Flow Categories as Exit Path Categories

Given a Morse function on a closed smooth manifold and a Smale gradient-like vector field adapted to it, one can construct a topological category called the flow category associated with this data. Its objects are the critical points of the function, and its morphisms are the broken trajectories of the vector field connecting critical points. The notion of a flow category encompasses the realm of Morse theory, and flow categories are also central objects in symplectic topology. Like any topological category, a flow category models an ∞-category, and a theorem of Cohen–Jones–Segal states that, in the Morse case, the homotopy type underlying this ∞-category is that of the manifold itself. However, different choices of Morse–Smale pairs on the same manifold can give rise to non-equivalent ∞-categories. After making these ideas more precise, I will present a result I obtained, which asserts that the ∞-category associated with the flow category of a Morse–Smale pair is equivalent to the exit-path ∞-category associated with the stratification of the manifold by the ascending manifolds of the critical points.