{
  "id": "2309.07258",
  "version": "v1",
  "published": "2023-09-13T18:48:31.000Z",
  "updated": "2023-09-13T18:48:31.000Z",
  "title": "On the integrability of Lie algebroids by diffeological spaces",
  "authors": [
    "Joel Villatoro"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "Lie's third theorem does not hold for Lie groupoids and Lie algebroids. In this article, we show that Lie's third theorem is valid within a specific class of diffeological groupoids that we call `singular Lie groupoids.' To achieve this, we introduce a subcategory of diffeological spaces which we call `quasi-etale.' Singular Lie groupoids are precisely the groupoid objects within this category, where the unit space is a manifold. Our approach involves the construction of a functor that maps singular Lie groupoids to Lie algebroids, extending the classical functor from Lie groupoids to Lie algebroids. We prove that the \\v{S}evera-Weinstein groupoid of an algebroid is an example of a singular Lie groupoid, thereby establishing Lie's third theorem in this context.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-13T18:48:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53D17",
      "57P05",
      "22A22"
    ],
    "keywords": [
      "lie algebroids",
      "diffeological spaces",
      "maps singular lie groupoids",
      "integrability",
      "establishing lies third theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}