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  "body": "\n## Definition\n\nWork over $\\Q$ with cochain differential graded algebras (dgas) that are graded-commutative: $ab = (-1)^{\\abs{a}\\abs{b}} ba$ for homogeneous $a$ and $b$.\nFor a graded vector space $V = \\bigoplus_{n \\ge 1} V^n$, let $\\Lambda V$ be the free graded-commutative algebra on $V$.\nIt is the tensor product of the [[symmetric algebra|YZCBSRCZC7]] on the even part and the [[exterior algebra|08HK38GCSM]] on the odd part.\nLet $\\Lambda^{\\ge 2} V$ be its ideal of decomposable elements, the span of products of at least two elements of $V$.\n\n1. A *Sullivan algebra* is a dga $(\\Lambda V, d)$ with $V = V^{\\ge 1}$ and an increasing chain of graded subspaces $V(0) \\subseteq V(1) \\subseteq \\cdots$ with union $V$.\n   The chain must satisfy $d(V(0)) = 0$ and $d(V(k)) \\subseteq \\Lambda V(k-1)$ for $k \\ge 1$.\n2. A Sullivan algebra is *minimal* if $d(V) \\subseteq \\Lambda^{\\ge 2} V$.\n3. Let $(A, d_A)$ be a graded-commutative dga with $H^0(A) = \\Q$.\n   A *Sullivan minimal model* of it is a minimal Sullivan algebra $(\\Lambda V, d)$ with a dga map $m \\colon (\\Lambda V, d) \\to (A, d_A)$ that is an isomorphism in cohomology.\n\nThe minimal model of a path-connected space $X$ is the minimal model of Sullivan's algebra $A_{PL}(X)$ of rational polynomial differential forms on $X$.\nThe cohomology of $A_{PL}(X)$ is the rational [[singular cohomology|6N7BC70CX0]] $H^*(X; \\Q)$.\n\n## Remarks\n\nIf $X$ is simply connected and each $H^n(X; \\Q)$ is finite-dimensional, then $V^n \\cong \\Hom(\\pi_n(X), \\Q)$ for all $n \\ge 2$, and $V^1 = 0$.\n\nExample: for $n$ odd, $(\\Lambda(x), 0)$ with $\\abs{x} = n$ is the minimal model of $S^n$.\nHere $\\Lambda(x) = \\Q \\oplus \\Q x$ since $x^2 = 0$ by graded commutativity, which is $H^*(S^n; \\Q)$.\n\nExample: for $n$ even, the minimal model of $S^n$ is $(\\Lambda(x, y), d)$ with $\\abs{x} = n$, $\\abs{y} = 2n - 1$, $dx = 0$ and $dy = x^2$.\nIts cohomology has basis $1, x$: each $x^k$ with $k \\ge 2$ equals $d(x^{k-2} y)$, and $d(x^k y) = x^{k+2} \\ne 0$.\n\nNon-example: $(\\Lambda(x, y), d)$ with $\\abs{y} = 1$, $\\abs{x} = 2$ and $dy = x$ is a Sullivan algebra but not minimal, since $dy = x$ is not decomposable.\nIt is acyclic in positive degrees, and its minimal model is $\\Q$.\n",
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