{
  "id": "2202.03659",
  "version": "v1",
  "published": "2022-02-08T05:28:15.000Z",
  "updated": "2022-02-08T05:28:15.000Z",
  "title": "Cellular cosheaf homology are cosheaf homology",
  "authors": [
    "Daisuke Kishimoto",
    "Yasutomo Yushima"
  ],
  "comment": "11 pages",
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "A cosheaf is the dual notion of a sheaf, but we cannot define its homology as the formal dual of sheaf cohomology, in general, because of the lack of the cosheafification. A cellular cosheaf is a contravariant functor from the face poset of a CW complex to the category of abelian groups. We show that given a cellular cosheaf $F$, there is a natural way to associate a cosheaf $\\widehat{F}$, for which we can define homology as the formal dual of sheaf cohomology, such that the Borel-Moore homology of $F$ is isomorphic to the homology of $\\widehat{F}$ whenever the underlying CW complex of $F$ is a simplicial complex.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-08T05:28:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55N30",
      "18F20"
    ],
    "keywords": [
      "cellular cosheaf homology",
      "cw complex",
      "formal dual",
      "sheaf cohomology",
      "dual notion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}