Changes to classicthesis and Adds part about applications

thesis/chapters/Appendices.tex

@@ -0,0 +1,4 @@
+
+\input{notes/Algebra}
+\input{notes/Free_CDGA}
+\input{notes/Model_Categories}

thesis/chapters/Applications_And_Further_Topics.tex

@@ -0,0 +1,64 @@
+
+\chapter{Rational Homotopy Groups Of The Spheres And Other Calculations}
+
+In this chapter we will calculate the rational homotopy groups of the spheres using minimal models. The minimal model for the sphere was already given, but we will quickly redo the calculation.
+
+\Proposition{}{
+	For odd $n$ the rational homotopy groups of $S^n$ are given by
+	$$ \pi_i(S^n) \tensor \Q \iso \begin{cases}
+		\Q, &\text{ if } i=n \\
+		0, &\text{ otherwise.}
+	\end{cases} $$
+}
+\Proof{
+	We know the cohomology of the sphere by classical results:
+	$$ H^i(S^n ; \Q) = \begin{cases}
+		\Q \cdot 1, &\text{ if } i = 0 \\
+		\Q \cdot x, &\text{ if } i = n \\
+		0, &\text{ otherwise,}
+	\end{cases}$$
+	where $x$ is a generator of degree $n$. Define $M_{S^n} = \Lambda(e)$ with $d(e) = 0$ and $e$ of degree $n$. Notice that since $n$ is odd, we get $e^2 = 0$. By taking a representative for $x$, we can give a map $M_{S^n} \to A(S^n)$, which is a weak equivalence.
+
+	Clearly $M_{S^n}$ is minimal, and hence it is a minimal model for $S^n$. By \CorollaryRef{minimal-cdga-homotopy-groups} and the main equivalence we have
+	$$ \pi_\ast(S^n) \tensor \Q = \pi_\ast(K(M_{S^n})) = \pi^\ast(M_{S^n})^\ast = \Q \cdot e^\ast. $$
+}
+
+\Proposition{}{
+	For even $n$ the rational homotopy groups of $S^n$ are given by
+	$$ \pi_i(S^n) \tensor \Q \iso \begin{cases}
+		\Q, &\text{ if } i = 2n-1 \\
+		\Q, &\text{ if } i = n \\
+		0, &\text{ otherwise.}
+	\end{cases} $$
+}
+\Proof{
+	Again since we know the cohomology of the sphere, we can construct its minimal model. Define $M_{S^n} = \Lambda(e, f)$ with $d(e) = 0, d(f) = e^2$ and $\deg{e} = n, \deg{f} = 2n-1$. Let $x \in H^n(S^n; \Q)$ be a generator and notice that $x^2 = 0$. This means that for a representative $x' \in A(S^n)$ of $x$ there exists an element $y \in A(S^n)$ such that $dy = x'^2$. Mapping $e$ and $f$ to $x'$ and $y$ respectively defines a quasi isomorphism $M_{S^n} \to A(S^n)$.
+
+	Again we can use \CorollaryRef{minimal-cdga-homotopy-groups} to directly conclude:
+	$$ \pi_i(S^n) \tensor \Q = \pi^i(M_{S^n})^\ast = \Q \cdot e^\ast \oplus \Q \cdot f^\ast. $$
+}
+
+The generators $e$ and $f$ in the last proof are related by the so callend \Def{Whitehead product}. The whitehead product is a bilinear map $\pi_p(X) \times \pi_q(X) \to \pi_{p+q-1}(X)$ satisfying a graded commutativity relation and a graded Jacobi relation, see \cite{felix}. If we define a \Def{Whitehead algebra} to be a graded vector space with such a map satisfying these relations, we can summarize the above two propositions as follows \cite{berglund}.
+
+\Corollary{}{
+	The rational homotopy groups of $S^n$ are given by
+	$$ \pi_\ast(S^n) \tensor \Q = \text{the free whitehead algebra on 1 generator}. $$
+}
+
+Together with the fact that all groups $\pi_i(S^n)$ are finitely generated (this was proven by Serre \cite{serre}) we can conclude that $\pi_i(S^n)$ is a finite group unless $i=n$ or $i=2n-1$ when $n$ is even. The fact that $\pi_i(S^n)$ are finitely generated can be proven by the Serre-Hurewicz theorems (\TheoremRef{serre-hurewicz}) when taking the Serre class of finitely generated abelian groups.
+
+The following result is already used in proving the main theorem. But using the main theorem it is an easy and elegant consequence.
+
+\Proposition{}{
+	For an Eilenberg-MacLane space of type $K(\Z, n)$ we have:
+	$$ H^\ast(K(\Z, n); \Q) \iso \Q[x], $$
+	i.e. the free graded commutative algebra on 1 generator.
+}
+\Proof{
+	By the existence theorem for minimal models, we know there is a minimal model $(\Lambda V, d) \we A(K(\Z, n))$. By calculating the homotopy groups we see
+	$$ {V^i}^\ast = \pi^i(\Lambda V)^\ast = \pi_i(K(\Z, n)) \tensor \Q = \begin{cases}
+		\Q, &\text{ if } i = n \\
+		0, &\text{ otherwise.}
+	\end{cases} $$
+	This means that $V$ is concentrated in degree $n$ and that the differential is trivial. Take a generator $x$ of degree $n$ such that $V = \Q \cdot x$ and conclude that the cohomology of the minimal model, and hence the cohomology of $K(\Z, n)$, is $H(\Lambda V, 0) = \Q[x]$.
+}

