commit 3c8b2a1fee2884d0c410ce133400bf890b8c3ea6 Author: Joshua Moerman Date: Tue Apr 29 15:31:06 2014 +0200 First two pages of algebraic definitions diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..c090668 --- /dev/null +++ b/.gitignore @@ -0,0 +1,5 @@ +.DS_Store + +*.pdf +build + diff --git a/thesis/Definitions.tex b/thesis/Definitions.tex new file mode 100644 index 0000000..c1ee755 --- /dev/null +++ b/thesis/Definitions.tex @@ -0,0 +1,62 @@ + +\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} + diff --git a/thesis/Makefile b/thesis/Makefile new file mode 100644 index 0000000..1d21e09 --- /dev/null +++ b/thesis/Makefile @@ -0,0 +1,26 @@ + +.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 diff --git a/thesis/preamble.tex b/thesis/preamble.tex new file mode 100644 index 0000000..e8b469e --- /dev/null +++ b/thesis/preamble.tex @@ -0,0 +1,88 @@ + +% 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}} diff --git a/thesis/references.bib b/thesis/references.bib new file mode 100644 index 0000000..5f12847 --- /dev/null +++ b/thesis/references.bib @@ -0,0 +1,25 @@ +@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} +} diff --git a/thesis/style.tex b/thesis/style.tex new file mode 100644 index 0000000..a36543b --- /dev/null +++ b/thesis/style.tex @@ -0,0 +1,9 @@ + +% lesser margins +\usepackage{geometry} +\geometry{a4paper} +\geometry{twoside=false} + +% no indent, but vertical spacing +\usepackage[parfill]{parskip} +\setlength{\marginparwidth}{2cm} diff --git a/thesis/thesis.tex b/thesis/thesis.tex new file mode 100644 index 0000000..5c3a1c4 --- /dev/null +++ b/thesis/thesis.tex @@ -0,0 +1,19 @@ +\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} \ No newline at end of file