{
  "id": "1506.07024",
  "version": "v1",
  "published": "2015-06-23T14:23:17.000Z",
  "updated": "2015-06-23T14:23:17.000Z",
  "title": "Amalgamation from categoricity in universal classes",
  "authors": [
    "Sebastien Vasey"
  ],
  "comment": "27 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove that, in universal classes, categoricity in a high-enough cardinal implies amalgamation. The proof stems from ideas of Adi Jarden and Will Boney. $\\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in a high-enough cardinal, then there exists $\\lambda$ such that: * $K_{\\ge \\lambda}$ has amalgamation. * $K_{\\ge \\lambda}$ is fully good (i.e. it admits a global forking-like notion). We obtain several categoricity transfers: $\\mathbf{Corollary}$ Let $K$ be a universal class. * If $K$ is categorical in a high-enough successor cardinal, then $K$ is categorical on a tail of cardinals. * Assume that an unpublished claim of Shelah holds, and that $2^{\\lambda} < 2^{\\lambda^+}$ for all cardinals $\\lambda$. If $K$ is categorical in a high-enough cardinal, then $K$ is categorical on a tail of cardinals. We work in a more general context than universal classes: abstract elementary classes which admit intersections, a notion introduced by Baldwin and Shelah.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-23T14:23:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C48",
      "03C45",
      "03C52",
      "03C55"
    ],
    "keywords": [
      "universal class",
      "categoricity",
      "high-enough cardinal implies amalgamation",
      "high-enough successor cardinal",
      "categorical"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150607024V"
    }
  }
}