{
  "aliases": null,
  "backlinks": [
    {
      "field": "specializes",
      "target": "E3EYXM7GJN"
    },
    {
      "field": "defines",
      "target": "K8YRDXE4WC"
    },
    {
      "field": "uses",
      "target": "VRP0HESPYD"
    }
  ],
  "body": "\n## Summary\n\nA pseudogroup on a [[topological space|0CQHFPGJ45]] is a collection of [[homeomorphisms|0QT00PB5G2]] between open subsets that is closed under composition, inversion, restriction and gluing.\nIt records the local symmetries of a space, which need not extend to global ones.\nPseudogroups are the transition rules of geometric structures: a smooth, complex or symplectic manifold is a space glued together by maps from the corresponding pseudogroup.\nThey also describe the transverse structure of foliations.\n\n## Explanations\n\n### L1 human\n\nA group of symmetries acts on a whole space at once.\nMany geometric questions only need symmetries that act on small pieces, such as a map from one disc to another.\nA pseudogroup collects such local maps, with the rule that the collection is stable under the obvious operations: composing, inverting, shrinking the domain and patching pieces together.\nFor example, all smooth maps with smooth inverses between open subsets of $\\R^n$ form a pseudogroup.\nA smooth manifold is a space whose charts change by maps of this pseudogroup.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "58H05",
      "57R30"
    ],
    "name": "Pseudogroup",
    "same_as": {
      "wikidata": "Q4382845",
      "wikipedia": "Pseudogroup"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Pseudogroup"
      }
    ],
    "tag": "3017J7P6JD"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/3017J7P6JD/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/3017J7P6JD.md",
  "msc": [
    "58H05",
    "57R30"
  ],
  "name": "Pseudogroup",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/301/3017J7P6JD.md",
  "status": null,
  "tag": "3017J7P6JD",
  "uri": "m2://3017J7P6JD",
  "url": "https://tamnd.github.io/math2/wiki/pseudogroup/",
  "writing": {
    "tag": "3017J7P6JD",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}