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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 8th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexI made starts on [[lexicographic order]] and on [[compactification]]. Lexicographic order was defined only for products of well-ordered families of linear orders (probably the most common type of application). I'm not very happy with the opening of [[compactification]].

   I made starts on lexicographic order and on compactification. Lexicographic order was defined only for products of well-ordered families of linear orders (probably the most common type of application).

   I’m not very happy with the opening of compactification.

2. • CommentRowNumber2.
   • CommentAuthorjcmckeown
   • CommentTimeSep 8th 2012
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   Author: jcmckeown
   Format: Markdowncompactification asks for topological ends; can we call these "components at infinity"?

   compactification asks for topological ends; can we call these "components at infinity"?

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 8th 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItex@Jesse (it's Jesse, right?): is that what some people call them? I can see why they might, but I can't recall ever seeing that in print. It reminds me that I sometimes like to think of points of a Stone-Cech compactification as "ideal points at infinity". The actual image I have for points of say $\beta(X)$, where $X$ is a discrete space and the points are ultrafilters on $X$, is of a tunneling microscope of infinite resolving power, able to penetrate down to the Planck length and beyond, where if the microscope is scanning a region $A \subset X$ and $B$ is a proper subset of $A$, the ultrafilter directs the microscope to home in on either $B$ or its relative complement $A - B$, and down you go into smaller and smaller regions, ad infinitum. By way of contrast, the set of ends of a discrete space is empty (not that anyone asked it to be otherwise).

   @Jesse (it’s Jesse, right?): is that what some people call them? I can see why they might, but I can’t recall ever seeing that in print.

   It reminds me that I sometimes like to think of points of a Stone-Cech compactification as “ideal points at infinity”. The actual image I have for points of say $\beta(X)$, where $X$ is a discrete space and the points are ultrafilters on $X$, is of a tunneling microscope of infinite resolving power, able to penetrate down to the Planck length and beyond, where if the microscope is scanning a region $A \subset X$ and $B$ is a proper subset of $A$, the ultrafilter directs the microscope to home in on either $B$ or its relative complement $A - B$, and down you go into smaller and smaller regions, ad infinitum. By way of contrast, the set of ends of a discrete space is empty (not that anyone asked it to be otherwise).

4. • CommentRowNumber4.
   • CommentAuthorjcmckeown
   • CommentTimeSep 8th 2012
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   Author: jcmckeown
   Format: MarkdownYes, it's Jesse; no, not that anyone used that phrase before, but rather that trying to remember how to define the ends of a space struck me as being "the same" as the construction of the "fundamental group(s) at infinity", which definitely *is* a current phrase. I suppose there ought generally to be a full "site at infinity", in which one can do all the point-agnostic topology he likes.

   Yes, it's Jesse; no, not that anyone used that phrase before, but rather that trying to remember how to define the ends of a space struck me as being "the same" as the construction of the "fundamental group(s) at infinity", which definitely is a current phrase. I suppose there ought generally to be a full "site at infinity", in which one can do all the point-agnostic topology he likes.

5. • CommentRowNumber5.
   • CommentAuthorjim_stasheff
   • CommentTimeSep 9th 2012
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   Author: jim_stasheff
   Format: Textwhat about the physspeak of `compactification' exemplified by compactifying R^2 to the torus?

   what about the physspeak of `compactification' exemplified by compactifying R^2 to the torus?
6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 9th 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItex@Jesse: I certainly like "components at infinity" as intuitive description (and words to that effect should go into an nLab article, I think); of course most of the compactifications I can think of right now all have that "point at infinity" flavor to them, the Stone-Cech compactification being the universal example. Can you say a little more (intuitively, if you like) what you mean by "full site at infinity"? @Jim: I guess you mean things like the quotient $\mathbb{R}^2 \to \mathbb{R}^/\mathbb{Z}^2$. IMO that should be sharply distinguished from the sorts of compactification I had in mind there. You think a note of disambiguation (wikipedia-speak) is in order?

   @Jesse: I certainly like “components at infinity” as intuitive description (and words to that effect should go into an nLab article, I think); of course most of the compactifications I can think of right now all have that “point at infinity” flavor to them, the Stone-Cech compactification being the universal example. Can you say a little more (intuitively, if you like) what you mean by “full site at infinity”?

