Continues with Kan extensions. Adds stub about sing cochain. Proves (A,K) to be Quillen Pair (adds diagrams).

.gitignore

@@ -5,3 +5,5 @@ build
 
 *sublime*
 
+*.bak
+

thesis/notes/A_K_Quillen_Pair.tex

@@ -0,0 +1,33 @@
+
+\subsection{$A$ and $K$ form a Quillen pair}
+
+We will prove that $A$ preserves cofibrations and trivial cofibrations. We only have to check this fact for the generating (trivial) cofibrations in $\sSet$. Note that the contravariance of $A$ means that a (trivial) cofibrations should be sent to a (trivial) fibration.
+
+\begin{lemma}
+	$A(i) : A(\Delta[n]) \to A(\del \Delta[n])$ is surjective.
+\end{lemma}
+\begin{proof}
+	Let $\phi \in A(\del \Delta[n])$ be an element of degree $k$, hence it is a map $\del \Delta[n] \to \Apl^k$. We want to extend this to the whole simplex. By the fact that $\Apl^k$ is Kan and contractible we can find a lift $\overline{\phi}$ in the following diagram showing the surjectivity.
+
+	\cimage[scale=0.5]{Extend_Boundary_Form}
+\end{proof}
+
+\begin{lemma}
+	$A(j) : A(\Delta[n]) \to A(\Lambda^n_k)$ is surjective and a quasi isomorphism.
+\end{lemma}
+\begin{proof}
+	As above we get surjectivity from the Kan condition. To prove that $A(j)$ is a quasi isomorphism we pass to the singular cochain complex and use that $C^\ast(j) : C^\ast(\Delta[n]) \we C^\ast(\Lambda^n_k)$ is a quasi isomorphism. Consider the following diagram and conclude that $A(j)$ is surjective and a quasi isomorphism.
+
+	\cimage[scale=0.5]{A_Preserves_WCof}
+\end{proof}
+
+Since $A$ is a left adjoint, it preserves all colimits and by functoriality it preserves retracts. From this we can conclude the following corollary.
+
+\begin{corollary}
+	$A$ preserves all cofibrations and all trivial cofibrations and hence is a left Quillen functor.
+\end{corollary}
+
+\begin{corollary}
+	$A$ and $K$ induce an adjunction on the homotopy categories:
+	$$ \Ho{\sSet} \leftadj \opCat{\Ho{\CDGA}}. $$
+\end{corollary}

thesis/notes/Algebra.tex

@@ -67,8 +67,6 @@ A differential graded module $(M, d)$ with $M_i = 0$ for all $i < 0$ is a \emph{
 	$$ d_{M \tensor N} = d_M \tensor \id_N + \id_M \tensor d_N. $$
 \end{definition}
 
-\todo{Prove that this is in fact a differential?}
-
 Finally we come to the definition of a differential graded algebra. This will be a graded algebra with a differential. Of course we want this to be compatible with the algebra structure, or stated differently: we want $\mu$ and $\eta$ to be chain maps.
 
 \begin{definition}
@@ -107,6 +105,8 @@ For differential graded algebras we can consider the (co)homology by forgetting
 	\todo{Maybe just state this?}
 \end{proof}
 
+Note that taking homology of a differential graded module (or algebra) is functorial. Whenever a map $f: M \to N$ of differential graded modules (or algebras) induces an isomorphism on homology, we say that $f$ is a \emph{quasi isomorphism}.
+
 \TODO{Discuss:
 \titem The Künneth theorem (especially in the case of fields)
 \titem The tensor algebra $T : Ch^\ast(\Q) \to \DGA_\Q$ and free cdga $\Lambda : Ch^\ast(\Q) \to \CDGA_\Q$

thesis/notes/Model_Categories.tex

@@ -88,7 +88,7 @@ Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equival
 
 \TODO{Maybe some basic propositions (refer to Dwyer \& Spalinski):
 \titem Over/under category (or simply pointed objects)
-\titem If a map has LLP/RLP wrt fib/cof, it is a cof/fib
+\titem If a map has LLP/RLP w.r.t. fib/cof, it is a cof/fib
 \titem Fibs are preserved under pullbacks/limits
 \titem Cofibrantly generated mod. cats.
 \titem Small object argument

thesis/notes/Model_Of_CDGA.tex

@@ -8,7 +8,7 @@ In this section we will define a model structure on CDGAs over a field $\k$ of c
 \begin{proposition}
 	There is a model structure on $\CDGA_\k$ where $f: A \to B$ is
 	\begin{itemize}
-		\item a \emph{weak equivalence} if $H(f)$ is an isomorphism,
+		\item a \emph{weak equivalence} if $f$ is a quasi isomorphism,
 		\item a \emph{fibration} if $f$ is an surjective and
 		\item a \emph{cofibration} if $f$ has the LLP w.r.t. trivial fibrations
 	\end{itemize}

thesis/notes/Polynomial_Forms.tex

@@ -5,7 +5,46 @@ There is a general way to construct functors from $\sSet$ whenever we have some
 
