Timeline for What properties make $[0,1]$ a good candidate for defining fundamental groups?
Current License: CC BY-SA 3.0
25 events
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| Jun 27, 2018 at 19:45 | comment | added | John Gowers | 'In order to define composition of paths in the naive way': in fact, you can get away with saying that the composition $p_1;p_2\colon J \to X$ should exist for any paths $p_1,p_2\colon J\to X$ such that $p_1(e_1)=p_2(e_0)$ and that it should be natural in the sense that if $f\colon X \to Y$ is continuous then $(f\circ p_1);(f\circ p_2)=f\circ(p_1;p_2)$. Your continuous map then arises as the composition of the two obvious paths in $J\vee J$, and naturality tells us that every composition can be defined using this map. | |
| S Apr 15, 2017 at 14:23 | history | suggested | Ascenso |
added fundamental group tag
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| Apr 15, 2017 at 14:04 | review | Suggested edits | |||
| S Apr 15, 2017 at 14:23 | |||||
| Aug 3, 2013 at 2:50 | vote | accept | Daniel Miller | ||
| Mar 7, 2013 at 5:25 | answer | added | Keith Penrod | timeline score: 6 | |
| Mar 6, 2013 at 13:24 | comment | added | Alexey Muranov | I have just asked a related question about topological characterization of [0, 1]: mathoverflow.net/questions/123760 | |
| Sep 16, 2012 at 9:59 | comment | added | o a | There is a paper by Drinfeld (and related papers by A.Besser and D.Grayson) arxiv.org/abs/math/0304064 where he dwells on the notion of the interval..i quote: We reformulate the definitions so that the following facts become obvious: (i) geometric realization commutes with finite projective limits (e.g., with Cartesian products); (ii) the geometric realization of a simplicial set (resp. cyclic set) is equipped with an action of the group of orientation preserving homeomor- phisms of the segment I := [0, 1] (resp. the circle S1 ) | |
| May 3, 2012 at 10:27 | comment | added | Ronnie Brown | I'd just like to put in a plea for Dedecker, Paul and Valderrama, Jerko, Graphes et cographes sur une cat\'egorie abstraite. Application `a l'homotopie, C. R. Acad. Sci. Paris S\'er. A-B, 262, 1966, A377--A380, for an early discussion of an "interval object". | |
| Mar 26, 2012 at 19:36 | answer | added | Valerio Capraro | timeline score: 4 | |
| Mar 26, 2012 at 17:29 | comment | added | Mike Shulman | Good point, Charles. The paper by Kennison that I mentioned in my answer also considers covering spaces as one way to define the fundamental group, and relates it to the Cech-style definition. | |
| Mar 26, 2012 at 15:48 | comment | added | Charles Rezk | You don't need the unit interval at all to define the fundamental group, at least for "nice spaces, since then you can characterize the fundamental group(oid) of $X$ in terms of its covering spaces. Unfortunately, I don't know how to recognize when a space is "nice" (e.g., locally path connected and semi-locally simply connected) without using the unit interval. | |
| Mar 26, 2012 at 14:50 | answer | added | Karol Szumiło | timeline score: 5 | |
| Mar 26, 2012 at 5:12 | answer | added | Mike Shulman | timeline score: 108 | |
| Mar 26, 2012 at 3:58 | answer | added | Tom Leinster | timeline score: 136 | |
| Mar 26, 2012 at 1:56 | comment | added | Jonathan Beardsley | The question supposes that fundamental groups were defined before the unit interval... | |
| Mar 26, 2012 at 1:26 | comment | added | David Roberts♦ | Arg, I took too long writing my answer, Benjamin Steinberg got the much shorter and punchier version out. | |
| Mar 26, 2012 at 1:06 | answer | added | David Roberts♦ | timeline score: 8 | |
| Mar 26, 2012 at 0:35 | comment | added | Benjamin Steinberg | I think really the circle is the correct object for defining the fundamental group. What is important is that it is a cogroup object in the category of pointed spaces if you allow coassociativity up to homotopy. | |
| Mar 25, 2012 at 23:32 | comment | added | Qiaochu Yuan | ncatlab.org/nlab/show/interval+object | |
| Mar 25, 2012 at 23:03 | comment | added | Anton Petrunin | I had a feeling that "right" definition of fundamental group should use pseudoarc instead of $[0,1]$. I am sure someone did it this way. | |
| Mar 25, 2012 at 22:47 | comment | added | Daniel Litt | ncatlab.org/nlab/show/cylinder+object | |
| Mar 25, 2012 at 22:43 | history | edited | Daniel Miller | CC BY-SA 3.0 |
added 90 characters in body
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| Mar 25, 2012 at 22:40 | comment | added | Daniel Miller | You're quite right. Edited to reflect the change. | |
| Mar 25, 2012 at 22:32 | comment | added | Tom Goodwillie | To define multiplication of paths, you don't need a homeomorphism. You just need a map from $J$ to that quotient (taking $e_0$ and $e_1$ to the right things). | |
| Mar 25, 2012 at 22:21 | history | asked | Daniel Miller | CC BY-SA 3.0 |