   @Jim: I guess you mean things like the quotient $\mathbb{R}^2 \to \mathbb{R}^/\mathbb{Z}^2$. IMO that should be sharply distinguished from the sorts of compactification I had in mind there. You think a note of disambiguation (wikipedia-speak) is in order?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeSep 9th 2012
   • (edited Sep 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexActually what is called "compactification" in physics is a process where one starts with data depending on Riemannian structures on a bundle $X \to B$ and then considers the behaviour of that data as the Riemannian volume of the fibers tends to 0. We have an entry on that, titled _[[Kaluza-Klein mechanism]]_. I would suggest we add a disambiguation pointer to that. The relation to the mathematical use of "compactification" comes from the fact that this process is usually of interest only when the fibers of that bundle are compact, as topological spaces. So sometimes in physics they may start with a bundle of noncompact fibers and then say "bah, let's rather start with a bundle with compact fibers". But this is not really "compactification" in the sense of maths. They will say things like "We compactify the theory on the torus" and all this means is that they consider a functional on the space of arbitrary Riemannian manifolds of some dimension restricted to those which happen to be torus-fiber bundles.

   Actually what is called “compactification” in physics is a process where one starts with data depending on Riemannian structures on a bundle $X \to B$ and then considers the behaviour of that data as the Riemannian volume of the fibers tends to 0. We have an entry on that, titled Kaluza-Klein mechanism. I would suggest we add a disambiguation pointer to that.

   The relation to the mathematical use of “compactification” comes from the fact that this process is usually of interest only when the fibers of that bundle are compact, as topological spaces. So sometimes in physics they may start with a bundle of noncompact fibers and then say “bah, let’s rather start with a bundle with compact fibers”. But this is not really “compactification” in the sense of maths. They will say things like “We compactify the theory on the torus” and all this means is that they consider a functional on the space of arbitrary Riemannian manifolds of some dimension restricted to those which happen to be torus-fiber bundles.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 9th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexThanks for the clarification, Urs. I'll add a sentence of disambiguation with a pointer to [[Kaluza-Klein mechanism]] to [[compactification]].

   Thanks for the clarification, Urs. I’ll add a sentence of disambiguation with a pointer to Kaluza-Klein mechanism to compactification.

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeSep 9th 2012
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   Author: TobyBartels
   Format: MarkdownItexOne nice thing about the term ‘end’ is that it allows one to say ‘end compactification’, like all of the others. (I've done that and added links.)

   One nice thing about the term ‘end’ is that it allows one to say ‘end compactification’, like all of the others. (I’ve done that and added links.)

10. • CommentRowNumber10.
    • CommentAuthorjcmckeown
    • CommentTimeSep 10th 2012
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    Author: jcmckeown
    Format: MarkdownItexYes, rhythm is a goodly part of poetics. On review, it seems that the noun I wanted was "locale" rather than "site". And by "full" I just meant "to be taken seriously" rather than any categorical jargon. The thing that looks most natural is to ask for the colimit $\colim_K \mathcal{O} (X - K)$ of the open set Heyting algebras, taken over compact subspaces $K$; I *think* this colimit should exist in HAs? Anyways, in that context one can talk of ultrafilters and covers and Čech nerves and all that. However, "homotopy at infinity" seems to have its arrows pointing the other way: the fundamental group based at a ray $\gamma$ is $$ \pi^e_1 (X, \gamma) = \lim_K\; \underset{t}{Lim} \pi_1( X - K, \gamma(t)) .$$ (the big-L $\Lim$ meaning we only want morphisms to the approximant for $t$ large enough). It should be straightforward to describe a singular complex at-infinity in the same style, but I don't really want to try, just now.

    Yes, rhythm is a goodly part of poetics.

    On review, it seems that the noun I wanted was “locale” rather than “site”. And by “full” I just meant “to be taken seriously” rather than any categorical jargon.

    The thing that looks most natural is to ask for the colimit $\colim_K \mathcal{O} (X - K)$ of the open set Heyting algebras, taken over compact subspaces $K$; I think this colimit should exist in HAs? Anyways, in that context one can talk of ultrafilters and covers and Čech nerves and all that.

