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This 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?

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You might be interested in the following question, which asks 'why' the standard simplices are standard:… – Eric Peterson Sep 5 '10 at 17:02
up vote 17 down vote accepted

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.)

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.

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And transitivity failing means that you can't get a fundamental group/groupoid, either. – Mike Shulman Sep 5 '10 at 17:18
Lack of symmetry is not a problem if you are only hoping to get a fundamental long category (different of course to the usual fundamental category of a directed space). Lack of transitivity, though, is fatal. – David Roberts Sep 5 '10 at 22:02

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 ≤ 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.

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So now I'm confused. The Sierpinski space suffers at least as much as the one-point-compactified long ray in that it does not have an involution swapping the endpoints. Or maybe that's not what you're using it for? – Theo Johnson-Freyd Sep 5 '10 at 18:18
In doing homotopy theory the simplicial way, you don't really need that swap of endpoints, do you, or that H-space structure, either? You get the group structure from the filling of horns. So, in addition to the usual cosimplicial face whose space number n is an n-simplex and another whose space number 1 is the Sierpinski space (and number n defined accordingly), maybe there is another whose number 1 space is that half-long thing? – Tom Goodwillie Sep 5 '10 at 18:46
Note that the category with one non-identity arrow between two objects has the same 'problem', but its geometric realisation is the interval - natural transformations of functors become homotopies of maps under |-|. – David Roberts Sep 5 '10 at 22:04
In one way, the Sierpinski space is nicer than the usual interval (though less interesting): it's a strict co-category, so homming out of it gives a strict “Sierpinski fundamental category”. (“Strict” here = “on-the-nose, not just up-to-[some-kind-of]-homotopy”.) Since it's “co-transitive”, it certainly isn't as bad as as the long line... – Peter LeFanu Lumsdaine Sep 6 '10 at 5:46
@Tom: Perhaps. But the "nerve" obtained from the Sierpinski cosimplicial space will not in general be a Kan complex; I think it'll actually be exactly the nerve of the "Sierpinski fundamental category" that Peter mentions. (The lack of an endpoint-swap just means that you have to keep track of directionality.) OTOH, since the long line is not cotransitive even up to homotopy, I wouldn't expect the "nerve" obtained from any such cosimplicial space to even be a quasicategory. – Mike Shulman Sep 8 '10 at 4:14

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.

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