{
  "id": "1705.00955",
  "version": "v1",
  "published": "2017-05-02T13:23:03.000Z",
  "updated": "2017-05-02T13:23:03.000Z",
  "title": "Persistent homology and microlocal sheaf theory",
  "authors": [
    "Masaki Kashiwara",
    "Pierre Schapira"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "We interpret some results of persistent homology and barcodes (in any dimension) with the language of microlocal sheaf theory. For that purpose we study the derived category of sheaves on a real finite-dimensional vector space V. By using the operation of convolution, we introduce a pseudo-distance on this category and prove in particular a stability result for direct images. Then we assume that V is endowed with a closed convex proper cone $\\gamma$ with non empty interior and study $\\gamma$-sheaves, that is, constructible sheaves with microsupport contained in the antipodal to the polar cone (equivalently, constructible sheaves for the $\\gamma$-topology). We prove that such sheaves may be approximated (for the pseudo-distance) by \"piecewise linear\" $\\gamma$-sheaves. Finally we show that these last sheaves are constant on stratifications by $\\gamma$-locally closed sets, an analogue of barcodes in higher dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-02T13:23:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55N99",
      "18A99",
      "35A27"
    ],
    "keywords": [
      "microlocal sheaf theory",
      "persistent homology",
      "real finite-dimensional vector space",
      "non empty interior",
      "closed convex proper cone"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}