{
  "id": "1609.07101",
  "version": "v1",
  "published": "2016-09-22T18:36:56.000Z",
  "updated": "2016-09-22T18:36:56.000Z",
  "title": "Superstability from categoricity in abstract elementary classes",
  "authors": [
    "Will Boney",
    "Rami Grossberg",
    "Monica M. VanDieren",
    "Sebastien Vasey"
  ],
  "comment": "16 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for a certain independence relation called nonsplitting. We generalize their result as follows: given an abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisfies a superstability property provided that the class is categorical and satisfies a weakening of amalgamation. This extends the Shelah-Villaveces result (the independence notion there was splitting) as well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the Shelah-Villaveces proof.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-22T18:36:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C48",
      "03C45",
      "03C52",
      "03C55"
    ],
    "keywords": [
      "abstract elementary class",
      "independence notion",
      "abstract notion",
      "shelah-villaveces result",
      "superstability property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}