{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "5ZNN348M6X"
    },
    {
      "field": "uses",
      "target": "5ZNN348M6X"
    },
    {
      "field": "uses",
      "target": "SXMPSYESTP"
    },
    {
      "field": "uses",
      "target": "TY60P5RE56"
    },
    {
      "field": "uses",
      "target": "VRP0HESPYD"
    },
    {
      "field": "specializes",
      "target": "HMRP135M6K"
    },
    {
      "field": "defines",
      "target": "KCP52TPREW"
    }
  ],
  "body": "\n## Summary\n\nA Lie groupoid is a [[groupoid|FH3C0T643F]] whose objects and arrows each form a [[smooth manifold|F45VQ47GPV]].\nAll its structure maps are smooth, and the source and target maps are submersions.\nIt generalises [[Lie groups|7JBRGCVJRB]], which are Lie groupoids with a single object, and also smooth group actions, foliations and principal bundles.\nLie groupoids give a common language for symmetries that vary from point to point, and they have Lie algebroids as their infinitesimal versions.\n\n## Explanations\n\n### L1 human\n\nA group describes symmetries of one object; a groupoid describes symmetries between many objects.\nAn arrow goes from one object to another, and two arrows can be composed when they meet end to end.\nIn a Lie groupoid both the objects and the arrows form smooth spaces, and composing arrows is a smooth operation.\nFor example, take as objects the points of a manifold $M$, and as arrows from $x$ to $y$ the pairs $(y, x)$.\nComposing $(z, y)$ with $(y, x)$ gives $(z, x)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22A22",
      "58H05"
    ],
    "name": "Lie groupoid",
    "same_as": {
      "wikidata": "Q6543824",
      "wikipedia": "Lie groupoid"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lie_groupoid"
      }
    ],
    "specializes": [
      "FH3C0T643F"
    ],
    "tag": "HNVW0E732J"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/HNVW0E732J/",
  "links": [
    {
      "field": "specializes",
      "target": "FH3C0T643F"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HNVW0E732J.md",
  "msc": [
    "22A22",
    "58H05"
  ],
  "name": "Lie groupoid",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HNV/HNVW0E732J.md",
  "status": null,
  "tag": "HNVW0E732J",
  "uri": "m2://HNVW0E732J",
  "url": "https://tamnd.github.io/math2/wiki/lie-groupoid/",
  "writing": {
    "tag": "HNVW0E732J",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}