thesis/chapters/Basics_Of_Rational_Homotopy_Theory.tex

@@ -0,0 +1,4 @@
+
+\input{notes/Basics}
+\input{notes/Serre}
+\input{notes/Rationalization}

thesis/chapters/Homotopy_Theory_CDGA.tex → thesis/chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory.tex

@@ -1,7 +1,5 @@
 
-\newcommand{\titleCDGA}{\texorpdfstring{$\CDGA_\k$}{CDGA}}
+\chapter{Homotopy Theory For cdga's}
-\section{Homotopy theory of \titleCDGA}
-\label{sec:model-of-cdga}
 
 Recall the following facts about cdga's over a ring $\k$:
 \begin{itemize}
@@ -13,11 +11,26 @@ Recall the following facts about cdga's over a ring $\k$:
 \end{itemize}
 In this chapter the ring $\k$ is assumed to be a field of characteristic zero.
 
-\subsection{Cochain models for the $n$-disk and $n$-sphere}
+\section{Cochain models for the $n$-disk and $n$-sphere}
 \input{notes/CDGA_Basic_Examples}
 
-\subsection{The Quillen model structure on \titleCDGA}
+\section{The Quillen model structure on \titleCDGA}
 \input{notes/Model_Of_CDGA}
 
-\subsection{Homotopy relations on \titleCDGA}
+\section{Homotopy relations on \titleCDGA}
 \input{notes/Homotopy_Relations_CDGA}
+
+
+\chapter{Polynomial Forms}
+\label{sec:cdga-of-polynomials}
+
+\section{CDGA of Polynomials}
+\input{notes/CDGA_Of_Polynomials}
+
+\section{Polynomial Forms on a Space}
+\label{sec:polynomial-forms}
+\input{notes/Polynomial_Forms}
+
+
+\input{notes/Minimal_Models}
+\input{notes/A_K_Quillen_Pair}

thesis/chapters/Introduction.tex

@@ -0,0 +1,29 @@
+
+\chapter{Introduction}
+
+Schrijf hier wat
+
+\section{Preliminaries and Notation}
+
+We assume the reader is familiar with category theory, basics from algebraic topology and the basics of simplicial sets. Some knowledge about differential graded algebra (or homological algebra) and model categories is assumed, but the reader may review this in the appendices.
+
+\begin{itemize}
+	\item $\k$ will denote an arbitrary commutative ring (or field, if indicated at the start of a section). Modules, tensor products, \dots are understood as $\k$-modules, tensor products over $\k$, \dots. If ambiguitity can occur notation will be explicit.
+	\item $\cat{C}$ will denote an arbitrary category.
+	\item $\cat{0}$ (resp. $\cat{1}$) will denote the initial (resp. final) objects in a category $\cat{C}$.
+	\item $\Hom_\cat{C}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. The subscript $\cat{C}$ is occasionally left out if the category is clear from the context.
+\end{itemize}
+
+Some categories:
+\begin{itemize}
+	\item $\Top$: category of topological spaces and continuous maps.
+	\item $\Ab$: category of abelian groups and group homomorphisms.
+	\item $\DELTA$: category of simplices (i.e. finite, non-empty ordinals) and order preserving maps.
+	\item $\sSet$: category of simplicial sets and simplicial maps (more generally we have the category of simplicial objects, $\cat{sC}$, for any category $\cat{C}$).
+	\item $\Ch{\k}, \CoCh{\k}$: category of non-negatively graded chain (resp. cochain) complexes and chain maps.
+	\item $\DGA_\k$: category of non-negatively differential graded algebras over $\k$ (these are cochain complexes with a multiplication) and graded algebra  maps. As a shorthand we will refer to such an object as \emph{dga}.
+	\item $\CDGA_\k$: the full subcategory of $\DGA_\k$ of commutative dga's (\emph{cdga}'s).
+\end{itemize}
+
+\tableofcontents
+\addcontentsline{toc}{section}{Contents}

