Adds more proofs about the equivalence

thesis/notes/A_K_Quillen_Pair.tex

@@ -149,4 +149,32 @@ and the second map is obtained by the map $A \to A(K(A))$ and using the bijectio
 	To conclude that $B \to A(K(B))$ is a weak equivalence, we wish to prove that the front face of the cube is a homotopy pushout, as the back face clearly is one. This is a consequence of the Eilenberg-Moore spectral sequence \cite{mccleary}.
 }
 
-Now we wish to use the previous lemma as an induction step for minimal models.
+Now we wish to use the previous lemma as an induction step for minimal models. Let $(\Lambda V, d)$ be some minimal algebra. Write $V(n+1) = V(n) \oplus V'$ and let $v \in V'$ of degree $\deg{v} = k$, then the minimal algebra $(\Lambda (V(n) \oplus \Q \cdot v), d)$ is the pushout in the following diagram, where $f$ sends the generator $c$ to $dv$.
+\begin{displaymath}
+	\xymatrix{
+	S(k) \arcof[d] \ar[r]^f \xypo & (\Lambda V(n), d) \ar[d] \\
+	T(k-1) \ar[r] & (\Lambda (V(n) \oplus \Q \cdot v), d)
+	}
+\end{displaymath}
+In particular if the vector space $V'$ is finitely generated, we can repeat this procedure for all basis elements (it does not matter in what order we do so, as $dv \in \Lambda V(n)$). So in this case, if $(\Lambda V(n), d) \to A(K(\Lambda V(n), d))$ is a weak equivalence, so is $(\Lambda V(n+1), d) \to A(K(\Lambda V(n+1), d))$
+
+\Corollary{}{
+	Let $(\Lambda V, d)$ be a $1$-connected minimal algebra with $V^i$ finite dimensional for all $i$. Then $(\Lambda V, d) \to A(K(\Lambda V, d))$ is a weak equivalence.
+}
+\Proof{
+	Note that if we want to prove the isomorphism $H^i(\Lambda V, d) \to H^i(A(K(\Lambda V, d)))$ it is enough to prove that $H^i(\Lambda V^{\leq i}, d) \to H^i(A(K(\Lambda V^{\leq i}, d)))$ is an isomorphism (as the elements of higher degree do not change the isomorphism). By the $1$-connectedness we can choose our filtration to respect the degree by \LemmaRef{1-reduced-minimal-model}.
+
+	Now $V(n)$ is finitely generated for all $n$ by assumption. By the inductive procedure above we see that $(\Lambda V(n), d) \to A(K(\Lambda V(n), d))$ is a weak equivalence for all $n$. Hence $(\Lambda V, d) \to A(K(\Lambda V, d))$ is a weak equivalence.
+}
+
+\todo{$X \to K(A(X))$}
+
+We have proven the following theorem.
+\Theorem{main-theorem}{
+	The functors $A$ and $K$ induce an equivalence of homotopy categories, when restricted to rational, $1$-connected objects of finite type. more formally, we have:
+	$$ \Ho(\sSet_1^{\Q,f}) \iso \Ho(\CDGA_{\Q,1,f}). $$
+
+	Furthermore, for any $1$-connected space $X$ of finite type, we have the following isomorphism of groups:
+	$$ \pi_i(X) \tensor \Q \iso {V^i}^\ast, $$
+	where $(\Lambda V, d)$ is the minimal model of $A(X)$.
+}

thesis/notes/Minimal_Models.tex

@@ -26,14 +26,14 @@ In this section we will discuss the so called minimal models. These cdga's enjoy
 
 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 of which the proof is left out, as we are not going to use it.
 
-\begin{lemma}
+\Lemma{1-reduced-minimal-model}{
 	Let $(A, d)$ be a cdga which is $1$-reduced, such that $A$ is free as cga and $d$ is decomposable. Then $(A, d)$ is a minimal algebra.
-\end{lemma}
-\begin{proof}
+}
+\Proof{
 	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:
 	$$ \deg{x} + \deg{y} = \deg{xy} = \deg{dv} = \deg{v} + 1 = n + 1. $$
 	As $A$ is $1$-reduced we have $\deg{x}, \deg{y} \geq 2$ and so by the above $\deg{x}, \deg{y} \leq n-1$. Conclude that $d(V(k)) \subset \Lambda(V(n-1))$.
-\end{proof}
+}
 
 The above definition 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.