{
  "id": "math/0009004",
  "version": "v3",
  "published": "2000-09-01T08:16:29.000Z",
  "updated": "2001-09-18T13:52:50.000Z",
  "title": "Higher fundamental functors for simplicial sets",
  "authors": [
    "Marco Grandis"
  ],
  "comment": "Revised version with minor changes 36 pages, 428 K",
  "journal": "Cahiers Topologie Geom. Differentielle Categ. 42 (2001), 101-136",
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "This is a sequel to a previous paper, developing an intrinsic, combinatorial homotopy theory for simplicial complexes; the latter form the cartesian closed subcategory of 'simple presheaves' in !Smp, the topos of symmetric simplicial sets, or presheaves on the category of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself, !Smp. As a crucial advantage, the fundamental groupoid functor !Smp --> Gpd is left adjoint to a natural functor Gpd --> !Smp, the symmetric nerve of a groupoid, and preserves all colimits - a strong van Kampen property. Similar results hold in all higher dimensions. Analogously, a notion of (non-reversible) directed homotopy can be developed in the ordinary simplicial topos Smp, with applications to image analysis as in the previous paper. We have now a 'homotopy n-category functor' Smp --> n-Cat, left adjoint to a nerve functor. This construction can be applied to various presheaf categories; the basic requirements seem to be: finite products of representables are finitely presentable and there is a representable 'standard interval'.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2001-09-18T13:52:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55U35",
      "55U10",
      "18B40"
    ],
    "keywords": [
      "higher fundamental functors",
      "simplicial sets",
      "left adjoint",
      "ordinary simplicial topos smp",
      "strong van kampen property"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......9004G"
    }
  }
}