In this section we will discuss the so called minimal models. These cdga's enjoy the property that we can easily prove properties inductively. Moreover it will turn out that weakly equivalent minimal models are actually isomorphic.
We will often say \Def{minimal model} or \Def{minimal algebra} to mean minimal Sullivan model or minimal Sullivan algebra. In many cases we can take the degree of the elements in $V$ to induce the filtration, as seen in the following lemma.
Let $V$ generate $A$. Take $V(n)=\bigoplus_{k=0}^n V^k$ (note that $V^0= V^1=0$). Since $d$ is decomposable we see that for $v \in V^n$: $d(v)= x \cdot y$ for some $x, y \in A$. Assuming $dv$ to be non-zero we can compute the degrees:
Minimal models admit very nice homotopy groups. Note that for a minimal algebra $\Lambda V$ there is a natural augmentation and the the differential is decomposable. Hence $Q \Lambda V$ is naturally isomorphic to $(V, 0)$. In particular the homotopy groups are simply given by $\pi^n(\Lambda V)= V^n$.
\DefinitionRef{minimal-algebra} is the same as in \cite{felix} without assuming connectivity. We find some different definitions of (minimal) Sullivan algebras in the literature. For example we find a definition using well orderings in \cite{hess}. The decomposability of $d$ also admits a different characterization (at least in the connected case). The equivalence of the definitions is expressed in the following two lemmas.\todo{to prove or not to prove}
A cdga $(\Lambda V, d)$ is a Sullivan algebra if and only if there exists a well order $J$ such that $V$ is generated by $v_j$ for $j \in J$ and $d v_j \in\Lambda V_{<j}$.
}
\Lemma{}{
Let $(\Lambda V, d)$ be a Sullivan algebra with $V^0=0$, then $d$ is decomposable if and only if there is a well order $J$ as above such that $i < j$ implies $\deg{v_i}\leq\deg{v_j}$.
}
It is clear that induction will be an important technique when proving things about (minimal) Sullivan algebras. We will first prove that minimal models always exist for $1$-connected cdga's and afterwards prove uniqueness.
Let $(A, d)$ be a $0$-connected cdga, then it has a Sullivan model $(\Lambda V, d)$. Furthermore if $(A, d)$ is $r$-connected with $r \geq1$ then $V^i =0$ for all $i \leq r$ and in particular $(\Lambda V, d)$ is minimal.
Note that the freeness introduces products such that the map $H(m_0) : H(\Lambda V(0))\to H(A)$ is \emph{not} an isomorphism. We will ``kill'' these defects inductively.
Suppose $V(k)$ and $m_k$ have been constructed. Consider the defect $\ker H(m_k)$ and let $\{[z_\alpha]\}_{\alpha\in A}$ be a basis for it. Define $V_{k+1}=\bigoplus_{\alpha\in A}\k\cdot v_\alpha$ with the degrees $\deg{v_\alpha}=\deg{z_\alpha}-1$.
Now extend the differential by defining $d(v_\alpha)= z_\alpha$. This step kills the defect, but also introduces new defects which will be killed later. Notice that $z_\alpha$ is a cocycle and hence $d^2 v_\alpha=0$, so $d$ is still a differential.
Since $[z_\alpha]$ is in the kernel of $H(m_k)$ we see that $m_k z_\alpha= d a_\alpha$ for some $a_\alpha$. Extend $m_k$ to $m_{k+1}$ by defining $m_{k+1}(v_\alpha)= a_\alpha$. Notice that $m_{k+1} d v_\alpha= m_{k+1} z_\alpha= d a_\alpha= d m_{k+1} v_\alpha$, so $m_{k+1}$ is a cochain map.
Complete the construction by taking the union: $V =\bigcup_k V(k)$. Clearly $H(m)$ is surjective, this was established in the first step. Now if $H(m)[z]=0$, then we know $z \in\Lambda V(k)$ for some stage $k$ and hence by construction is was killed, i.e. $[z]=0$. So we see that $m$ is a quasi isomorphism and by construction $(\Lambda V, d)$ is a Sullivan algebra.
\todo{Rewrite this section} Now assume that $(A, d)$ is $r$-connected ($r \geq1$), this means that $H^i(A)=0$ for all $1\leq i \leq r$, and so $V(0)^i =0$ for all $i \leq r$. Now $H(m_0)$ is injective on $\Lambda^{\leq1} V(0)$, and so the defects are in $\Lambda^{\geq2} V(0)$ and have at least degree $2(r+1)$. This means two things in the first inductive step of the construction. First, the newly added elements have decomposable differential. Secondly, these elements are at least of degree $2(r+1)-1$. After adding these elements, the new defects are in $\Lambda^{\geq2} V(1)$ and have at least degree $2(2(r+1)-1)$. We see that as the construction continues, the degrees of adjoined elements go up. Hence $V^i =0$ for all $i \leq r$ and by \LemmaRef{1-reduced-minimal-model}\todo{does not apply}$(\Lambda V, d)$ is minimal.
