject, but it takes effort - I think
>> you'll agree - to work with the hyperspaces in a pointwise manner.
>>
>> Do you think there's a less derogatory term for the style of reasoning
>> without points?
>>
>> All the best,
>>
>> Steve.

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• ```* categories: Re: complete Galois groups
2023-01-28 10:48       ` categories: complete Galois groups Clemens Berger
@ 2023-01-30 17:34         ` Eduardo J. Dubuc
0 siblings, 0 replies; 19+ messages in thread
From: Eduardo J. Dubuc @ 2023-01-30 17:34 UTC (permalink / raw)
To: Clemens Berger, Pedro Resende; +Cc: Steven Vickers, categories list

On 28/01/2023 07:48, Clemens Berger wrote:
> How can we characterise ``intrinsically'' toposes that are of the
> form BG for a complete Galois group G

An observation that I do not know if of any help:

What you want is how to characterize a pointed atomic (i.e. connected,
locally connected, Boolean) topos p: G ---> Ens such that the morphism
lAut(p) ---> Aut(p) is an isomorphism (where lAut is the localic group
of automorphism).

(recall that this localic group is explicitely constructed in my article
"Localic Galois Theory")

all the best  Eduardo

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• `[parent not found: <LNXP265MB1049E00AEC9CE5BE1233CCEF95C69@LNXP265MB1049.GBRP265.PROD.OUTLOOK.COM>]`
• ```* Re: Terminology for point-free topology?
@ 2023-01-22 21:32   ` Vaughan Pratt
0 siblings, 0 replies; 19+ messages in thread
From: Vaughan Pratt @ 2023-01-22 21:32 UTC (permalink / raw)
To: Steven Vickers; +Cc: categories

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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• `[parent not found: <CAL7kZqCz081nyFQuo_QFfaGzGe+UBORJHOZWsO9VgNzpDJ9_Gw@mail.gmail.com>]`
• ```* Re:  Terminology for point-free topology?
@ 2023-01-23 13:25   ` Steven Vickers
0 siblings, 0 replies; 19+ messages in thread
From: Steven Vickers @ 2023-01-23 13:25 UTC (permalink / raw)
To: Vaughan Pratt; +Cc: categories

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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• `[parent not found: <LNXP265MB104912A7940157738582CE2595C89@LNXP265MB1049.GBRP265.PROD.OUTLOOK.COM>]`
• ```* categories: Re: Terminology for point-free topology?
@ 2023-01-23 23:17   ` Vaughan Pratt
0 siblings, 0 replies; 19+ messages in thread
From: Vaughan Pratt @ 2023-01-23 23:17 UTC (permalink / raw)
To: Steven Vickers; +Cc: categories

" 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 → 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 = x with the unit circle in the upper right
quadrant would presumably be the dense point at (1,1)/√2.

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)/√2 itself with the generic dense point located there, I no longer 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 <s.j.vickers.1@bham.ac.uk>
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 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
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

• `[parent not found: <CAL7kZqAPzmzf=wt=qKNBjjeb_dGtG4eDu7tv5Eku-AVZD7wWtw@mail.gmail.com>]`
• ```* categories: Re:  Terminology for point-free topology?
@ 2023-01-24 11:45   ` Steven Vickers
0 siblings, 0 replies; 19+ messages in thread
From: Steven Vickers @ 2023-01-24 11:45 UTC (permalink / raw)
To: Vaughan Pratt; +Cc: categories

Dear Vaughan,

The idea of generic point as dense point looks misconceived to me. I'll say  more below. First, however, can I check we're in line with intro/extrovert? When I discussed the circle, was I using them with the same metaphorical content as you had in mind?

More below.

Steve.

________________________________
From: Vaughan Pratt <pratt@cs.stanford.edu>
Sent: Monday, January 23, 2023 11:17 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?

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 → k.

A good picture for internal logic in E is often to imagine E as sheaves over some B, with the objects as local homeomorphisms to B. That might sound a  bit point-set, but Joyal and Tierney showed how to make sense of it for general E. Then geometric constructions, such as binary product a x x, are calculated fibrewise. The subobject classifier k is not geometric, but your maps r are just subobjects of a x x, and those too can be seen fibrewise.

Actually, frames aren't geometric either, but J&T showed that the internal frames correspond to more general bundles over B. When you build in the right conditions on r, it should correspond to a bundle map, from the local homeomorphism for a to the more general bundle for x. That is why we're approximating a bundle by a local homeomorphism. There is a best possible approximation, the discrete coreflection, or the "set of points" as calculated internally in E, but my example with B = Sierpinski shows it can be badly deficient even for straightforward bundles.

I must stress that the ability to treat bundles as internal spaces is a wonderful feature of point-free topology, something to be treasured, so one shouldn't be tempted to treat such straightforward bundles as pathological on  the grounds of their "non-spatiality".

"In fact the generic point in the topos of sheaves, on its own, is enough for most purposes."

Would that be the dense point?

No.

You'll see more discussion of this in my arXiv notes, but perhaps I can explain the generic point in computer science terms as a formal parameter.

Suppose you're working in a programming language L, and you write a procedure with a formal parameter x of type T. Within the scope of the declaration  x:T, you are no longer working in pure L, but in L with an element of T freely adjoined. Call it L[x:T]. The freeness lies in the fact that whatever you construct with x can be transported, by substitution, to any actual parameter a:T.

Note that a may itself be in the scope of some other formal parameter, hence in some L[x':T']. Substituting a for x transports any construction in L[x:T] into one in L[x':T'], so, in some sense you get a homomorphism L[x:T] -> L[x':T'].

Now think of x as "generic", in that it has no properties other than what follows from being of type T, and the actual parameters a as being more specific. The same idea applies to the generic point in a classifying topos.

Instead of types and elements, we have geometric theories and models. For instance T might be the geometric theory of Dedekind sections, and then a model is a real number. The classifying topos Set[x:T] is the geometric mathematics of Set with a model of T freely adjoined - that is pretty much what the universal property of classifying toposes tells us. If we have a model a of T, and it could be in some other topos, say Set[x':T'], then by substitution we get a functor Set[x:T] -> Set[x':T'] that preserves geometric structure (essentially: colimits and finite limits). It has a right adjoint, and hence we get a geometric morphism. That's a generalized point of Set[x:T] - its generalized points are equivalent to models of T.

Note also that, for a locale, that classifying topos is equivalent to the topos of sheaves.

I hope this helps clarify the fact that the generic point does not live in Set, but in a different topos Set[x:T] that is more or less a syntactic construct. Its sufficiency as a point on its own lies in the fact that it can be instantiated as any other more specific point.

I can't see any way in which it is helpful to see it as a "dense" point. (Are you trying to think of it as a "thickened" point where a circle meets a tangent? I don't think that analogy goes anywhere.)

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```end of thread, other threads:[~2023-02-01 20:41 UTC | newest]

Thread overview: 19+ messages (download: mbox.gz / follow: Atom feed)
2023-01-16 11:50 Terminology for point-free topology? 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
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