{
  "id": "1308.5517",
  "version": "v1",
  "published": "2013-08-26T08:55:07.000Z",
  "updated": "2013-08-26T08:55:07.000Z",
  "title": "The infinite random simplicial complex",
  "authors": [
    "Andrew Brooke-Taylor",
    "Damiano Testa"
  ],
  "comment": "33 pages",
  "categories": [
    "math.LO",
    "math.CO"
  ],
  "abstract": "We study the Fraisse limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fraisse limits of structures for finite languages carry across to this important example. We introduce the notion of a local class, with the class of finite simplicial complexes as an archetypal example, and in this general context prove the existence of a 0-1 law and other basic model-theoretic results. Constraining to the case where all relations are symmetric, we show that every direct limit of finite groups, and every metrizable profinite group, appears as a subgroup of the automorphism group of the Fraisse limit. Finally, for the specific case of simplicial complexes, we show that the geometric realisation is topologically surprisingly simple: despite the combinatorial complexity of the Fraisse limit, its geometric realisation is homeomorphic to the infinite simplex.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-26T08:55:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C13",
      "05E45",
      "60F20",
      "20B27"
    ],
    "keywords": [
      "infinite random simplicial complex",
      "fraisse limit",
      "finite simplicial complexes",
      "geometric realisation",
      "basic model-theoretic results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.5517B"
    }
  }
}