Adds a part on augmented cdgas

thesis/chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory.tex

@@ -31,6 +31,9 @@ In this chapter the ring $\k$ is assumed to be a field of characteristic zero. I
 \section{Homotopy relations on \titleCDGA}
 \input{notes/Homotopy_Relations_CDGA}
 
+\section{Homotopy theory of augmented cdga's}
+\input{notes/Homotopy_Augmented_CDGA}
+
 \section{Homotopy groups of cdga's}
 \input{notes/Homotopy_Groups_CDGA}
 

thesis/notes/Homotopy_Augmented_CDGA.tex

@@ -0,0 +1,23 @@
+
+Recall that an augmented cdga is a cdga $A$ with an algebra map $A \tot{\counit} \k$ (this implies that $\counit \unit = \id$). This is precisely the dual notion of a pointed space. We will use the general fact that if $\cat{C}$ is a model category, then the over (resp. under) category $\cat{C} / A$ (resp. $A / \cat{C}$) for any object $A$ admit an induced model structure. In particular, the category of augmented cdga's (with augmentation preserving maps) has a model structure with the fibrations, cofibrations and weak equilavences as above.
+
+Although the model structure is completely induced, it might still be fruitful to discuss the right notion of a homotopy for augmented cdga's. Consider the following pullback of cdga's:
+\[ \xymatrix{
+	\Lambda(t, dt) \overline{\tensor} A \ar[r] \xypb \ar[d] & \Lambda(t, dt) \tensor A \ar[d] \\
+	\k \ar[r] & \k \tensor \Lambda(t, dt)
+}\]
+The pullback is the subspace of elements $x \tensor a$ in $\Lambda(t, dt) \tensor A$ such that $\counit(a) \cdot x \in \k$. Note that this construction is dual to a construction on topological spaces: in order to define a homotopy which is constant on the point $x_0$, we define the homotopy to be a map from a quotient ${X \times I} / {x_0 \times I}$.
+\Definition{homotopy-augmented}{
+	Two maps $f, g: A \to X$ between augmented cdga's are said to be \emph{homotopic} if there is a map
+	$$h : A \to \Lambda(t, dt) \overline{\tensor} X$$
+	such that $d_0 h = g$ and $d_1 h = f$.
+}
+
+In the next section homotopy groups of augmented cdga's will be defined. In order to define this we first need another tool.
+\Definition{indecomposables}{
+		Define the \Def{augmentation ideal} of $A$ as $\overline{A} = \ker \counit$. Define the \Def{cochain complex of indecomposables} of $A$ as $QA = \overline{A} / \overline{A} \cdot \overline{A}$.
+} 
+
+The first observation one should make is that $Q$ is a functor from algebras to modules (or differential algebras to differential modules) which is particularly nice for free algebras, as we have that $Q \Lambda V = V$ for any (differential) module $V$.
+
+\todo{tensor}

thesis/notes/Homotopy_Groups_CDGA.tex

@@ -1,10 +1,8 @@
 
-As the eventual goal is to compare the homotopy theory of spaces with the homotopy theory of cdga's, it is natural to investigate an analogue of homotopy groups in the category of cdga's. In topology we can only define homotopy groups on pointed spaces, dually we will consider augmented cdga's in this section. Recall that an augmented cdga is a cdga $A$ with an algebra map $A \tot{\counit} \k$ such that $\counit \unit = \id$.
+As the eventual goal is to compare the homotopy theory of spaces with the homotopy theory of cdga's, it is natural to investigate an analogue of homotopy groups in the category of cdga's. In topology we can only define homotopy groups on pointed spaces, dually we will consider augmented cdga's in this section.
 
