10

In homotopy theory, a homotopy between two functions $f\ g : A \to B$ is a function $X : [0, 1] \to A \to B$ such that $X(0) = f$ and $X(1) = g$. How do we know that the real interval $[0, 1]$ captures the right notion of continuity? I find it unintuitive that such an elementary concept necessarily depends on the real numbers, which seem much less fundamental than homotopies. From the perspective of homotopy type theory, this seems like a strange dependency.

lily
  • 3,727
  • 1
    Also relevant: http://mathoverflow.net/q/92206/49 – Mike Shulman Aug 22 '16 at 17:50
  • Incidentally, I think the POV of homotopy-type-theory is that $[0,1]$ is homotopy equivalent to a point, and all functions $* \to B^A$ are equivalent. The use of a path is just a technical detail about how homotopy types are modeled by spaces. –  Jan 06 '17 at 10:33

1 Answers1

5

It is ordered, connected, and has both endpoints, rather desirable properties. See https://mathoverflow.net/a/123797/13113 for a full characterization.

An alternative characterization comes from https://math.stackexchange.com/a/1824082/14972 that considers generalized paths from compact Hausdorff spaces with two marked points, and shows that the interval is the minimal universal path.

Community
  • 1