Re: Terminology for point-free topology?
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/ ]
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/ ]
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/ ]
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/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Martin Hyland -
Pedro Resende -
ptj＠maths.cam.ac.uk -
Steven Vickers -
Wesley Phoa