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 $a \leq b$?



Constructing ($\infty, 1)$-category from Morse theory on a manifold
This problem has a long history going back at least to the paper of Cohen, Jones and Segal (available here: [http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0883-04.pdf](http://www.kurims.kyoto-u.ac.jp/%7Ekyodo/kokyuroku/contents/pdf/0883-04.pdf)).


In terms of your first definition,
the answer is that if $f:M \to \Bbb R$ is a Morse-Smale function on a closed Riemannian manifold, then there is indeed a topological category whose objects are the critical points and where morphisms are given by sequences of contiguous flow lines connecting the critical points.


There are a number of formidable technical difficulties in getting this to work since the composition law needs to be continuous and associative. Many of the papers in this area contain gaps.


However, I would like to unabashedly credit Lizhen Qin (my student) with settling this question. See the papers


<https://arxiv.org/abs/1107.5527>


<https://arxiv.org/abs/1102.2838>


<https://arxiv.org/abs/1012.3643>


