@ -39,24 +39,20 @@ The requirement that there exists a filtration can be replaced by a stronger sta
\section{Existence}
\section{Existence}
\begin{theorem}
\begin{theorem}
Let $(A, d)$ be an $0$-connected cdga, then it has a Sullivan model $(\Lambda V, d)$. Furthermore if $(A, d)$ is $r$-connected,$V^i =0$ for all $i \leq r$.
Let $(A, d)$ be an $0$-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.
\end{theorem}
\end{theorem}
\begin{proof}
\begin{proof}
Start by setting $V(0)= H^{\geq1}(A)$ and $d =0$. This extends to a morphism $m_0 : (\Lambda V(0), 0)\to(A, d)$.
Start by setting $V(0)= H^{\geq1}(A)$ and $d =0$. This extends to a morphism $m_0 : (\Lambda V(0), 0)\to(A, d)$.
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.
Note that the freeness introduces products such that the map $H(m_0) : H(\Lambda V(0))\to H(A)$ is not an isomorphism. We will ``kill'' these defects inductively.
Suppose $V(k)$ and $m_k$ are 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 $\deg{v_\alpha}=\deg{z_\alpha}-1$.
Suppose $V(k)$ and $m_k$ are 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.
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 $d^2 v_\alpha=0$.
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.
Now take $V(k+1)= V(k)\oplus V_{k+1}$.
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 $m_{k+1} d v_\alpha= m_{k+1} z_\alpha= d a_\alpha= d m_{k+1} v_\alpha$.
Now take $V(k+1)= V(k)\oplus V_{k+1}$. We already proved that $d$ is indeed a differential, and that $m_{k+1}$ is indeed a chain map.
Complete the construction by taking the union: $V =\bigcup_k V(k)$. Clearly $H(m)$ is surjective, this was establsihed 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.
Complete the construction by taking the union: $V =\bigcup_k V(k)$. Clearly $H(m)$ is surjective, this was establsihed 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{minimality for $1$-connected}
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 the previous lemma $(\Lambda V, d)$ is minimal.
\end{proof}
\end{proof}
@ -65,14 +61,14 @@ The requirement that there exists a filtration can be replaced by a stronger sta
Before we state the uniqueness theorem we need some more properties of minimal models.
Before we state the uniqueness theorem we need some more properties of minimal models.
\begin{lemma}
\begin{lemma}
Sullivan algebras are cofibrant.
Sullivan algebras are cofibrant and the inclusions $(\Lambda V(k), d)\to(\Lambda V(k+1), d)$ are cofibrations.
\end{lemma}
\end{lemma}
\begin{proof}
\begin{proof}
Consider the following lifting problem, where $\Lambda V$ is a Sullivan algebra.
Consider the following lifting problem, where $\Lambda V$ is a Sullivan algebra.
\cimage[scale=0.5]{Sullivan_Lifting}
\cimage[scale=0.5]{Sullivan_Lifting}
By the left adjointness of $\Lambda$ we only have to specify a map $\phi: V \to X$ such that $p \circ\phi= g$. We will do this by induction.
By the left adjointness of $\Lambda$ we only have to specify a map $\phi: V \to X$ 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 cofibrations.
\begin{itemize}
\begin{itemize}
\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 $\{v_\alpha\}$ 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
\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
@ -90,16 +86,18 @@ Before we state the uniqueness theorem we need some more properties of minimal m
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 $d x' = d \overline{x}- d d a = d \overline{x}= x$.
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 $d x' = d \overline{x}- d d a = d \overline{x}= x$.
\end{proof}
\end{proof}
\begin{lemma}
\Lemma{minimal-model-bijection}{
Let $f: X \we Y$ be a weak equivalence between cdga's and $M$ a minimal model for $X$. Then $f$ induces an bijection:
Let $f: X \we Y$ be a weak equivalence between cdga's and $M$ a minimal algebra. Then $f$ induces an bijection:
$$ f_\ast: [M, X]\tot{\iso}[M, Y]. $$
$$ f_\ast: [M, X]\tot{\iso}[M, Y]. $$
\end{lemma}
}
\begin{proof}
\begin{proof}
If $f$ is surjective this follows from the fact that $M$ is cofibrant and $f$ being a trivial fibration (see \cite[lemma 4.9]{dwyer}).
If $f$ is surjective this follows from the fact that $M$ is cofibrant and $f$ being a trivial fibration, see \CorollaryRef{cdga_homotopy_properties}.
In general we can construct a cdga $Z$ and trivial fibrations $X \to Z$ and $Y \to Z$ inducing bijections:
\todo{Put this surjectivity trick in a lemma} In general we will reduce to the surjective case. Let $C$ be any cochain complex and define $\delta C^k = C^{k-1}$. Now $(C \oplus\delta C, \delta)$ is again a cochain complex and there is a surjective map to $C$. Define $E(Y)=\Lambda(Y \oplus\delta Y, \delta)$ (we consider $Y$ as a cochain complex). We obtain:
$$[M, X]\tot{\iso}[M, Z]\toti{\iso}[M, Y], $$
$$ f: X \tot{x \mapsto x \tensor1} X \tensor E(Y)\tot{x \tensor y \mapsto f(x)\cdot y} Y. $$
compatible with $f_\ast$. \cite[Proposition 12.9]{felix}.
Now the first map has a left inverse. We have two trivial fibrations $E(Y)\to X$ and $E(Y)\to Y$. This induces
$$[M, X]\toti{\iso}[M, E(Y)]\tot{\iso}[M, Y], $$
compatible with $f_\ast$.
\end{proof}
\end{proof}
\begin{lemma}
\begin{lemma}
@ -117,6 +115,20 @@ Before we state the uniqueness theorem we need some more properties of minimal m
By the previous lemmas 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.
By the previous lemmas 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.
\end{proof}
\end{proof}
The assignment of $X$ to 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:
Joshua Moerman
10 years ago