{
  "id": "2205.15275",
  "version": "v1",
  "published": "2022-05-30T17:36:56.000Z",
  "updated": "2022-05-30T17:36:56.000Z",
  "title": "Categorification of Extended Persistence Diagrams",
  "authors": [
    "Ulrich Bauer",
    "Benedikt Fluhr"
  ],
  "comment": "66 pages + 12 pages appendix, 14 figures",
  "categories": [
    "math.AT",
    "cs.CG",
    "math.KT"
  ],
  "abstract": "The extended persistence diagram introduced by Cohen-Steiner, Edelsbrunner, and Harer is an invariant of real-valued continuous functions, which are $\\mathbb{F}$-tame in the sense that all open interlevel sets have degree-wise finite-dimensional cohomology with coefficients in a fixed field $\\mathbb{F}$. We show that relative interlevel set cohomology (RISC), which is based on the Mayer--Vietoris pyramid by Carlsson, de Silva, and Morozov, categorifies this invariant. More specifically, we define an abelian Frobenius category $\\mathrm{pres}(\\mathcal{J})$ of presheaves, which are presentable in some sense, such that for an $\\mathbb{F}$-tame function $f \\colon X \\rightarrow \\mathbb{R}$ its RISC $h(f)$ is an object of $\\mathrm{pres}(\\mathcal{J})$ and moreover, the extended persistence diagram of $f$ uniquely determines - and is determined by - the corresponding element $[h(f)] \\in K_0 (\\mathrm{pres}(\\mathcal{J}))$ in the Grothendieck group $K_0 (\\mathrm{pres}(\\mathcal{J}))$ of the abelian category $\\mathrm{pres}(\\mathcal{J})$. As an intermediate step we show that $\\mathrm{pres}(\\mathcal{J})$ is the Abelianization of the (localized) category of complexes of $\\mathbb{F}$-linear sheaves on $\\mathbb{R}$, which are tame in the sense that sheaf cohomology of any open interval is finite-dimensional in each degree. This yields a close link between derived level set persistence by Curry, Kashiwara, and Schapira and the categorification of extended persistence diagrams.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-30T17:36:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55N31",
      "55N30",
      "16G20",
      "16G70",
      "16E20",
      "06B99",
      "62R40"
    ],
    "keywords": [
      "extended persistence diagram",
      "categorification",
      "abelian frobenius category",
      "relative interlevel set cohomology",
      "derived level set persistence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 66,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}