{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.27112",
  "author_total": 1,
  "authors": [
    {
      "name": "Colin Fourel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0a0f1c6qj622s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.27112",
  "fulltext_url": null,
  "id": "0a0f1c6qj622s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.27112",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AT"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "arxiv",
      "math.SG"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7162521563",
  "pdf_url": "https://arxiv.org/pdf/2605.27112",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0a0f1c6qj622s.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2605.27112",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.27112"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Morse flow categories as exit path categories",
  "title_text": "Morse flow categories as exit path categories",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:42Z",
  "url": "https://research.draconitic.com/paper/0a0f1c6qj622s",
  "venue": "arXiv",
  "versions": []
}