{
  "id": "1206.2941",
  "version": "v2",
  "published": "2012-06-13T20:57:34.000Z",
  "updated": "2012-09-26T15:33:21.000Z",
  "title": "On the homotopy theory of Grothendieck \\infty-groupoids",
  "authors": [
    "Dimitri Ara"
  ],
  "comment": "58 pages, v2: revised according to referee's comments, in particular: paragraph headings added, Remark 1.13 added, Section 3 partially rewritten",
  "journal": "Journal of Pure and Applied Algebra 217(7) (2013), 1237-1278",
  "doi": "10.1016/j.jpaa.2012.10.010",
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "We present a slight variation on a notion of weak \\infty-groupoid introduced by Grothendieck in Pursuing Stacks and we study the homotopy theory of these \\infty-groupoids. We prove that the obvious definition for homotopy groups of Grothendieck \\infty-groupoids does not depend on any choice. This allows us to give equivalent characterizations of weak equivalences of Grothendieck \\infty-groupoids, generalizing a well-known result for strict \\infty-groupoids. On the other hand, given a model category M in which every object is fibrant, we construct, following Grothendieck, a fundamental \\infty-groupoid functor \\Pi_\\infty from M to the category of Grothendieck \\infty-groupoids. We show that if X is an object of M, then the homotopy groups of \\Pi_\\infty(X) and of X are canonically isomorphic. We deduce that the functor \\Pi_\\infty respects weak equivalences.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-09-26T15:33:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18D05",
      "18G55",
      "55P15",
      "55Q05"
    ],
    "keywords": [
      "homotopy theory",
      "grothendieck",
      "homotopy groups",
      "respects weak equivalences",
      "well-known result"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.2941A"
    }
  }
}