{
  "id": "2006.13150",
  "version": "v1",
  "published": "2020-06-23T16:49:07.000Z",
  "updated": "2020-06-23T16:49:07.000Z",
  "title": "Thickening of the diagonal, interleaving distance and Fourier-Sato transform",
  "authors": [
    "Francois Petit",
    "Pierre Schapira"
  ],
  "categories": [
    "math.AT",
    "math.AG"
  ],
  "abstract": "Given a \"good\" metric space $X$ we construct an interleaving distance $\\mathrm{dist}_X$ on the bounded derived category of abelian sheaves on $X$. Our main tool is the family of kernels associated with thickenings of the diagonal. Complete Riemannian manifolds having a strictly positive convexity radius are examples of good metric spaces. We prove a kind of (proper and non proper) stability theorem in this framework and also define the notion of a Lipschitz kernel on $Y\\times X$ which will be proved to define a Lipschitz map for the interleaving distances. Finally, we show that the Fourier-Sato transform of sheaves on Euclidian spheres may be obtained as a thickening of the diagonal and thus defines an isometry. We also obtain a similar result for the Radon transform of sheaves on projective spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-23T16:49:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55N99",
      "18A99",
      "35A27"
    ],
    "keywords": [
      "interleaving distance",
      "fourier-sato transform",
      "metric space",
      "thickening",
      "complete riemannian manifolds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}