{
  "id": "2302.01319",
  "version": "v1",
  "published": "2023-02-02T18:45:30.000Z",
  "updated": "2023-02-02T18:45:30.000Z",
  "title": "A classification of the Wadge hierarchies on zero-dimensional Polish spaces",
  "authors": [
    "Raphaël Carroy",
    "Luca Motto Ros",
    "Salvatore Scamperti"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We provide a complete classification, up to order-isomorphism, of all possible Wadge hierarchies on zero-dimensional Polish spaces using (essentially) countable ordinals as complete invariants. We also observe that although our assignment of invariants is very simple and there are only $\\aleph_1$-many equivalence classes, the above classification problem is quite complex from the descriptive set-theoretic point of view: in particular, there is no Borel procedure to determine whether two zero-dimensional Polish spaces have isomorphic Wadge hierarchies. All results are based on a complete and explicit description of the Wadge hierarchy on an arbitrary zero-dimensional Polish space, depending on its topological properties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-02T18:45:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15",
      "54E50"
    ],
    "keywords": [
      "wadge hierarchy",
      "isomorphic wadge hierarchies",
      "arbitrary zero-dimensional polish space",
      "borel procedure",
      "classification problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}