Moves everything to notes, as there is no intended order yet. Adds some notes.

.gitignore

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

thesis/notes/Algebra.tex

@@ -33,7 +33,7 @@ Recall that the tensor product of modules distributes over direct sums. This def
 	$$ (f \tensor g)(a \tensor x) = (-1)^{\deg{a}\deg{g}} \cdot f(a) \tensor g(x). $$
 \end{definition}
 
-The sign is due to \emph{Koszuls sign convention}: whenever two elements next to each other are swapped (in this case $g$ and $a$) a minus sign appears if both elements are of odd degree. More formally we can define a swap map
+The sign is due to \emph{Koszul's sign convention}: whenever two elements next to each other are swapped (in this case $g$ and $a$) a minus sign appears if both elements are of odd degree. More formally we can define a swap map
 $$ \tau : A \tensor B \to B \tensor A : a \tensor b \mapsto (-1)^{\deg{a}\deg{b}} b \tensor a. $$
 
 The graded modules together with graded maps of degree $0$ form the category $\grMod{\k}$ of graded modules. From now on we will simply refer to maps instead of graded maps. Together with the tensor product and the ground ring, $(\grMod{\k}, \tensor, \k)$ is a symmetric monoidal category (with the symmetry given by $\tau$). This now dictates the definition of a graded algebra.
@@ -78,11 +78,11 @@ Finally we come to the definition of a differential graded algebra. This will be
 
 \todo{Define the notion of derivation?}
 
-It is not hard to see that this definition precisely defines the monoidal objects in the category of differential graded modules. The category of DGAs will be denoted by $\DGA_\k$, the category of commutative DGAs (CDGAs) will be denoted by $\CDGA_\k$. If no confusion can arise, the ground ring $\k$ will be suppressed in this notation.
+It is not hard to see that this definition precisely defines the monoidal objects in the category of differential graded modules. The category of dga's will be denoted by $\DGA_\k$, the category of commutative dga's (cdga's) will be denoted by $\CDGA_\k$. If no confusion can arise, the ground ring $\k$ will be suppressed in this notation.
 
 Let $M$ be a DGA, just as before $M$ is called a \emph{chain algebras} if $M_i = 0$ for $i < 0$. Similarly if $M^i = 0$ for all $i < 0$, then $M$ is a \emph{cochain algebra}.
 
-\todo{The notation $\CDGA$ seem to refer to cochain algebras in literature and not arbitrary CDGAs.}
+\todo{The notation $\CDGA$ seem to refer to cochain algebras in literature and not arbitrary cdga's.}
 
 \subsection{Homology}
 
@@ -110,7 +110,7 @@ For differential graded algebras we can consider the (co)homology by forgetting
 \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 $\Lambda : Ch^\ast(\Q) \to \CDGA_\Q$
-\titem Coalgebras and Hopfalgebras?
+\titem Coalgebras and Hopf algebras?
 \titem Define reduced/connected differential graded things
 \titem Singular (co)homology as a quick example?
 }