thesis/chapters/Polynomial_Forms.tex

@@ -1,10 +0,0 @@
-
-\section{Polynomial Forms}
-\label{sec:cdga-of-polynomials}
-
-\subsection{CDGA of Polynomials}
-\input{notes/CDGA_Of_Polynomials}
-
-\subsection{Polynomial Forms on a Space}
-\label{sec:polynomial-forms}
-\input{notes/Polynomial_Forms}

thesis/notes/A_K_Quillen_Pair.tex

@@ -1,5 +1,5 @@
 
-\section{\texorpdfstring{$A$}{A} and \texorpdfstring{$K$}{K} form a Quillen pair}
+\chapter{\texorpdfstring{$A$}{A} and \texorpdfstring{$K$}{K} form a Quillen pair}
 \label{sec:a-k-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.
@@ -34,7 +34,7 @@ Since $A$ is a left adjoint, it preserves all colimits and by functoriality it p
 \end{corollary}
 
 
-\subsection{Homotopy groups of \texorpdfstring{$K(A)$}{K(A)}}
+\section{Homotopy groups of \texorpdfstring{$K(A)$}{K(A)}}
 We are after an equivalence of homotopy categories, so it is natural to ask what the homotopy groups of $K(A)$ are for a cdga $A$. In order to do so, we will define homotopy groups of cdga's directly and compare the two notions.
 
 Recall that an augmented cdga is a cdga $A$ with an algebra map $A \tot{\counit} \k$ such that $\counit \unit = \id$.
@@ -84,7 +84,7 @@ We get a particularly nice result for minimal cdga's, because the functor $Q$ is
 }
 
 
-\subsection{Equivalence on rational spaces}
+\section{Equivalence on rational spaces}
 For the equivalence of rational spaces and cdga's we need that the unit and counit of the adjunction are in fact weak equivalences. More formally we want the following maps to be weak equivalences:
 $$ X \to K(A(X)) \text{ for any rational space $X \in \sSet$ of finite type}, $$
 $$ A \to A(K(A)) \text{ for any $A \in \CDGA_\Q$ of finite type}. $$

thesis/notes/Algebra.tex

@@ -1,10 +1,10 @@
 
-\section{Differential Graded Algebra}
+\chapter{Differential Graded Algebra}
 \label{sec:algebra}
 
 In this section $\k$ will be any commutative ring. We will recap some of the basic definitions of commutative algebra in a graded setting. By \emph{linear}, \emph{module}, \emph{tensor product}, etc\dots we always mean $\k$-linear, $\k$-module, tensor product over $\k$, etc\dots.
 
-\subsection{Graded algebra}
+\section{Graded algebra}
 
 \begin{definition}
 	A \emph{graded module} $M$ is a family of modules $\{M_n\}_{n\in\Z}$. An element $x \in M_n$ is called a \emph{homogeneous element} and said to be of \emph{degree $\deg{x} = n$}. We will often identify $M = \bigoplus_{n \in \Z} M_n$.
@@ -54,7 +54,7 @@ Again these objects and maps form a category, denoted as $\grAlg{\k}$. We will d
 \end{definition}
 
