{
  "id": "1205.6308",
  "version": "v2",
  "published": "2012-05-29T09:20:29.000Z",
  "updated": "2013-04-24T12:45:32.000Z",
  "title": "Extensions of Picard 2-Stacks and the cohomology groups Ext^i of length 3 complexes",
  "authors": [
    "Cristiana Bertolin",
    "Ahmet Emin Tatar"
  ],
  "comment": "2 New Appendix: in the first Appendix we compute a long exact sequence involving the homotopy groups of an extension of Picard 2-stacks, and in the second Appendix we sketch the proof that the fibered sum of Picard 2-stacks satisfies the universal property",
  "journal": "Ann. Mat. Pura Appl. 193 (2014), no. 1, pp. 291--315",
  "doi": "10.1007/s10231-013-0347-5",
  "categories": [
    "math.AG",
    "math.CT"
  ],
  "abstract": "The aim of this paper is to define and study the 3-category of extensions of Picard 2-stacks over a site S and to furnish a geometrical description of the cohomology groups Ext^i of length 3 complexes of abelian sheaves. More precisely, our main Theorem furnishes (1) a parametrization of the equivalence classes of objects, 1-arrows, 2-arrows, and 3-arrows of the 3-category of extensions of Picard 2-stacks by the cohomology groups Ext^i, and (2) a geometrical description of the cohomology groups Ext^i of length 3 complexes of abelian sheaves via extensions of Picard 2-stacks. To this end, we use the triequivalence between the 3-category of Picard 2-stacks and the tricategory T^[-2,0](S) of length 3 complexes of abelian sheaves over S introduced by the second author in arXiv:0906.2393, and we define the notion of extension in this tricategory T^[-2,0](S), getting a pure algebraic analogue of the 3-category of extensions of Picard 2-stacks. The calculus of fractions that we use to define extensions in the tricategory T^[-2,0](S) plays a central role in the proof of our Main Theorem.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-04-24T12:45:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18G15"
    ],
    "keywords": [
      "cohomology groups",
      "abelian sheaves",
      "pure algebraic analogue",
      "main theorem furnishes",
      "geometrical description"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.6308B"
    }
  }
}