thesis/notes/CDGA_Basic_Examples.tex

@@ -0,0 +1,51 @@
+
+\subsection{Cochain models for the $n$-disk and $n$-sphere}
+We will first define some basic cochain complexes which model the $n$-disk and $n$-sphere. $D(n)$ is the cochain complex generated by one element $b \in D(n)^n$ and its differential $c = d(b) \in D(n)^{n+1}$. $S(n)$ is the cochain complex generated by one element $a \in S(n)^n$ which differential vanishes (i.e. $da = 0$). In other words:
+
+$$ D(n) = ... \to 0 \to \k \to \k \to 0 \to ... $$
+$$ S(n) = ... \to 0 \to \k \to 0 \to 0 \to ... $$
+
+Note that $D(n)$ is acyclic for all $n$, or put in different words: $j_n : 0 \to D(n)$ is a quasi isomorphism. The sphere $S(n)$ has exactly one non-trivial cohomology group $H^n(S(n)) = \k \cdot [a]$. There is an injective function $i_n : S(n+1) \to D(n)$, sending $a$ to $c$. The maps $j_n$ and $i_n$ play the following important role in the model structure of cochain complexes:
+
+\begin{claim}
+	The set $I = \{i_n : S(n+1) \to D(n) \I n \in \N\}$ generates all cofibrations and the set $J = \{j_n : 0 \to D(n) \I n \in \N\}$ generates all trivial cofibrations.
+\end{claim}
+
+The proof is omitted but can be found in different sources \todo{Cite sources}. In the next section we will prove a similar result for cdga's, so the reader can also refer to that proof.
+
+$S(n)$ plays a another special role: maps from $S(n)$ to some cochain complex $X$ correspond directly to elements in the kernel of $\restr{d}{X^n}$. Any such map is null-homotopic precisely when the corresponding elements in the kernel is a coboundary. So there is a natural isomorphism: $\Hom(S(n), X) / ~ \iso H^n(X)$. So the cohomology groups can be considered as honest homotopy groups.
+
+By using the free cdga functor we can turn these cochain complexes into cdga's $\Lambda(D(n))$ and $\Lambda(S(n))$. So $\Lambda(D(n))$ consists of linear combinations of $b^n$ and $c b^n$ when $n$ is even, and $c^n b$ and $c^n$ when $n$ is odd. In both cases we can compute the differentials using the Leibniz rule:
+$$ d(b^n) = n \cdot c b^{n-1} $$
+$$ d(c b^n) = 0 $$
+
+$$ d(c^n b) = c^{n+1} $$
+$$ d(c^n) = 0 $$
+
+Those cocycles are in fact coboundaries (remember that $\k$ is a field of characteristic $0$):
+$$ c b^n = \frac{1}{n} d(b^{n+1}) $$
+$$ c^n = d(b c^{n-1}) $$
+
+There are no additional cocycles in $\Lambda(D(n))$ besides the constants and $c$. So we conclude that $\Lambda(D(n))$ is acyclic as an algebra. In other words $\Lambda(j_n): \k \to \Lambda D(n)$ is a quasi isomorphism.
+
+The situation for $\Lambda S(n)$ is easier: when $n$ is even it is given by polynomials in $a$, if $n$ is odd it is an exterior algebra (i.e. $a^2 = 0$). Again the sets $\Lambda(I) = \{ \Lambda(i_n) : \Lambda S(n+1) \to \Lambda D(n) \I n \in \N\}$ and $\Lambda(J) = \{ \Lambda(j_n) : \k \to \Lambda D(n) \I n \in \N\}$ play an important role.
+
+\begin{theorem}
+	The sets $\Lambda(I)$ and $\Lambda(J)$ generate a model structure on $\CDGA_\k$ where:
+	\begin{itemize}
+		\item weak equivalences are quasi isomorphisms,
+		\item fibrations are (degree wise) surjective maps and
+		\item cofibrations are maps with the left lifting property against trivial fibrations.
+	\end{itemize}
+\end{theorem}
+
+We will prove this theorem in the next section. Note that the functors $\Lambda$ and $U$ thus form a Quillen pair with this model structure.
+
+\subsection{Why we need $\Char{\k} = 0$ for algebras}
+The above Quillen pair $(\Lambda, U)$ fails to be a Quillen pair if $\Char{\k} = p \neq 0$. We will show this by proving that the maps $\Lambda(j_n)$ are not weak equivalences for even $n$. Consider $b^p \in D(n)$, then by the Leibniz rule:
+$$ d(b^p) = p \cdot c b^{p-1} = 0. $$
+So $b^p$ is a cocycle. Now assume $b^p = dx$ for some $x$ of degree $pn - 1$, then $x$ contains a factor $c$ for degree reasons. By the calculations above we see that any element containing $c$ has a trivial differential or has a factor $c$ in its differential, contradicting $b^p = dx$. So this cocycle is not a coboundary and $\Lambda D(n)$ is not acyclic.
+
+
+
+