    However, “homotopy at infinity” seems to have its arrows pointing the other way: the fundamental group based at a ray $\gamma$ is

    $$\pi^e_1 (X, \gamma) = \lim_K\; \underset{t}{Lim} \pi_1( X - K, \gamma(t)) .$$

    (the big-L $\Lim$ meaning we only want morphisms to the approximant for $t$ large enough).

    It should be straightforward to describe a singular complex at-infinity in the same style, but I don’t really want to try, just now.

11. • CommentRowNumber11.
    • CommentAuthorTim_Porter
    • CommentTimeSep 10th 2012
    • (edited Sep 10th 2012)
    • PermaLink
    Author: Tim_Porter
    Format: MarkdownItexA lot of this fits well with [[proper homotopy theory]. but there the better version of ends is via strong ends / Waldhausen ends, in which you take a homotopy limit and then use $\pi_1$ or whatever. (NB. I was not following this thread, until now and have not read all 10 entries, so may not be saying anything new.:-)) (especially see [here](http://ncatlab.org/nlab/show/proper+homotopy+theory#ends_12)) > It should be straightforward to describe a singular complex at-infinity in the same style, but I don't really want to try, just now. That has been done. See the Handbook of Algebraic Topology article on Proper Homotopy and the references therein. No one seems to have examined if there is a useful quasi-category structure on them (because there are at least two if I remember rightly.)

    A lot of this fits well with [[proper homotopy theory]. but there the better version of ends is via strong ends / Waldhausen ends, in which you take a homotopy limit and then use $\pi_1$ or whatever. (NB. I was not following this thread, until now and have not read all 10 entries, so may not be saying anything new.:-)) (especially see here)

    It should be straightforward to describe a singular complex at-infinity in the same style, but I don’t really want to try, just now.

    That has been done. See the Handbook of Algebraic Topology article on Proper Homotopy and the references therein. No one seems to have examined if there is a useful quasi-category structure on them (because there are at least two if I remember rightly.)

12. • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeNov 2nd 2012
    • (edited Nov 2nd 2012)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexI have expanded [[compactification]] in a way which includes the geometric cases (like DM and wonderful compactifications in algebraic geometry), which are not only topological as well as the compactifications of the fiber which includes the physical cases, I think. Please check. I see above mention of "end compactification", the link in [[compactification]] does not work. Urs says > Actually what is called "compactification" in physics is a process where one starts with data depending on Riemannian structures on a bundle $X \to B$ and then considers the behaviour of that data as the Riemannian volume of the fibers tends to 0. Still, one usually works with noncompact fiber at the begining, like a line fiber, often for simplicity. Then one takes it compactified, say to a circle, plus one takes that the size of the fiber to be small. Not really a limiting small, but compact of finite and small size. This is about the same but it fits with mathematical meaning once fiberwise taken and with Riemann structure accounted for. I think these things should be in the entry [[compactification]], you are free to improve what I wrote. I mean Kaluza/Klein is not the only case of compactification in physics so better the general description be under fiber compactification in the entry [[compactification]].

    I have expanded compactification in a way which includes the geometric cases (like DM and wonderful compactifications in algebraic geometry), which are not only topological as well as the compactifications of the fiber which includes the physical cases, I think. Please check.

    I see above mention of “end compactification”, the link in compactification does not work.

    Urs says

    Actually what is called “compactification” in physics is a process where one starts with data depending on Riemannian structures on a bundle $X \to B$ and then considers the behaviour of that data as the Riemannian volume of the fibers tends to 0.

    Still, one usually works with noncompact fiber at the begining, like a line fiber, often for simplicity. Then one takes it compactified, say to a circle, plus one takes that the size of the fiber to be small. Not really a limiting small, but compact of finite and small size. This is about the same but it fits with mathematical meaning once fiberwise taken and with Riemann structure accounted for. I think these things should be in the entry compactification, you are free to improve what I wrote. I mean Kaluza/Klein is not the only case of compactification in physics so better the general description be under fiber compactification in the entry compactification.

13. • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeNov 2nd 2012
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItex>I see above mention of "end compactification", the link in [[compactification]] does not work. Nobody here has written an article on it, but [Wikipedia has one](https://en.wikipedia.org/wiki/end_compactification).

    I see above mention of “end compactification”, the link in compactification does not work.

    Nobody here has written an article on it, but Wikipedia has one.