Before we state the uniqueness theorem we need some more properties of minimal models. In this section we will use some general facts about model categories.
By the left adjointness of $\Lambda$ we only have to specify a map $\phi: V \to X$ which commutes with the differential such that $p \circ\phi= g$. We will do this by induction. Note that the induction step proves precisely that $(\Lambda V(k), d)\to(\Lambda V(k+1), d)$ is a cofibration.
\item Suppose $\{v_\alpha\}$ is a basis for $V(0)$. Define $V(0)\to X$ by choosing preimages $x_\alpha$ such that $p(x_\alpha)= g(v_\alpha)$ ($p$ is surjective). Define $\phi(v_\alpha)= x_\alpha$.
\item Suppose $\phi$ has been defined on $V(n)$. Write $V(n+1)= V(n)\oplus V'$ and let $\{v_\alpha\}$ be a basis for $V'$. Then $dv_\alpha\in\Lambda V(n)$, hence $\phi(dv_\alpha)$ is defined and
$$ d \phi d v_\alpha=\phi d^2 v_\alpha=0$$
$$ p \phi d v_\alpha= g d v_\alpha= d g v_\alpha. $$
Now $\phi d v_\alpha$ is a cycle and $p \phi d v_\alpha$ is a boundary of $g v_\alpha$. By the following lemma there is a $x_\alpha\in X$ such that $d x_\alpha=\phi d v_\alpha$ and $p x_\alpha= g v_\alpha$. The former property proves that $\phi$ is a chain map, the latter proves the needed commutativity $p \circ\phi= g$.
\end{itemize}
\end{proof}
\begin{lemma}
Let $p: X \to Y$ be a trivial fibration, $x \in X$ a cycle, $p(x)\in Y$ a boundary of $y' \in Y$. Then there is a $x' \in X$ such that
$$ dx' = x \quad\text{ and }\quad px' = y'. $$
\end{lemma}
\begin{proof}
We have $p^\ast[x]=[px]=0$, since $p^\ast$ is injective we have $x = d \overline{x}$ for some $\overline{x}\in X$. Now $p \overline{x}= y' + db$ for some $b \in Y$. Choose $a \in X$ with $p a = b$, then define $x' =\overline{x}- da$. Now check the requirements: $p x' = p \overline{x}- p a = y'$ and $d x' = d \overline{x}- d d a = d \overline{x}= x$.
Since both $M$ and $M'$ are minimal, they are cofibrant and so the weak equivalence is a strong homotopy equivalence (\CorollaryRef{cdga_homotopy_properties}). And so the induced map $\pi^n(\phi) : \pi^n(M)\to\pi^n(M')$ is an isomorphism (\LemmaRef{cdga-homotopic-maps-equal-pin}).
Since $M$ (resp. $M'$) is free as a cga's, it is generated by some graded vector space $V$ (resp. $V'$). By an earlier remark the homotopy groups were easy to calculate and we conclude that $\phi$ induces an isomorphism from $V$ to $V'$:
By induction on the degree one can prove that $\phi$ needs to be surjective and hence is a fibration. By the lifting property we can find a right inverse $\psi$, which is then injective and a weak equivalence. Now the above argument also proves that $\psi$ is surjective. Conclude that $\psi$ is an isomorphism and $\phi$, being its right inverse, is an isomorphism as well.
Let $m: (M, d)\we(A, d)$ and $m': (M', d')\we(A, d)$ be two minimal models for $A$. Then there is an isomorphism $\phi(M, d)\tot{\iso}(M', d')$ such that $m' \circ\phi\eq m$.
By \CorollaryRef{minimal-model-bijection} we have $[M', M]\iso[M', A]$. By going from right to left we get a map $\phi: M' \to M$ such that $m' \circ\phi\eq m$. On homology we get $H(m')\circ H(\phi)= H(m)$, proving that (2-out-of-3) $\phi$ is a weak equivalence. The previous lemma states that $\phi$ is then an isomorphism.
The assignment to $X$ of its minimal model $M_X =(\Lambda V, d)$ can be extended to morphisms. Let $X$ and $Y$ be two cdga's and $f: X \to Y$ be a map. By considering their minimal models we get the following diagram.
Now by \LemmaRef{minimal-model-bijection} we get a bijection ${m_Y}_\ast^{-1} : [M_X, Y]\iso[M_X, M_Y]$. This gives a map $M(f)={m_Y}_\ast^{-1}(f m_X)$ from $M_X$ to $M_Y$. Of course this does not define a functor of cdga's as it is only well defined on homotopy classes. However it is clear that it does define a functor on the homotopy category of cdga's.
\Corollary{}{
The assignment $X \mapsto M_X$ defines a functor $M: \Ho(\CDGA^1_\Q)\to\Ho(\CDGA^1_\Q)$. Moreover, since the minimal model is weakly equivalent, $M$ gives an equivalence of categories:
