2026/05/26 by Colin Fourel
Mathematics · #math.AT #math.CT #math.SG
We prove that the topological flow category M arising from a Morse-Smale pair (f,ξ) on a smooth closed manifold X is equivalent, as an ∞-category, to Lurie's ∞-category SingA(X) of exit paths in X with respect to the stratification by the stable manifolds of ξ. The objects of M are the critical points of f, and for every pair of critical points, the space of morphisms of 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 M, denoted N(M), and SingA(X). The n-simplices of N(M) are homotopy coherent diagrams of n composable morphisms of M; we introduce the notion of unbroken diagram, yielding an ∞-subcategory of N(M), which we refer to as the flow coherent nerve of 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 ∞-categories that satisfies certain conditions. Our theorem has implications regarding constructible sheaves and the description of homotopy types in terms of flow categories.