{
  "id": "1610.10085",
  "version": "v1",
  "published": "2016-10-31T19:41:44.000Z",
  "updated": "2016-10-31T19:41:44.000Z",
  "title": "Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem",
  "authors": [
    "Ulrich Bauer",
    "Michael Lesnick"
  ],
  "comment": "9 pages",
  "categories": [
    "math.AT",
    "cs.CG",
    "math.CT"
  ],
  "abstract": "Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski, and Mrozek suggests an equivalent description of barcodes as functors R -> Mch, where R is the poset category of real numbers and Mch is the category whose objects are sets and whose morphisms are matchings (i.e., partial injective functions). Such functors form a category Mch^R whose morphisms are the natural transformations. Thus, this interpretation of barcodes gives us a hitherto unstudied categorical structure on barcodes. The aim of this note is to show that this categorical structure leads to surprisingly simple reformulations of both the well-known stability theorem for persistent homology and a recent generalization called the induced matching theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-31T19:41:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13P20",
      "55U99"
    ],
    "keywords": [
      "persistence diagrams",
      "categorification",
      "persistent homology",
      "well-known stability theorem",
      "topological data analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}