Long line fundamental groupoid - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:13:45Z http://mathoverflow.net/feeds/question/37800 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37800/long-line-fundamental-groupoid Long line fundamental groupoid Steven Gubkin 2010-09-05T16:07:31Z 2010-09-05T18:34:01Z <p><a href="http://mathoverflow.net/questions/37792/a-possible-generalization-of-the-homotopy-groups" rel="nofollow">This</a> question got me thinking about what makes the fundamental group (or groupoid) tick. What is so special about the circle? As another possible candidate for generalization, what about taking the one point compactification of the long closed ray R and thinking about homotopy theory with R in place of the interval? Would a theory of "long homotopy" arise? As a follow up, if this doesn't work, are there any other interesting instances of replacing the unit interval with another topological space and getting an interesting homotopy theory out of it? If not is there some characterization of the interval as the unique space which induces a nice homotopy theory?</p> http://mathoverflow.net/questions/37800/long-line-fundamental-groupoid/37802#37802 Answer by Robin Chapman for Long line fundamental groupoid Robin Chapman 2010-09-05T16:33:35Z 2010-09-05T16:33:35Z <p>The compactified long closed ray $\overline R$ will have two endpoints, but these are distinguishable. One has a neighbourhood homeomorphic to $[0,1)$ and the other doesn't. This scuppers "long homotopy" being a symmetric relation. (Also the transitivity would fail too.)</p> <p>The standard notion of homotopy relies on the interval $I$ having distinguished points $0$ and $1$, there being a self map of $I$ swapping $0$ and $1$, and there being a map from $I\coprod I/\sim$ to $I$ where $\sim$ is the equivalence relation identifying the $1$ in the first component to the $0$ in the second. These maps have to satisfy various formal properties. There is no continuous map of $\overline R$ swapping its "endpoints", so we can't mimic the classical notion of homotopy.</p> http://mathoverflow.net/questions/37800/long-line-fundamental-groupoid/37807#37807 Answer by Mike Shulman for Long line fundamental groupoid Mike Shulman 2010-09-05T17:23:33Z 2010-09-05T17:23:33Z <p>Especially when doing topos theory, one sometimes uses the Sierpinski space (the two-point space with one open point) as a sort of "directed interval." This is convenient because "Sierpinski homotopies" are exactly the 2-cells in the 2-category of topoi. For topological spaces regarded as (their sheaf) topoi, such 2-cells are the pointwise &le; relation in the specialization ordering. I think I recall that "geometric realization" relative to the Sierpinski interval is important too, perhaps it can be identified with some sort of descent.</p> http://mathoverflow.net/questions/37800/long-line-fundamental-groupoid/37814#37814 Answer by Cap for Long line fundamental groupoid Cap 2010-09-05T18:34:01Z 2010-09-05T18:34:01Z <p>Perhaps the article of J. Cannon and G. Conner, "The big fundamental group, big Hawaiian earrings, and the big free groups", will interest you. I believe they work with just what you said, the one-point compactification of the long closed ray, or something very similar.</p>