 
-\subsection{Differential graded algebra}
+\section{Differential graded algebra}
 
 \begin{definition}
 	A \emph{differential graded module} $(M, d)$ is a graded module $M$ together with a map $d: M \to M$ of degree $-1$, called a \emph{differential}, such that $dd = 0$. A map $f: M \to N$ is a \emph{chain map} if it is compatible with the differential, i.e. $d_N f = f d_M$.
@@ -83,7 +83,7 @@ Let $M$ be a DGA, just as before $M$ is called a \emph{chain algebras} if $M_i =
 \todo{The notation $\CDGA$ seem to refer to cochain algebras in literature and not arbitrary cdga's.}
 
 
-\subsection{Homology}
+\section{Homology}
 
 Whenever we have a differential graded module we have $d \circ d = 0$, or put in other words: the image of $d$ is a submodule of the kernel of $d$. The quotient of the two graded modules will be of interest.
 
@@ -115,7 +115,7 @@ Note that taking homology of a differential graded module (or algebra) is functo
 \end{definition}
 
 
-\subsection{Classical results}
+\section{Classical results}
 
 We will give some classical known results of algebraic topology or homological algebra. Proofs of these theorems can be found in many places. \todo{cite at least 1 place}
 

thesis/notes/Basics.tex

@@ -1,5 +1,5 @@
 
-\section{Rational homotopy theory}
+\chapter{Rational homotopy theory}
 \label{sec:basics}
 
 In this section we will state the aim of rational homotopy theory. Moreover we will recall classical theorems from algebraic topology and deduce rational versions of them.
@@ -30,7 +30,7 @@ Note that a weak equivalence (and hence also a homotopy equivalence) is always a
 
 We will later see that any space admits a rationalization. The theory of rational homotopy theory is then the study of the homotopy category $\Ho_\Q(\Top) \iso \Ho(\Top_\Q)$, which is on its own turn equivalent to $\Ho(\sSet_\Q) \iso \Ho_\Q(\sSet)$.
 
-\subsection{Classical results from algebraic topology}
+\section{Classical results from algebraic topology}
 
 We will now recall known results from algebraic topology, without proof. One can find many of these results in basic text books, such as \cite{may, dold}. We do not assume $1$-connectedness here.
 
@@ -68,7 +68,7 @@ The following two theorems can be found in textbooks about homological algebra s
 	where $H(X; A)$ and $H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded. \todo{Geef algebraische versie voor ketencomplexen}
 }
 
-\subsection{Immediate results for rational homotopy theory}
+\section{Immediate results for rational homotopy theory}
 
 The latter two theorems have a direct consequence for rational homotopy theory. By taking $A = \Q$ we see that the torsion groups vanish. We have the immediate corollary.
 

thesis/notes/CDGA_Basic_Examples.tex

@@ -40,7 +40,7 @@ The situation for $\Lambda S(n)$ is easier: when $n$ is even it is given by poly
 
 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.
 
-\subsubsection{Why we need $\Char{\k} = 0$ for algebras}
+\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 $b^p = d x$ for some $x$ of degree $p n - 1$, 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 $b^p =  d x$. So this cocycle is not a coboundary and $\Lambda D(n)$ is not acyclic.

thesis/notes/Free_CDGA.tex

@@ -1,5 +1,5 @@
 
-\subsection{The free cdga}
+\section{The free cdga}
 \label{sec: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.

thesis/notes/Minimal_Models.tex

@@ -1,5 +1,5 @@
 
-\section{Minimal models}
+\chapter{Minimal models}
 \label{sec:minimal-models}
 
 In this section we will discuss the so called minimal models. These are cdga's with the property that a quasi isomorphism between them is an actual isomorphism.
@@ -34,7 +34,7 @@ The requirement that there exists a filtration can be replaced by a stronger sta
 \end{proof}
 
 
-\subsubsection{Existence}
+\section{Existence}
 
 \begin{theorem}
 	Let $(A, d)$ be an $1$-connected cdga, then it has a minimal model.
@@ -56,7 +56,7 @@ The requirement that there exists a filtration can be replaced by a stronger sta
 \end{proof}
 
 
-\subsubsection{Uniqueness}
+\section{Uniqueness}
 
 Before we state the uniqueness theorem we need some more properties of minimal models.
 
