First two pages of algebraic definitions

.gitignore

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+.DS_Store
+
+*.pdf
+build
+

thesis/Definitions.tex

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+
+\section{Definitions}
+\label{sec:definitions}
+
+\subsection{Graded 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}, \dots we always mean $\k$-linear, $\k$-module, tensor product over $\k$, \dots.
+
+\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{homogenous element} and said to be of \emph{degree $\deg{x} = n$}. We will often identify $M = \bigoplus_{n \in \Z} M_n$.
+\end{definition}
+
+For an arbitrary module $M$ we can consider the graded module $M[0]$ \emph{concentrated in degree $0$} defined by setting $M[0]_0 = M$ and $M[0]_n = 0$ for $i \neq 0$. If clear from the context we will denote this graded module by $M$. In particular $\k$ is a graded module concentrated in degree $0$.
+
+\begin{definition}
+	A linear map $f: M \to N$ between graded modules is \emph{graded of degree $p$} if it respects the grading, i.e. $\restr{f}{M_n} : M_n \to N_{n+p}$.
+\end{definition}
+
+\begin{definition}
+	The graded maps $f: M \to N$ between graded modules can be arranged in a graded module by defining:
+	$$ \Hom{gr}{M}{N}_n = \{ f: M \to N \I f \text{ is graded of degree } n \}. $$
+\end{definition}
+
+Note that not all linear maps can be decomposed into a sum of graded maps. In other words $\Hom{gr}{M}{N} \subset \Hom{}{M}{N}$ might not be equal.
+
+Recall that the tensor product of modules distributes over direct sums. So if $M = \bigoplus_{n \in \Z} M_n$ and $N = \bigoplus_{n \in \Z} N_n$, then
+$$ M \tensor N \iso \bigoplus_{n \in Z} \bigoplus_{m \in Z} M_m \tensor N_n \iso \bigoplus_{n \in Z} \bigoplus_{i + j = n} M_i \tensor N_j. $$
+This defines a natural grading on the tensor product.
+
+\begin{definition}
+	The graded tensor product is defined as:
+	$$ (M \tensor N)_n = \bigoplus_{i + j = n} M_i \tensor N_j. $$
+\end{definition}
+
+The graded modules together with graded maps of degree $0$ form the category $\grMod{\k}$ of graded modules. Together with the tensor product and the ground ring, $(\grMod{\k}, \tensor, \k)$ is a monoidal category. This now dictates the definition of a graded algebra.
+
+\begin{definition}
+	A \emph{graded algebra} consists of a graded module $A$ together with two graded maps of degree $0$:
+	$$ \mu: A \tensor A \to A \quad\text{ and }\quad \eta: k \to A $$
+	such that $\mu$ is associative and $\eta$ is a unit for $\mu$.
+
+	A graded map between two graded algebra will be called \emph{graded algebra map} if the map is compatible with the multiplication and unit.
+\end{definition}
+
+Again these objects form a category, denoted as $\grAlg{\k}$.
+
+\begin{definition}
+	A graded algebra $A$ is \emph{commutative} if for all $x, y \in A$
+	$$ xy = (-1)^{\deg{x}\deg{y}}yx. $$
+\end{definition}
+
+
+\subsection{Differential graded algebra}
+Now define differentials... and the categories $\cat{DGA}_\k, \cat{CGDA}_\k$.
+
+Note that a monoidal object of differential graded modules is the same as a graded algebra with a differential.
+
+Conclude with (co)chain complexes and (co)chain (co)algebras.
+
+
+\subsection{Model categories}
+

thesis/Makefile

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+
+.PHONY: thesis fast images
+
+# We don want to pollute the root dir, so we use a build dir
+# http://tex.stackexchange.com/questions/12686/how-do-i-run-bibtex-after-using-the-output-directory-flag-with-pdflatex-when-f
+thesis:
+	mkdir -p build
+	cp references.bib build/
+	pdflatex -output-directory=build thesis.tex
+	cd build; bibtex thesis
+	pdflatex -output-directory=build thesis.tex
+	pdflatex -output-directory=build thesis.tex
+	cp build/thesis.pdf ./
+
+fast:
+	mkdir -p build
+	pdflatex -output-directory=build thesis.tex
+	cp build/thesis.pdf ./
+
+images:
+	mkdir -p build
+	pdflatex -output-directory=build images.tex
+	pdflatex -output-directory=build images.tex
+	scp build/images.pdf moerman@stitch.science.ru.nl:~/wvlt_images.pdf
+	ssh moerman@stitch.science.ru.nl 'pdf2svg wvlt_images.pdf wvlt_images.svg'
+	scp moerman@stitch.science.ru.nl:~/wvlt_images.svg ./images.svg

