{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "V9079C40AZ"
    }
  ],
  "body": "\n## Summary\n\nA Sullivan minimal model is a free graded-commutative differential graded algebra over $\\Q$ with a decomposable differential.\nIt models the rational homotopy type of a space.\nEvery simply connected space of finite type has a minimal model, unique up to isomorphism.\nIts generators in degree $n$ form the dual of the rational homotopy group $\\pi_n(X) \\otimes \\Q$, and its cohomology is the rational cohomology of $X$.\nSo the minimal model is the main computational tool of [[rational homotopy theory|8G9QPMBT7R]].\n\n## Explanations\n\n### L1 human\n\nThe minimal model describes a space by a list of generators, one batch for each degree.\nA rule $d$ says how each generator is \"killed\" by products of earlier ones.\nThe generators count the rational homotopy groups, and the rule records how they interact.\nFor the sphere $S^2$, one needs a generator $x$ in degree $2$ for the class of the sphere and a generator $y$ in degree $3$ with $dy = x^2$.\nThe second generator kills the square $x^2$, which must vanish in the cohomology of $S^2$, and it matches the rational homotopy group $\\pi_3(S^2) \\otimes \\Q \\cong \\Q$ given by the Hopf map.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "55P62"
    ],
    "name": "Sullivan minimal model",
    "sources": [
      {
        "locator": "Sullivan algebras",
        "url": "https://en.wikipedia.org/wiki/Rational_homotopy_theory"
      }
    ],
    "tag": "3QQ2PBGY3N"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/3QQ2PBGY3N/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/3QQ2PBGY3N.md",
  "msc": [
    "55P62"
  ],
  "name": "Sullivan minimal model",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3QQ/3QQ2PBGY3N.md",
  "status": null,
  "tag": "3QQ2PBGY3N",
  "uri": "m2://3QQ2PBGY3N",
  "url": "https://math2.tamnd.com/wiki/sullivan-minimal-model/",
  "writing": {
    "tag": "3QQ2PBGY3N",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}