[Submitted on 26 May 2026 (v1), last revised 23 Jul 2026 (this version, v3)]

View PDF

Abstract:We prove that the topological flow category $\mathcal{M}$ arising from a Morse-Smale pair $(f,\xi)$ 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 $\xi$.
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 $\xi$ 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.

Submission history

From: Colin Fourel [view email]
[v1] Tue, 26 May 2026 14:52:01 UTC (189 KB)
[v2] Tue, 2 Jun 2026 09:29:28 UTC (189 KB)
[v3] Thu, 23 Jul 2026 13:08:11 UTC (191 KB)

Current browse context:

math.AT

Bookmark

Bibliographic and Citation Tools

Code, Data and Media Associated with this Article

Demos

Recommenders and Search Tools

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.