thesis/preamble.tex

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+
+% clickable tocs
+\usepackage{hyperref}
+
+% floating figures
+\usepackage{float}
+
+\usepackage{listings}
+
+\usepackage{tikz}
+\usetikzlibrary{matrix, arrows, decorations}
+\tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto}
+
+\usepackage{graphicx}
+\graphicspath{ {./images/} }
+\usepackage{caption}
+\usepackage{subcaption}
+
+% Matrices have a upper bound for its size
+\setcounter{MaxMatrixCols}{20}
+
+% Remove trailing `contents` after toc
+\renewcommand{\contentsname}{}
+
+% for the fib arrow
+\usepackage{amssymb}
+
+% mathbb for lowercase
+\usepackage{bbm}
+
+% for slanted text/symbols
+\usepackage{slantsc}
+
+\DeclareMathOperator*{\colim}{colim}
+\DeclareMathOperator*{\tensor}{\otimes}
+\DeclareMathOperator*{\bigtensor}{\bigotimes}
+
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\Np}{{\mathbb{N}^{>0}}}
+\newcommand{\Z}{\mathbb{Z}}
+\newcommand{\R}{\mathbb{R}}
+\renewcommand{\k}{\mathbbm{k}}
+
+\newcommand{\cat}[1]{\mathbf{#1}}
+\newcommand{\Set}{\cat{Set}}
+\newcommand{\sSet}{\cat{sSet}}
+\newcommand{\Top}{\cat{Top}}
+\newcommand{\DELTA}{\cat{\Delta}}
+\newcommand{\grMod}[1]{\cat{gr-{#1}Mod}}
+\newcommand{\grAlg}[1]{\cat{gr-{#1}Alg}}
+
+\newcommand{\Hom}[3]{\mathbf{Hom}_{#1}(#2, #3)}
+\newcommand{\id}{\mathbf{id}}
+
+\newcommand{\I}{\,\mid\,}
+\newcommand{\del}{\partial}				% boundary
+\newcommand{\iso}{\cong}				% isomorphic
+\newcommand{\eq}{\sim}					% homotopic
+\newcommand{\tot}[1]{\xrightarrow{\,\,{#1}\,\,}}	% arrow with name
+\newcommand{\mapstot}[1]{\xmapsto{\,\,{#1}\,\,}}	% mapsto with name
+\newcommand{\cof}{\hookrightarrow}		% cofibration
+\newcommand{\fib}{\twoheadrightarrow}	% fibration
+\newcommand{\we}{\tot{\simeq}}			% weak equivalence
+\renewcommand{\deg}[1]{|{#1}|}
+
+\newcommand\restr[2]{{% we make the whole thing an ordinary symbol
+  \left.\kern-\nulldelimiterspace % automatically resize the bar with \right
+  #1 % the function
+  \vphantom{\big|} % pretend it's a little taller at normal size
+  \right|_{#2} % this is the delimiter
+  }}
+
+\newcommand{\todo}[1]{
+	\addcontentsline{tdo}{todo}{\protect{#1}}
+	$\ast$ \marginpar{\tiny $\ast$ #1}
+}
+
+\theoremstyle{plain}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{example}[theorem]{Example}
+
+\newcommand*{\thead}[1]{\multicolumn{1}{c}{\bfseries #1}}

thesis/references.bib

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+@book{felix,
+	title={Rational homotopy theory},
+	author={F{\'e}lix, Yves and Halperin, Steve and Thomas, Jean-Claude},
+	volume={205},
+	year={2001},
+	publisher={Springer}
+}
+
+@book{bous,
+	title={On PL de Rham theory and rational homotopy type},
+	author={Bousfield, Aldridge Knight and Gugenheim, Victor KAM},
+	volume={179},
+	year={1976},
+	publisher={American Mathematical Soc.}
+}
+
+@article{hess,
+	title={Rational homotopy theory: a brief introduction},
+	author={Hess, Kathryn and others},
+	journal={Contemporary Mathematics},
+	volume={436},
+	pages={175},
+	year={2007},
+	publisher={Providence, RI; American Mathematical Society; 1999}
+}

thesis/style.tex

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

thesis/thesis.tex

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+\documentclass[a4paper, 11pt]{amsart}
+
+\input{style}
+\input{preamble}
+
+\title{Rational Homotopy Theory}
+\author{Joshua Moerman}
+
+\begin{document}
+
+\maketitle
+
+\input{Definitions} \newpage
+
+\nocite{*}
+\bibliographystyle{alpha}
+\bibliography{references}
+
+\end{document}