{
  "id": "1303.3255",
  "version": "v2",
  "published": "2013-03-13T19:40:51.000Z",
  "updated": "2014-12-17T19:59:07.000Z",
  "title": "Sheaves, Cosheaves and Applications",
  "authors": [
    "Justin Curry"
  ],
  "comment": "v2: 307 pages, 68 figures, includes minor edits of version 1 and is expanded with additional results, accepted as a doctoral thesis at the University of Pennsylvania. v1: 188 pages, 47 figures, work in progress",
  "categories": [
    "math.AT",
    "math.RT"
  ],
  "abstract": "This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular sheaves and cosheaves, which are finite families of vector spaces and maps parametrized by a cell complex. We develop cellular (co)sheaves as a new tool for topological data analysis, network coding and sensor networks. A foundation for multi-dimensional level-set persistent homology is laid via constructible cosheaves, which are equivalent to representations of MacPherson's entrance path category. By proving a van Kampen theorem, we give a direct proof of this equivalence. A cosheaf version of the i'th derived pushforward of the constant sheaf along a definable map is constructed directly as a representation of this category. We go on to clarify the relationship of cellular sheaves to cosheaves by providing a formula that defines a derived equivalence, which in turn recovers Verdier duality. Compactly-supported sheaf cohomology is expressed as the coend with the image of the constant sheaf through this equivalence. The equivalence is further used to establish relations between sheaf cohomology and a herein newly introduced theory of cellular sheaf homology. Inspired to provide fast algorithms for persistence, we prove that the derived category of cellular sheaves over a 1D cell complex is equivalent to a category of graded sheaves. Finally, we introduce the interleaving distance as an extended pseudo-metric on the category of sheaves. We prove that global sections partition the space of sheaves into connected components. We conclude with an investigation into the geometry of the space of constructible sheaves over the real line, which we relate to the bottleneck distance in persistence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-13T19:40:51.000Z",
      "abstract": "This note advertises the theory of cellular sheaves and cosheaves, which are devices for conducting linear algebra parametrized by a cell complex. The theory is presented in a way that is meant to be read and appreciated by a broad audience, including those who hope to use the theory in applications across science and engineering disciplines. We relay two approaches to cellular cosheaves. One relies on heavy stratification theory and MacPherson's entrance path category. The other uses the Alexandrov topology on posets. We develop applications to persistent homology, network coding, and sensor networks to illustrate the utility of the theory. The driving computational force is cellular cosheaf homology and sheaf cohomology. However, to interpret this computational theory, we make use of the Remak decomposition into indecomposable representations of the cell category. The computational formula for cellular cosheaf homology is put on the firm ground of derived categories. This leads to an internal development of the derived perspective for cell complexes. We prove a conjecture of MacPherson that says cellular sheaves and cosheaves are derived equivalent. Although it turns out to be an old result, our proof is more explicit than other proofs and we make clear that Poincar\\'e-Verdier duality should be viewed as an exchange of sheaves and cosheaves. The existence of enough projectives allows us to define a new homology theory for sheaves on posets and we establish some classical duality results in this setting. Finally, we make use of coends as a generalized tensor product to phrase compactly supported sheaf cohomology as the pairing with the image of the constant sheaf through the derived equivalence.",
      "comment": "188 pages, 47 figures, work in progress",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-17T19:59:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "applications",
      "compactly supported sheaf cohomology",
      "cellular cosheaf homology",
      "macphersons entrance path category",
      "computational"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 307,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.3255C"
    }
  }
}