Sullivan minimal model

Definition

Work over $\Q$ with cochain differential graded algebras (dgas) that are graded-commutative: $ab = (-1)^{\abs{a}\abs{b}} ba$ for homogeneous $a$ and $b$. For a graded vector space $V = \bigoplus_{n \ge 1} V^n$, let $\Lambda V$ be the free graded-commutative algebra on $V$. It is the tensor product of the symmetric algebra on the even part and the exterior algebra on the odd part. Let $\Lambda^{\ge 2} V$ be its ideal of decomposable elements, the span of products of at least two elements of $V$.

1. 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 $V$. The chain must satisfy $d(V(0)) = 0$ and $d(V(k)) \subseteq \Lambda V(k-1)$ for $k \ge 1$.
2. A Sullivan algebra is minimal if $d(V) \subseteq \Lambda^{\ge 2} V$.
3. Let $(A, d_A)$ be a graded-commutative dga with $H^0(A) = \Q$. 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.

The 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 $X$. The cohomology of $A_{PL}(X)$ is the rational singular cohomology $H^*(X; \Q)$.