@@ -114,7 +114,7 @@ Before we state the uniqueness theorem we need some more properties of minimal m
 \end{proof}
 
 
-\subsection{The minimal model of the sphere}
+\section{The minimal model of the sphere}
 We know from singular cohomology that the cohomology ring of a $n$-sphere is $\Z[X] / (X^2)$. This allows us to construct a minimal model for $S^n$.
 \Definition{minimal-model-sphere}{
 	Define $A(n)$ to be the cdga defined as

thesis/notes/Model_Categories.tex

@@ -1,5 +1,5 @@
 
-\section{Model categories}
+\chapter{Model categories}
 \label{sec:model_categories}
 
 As this thesis considers different categories, each with its own homotopy theory, it is natural to use Quillen's formalism of model categories. Not only gives this the right definition of the associated homotopy category, it also gives existence of lifts and lifts of homotopies.
@@ -119,7 +119,7 @@ In this thesis we often restrict to $1$-connected spaces. The full subcategory $
 	have no coequalizer and respectively no equalizer in $\Top_r$.
 }
 
-\subsection{Homotopies}
+\section{Homotopies}
 So far we have only seen equivalences between objects of the category. We can, however, also define homotopy relations between maps (as we are used to in $\Top$). There are two such construction, which will coincide on nice objects. We will only state the definitions and important results. One can find proofs of these results in \cite{dwyer}. Throughout this section we silently work with a fixed model category $\cat{C}$.
 
 \newcommand{\cylobj}[1]{Cyl_{#1}}
@@ -231,10 +231,10 @@ The two notions (left resp. right homotopy) agree on nice objects. Hence in this
 }
 
 
-\subsection{The Homotopy Category \texorpdfstring{$\Ho(\cat{C})$}{Ho(C)}}
+\section{The Homotopy Category \texorpdfstring{$\Ho(\cat{C})$}{Ho(C)}}
 A model category induces a homotopy category $\Ho(\cat{C})$, in which weak equivalences are isomorphisms and homotopic maps are equal. This category only depends on the category $\cat{C}$ and the class of weak equivalences.
 \todo{Definition etc}
 
-\subsection{Quillen pairs}
+\section{Quillen pairs}
 In order to relate model categories and their associated homotopy categories we need a notion of maps between them. We want the maps such that they induce maps on the homotopy categories.
 \todo{Definition etc}

thesis/notes/Polynomial_Forms.tex

@@ -23,7 +23,7 @@ In our case where $F = \Apl$ and $\cat{C} = \CDGA_\k$ we get:
 \cdiagram{Apl_Extension}
 
 
-\subsubsection{The cochain complex of polynomial forms}
+\subsection{The cochain complex of polynomial forms}
 
 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:
 
@@ -40,7 +40,7 @@ where the addition, multiplication and differential are defined pointwise. Concl
 \end{align*}
 
 
-\subsubsection{The singular cochain complex}
+\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 \todo{Choose: normalized or not?}:
 $$ C_n = C^\ast(\Delta^n; \k). $$
@@ -51,7 +51,7 @@ The inclusion maps $d^i : \Delta^n \to \Delta^{n+1}$ and the maps $s^i: \Delta^n
 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.
 
 
-\subsubsection{Integration and Stokes' theorem for polynomial forms}
+\subsection{Integration and Stokes' theorem for polynomial forms}
 
 In this section we will prove that the singular cochain complex is quasi isomorphic to the cochain complex of polynomial forms. In order to do so we will define an integration map $\int_n : \Apl_n^n \to \k$, which will induce a map $\oint_n : \Apl_n \to C_n$. For the simplices $\Delta[n]$ we already showed the cohomology agrees by the acyclicity of $\Apl_n = A(\Delta[n])$ (\LemmaRef{apl-acyclic}).
 

thesis/notes/Rationalization.tex

@@ -1,10 +1,10 @@
 
-\section{Rationalizations}
+\chapter{Rationalizations}
 \label{sec:rationalizations}
 
 In this section we will prove the existence of rationalizations $X \to X_\Q$. We will do this in a cellular way. The $n$-spheres play an important role here, so their rationalizations will be discussed first. Again spaces (except for $S^1$) are assumed to be $1$-connected.
 
