[This](https://mathoverflow.net/questions/37792/a-possible-generalization-of-the-homotopy-groups) 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?



Long line fundamental groupoid
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.


