{
  "id": "2006.14852",
  "version": "v1",
  "published": "2020-06-26T08:19:01.000Z",
  "updated": "2020-06-26T08:19:01.000Z",
  "title": "Boolean valued models, presheaves, and étalé spaces",
  "authors": [
    "Moreno Pierobon",
    "Matteo Viale"
  ],
  "categories": [
    "math.LO",
    "math.CT"
  ],
  "abstract": "Boolean valued models for a signature $\\mathcal{L}$ are generalizations of $\\mathcal{L}$-structures in which we allow the $\\mathcal{L}$-relation symbols to be interpreted by boolean truth values; for example for elements $a,b\\in\\mathcal{M}$ with $\\mathcal{M}$ a $\\mathsf{B}$-valued $\\mathcal{L}$-structure for some boolean algebra $\\mathsf{B}$, $(a=b)$ may be neither true nor false, but get an intermediate truth value in $\\mathsf{B}$. In this paper we expand and relate the work of Mansfield and others on the semantics of boolean valued models, and of Munro and others on the adjunctions between $\\mathsf{B}$-valued models and $\\mathsf{B}^+$-presheaves for a boolean algebra $\\mathsf{B}$. In particular we give an exact topological characterization (the so called \\emph{fullness property}) of which boolean valued models satisfy \\L o\\'s theorem (i.e. the general form of the forcing theorem which Cohen -- Scott, Solovay, Vopenka -- established for the special case given by the forcing method in set theory). We also give an exact categorial characterization of which presheaves correspond to \\emph{full} boolean valued models in terms of the structure of global sections of their associated \\'etal\\'e space. To do so we introduce a slight variant of the sheafification process of a given presheaf by means of its embedding into an \\'etal\\'e space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-26T08:19:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E40",
      "18A40",
      "18F20",
      "03C90",
      "03C20"
    ],
    "keywords": [
      "boolean algebra",
      "etale space",
      "boolean truth values",
      "intermediate truth value",
      "boolean valued models satisfy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}