-\subsection{Construction of \texorpdfstring{$S^n_\Q$}{SnQ}}
+\section{Construction of \texorpdfstring{$S^n_\Q$}{SnQ}}
 Fix $n>0$ we will construct the rationalization in stages, where at each stage we wedge a sphere and then glue a $n+1$-cell to ``invert'' some element in the $n$th homotopy group. At each stage the space will be homotopy equivalent to $S^n$.
 
 \todo{Put this in a lemma. And make it more readable.}
@@ -63,7 +63,7 @@ The \Def{rational disk} is now defined as cone of the rational sphere: $D^{n+1}_
 }
 \todo{Add: unique up to homotopy. A homotopy extends to a homotopy.}
 
-\subsection{Rationalizations of arbitrary spaces}
+\section{Rationalizations of arbitrary spaces}
 Having rational cells we wish to replace the cells in a CW complex $X$ by the rational cells to obtain a rationalization.
 
 \Lemma{rationalization-CW}{
@@ -109,7 +109,7 @@ We already mentioned in the first section that for rational spaces the notions o
 	The homotopy category of $1$-connected rational spaces is equivalent to the rational homotopy category of $1$-connected spaces.
 }
 
-\subsection{Other constructions}
+\section{Other constructions}
 There are others ways to obtain a rationalization. One of them relies on the observations that it is easy to rationalize Eilenberg-MacLane spaces.
 
 \Lemma{rationalization-em-space}{

thesis/notes/Serre.tex

@@ -1,5 +1,5 @@
 
-\section{Serre theorems mod \texorpdfstring{$C$}{C}}
+\chapter{Serre theorems mod \texorpdfstring{$\C$}{C}}
 \label{sec:serre}
 
 In this section we will prove the Whitehead and Hurewicz theorems in a rational context. Serre proved these results in \cite{serre}. In his paper he considered homology groups `modulo a class of abelian groups'. In our case of rational homotopy theory, this class will be the class of torsion groups.

thesis/preamble.tex

@@ -1,4 +1,7 @@
 
+% normally included with amsart
+\usepackage{amsmath, amsthm}
+
 % font with unicode support
 \usepackage{fontspec}
 
@@ -13,7 +16,7 @@
 \usepackage{float}
 
 % for appendices
-\usepackage[toc,page]{appendix}
+% \usepackage[toc,page]{appendix}
 
 % for multiple cites
 \usepackage{cite}
@@ -31,14 +34,11 @@
 \usepackage{caption}
 \usepackage{subcaption}
 
-% Matrices have a upper bound for its size
-\setcounter{MaxMatrixCols}{20}
-
 % for the fib arrow
 \usepackage{amssymb}
 
 % mathbb for lowercase bbs
-\usepackage{bbm}
+\usepackage{dsfont}
 
 % Some basic objects
 \newcommand{\N}{\mathbb{N}}				% natural numbers
@@ -46,7 +46,7 @@
 \newcommand{\Z}{\mathbb{Z}}				% integers
 \newcommand{\R}{\mathbb{R}}				% reals
 \newcommand{\Q}{\mathbb{Q}}				% rationals
-\renewcommand{\k}{\mathbbm{k}}			% default ground ring
+\renewcommand{\k}{\mathds{k}}			% default ground ring
 
 % Basic category stuff
 \newcommand{\cat}[1]{\mathbf{#1}}		% the category of ...
@@ -117,6 +117,8 @@
 \renewcommand{\C}{\mathcal{C}}			% Serre mod C class
 \newcommand{\Apl}[0]{{A_{PL}}}			% Apl simplicial set of polynomials
 
+\newcommand{\titleCDGA}{\texorpdfstring{$\CDGA$}{CDGA}}
+
 % restriction of a function
 \newcommand\restr[2]{{% we make the whole thing an ordinary symbol
   \left.\kern-\nulldelimiterspace % automatically resize the bar with \right

thesis/references.bib

@@ -1,3 +1,11 @@
+@misc{berglund,
+	title={Rational Homotopy Theory},
+	author={Berglund, Alexander},
+	year={2012},
+	institution = {University of Copenhagen},
+	howpublished = {University Lecture}
+}
+
 @book{bousfield,
 	title={On PL de Rham theory and rational homotopy type},
 	author={Bousfield, Aldridge Knight and Gugenheim, Victor KAM},

thesis/style.tex

@@ -1,9 +1,9 @@
 
 % lesser margins
-\usepackage{geometry}
+% \usepackage{geometry}
-\geometry{a4paper}
+% \geometry{a4paper}
-\geometry{twoside=false}
+% \geometry{twoside=false}
 