 Given a category $\cat{C}$ and a functor $F: \DELTA \to \cat{C}$, then define the following on objects:
 \begin{align*}
-	F_!(X)      &= \colim_{\Delta[n] \to X} F[n] &\quad X \in \sSet \\
+	F_!(X)      &= \colim_{\Delta[n] \to X} F[n] & X \in \sSet \\
-	F^\ast(C)_n &= \Hom_{\cat{C}}(F[n], Y)       &\quad C \in \cat{C}
+	F^\ast(C)_n &= \Hom_{\cat{C}}(F[n], Y)       & C \in \cat{C}
 \end{align*}
-A simplicial map $X \to Y$ induces a map of the diagrams of which we take colimits. Applying $F$ on these diagrams, make it clear that $F_!$ is functorial.
+A simplicial map $X \to Y$ induces a map of the diagrams of which we take colimits. Applying $F$ on these diagrams, make it clear that $F_!$ is functorial. Secondly we see readily that $F^\ast$ is functorial. By using the definition of colimit and the Yoneda lemma (Y) we can prove that $F_!$ is left adjoint to $F^\ast$:
+
+\begin{align*}
+	\Hom_\cat{C}(F_!(X), Y) &\iso \Hom_\cat{C}(\colim_{\Delta[n] \to X} F[n], Y) \iso \lim_{\Delta[n] \to X} \Hom_\cat{C}(F[n], Y) \iso \lim_{\Delta[n] \to X} F^\ast(Y)_n \\
+	&\stackrel{\text{Y}}{\iso} \lim_{\Delta[n] \to X} \Hom_\sSet(\Delta[n], F^\ast(Y)) \iso \Hom_\sSet(\colim_{\Delta[n] \to X} \Delta[n], F^\ast(Y)) \\
+	&\iso \Hom_\sSet(X, F^\ast(Y)).
+\end{align*}
+
+Furthermore we have $F_! \circ \Delta[-] = F$. In short we have the following:
+
+\cimage[scale=0.5]{Kan_Extension}
+
+In our case where $F = \Apl$ and $\cat{C} = \CDGA_\k$ we get:
+
+\cimage[scale=0.5]{Apl_Extension}
+
+In our case we take the opposite category, so the definition of $A$ is in terms of a limit instead of colimit. This allows us to give a nicer description:
+
+\begin{align*}
+	A(X) &= \lim_{\Delta[n] \to X} \Apl_n \stackrel{Y}{\iso} \lim_{\Delta[n] \to X} \Hom_\sSet(\Delta[n], \Apl) \iso \Hom_\sSet(\colim_{\Delta[n] \to X}\Delta[n], \Apl) \\
+	&= \Hom_\sSet(X, \Apl),
+\end{align*}
+
+where the addition, multiplication and differential are defined pointwise. Conclude that we have the following contravariant functors (which form an adjoint pair):
+
+\begin{align*}
+	A(X) &= \Hom_\sSet(X, \Apl) & X \in \sSet \\
+	K(C)_n &= \Hom_{\CDGA_\k}(C, \Apl_n) & C \in \CDGA_\k.
+\end{align*}
+
+
+\subsection{The singular cochain complex}
+
+Another way to model the $n$-simplex is by the singular cochain complex associated to the topological $n$-simplices. Define the following (non-commutative) dga's:
+$$ C_n = C^\ast(\Delta^n; \k). $$
+The inclusion maps $d^i : \Delta^n \to \Delta^{n+1}$ and the maps $s^i: \Delta^n \to \Delta^{n-1}$ induce face and degeneracy maps on the dga's $C_n$, turning $C$ into a simplicial dga. Again we can extend this to functors by Kan extensions
+
+\cimage[scale=0.5]{C_Extension}
+
+where the left adjoint is precisely the functor $C^\ast$ as noted in \cite{felix}. We will relate $\Apl$ and $C$ in order to obtain a natural quasi isomorphism $A(X) \we C^\ast(X)$ for every $X \in \sSet$. Furthermore this map preserves multiplication on the homology algebras.

thesis/preamble.tex

@@ -42,6 +42,7 @@
 \newcommand{\opCat}[1]{{#1}^{\text{op}}}% opposite category
 \newcommand{\Hom}{\mathbf{Hom}}
 \newcommand{\id}{\mathbf{id}}
+\newcommand{\Ho}[1]{\cat{Ho(#1)}}
 
 % Categories
 \newcommand{\Set}{\cat{Set}}			% sets
@@ -123,3 +124,10 @@
 
 % headings for a table
 \newcommand*{\thead}[1]{\multicolumn{1}{c}{\bfseries #1}}
+
+% simple way to center an image
+\newcommand{\cimage}[2][]{
+	\begin{center}
+	\includegraphics[#1]{#2}
+	\end{center}
+}

thesis/thesis.tex

@@ -1,4 +1,4 @@
-\documentclass[a4paper, 12pt, draft]{amsart}
+\documentclass[a4paper, 12pt]{amsart}
 
 \input{style}
 \input{preamble}
@@ -26,6 +26,7 @@ Some general notation: \todo{leave this out, or define somewhere else?}
 \input{notes/Model_Of_CDGA}			\vspace{2cm}
 \input{notes/CDGA_Of_Polynomials}	\vspace{2cm}
 \input{notes/Polynomial_Forms}		\vspace{2cm}
+\input{notes/A_K_Quillen_Pair}		\vspace{2cm}
 
 % \listoftodos