TITLE: What properties make $[0,1]$ a good candidate for defining fundamental groups?
QUESTION [109 upvotes]: The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in  J$. There is an obvious generalization of the definition of homotopic maps. Suppose we have selected $(J,e_0,e_1)\in \mathfrak{Top}_2$. We could say that two continuous maps $f,g:X\to Y$ are "$J$-homotopic"  if there is a continuous map $h:X\times J\to Y$ such that $h(x,e_0) = f(x)$ and $h(x,e_1) = g(x)$. We could then define $\pi_1 (X,x)$ to be the set of continuous maps $f:J\to X$ satisfying $f(e_0)=f(e_1)=x$, with $J$-homotopic maps identified. Here  in order to define composition of paths in the naive way, we need to have picked some homeomorphism continuous map from $J$ to $(\{0\}\times J\cup \{1\}\times J)/((\{0\},e_1) = (1,e_0))$, taking $e_0$ to $e_0\times 0$ and $e_1$ to $e_1\times 1$. I have two questions:

Can $([0,1],0,1)$ be characterized as an object in $\mathfrak{Top}_2$ in a purely categorical manner?
When is $\pi_1 (X,x)$ a group? For that matter, when is $\pi_1 (X,x)$ associative?

Essentially, the question comes down to: what properties of $[0,1]$ are needed in order to do homotopy theory?

REPLY [4 votes]: You may be interested in a paper entitled "The Big Fundamental Group, Big Hawaiian Earrings, and Big Free Groups" by J. Cannon and G. Conner.  In this paper, they talk about "big intervals" and use them to define a "big fundamental group".  A "big interval" is a totally-ordered set (with the order topology) which is compact and connected (equivalently, a linear continuum with first and last point).  It is true that [0,1] is terminal in the category of (non-degenerate) big intervals.  (Which also means that it is initial since for big intervals I and J, a monomorphism $I\to J$ exists precisely when an epimorphism $J\to I$ exists).