 % no indent, but vertical spacing
-\usepackage[parfill]{parskip}
+% \usepackage[parfill]{parskip}
-\setlength{\marginparwidth}{2cm}
+% \setlength{\marginparwidth}{2cm}

thesis/thesis.tex

@@ -1,60 +1,34 @@
-\documentclass[a4paper, 12pt]{amsart}
+\documentclass[a4paper,12pt,footinclude=true,headinclude=true,oneside,dottedtoc]{scrbook}
+\usepackage[parts,drafting,eulerchapternumbers,beramono,eulermath]{classicthesis}
+\setcounter{tocdepth}{0} % parts, chapters
 
-\input{style}
 \input{preamble}
 
 \title{Rational Homotopy Theory}
 \author{Joshua Moerman}
 
 \begin{document}
+\pagenumbering{roman}
 
 \maketitle
-{\bf \today}
+
-
+\include{chapters/Introduction}
-\section*{Contents}
+
-\tableofcontents
+\pagenumbering{arabic}
-
+\part{Basics Of Rational Homotopy Theory}
-\section*{Preliminaries}
+\include{chapters/Basics_Of_Rational_Homotopy_Theory}
-We assume the reader is familiar with category theory, basics from algebraic topology and the basics of simplicial sets. Some knowledge about differential graded algebra (or homological algebra) and model categories is assumed, but the reader may review this in the appendices.
+
-
+\part{CDGAs As Algebraic Models}
-Some notation:
+\include{chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory}
-\begin{itemize}
+
-	\item $\k$ will denote an arbitrary commutative ring (or field, if indicated at the start of a section). Modules, tensor products, \dots are understood as $\k$-modules, tensor products over $\k$, \dots. If ambiguitity can occur notation will be explicit.
+\part{Applications and Further Topics}
-	\item $\cat{C}$ will denote an arbitrary category.
+\include{chapters/Applications_And_Further_Topics}
-	\item $\cat{0}$ (resp. $\cat{1}$) will denote the initial (resp. final) objects in a category $\cat{C}$.
+
-	\item $\Hom_\cat{C}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. The subscript $\cat{C}$ is occasionally left out if the category is clear from the context.
+\appendix
-\end{itemize}
+\part{Appendices}
-
+\include{chapters/Appendices}
-Some categories:
+
-\begin{itemize}
+\listoftodos
-	\item $\Top$: category of topological spaces and continuous maps.
-	\item $\Ab$: category of abelian groups and group homomorphisms.
-	\item $\DELTA$: category of simplices (i.e. finite, non-empty ordinals) and order preserving maps.
-	\item $\sSet$: category of simplicial sets and simplicial maps (more generally we have the category of simplicial objects, $\cat{sC}$, for any category $\cat{C}$).
-	\item $\Ch{\k}, \CoCh{\k}$: category of non-negatively graded chain (resp. cochain) complexes and chain maps.
-	\item $\DGA_\k$: category of non-negatively differential graded algebras over $\k$ (these are cochain complexes with a multiplication) and graded algebra  maps. As a shorthand we will refer to such an object as \emph{dga}.
-	\item $\CDGA_\k$: the full subcategory of $\DGA_\k$ of commutative dga's (\emph{cdga}'s).
-\end{itemize}
-
-\newcommand{\myinput}[1]{\include{#1}}
-
-\addtocontents{toc}{\protect\setcounter{tocdepth}{2}}
-\myinput{notes/Basics}
-\myinput{notes/Serre}
-\myinput{notes/Rationalization}
-\myinput{chapters/Homotopy_Theory_CDGA}
-\myinput{chapters/Polynomial_Forms}
-\myinput{notes/Minimal_Models}
-\myinput{notes/A_K_Quillen_Pair}
-
-\addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
-\begin{appendices}
-	\input{notes/Algebra}
-	\input{notes/Free_CDGA}
-	\include{notes/Model_Categories}
-\end{appendices}
-
-% \listoftodos
 
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