thesis/notes/CDGA_Of_Polynomials.tex

@@ -0,0 +1,71 @@
+
+\subsection{CDGA of Polynomials}
+
+\newcommand{\Apl}[0]{{A_{PL}}}
+
+We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows for simplicial methods. In the following definition one should be reminded of the topological $n$-simplex defined as convex span.
+
+\begin{definition}
+	For all $n \in \N$ define the following cdga:
+	$$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, dx_0, \ldots, dx_n)}{(\sum_{i=0}^n) x_i - 1, \sum_{i=0}^n dx_i)} $$
+	So it is the free cdga with $n+1$ generators and their differentials such that $\sum_{i=0}^n x_i = 1$ and in order to be well behaved $\sum_{i=0}^n dx_i = 0$.
+\end{definition}
+
+Note that the inclusion $\Lambda(x_1, \ldots, x_n, dx_1, \ldots, dx_n) \to \Apl_n$ is an isomorphism of cdga's. So $\Apl_n$ is free and (algebra) maps from it are determined by their images on $x_i$ for $i = 1, \ldots, n$ (also note that this determines the images for $dx_i$). This fact will be used throughout.
+
+These cdga's will assemble into a simplicial cdga when we define the face and degeneracy maps as follows ($j = 1, \ldots, n$):
+
+$$ d_i(x_j) = \begin{cases}
+	x_{j-1}, &\text{ if } i < j \\
+	0,       &\text{ if } i = j \\
+	x_j,     &\text{ if } i > j
+\end{cases} \qquad d_i : \Apl_n \to \Apl_{n-1} $$
+$$ s_i(x_j) = \begin{cases}
+	x_{j+1},       &\text{ if } i < j \\
+	x_j + x_{j+1}, &\text{ if } i = j \\
+	x_j,           &\text{ if } i > j	
+\end{cases} \qquad s_i : \Apl_n \to \Apl_{n+1} $$
+
+One can check that $\Apl \in \simplicial{\CDGA_\k}$. We will denote the subspace of homogeneous elements of degree $k$ as $\Apl^k \in \simplicial{\Mod{\k}}$, this is indeed a simplicial $\k$-module as the maps $d_i$ and $s_i$ are graded maps of degree $0$.
+
+\begin{lemma}
+	$\Apl^k$ is contractible.
+\end{lemma}
+\begin{proof}
+	We will prove this by defining an extra degeneracy $s: \Apl_n \to \Apl_{n+1}$. Define for $i = 1, \ldots, n$:
+	\begin{align*}
+		s(1) &= (1-x_0)^2 \\
+		s(x_i) &= (1-x_0) \cdot x_{i+1}
+	\end{align*}
+	Extend on the differentials and multiplicatively on $\Apl_n$. As $s(1) \neq 1$ this map is not an algebra map, however it well-defined as a map of cochain complexes. In particular when restricted to degree $k$ we get a linear map:
+	$$ s: \Apl^k_n \to \Apl^k_{n+1}. $$
+	Proving the necessary properties of an extra degeneracy is fairly easy. For $n \geq 1$ we get (on generators):
+	\begin{align*}
+		d_0 s(1)   &= d_0 (1 - x_0)^2 = (1 - 0) \cdot (1 - 0) = 1 \\
+		d_0 s(x_i) &= d_0((1-x_0)x_{i+1}) = d_0(1-x_0) \cdot x_i  \\
+		           &= (1-0) \cdot x_i = x_i
+	\end{align*}
+	So $d_0 s = \id$.
+	\begin{align*}
+		d_{i+1} s(1) &= d_{i+1} (1 - x_0)^2 = d_{i+1} (\sum_{j=1}^n x_j)^2 \\
+		             &= (\sum_{j=1}^{n-1} x_j)^2 = (1-x_0)^2 = s d_i(1)    \\
+		d_{i+1} s(x_j) &= d_{i+1}(1-x_0) d_{i+1}(x_j) = (1-x_0) d_i(x_{j+1}) = s d_i (x_j)
+	\end{align*}
+	So $d_{i+1} s = s d_i$. Similarly $s_{i+1} s = s s_i$. And finally for $n=0$ we have $d_1 s = 0$.
+
+	So we have an extra degeneracy $s: \Apl^k \to \Apl^k$, and hence (see for example \cite{goerss}) we have that $\Apl^k$ is contractible. As a consequence $\Apl \to \ast$ is a weak equivalence.
+\end{proof}
+
+\begin{lemma}
+	$\Apl_n^k$ is a Kan complex.
+\end{lemma}
+\begin{proof}
+	By the simple fact that $\Apl_n^k$ is a simplicial group, it is a Kan complex \cite{goerss}.
+\end{proof}
+
+\begin{corollary}
+	$\Apl^k \to \ast$ is a trivial fibration in the standard model structure on $\sSet$.
+\end{corollary}
+
+
+

