Morse flow categories as exit path categories

Abstract

We prove that the topological flow category $\mathcal{M}$ arising from a Morse-Smale pair $(f,ξ)$ on a smooth closed manifold $X$ is equivalent, as an $\infty$-category, to Lurie's $\infty$-category $\mathrm{Sing}_A(X)$ of exit paths in $X$ with respect to the stratification by the stable manifolds of $ξ$. The objects of $\mathcal{M}$ are the critical points of $f$, and for every pair of critical points, the space of morphisms of $\mathcal{M}$ between these is the space of possibly broken trajectories of $ξ$ connecting them; it can be identified up to homotopy with the space of unbroken ones. The latter maps naturally to the space of exit paths connecting these critical points; we prove this map to be a weak homotopy equivalence. Then, we combine these ingredients with several others to construct a zigzag of equivalences between the homotopy coherent nerve of $\mathcal{M}$, denoted $\mathcal{N}(\mathcal{M})$, and $\mathrm{Sing}_A(X)$. The $n$-simplices of $\mathcal{N}(\mathcal{M})$ are homotopy coherent diagrams of $n$ composable morphisms of $\mathcal{M}$; we introduce the notion of unbroken diagram, yielding an $\infty$-subcategory of $\mathcal{N}(\mathcal{M})$, which we refer to as the flow coherent nerve of $\mathcal{M}$. The simplices of the latter give rise to stratified maps out of a family of stratified cubes, into $X$. We organize this family into a functor from the category of finite ordered sequences of critical points, to the category of $A$-stratified topological spaces, and we prove a comparison result with the usual stratified geometric realization functor. We finally use a theorem of Tanaka that associates a functor to a semi-simplicial map between $\infty$-categories that satisfies certain conditions. Our theorem has implications regarding constructible sheaves and the description of homotopy types in terms of flow categories.