 \Definition{cdga-homotopy-groups}{
-	Define the \Def{augmentation ideal} of $A$ as $\overline{A} = \ker \counit$. Define the \Def{cochain complex of indecomposables} of $A$ as $QA = \overline{A} / \overline{A} \cdot \overline{A}$.
+	The \Def{homotopy groups of an augmented cdga} $A$ are
-
-	Now define the \Def{homotopy groups of a cdga} $A$ as
 	$$ \pi^i(A) = H^i(QA). $$
 }
 
@@ -15,7 +13,7 @@ This construction is functorial and, as the following lemma shows, homotopy inva
 	$$ f_\ast = g_\ast : \pi_\ast(A) \to \pi_\ast(B). $$
 }
 \Proof{
-	Let $\phi: A \to B$ be a map of algebras. Then clearly we get an induced map $\overline{A} \to \overline{B}$ as $\phi$ preserves the augmentation. By composition we get a map $\phi': \overline{A} \to Q(B)$ for which we have $\phi'(xy) = \phi'(x)\phi'(y) = 0$. So it induces a map $Q(\phi): Q(A) \to Q(B)$. By functoriality of taking homology we get $f_\ast : \pi^n(A) \to \pi^n(B)$.
+	Let $\phi: A \to B$ be a map of algebras. Then clearly we get an induced map $\overline{A} \to \overline{B}$ as $\phi$ preserves the augmentation. By composition we get a map $\phi': \overline{A} \to Q(B)$ for which we have $\phi'(xy) = \phi'(x)\phi'(y) = 0$. So it induces a map $Q(\phi): Q(A) \to Q(B)$. By functoriality of taking homology we get $f_\ast : \pi^n(A) \to \pi^n(B)$. \todo{functoriality is redundant with previous section}
 
 	Now if $f$ and $g$ are homotopic, then there is a homotopy $h: A \to \Lambda(t, dt) \tensor B$. By the Künneth theorem we have:
 	$$ {d_0}_\ast = {d_1}_\ast : H(\Lambda(t, dt) \tensor Q(B)) \to H(Q(B)). $$

thesis/notes/Homotopy_Relations_CDGA.tex

@@ -38,8 +38,10 @@ The results from model categories immediately imply the following results. \todo
 		\item Let $i: A \to B$ be a trivial cofibration, then the induced map $i^\ast: [B, X] \to [A, X]$ is a bijection.
 		\item Let $p: X \to Y$ be a trivial fibration, then the induced map $p_\ast: [A, X] \to [A, Y]$ is a bijection.
 		\item Let $A$ and $X$ both be cofibrant, then $f: A \we X$ is a weak equivalence if and only if $f$ is a strong homotopy equivalence. Moreover, the two induced maps are bijections:
-		$$ f_\ast: [Z, A] \tot{\iso} [Z, X], $$
+		\begin{align*}
-		$$ f^\ast: [X, Z] \tot{\iso} [A, X]. $$
+			f_\ast: [Z, A] &\tot{\iso} [Z, X], \\
+			f^\ast: [X, Z] &\tot{\iso} [A, X].
+		\end{align*}
 		\todo{De eerste werkt ook als $i$ gewoon een w.e. is. (Gebruik factorizatie.)}
 	\end{itemize}
 }
@@ -48,5 +50,10 @@ The results from model categories immediately imply the following results. \todo
 	Let $f, g: A \to X$ be two homotopic maps, then $H(f) = H(g): HA \to HX$.
 }
 \Proof{
-	We only need to consider $H(d_0)$ and $H(d_1)$. \todo{Bewijs afmaken}
+	Let $h$ be the homotopy such that $f = d_1 h$ and $g = d_0 h$. By the Künneth theorem we get the following commuting square for $i = 0, 1$:
+	\[ \xymatrix{
+		H(\Lambda(t, dt)) \tensor H(A) \ar[r]^-{d_i \tensor \id} \ar[d]^-{\iso} & \k \tensor H(A) \ar[d]^-{\iso} \\
+		H(\Lambda(t, dt) \tensor A) \ar[r]^-{d_i} & H(\k \tensor A)
+	} \]
+	Now we know that $H(d_0) = H(d_1) : H(\Lambda(t, dt)) \to \k$ as $\Lambda(t, dt)$ is acyclic and the induced map send $1$ to $1$. So the two bottom maps in the diagram are equal as well. Now we conclude $H(f) = H(d_1)H(h) = H(d_0)H(h) = H(g)$.
 }