thesis/notes/Free_CDGA.tex

@@ -0,0 +1,50 @@
+
+\subsection{The free cdga}
+
+Just as in ordinary linear algebra we can form an algebra from any graded module. Furthermore we will see that a differential induces a derivation.
+
+\begin{definition}
+	The \emph{tensor algebra} of a graded module $M$ is defined as
+	$$ T(M) = \bigoplus_{n\in\N} M^{\tensor n}, $$
+	where $M^{\tensor 0} = \k$. An element $m = m_1 \tensor \ldots \tensor m_n$ has a \emph{word length} of $n$ and its degree is $\deg{m} = \sum_{i=i}^n \deg{m_i}$. The multiplication is given by the tensor product (note that the bilinearity follows immediately).
+\end{definition}
+
+Note that this construction is functorial and it is free in the following sense.
+
+\begin{lemma}
+	Let $M$ be a graded module and $A$ a graded algebra.
+	\begin{itemize}
+		\item A graded map $f: M \to A$ of degree $0$ extends uniquely to an algebra map $\overline{f} : TM \to A$.
+		\item A differential $d: M \to M$ extends uniquely to a derivation $d: TM \to TM$.
+	\end{itemize}
+\end{lemma}
+
+\begin{corollary}
+	Let $U$ be the forgetful functor from graded algebras to graded modules, then $T$ and $U$ form an adjoint pair:
+	$$ T: \grMod{\k} \leftadj \grAlg{\k} $$
+	Moreover it extends and restricts to
+	$$ T: \dgMod{\k} \leftadj \dgAlg{\k} $$
+	$$ T: \CoCh{\k} \leftadj \DGA{\k} $$
+\end{corollary}
+
+As with the symmetric algebra and exterior algebra of a vector space, we can turn this graded tensor algebra in a commutative graded algebra.
+
+\begin{definition}
+	Let $A$ be a graded algebra and define
+	$$ I = < ab - (-1)^{\deg{a}\deg{b}}ba \I a,b \in A >. $$
+	Then $A / I$ is a commutative graded algebra.
+
+	For a graded module $M$ we define the \emph{free commutative graded algebra} as
+	$$ \Lambda(M) = TM / I $$
+\end{definition}
+
+Again this extends to differential graded modules (i.e. the ideal is preserved by the derivative) and restricts to cochain complexes.
+
+\begin{lemma}
+	We have the following adjunctions:
+	$$ \Lambda: \grMod{\k} \leftadj \grAlg{\k}^{comm} $$
+	$$ \Lambda: \dgMod{\k} \leftadj \dgAlg{\k}^{comm} $$
+	$$ \Lambda: \CoCh{\k} \leftadj \CDGA_\k $$
+\end{lemma}
+
+We can now easily construct cdga's by specifying generators and their differentials.

thesis/notes/Polynomial_Forms.tex

@@ -0,0 +1,11 @@
+
+\subsection{Polynomial Forms}
+
+There is a general way to construct functors from $\sSet$ whenever we have some simplicial object. In our case we have the simplicial cdga $\Apl$ (which is nothing more than a functor $\opCat{\DELTA} \to \CDGA$) and we want to extend to a contravariant functor $\sSet \to \CDGA_\k$. This will be done via Kan extensions.
+
+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^\ast(C)_n &= \Hom_{\cat{C}}(F[n], Y)       &\quad 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.

thesis/preamble.tex

@@ -39,23 +39,34 @@
 
 % Basic category stuff
 \newcommand{\cat}[1]{\mathbf{#1}}		% the category of ...
+\newcommand{\opCat}[1]{{#1}^{\text{op}}}% opposite category
 \newcommand{\Hom}{\mathbf{Hom}}
 \newcommand{\id}{\mathbf{id}}
 
