Some small fixups

thesis/notes/Algebra.tex

@@ -2,7 +2,7 @@
 \section{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.
+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}
 

thesis/notes/CDGA_Basic_Examples.tex

@@ -1,5 +1,5 @@
 
-\subsection{Cochain models for the $n$-disk and $n$-sphere}
+\section{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:
 
 $$ D(n) = ... \to 0 \to \k \to \k \to 0 \to ... $$

thesis/notes/CDGA_Of_Polynomials.tex

@@ -1,5 +1,5 @@
 
-\subsection{CDGA of Polynomials}
+\section{CDGA of Polynomials}
 
 \newcommand{\Apl}[0]{{A_{PL}}}
 
@@ -7,7 +7,7 @@ We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows f
 
 \begin{definition}
 	For all $n \in \N$ define the following cdga:
-	$$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, dx_0, \ldots, dx_n)}{(\sum_{i=0}^n) x_i - 1, \sum_{i=0}^n dx_i)} $$
+	$$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, dx_0, \ldots, dx_n)}{(\sum_{i=0}^n x_i - 1, \sum_{i=0}^n dx_i)} $$
 	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}
 

thesis/notes/Free_CDGA.tex

@@ -21,10 +21,10 @@ Note that this construction is functorial and it is free in the following sense.
 
 \begin{corollary}
 	Let $U$ be the forgetful functor from graded algebras to graded modules, then $T$ and $U$ form an adjoint pair:
-	$$ T: \grMod{\k} \leftadj \grAlg{\k} $$
+	$$ T: \grMod{\k} \leftadj \grAlg{\k} :U $$
 	Moreover it extends and restricts to
-	$$ T: \dgMod{\k} \leftadj \dgAlg{\k} $$
-	$$ T: \CoCh{\k} \leftadj \DGA{\k} $$
+	$$ T: \dgMod{\k} \leftadj \dgAlg{\k} :U $$
+	$$ T: \CoCh{\k} \leftadj \DGA{\k} :U $$
 \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.
@@ -42,9 +42,9 @@ Again this extends to differential graded modules (i.e. the ideal is preserved b
 
 \begin{lemma}
 	We have the following adjunctions:
-	$$ \Lambda: \grMod{\k} \leftadj \grAlg{\k}^{comm} $$
-	$$ \Lambda: \dgMod{\k} \leftadj \dgAlg{\k}^{comm} $$
-	$$ \Lambda: \CoCh{\k} \leftadj \CDGA_\k $$
+	$$ \Lambda: \grMod{\k} \leftadj \grAlg{\k}^{comm} :U $$
+	$$ \Lambda: \dgMod{\k} \leftadj \dgAlg{\k}^{comm} :U $$
+	$$ \Lambda: \CoCh{\k} \leftadj \CDGA_\k :U $$
 \end{lemma}
 
 We can now easily construct cdga's by specifying generators and their differentials.

thesis/notes/Minimal_Models.tex

@@ -7,7 +7,7 @@ In this section we will discuss the so called minimal models. These are cdga's w
 	An cdga $(A, d)$ is a \emph{Sullivan algebra} if
 	\begin{itemize}
 		\item $(A, d)$ is quasi-free (or semi-free), i.e. $A = \Lambda V$ is free as a cdga, and
-		\item $V$ has a filtration $V(0) \subset V(1) \subset \cdots \subset \bigcup{k \in \N} V(k) = V$ such that $d(V(k)) \subset \Lambda V(k-1)$.
+		\item $V$ has a filtration $V(0) \subset V(1) \subset \cdots \subset \bigcup_{k \in \N} V(k) = V$ such that $d(V(k)) \subset \Lambda V(k-1)$.
 	\end{itemize}
 
 	An cdga $(A, d)$ is a \emph{minimal (Sullivan) algebra} if in addition
@@ -27,7 +27,7 @@ The requirement that there exists a filtration can be replaced by a stronger sta
 	Let $(A, d)$ be a cdga which is $1$-reduced, quasi-free and with a decomposable differential. Then $(A, d)$ is a minimal algebra.
 \end{lemma}
 \begin{proof}
-	Take $V(n) = \bigoplus_{k=0}^n V^k$ (note that $V^0 = v^1 = 0$). Since $d$ is decomposable we see that for $v \in V^n$: $d(v) = x \cdot y$ for some $x, y \in A$. Assuming $dv$ to be non-zero we can compute the degrees:
+	Let $V$ generate $A$. Take $V(n) = \bigoplus_{k=0}^n V^k$ (note that $V^0 = V^1 = 0$). Since $d$ is decomposable we see that for $v \in V^n$: $d(v) = x \cdot y$ for some $x, y \in A$. Assuming $dv$ to be non-zero we can compute the degrees:
 	$$ \deg{x} + \deg{y} = \deg{xy} = \deg{dv} = \deg{v} + 1 = n + 1. $$
 	As $A$ is $1$-reduced we have $\deg{x}, \deg{y} \geq 2$ and so by the above $\deg{x}, \deg{y} \leq n-1$. Conclude that $d(V(k)) \subset \Lambda(V(n-1))$.
 \end{proof}
@@ -67,7 +67,7 @@ Before we state the uniqueness theorem we need some more properties of minimal m
 
