Adds some details to the model structure (but still not all)

thesis/notes/Model_Of_CDGA.tex

@@ -32,7 +32,17 @@ Note that with these classes, every cdga is a fibrant object.
 	[MC3] All three classes are closed under retracts
 \end{lemma}
 \begin{proof}
-	\todo{Diagrammen en uitschrijven}
+	For the class of weak equivelances and fibrations this follows easily from basic category theory. For cofibrations we consider the following diagram where the horizontal compositions are identities:
+	\[ \xymatrix{
+		A' \ar[r] \ar[d]^g & A \ar[r] \arcof[d]^f & A' \ar[d]^g \\
+		B' \ar[r] & B \ar[r] & B'
+	}\]
+	We need to prove that $g$ is a cofibration, so for any lifting problem with a trivial fibration we need to find a lift. We are in the following situation:
+		\[ \xymatrix{
+		A' \ar[r] \ar[d]^g & A \ar[r] \arcof[d]^f & A' \ar[r] \ar[d]^g & X \artfib[d] \\
+		B' \ar[r] & B \ar[r] & B' \ar[r] & Y
+	}\]
+	Now we can find a lift starting at $B$, since $f$ is a cofibration. By precomposition we obtain a lift $B' \to X$.
 \end{proof}
 
 Next we will prove the factorization property [MC5]. We will do this by Quillen's small object argument. When proved, we get an easy way to prove the missing lifting property of [MC4]. For the Quillen's small object argument we use classes of generating cofibrations.
@@ -59,7 +69,9 @@ Next we will prove the factorization property [MC5]. We will do this by Quillen'
 	The class of (trivial) cofibrations is saturated.
 \end{lemma}
 \begin{proof}
-	\todo{prove this}
+	We need to prove that the classes are closed under retracts (this is already done), pushouts and transfinite compositions. For the class of cofibrations, this is easy as they are defined by the LLP and colimits behave nice with respect to such classes. 
+
+	However the case of trivial cofibrations does not follow immediately, as we still need to prove that quasi isomorphisms behave as such.\todo{THIS IS HARD}
 \end{proof}
 
 As a consequence of the above two lemmas, the class generated by $I$ is contained in the class of trivial cofibrations. Similarly the class generated by $J$ is contained in the class of cofibrations. We also have a similar lemma about (trivial) fibrations.
@@ -77,7 +89,7 @@ As a consequence of the above two lemmas, the class generated by $I$ is containe
 	If $p: X \to Y$ has the RLP w.r.t. $J$ then $p$ is a trivial fibration.
 \end{lemma}
 \begin{proof}
-	\todo{Even bewijzen}
+	\todo{bewijzen}
 \end{proof}
 
 We can use Quillen's small object argument with these sets. The argument directly proves the following lemma. Together with the above lemmas this translates to the required factorization.
@@ -86,7 +98,7 @@ We can use Quillen's small object argument with these sets. The argument directl
 	A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is in the class generated by $I$ and $p$ has the RLP w.r.t. $I$.
 \end{lemma}
 \begin{proof}
-	Quillen's small object argument. \todo{Definieer wat ``small'' betkent en geef een referentie}
+	This follows from Quillen's small object argument. \todo{Definieer wat ``small'' betkent en geef een referentie}
 \end{proof}
 
 \begin{corollary}
@@ -108,4 +120,19 @@ where $i$ is the obvious inclusion $i(a) = a \tensor 1$ and $p$ maps (products o
 	[MC5b] A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is a cofibration and $p$ a trivial fibration.
 \end{corollary}
 
-\todo{Bewijs [MC4].}
+\begin{lemma}
+	[MC4]
+\end{lemma}
+\Proof{
+	One part is already established by definition (cofibrations are defined by an LLP). It remains to show that we can lift in the following situation:
+	\[\xymatrix{
+		A \ar[r] \artcof[d]^f & X \arfib[d] \\
+		B \ar[r] & Y
+	}\]
+	Now factor $f = pi$, where $p$ is a fibration and $i$ a trivial cofibration. By the 2-out-of-3 property $p$ is also a weak equivalence and we can find a lift in the following diagram:
+	\[\xymatrix{
+		A \ar[r]^i \arcof[d]^f & Z \artfib[d]^p \\
+		B \ar[r]^\id \ar@{-->}[ur] & B
+	}\]
+	This defines $f$ as a retract of $i$. Now we know that $i$ has the LLP w.r.t. fibrations (by the small object argument above), hence $f$ has the LLP w.r.t. fibrations as well.
+}