 % Categories
 \newcommand{\Set}{\cat{Set}}			% sets
-\newcommand{\sSet}{\cat{sSet}}			% simplicial sets
 \newcommand{\Top}{\cat{Top}}			% topological spaces
 \newcommand{\DELTA}{\cat{\Delta}}		% the simplicial cat
+\newcommand{\simplicial}[1]{\cat{s{#1}}}% simplicial objects
+\newcommand{\sSet}{\simplicial{\Set}}	% simplicial sets
+\newcommand{\Mod}[1]{\cat{{#1}Mod}}		% modules over a ring
+\newcommand{\Alg}[1]{\cat{{#1}Alg}}		% algebras over a ring
 \newcommand{\grMod}[1]{\cat{gr\mbox{-}{#1}Mod}}	% graded modules over a ring
 \newcommand{\grAlg}[1]{\cat{gr\mbox{-}{#1}Alg}}	% graded algebras over a ring
-\newcommand{\DGA}{\cat{DGA}}			% differential graded algebras
-\newcommand{\CDGA}{\cat{CDGA}}			% commutative dgas
+\newcommand{\dgMod}[1]{\cat{dg\mbox{-}{#1}Mod}}	% differential graded modules over a ring
+\newcommand{\dgAlg}[1]{\cat{dg\mbox{-}{#1}Alg}}	% differential graded algebras over a ring
+\newcommand{\Ch}[1]{\cat{Ch_{n\geq0}({#1})}}	% chain complexes
+\newcommand{\CoCh}[1]{\cat{Ch^{n\geq0}({#1})}}	% cochain complexes
+\newcommand{\DGA}{\cat{DGA}}			% cochain algebras
+\newcommand{\CDGA}{\cat{CDGA}}			% commutative cochain algebras
 
 \newcommand{\cof}{\hookrightarrow}		% cofibration
 \newcommand{\fib}{\twoheadrightarrow}	% fibration
 \newcommand{\we}{\tot{\simeq}}			% weak equivalence
 
+%\newcommand{\leftadj}{\ooalign{\hss\rightleftarrows\hss\cr\bot}}
+\newcommand{\leftadj}{\rightleftarrows}
+
 % Notation and operators
 \newcommand{\I}{\,\mid\,}				% seperator in set notation
 \newcommand{\del}{\partial}				% boundary
@@ -68,6 +79,7 @@
 \DeclareMathOperator*{\tensor}{\otimes}
 \DeclareMathOperator*{\bigtensor}{\bigotimes}
 \renewcommand{\deg}[1]{{|{#1}|}}
+\newcommand{\Char}[1]{char({#1})}
 
 % restriction of a function
 \newcommand\restr[2]{{% we make the whole thing an ordinary symbol
@@ -102,6 +114,7 @@
 \newtheorem{proposition}[theorem]{Proposition}
 \newtheorem{lemma}[theorem]{Lemma}
 \newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{claim}[theorem]{Claim}
 
 \theoremstyle{definition}
 \newtheorem{definition}[theorem]{Definition}

thesis/references.bib

@@ -23,3 +23,17 @@
 	year={2007},
 	publisher={Providence, RI; American Mathematical Society; 1999}
 }
+
+@article{goerss,
+	title={Simplicial Homotopy Theory},
+	author={Goerss, PG and Jardine, JF},
+	publisher={Birkh{\"a}user},
+	year={1999}
+}
+
+@book{griffiths,
+  title={Rational homotopy theory and differential forms},
+  author={Griffiths, Phillip A and Morgan, John W},
+  year={2013},
+  publisher={Birkh{\"a}user}
+}

thesis/thesis.tex

@@ -19,9 +19,13 @@ Some general notation: \todo{leave this out, or define somewhere else?}
 
 \vspace{1cm}
 
-\input{1_Algebra}    \vspace{2cm}
-\input{2_Model_Cats} \vspace{2cm}
-\input{CDGA_Model}   \vspace{2cm}
+\input{notes/Algebra}				\vspace{2cm}
+\input{notes/Free_CDGA}				\vspace{2cm}
+\input{notes/CDGA_Basic_Examples}	\vspace{2cm}
+\input{notes/Model_Categories}		\vspace{2cm}
+\input{notes/Model_Of_CDGA}			\vspace{2cm}
+\input{notes/CDGA_Of_Polynomials}	\vspace{2cm}
+\input{notes/Polynomial_Forms}		\vspace{2cm}
 
 % \listoftodos