 	\cimage[scale=0.5]{Sullivan_Lifting}
 
-	By the left adjointness of $\Lambda$ we only have to specify a map $\phi: V \to X$ sucht that $p \circ \phi = g$. We will do this by induction.
+	By the left adjointness of $\Lambda$ we only have to specify a map $\phi: V \to X$ such that $p \circ \phi = g$. We will do this by induction.
 	\begin{itemize}
 		\item Suppose $\{v_\alpha\}$ is a basis for $V(0)$. Define $V(0) \to X$ by choosing preimages $x_\alpha$ such that $p(x_\alpha) = g(v_\alpha)$ ($p$ is surjective). Define $\phi(v_\alpha) = x_\alpha$.
 		\item Suppose $\phi$ has been defined on $V(n)$. Write $V(n+1) = V(n) \oplus V'$ and let $\{v_\alpha\}$ be a basis for $V'$. Then $dv_\alpha \in \Lambda V(n)$, hence $\phi(dv_\alpha)$ is defined and
@@ -101,7 +101,7 @@ Before we state the uniqueness theorem we need some more properties of minimal m
 	Let $\phi: (M, d) \we (M', d')$ be a weak equivalence between minimal algebras. Then $\phi$ is an isomorphism.
 \end{lemma}
 \begin{proof}
-	Let $M$ and $M'$ be generated by $V$ and $V'$. Then $\phi$ induces a weak equivalence on the linear part $\phi_0: V \we V'$ \cite[Theorem 1.5.10]{loday}. Since the differentials are decomposable, their linear part vanishes. So we see that $\phi_0: (V, 0) \tot{\iso} (V', 0)$ is an isomorphism.
+	Let $M$ and $M'$ be generated by $V$ and $V'$. Then $\phi$ induces a weak equivalence on the linear part $\phi_0: V \we V'$ \cite[Theorem 1.5.2]{loday}. Since the differentials are decomposable, their linear part vanishes. So we see that $\phi_0: (V, 0) \tot{\iso} (V', 0)$ is an isomorphism.
 	Conclude that $\phi = \Lambda \phi_0$ is an isomorphism.
 \end{proof}
 
@@ -109,5 +109,5 @@ Before we state the uniqueness theorem we need some more properties of minimal m
 	Let $m: (M, d) \we (A, d)$ and $m': (M', d') \we (A, d)$ be two minimal models. Then there is an isomorphism $\phi (M, d) \tot{\iso} (M', d')$ such that $m' \circ \phi \eq m$.
 \end{theorem}
 \begin{proof}
-	By the previous lemmas we have $[M', M] \iso [M', A]$. By going from right to elft we get a map $\phi: M' \to M$ such that $m' \circ \phi \eq m$. On homology we get $H(m') \circ H(\phi) = H(m)$, proving that (2-out-of-3) $\phi$ is a weak equivalence. The previous lemma states that $\phi$ is then an isomorphism.
+	By the previous lemmas we have $[M', M] \iso [M', A]$. By going from right to left we get a map $\phi: M' \to M$ such that $m' \circ \phi \eq m$. On homology we get $H(m') \circ H(\phi) = H(m)$, proving that (2-out-of-3) $\phi$ is a weak equivalence. The previous lemma states that $\phi$ is then an isomorphism.
 \end{proof}

thesis/preamble.tex

@@ -24,9 +24,6 @@
 % Matrices have a upper bound for its size
 \setcounter{MaxMatrixCols}{20}
 
-% Remove trailing `contents` after toc
-\renewcommand{\contentsname}{}
-
 % for the fib arrow
 \usepackage{amssymb}
 
@@ -51,7 +48,7 @@
 % Categories
 \newcommand{\Set}{\cat{Set}}			% sets
 \newcommand{\Top}{\cat{Top}}			% topological spaces
-\newcommand{\DELTA}{\cat{\Delta}}		% the simplicial cat
+\newcommand{\DELTA}{\boldsymbol{\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

thesis/style.tex

@@ -7,3 +7,6 @@
 % no indent, but vertical spacing
 \usepackage[parfill]{parskip}
 \setlength{\marginparwidth}{2cm}
+
+% skip subsections in toc
+\setcounter{tocdepth}{1}

thesis/thesis.tex

@@ -9,7 +9,13 @@
 \begin{document}
 
 \maketitle
+\begin{center}
+	{\bf \today}
+\end{center}
+
+\vspace{2cm}
 \tableofcontents
+\vspace{2cm}
 
 Some general notation: \todo{leave this out, or define somewhere else?}
 \begin{itemize}
@@ -17,9 +23,7 @@ Some general notation: \todo{leave this out, or define somewhere else?}
 	\item $\Hom_\cat{C}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$.
 \end{itemize}
 
-\vspace{1cm}
-
-\newcommand{\myinput}[1]{\input{#1} \vspace{2cm}}
+\newcommand{\myinput}[1]{\include{#1}}
 
 \myinput{notes/Algebra}
 \myinput{notes/Free_CDGA}