Adds a section about the homotopy relation on cdga

thesis/notes/Model_Of_CDGA.tex

@@ -17,6 +17,8 @@ In this section we will define a model structure on CDGAs over a field $\k$ of c
 
 We will prove the different axioms in the following lemmas. First observe that the classes as defined above are indeed closed under multiplication and contain all isomorphisms.
 
+Note that with these classes, every cdga is a fibrant object.
+
 \begin{lemma}
 	[MC1] The category has all finite limits and colimits.
 \end{lemma}
@@ -109,3 +111,51 @@ where $i$ is the obvious inclusion $i(a) = a \tensor 1$ and $p$ maps (products o
 \end{corollary}
 
 
+\subsection{Homotopy relation on \texorpdfstring{$\CDGA_\k$}{CDGA}}
+Although the abstract theory of model categories gives us tools to construct a homotopy relation (\DefinitionRef{homotopy}), it is useful to have a concrete notion of homotopic maps.
+
+Consider the free cdga on one generator $\Lambda(t, dt)$, this can be thought of as the (dual) unit interval. Indeed there is an isomorphism $\Lambda(t, dt) \iso \Apl_1$ and so we have maps for the two endpoint: $d_0, d_1: \Lambda(t, dt) \to \k \iso \Apl_0$. Given a cdga $X$ we will consider $d_0, d_1: \Lambda(t, dt) \tensor X \to \k \tensor X \iso X$.
+
+\Definition{cdga_homotopy}{
+	We call $f, g: A \to X$ homotopic ($f \simeq g$) if there is a map
+	$$ h: A \to \Lambda(t, dt) \tensor X, $$
+	such that $d_0 h = g$ and $d_1 h = f$.
+}
+
+In terms of model categories, such a homotopy is a right homotopy and the object $\Lambda(t, dt) \tensor X$ is a path object for $X$. We can easily see that it is a very good path object, first note that $\Lambda(t, dt) \tensor X \to X \oplus X$ is surjective (for $(x, y) \in X \oplus X$ take $t \tensor x + 1 \tensor y$), secondly $\Apl_0 \to \Apl_1$ is a cofibration and so is $X \to \Lambda(t, dt) \tensor X$.
+
+Clearly we have that $f \simeq g$ implies $f \simeq^r g$ (see \DefinitionRef{right_homotopy}), however the converse need not be true.
+
+\Lemma{cdga_homotopy}{
+	If $A$ is a cofibrant cdga and $f \simeq^r g: A \to X$, then $f \simeq g$ in the above sense.
+}
+\Proof{
+	Because $A$ is cofibrant, there is a very good homotopy $H$. Consider a lifting problem to construct a map $Path_X \to \Lambda(t, dt) \tensor X$.
+}
+
+\Corollary{cdga_homotopy_eqrel}{
+	For cofibrant $A$, $\simeq$ defines a equivalence relation.
+}
+\Definition{cdga_homotopy_classes}{
+	For cofibrant $A$ define the set of equivalence classes as:
+	$$ [A, X] = \Hom_{\CDGA_\k}(A, X) / \simeq. $$
+}
+
+The results from model categories immediately imply the following results.
+\Corollary{cdga_homotopy_properties}{
+	Let $A$ be cofibrant.
+	\begin{itemize}
+		\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:
+		$$ f_\ast: [Z, A] \tot{\iso} [Z, X], $$
+		$$ f^\ast: [X, Z] \tot{\iso} [A, X]. $$
+	\end{itemize}
+}
+
+\Lemma{cdga_homotopy_homology}{
+	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 $H(d_1)$.
+}

thesis/preamble.tex

@@ -179,6 +179,8 @@
 \newcommand{\LemmaRef}{\RefTemp{Lemma}{lem}}
 \newcommand{\CorollaryRef}{\RefTemp{Corollary}{cor}}
 
+\newcommand{\DefinitionRef}{\RefTemp{Definition}{def}}
+
 % headings for a table
 \newcommand*{\thead}[1]{\multicolumn{1}{c}{\bfseries #1}}