It is well known that a topological space is more or less the same as an $\infty$-groupoid. I'm wondering if there is an analogous construction which starts with a manifold endowed with Morse theory (either a function or a gradient-like vector field) and leads to an $(\infty, 1)$-category. The idea should be that morphisms are paths going "downwards".

One can easily come up with at least two definitions, which are obviously non-equivalent: the first one is taking critical points as objects and the space of gradient trajectories as morphisms, and the second one is taking the set of all points as objects and nondecreasing continuous paths as morphisms.

So this boils down to the following: is there any precise sense in which these categories are equivalent? And a minimal question is:

Is there any precise sense in which the two following categories are equivalent - the category with objects $0$ and $1$ and one non-identity morphism from $0$ to $1$, and the category with set of objects $[0, 1]$ and unique morphism between $a$ and $b$ iff $a \leq b$?