Timeline for Is there an algebraic approach to metric spaces?
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Sep 27, 2015 at 22:24 | comment | added | Joel David Hamkins | @AntonFetisov I would urge you to abandon the `evil' terminology, as it has been in other similar usages in category theory, on account of its general unhelpfulness in assisting communication between mathematicians of different viewpoints. | |
| Apr 7, 2013 at 4:18 | answer | added | Włodzimierz Holsztyński | timeline score: 2 | |
| Apr 23, 2012 at 6:32 | answer | added | Nik Weaver | timeline score: 24 | |
| Apr 5, 2012 at 8:41 | vote | accept | Mark | ||
| Apr 3, 2012 at 19:04 | comment | added | Anton Fetisov | @Mark, unfortunately, I don't know any such references. As far as I have worked with this paper, it looks just like some categorical folklore, often cited in the appropriate literature, but not really developed further. Citation search also doesn't really help. | |
| Apr 3, 2012 at 4:22 | comment | added | Misha | @Mark: PS. Of course, Greg Kuperberg would be the best person to ask since I could be misinterpreting his results with Weaver. | |
| Apr 3, 2012 at 4:20 | comment | added | Misha | @Mark: Kuperberg and Weaver define "algebraic" description of metric spaces to consist of certain algebras with filtrations. Once you accept this premise, such filtered algebra, I think, allows you to recover the metric space, see page 9-10 of their paper that I linked. Alternatively, one can consider algebra of Lipschitz functions with a family of additive filtrations given by sets $\{f: \|f-d_x\|_{\infty}\le t\}, t\in [0,\infty]$ (where you remember only the resulting partial order). From this ordered algebra, one can recover metric up to homothety. | |
| Apr 3, 2012 at 0:26 | answer | added | MTS | timeline score: 11 | |
| Apr 2, 2012 at 22:40 | comment | added | Mark | @Anton: I agree. I got the impression that the study of distance-decreasing (dd) maps is more natural than that of Lipschitz maps. After all, a dd bijection with a dd inverse is an isometry while in the Lipschitz case we get a bi-Lipschitz mapping (a quasi-isometry in the terminology of Weaver). The paper you cite by Lawvere seems to support this view, but I find it difficult to read because of the categorical-theoretic language involved. Do you know of a text which presents Lawvere's theory in a way which is more oriented towards an analyst's perspective? Thanks again. | |
| Apr 2, 2012 at 22:37 | comment | added | Mark | @Toby: that's true, though I don't know how natural Weaver's constructions look after this change of metric. As Anton mentioned in his comment, it seems somewhat arbitrary. | |
| Apr 2, 2012 at 5:50 | comment | added | Toby Bartels | I haven't read Weaver's book, but if you have a classification up to isometry for metric spaces of diameter less than $ 2 $, why can't you turn this into a classification up to isometry of all metric spaces? Specifically, given a metric space $ ( X , d ) $, consider the metric space $ ( X , d ' ) $, where $ d ' ( x , y ) : = \frac { \textstyle 2 d ( x , y ) } { \textstyle d ( x , y ) + 1 } $. Then $ d ' $ is also a metric, we can recover $ d $ from $ d ' $, and $ ( X , d ' ) $ has diameter less than $ 2 $, so it can be recovered up to isometry from some algebra in Weaver's book. | |
| Apr 2, 2012 at 2:08 | comment | added | Anton Fetisov | I didn't look into intricacies of his book, but it seems to me filled with arbitrary choices, like diameter-2-space, because he wants to add basepoint with distance 1 to set. Arbitrary constructions are evil. Just as arbitrary, to my mind, is the choice of metric on $Lip$ - $min(\Vert f \Vert_\infty, L(f))$. In the paper I cite natural metric on the function-space is sup-metric. Also, in the theorem on function spaces we require not just bijectivity, but actually isometry between function spaces. Clearly it is a stronger condition. I don't see at the moment any other deviations from expected. | |
| Apr 1, 2012 at 23:39 | history | edited | Mark | CC BY-SA 3.0 |
addressing the existing answers
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| Apr 1, 2012 at 15:01 | answer | added | Anton Fetisov | timeline score: 8 | |
| Mar 31, 2012 at 17:42 | answer | added | Jon Bannon | timeline score: 12 | |
| Mar 31, 2012 at 17:21 | answer | added | J. Alejandro Chávez-Domínguez | timeline score: 36 | |
| Mar 31, 2012 at 14:56 | answer | added | Misha | timeline score: 16 | |
| Mar 31, 2012 at 14:24 | history | asked | Mark | CC BY-SA 3.0 |