Browse Source

Adds xymatrix diagrams

master
Joshua Moerman 10 years ago
parent
commit
b784784eae
  1. 16
      thesis/Makefile
  2. 9
      thesis/diagrams/Model_Cats_Pushouts.tex
  3. 4
      thesis/diagrams/Model_Liftproblem.tex
  4. 9
      thesis/diagrams/Model_Replacements.tex
  5. 11
      thesis/diagrams/Topr_No_Coequalizer.tex
  6. BIN
      thesis/images/Model_Cats_Pushouts.png
  7. BIN
      thesis/images/Topr_No_Coequalizer.png
  8. BIN
      thesis/images/Topr_No_Equalizer.png
  9. 49
      thesis/notes/Model_Categories.tex
  10. 23
      thesis/preamble.tex
  11. 8
      thesis/test_diagram.sh
  12. 14
      thesis/test_diagram.tex

16
thesis/Makefile

@ -1,10 +1,9 @@
.PHONY: thesis fast images .PHONY: thesis fast images dirs
# We don want to pollute the root dir, so we use a build dir # 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 # http://tex.stackexchange.com/questions/12686/how-do-i-run-bibtex-after-using-the-output-directory-flag-with-pdflatex-when-f
thesis: thesis: dirs
mkdir -p build
cp references.bib build/ cp references.bib build/
xelatex -file-line-error -output-directory=build thesis.tex xelatex -file-line-error -output-directory=build thesis.tex
cd build; bibtex thesis cd build; bibtex thesis
@ -12,15 +11,18 @@ thesis:
xelatex -file-line-error -output-directory=build thesis.tex xelatex -file-line-error -output-directory=build thesis.tex
cp build/thesis.pdf ./ cp build/thesis.pdf ./
fast: fast: dirs
mkdir -p build
xelatex -file-line-error -output-directory=build thesis.tex xelatex -file-line-error -output-directory=build thesis.tex
cp build/thesis.pdf ./ cp build/thesis.pdf ./
images: images: dirs
mkdir -p build
xelatex -file-line-error -output-directory=build images.tex xelatex -file-line-error -output-directory=build images.tex
xelatex -file-line-error -output-directory=build images.tex xelatex -file-line-error -output-directory=build images.tex
scp build/images.pdf moerman@stitch.science.ru.nl:~/wvlt_images.pdf 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' ssh moerman@stitch.science.ru.nl 'pdf2svg wvlt_images.pdf wvlt_images.svg'
scp moerman@stitch.science.ru.nl:~/wvlt_images.svg ./images.svg scp moerman@stitch.science.ru.nl:~/wvlt_images.svg ./images.svg
dirs:
mkdir -p build
mkdir -p build/notes
mkdir -p build/diagrams

9
thesis/diagrams/Model_Cats_Pushouts.tex

@ -0,0 +1,9 @@
\xymatrix{
A \ar[r] \arcof[d]_{(\simeq)}^i & C \ar[d]^j \\
B \ar[r] & P
}
\qquad
\xymatrix{
P \ar[r] \ar[d]^q & X \arfib[d]_{(\simeq)}^p \\
Z \ar[r] & Y
}

4
thesis/diagrams/Model_Liftproblem.tex

@ -0,0 +1,4 @@
\xymatrix{
A \ar[d]^i \ar[r] & X \ar[d]^p \\
B \ar[r] & Y
}

9
thesis/diagrams/Model_Replacements.tex

@ -0,0 +1,9 @@
\xymatrix{
\cat{0} \ar[rr] \arcof[dr] & & X \\
& X^{cof} \artfib[ur] &
}
\qquad
\xymatrix{
X \ar[rr] \artcof[dr] & & \cat{1} \\
& X^{fib} \arfib[ur] &
}

11
thesis/diagrams/Topr_No_Coequalizer.tex

@ -0,0 +1,11 @@
\xymatrixcolsep{7pc}
\xymatrix{
\ast \ar@/^/[r]^{\ast \mapsto 0}
\ar@/_/[r]_{\ast \mapsto 1} & I
}
\quad
\xymatrixcolsep{7pc}
\xymatrix{
I \ar@/^/[r]^{x \mapsto (x, sin(\pi x))}
\ar@/_/[r]_{x \mapsto (x, -sin(\pi x))} & \R^2
}

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thesis/images/Model_Cats_Pushouts.png

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thesis/images/Topr_No_Coequalizer.png

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thesis/images/Topr_No_Equalizer.png

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49
thesis/notes/Model_Categories.tex

@ -19,20 +19,7 @@
\item[MC2] If $f$, $g$ and $fg$ are maps such that two of them are weak equivalences, then so it the third. This is called the \emph{2-out-of-3} property. \item[MC2] If $f$, $g$ and $fg$ are maps such that two of them are weak equivalences, then so it the third. This is called the \emph{2-out-of-3} property.
\item[MC3] All three classes of maps are closed under retracts\todo{Either draw the diagram or define a retract earlier}. \item[MC3] All three classes of maps are closed under retracts\todo{Either draw the diagram or define a retract earlier}.
\item[MC4] In any commuting square as follows where $i \in \Cof$ and $p \in \Fib$, \item[MC4] In any commuting square as follows where $i \in \Cof$ and $p \in \Fib$,
\begin{center} \cdiagram{Model_Liftproblem}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes]{
A & X \\
B & Y \\
};
\path[->] (m-1-1) edge (m-1-2);
\path[->] (m-2-1) edge (m-2-2);
\path[->] (m-1-1) edge node[auto] {$i$} (m-2-1);
\path[->] (m-1-2) edge node[auto] {$p$} (m-2-2);
\end{tikzpicture}
\end{center}
there exist a lift $h: B \to Y$ if either there exist a lift $h: B \to Y$ if either
\begin{itemize} \begin{itemize}
@ -63,30 +50,7 @@
Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equivalent fibrant object $X^{fib}$ and by [MC5b] by a weakly equivalent cofibrant object $X^{cof}$, as seen in the following diagram: Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equivalent fibrant object $X^{fib}$ and by [MC5b] by a weakly equivalent cofibrant object $X^{cof}$, as seen in the following diagram:
\begin{center} \cdiagram{Model_Replacements}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes]{
\cat{0} & & X \\
& X^{cof} & \\
};
\path[->] (m-1-1) edge (m-1-3);
\path[right hook->] (m-1-1) edge (m-2-2);
\path[->>] (m-2-2) edge node[auto] {$ \simeq $} (m-1-3);
\end{tikzpicture}\quad
\begin{tikzpicture}
\matrix (m) [matrix of math nodes]{
X & & \cat{1} \\
& X^{fib} & \\
};
\path[->] (m-1-1) edge (m-1-3);
\path[right hook->] (m-1-1) edge node[auto] {$ \simeq $} (m-2-2);
\path[->>] (m-2-2) edge (m-1-3);
\end{tikzpicture}
\end{center}
The fourth axiom actually characterizes the classes of (trivial) fibrations and (trivial) cofibrations. We will abbreviate left lifting property with LLP and right lifting property with RLP. We will not prove these statements, but only expose them because we use them throughout this thesis. One can find proofs in \cite{dwyer, may}. The fourth axiom actually characterizes the classes of (trivial) fibrations and (trivial) cofibrations. We will abbreviate left lifting property with LLP and right lifting property with RLP. We will not prove these statements, but only expose them because we use them throughout this thesis. One can find proofs in \cite{dwyer, may}.
@ -105,11 +69,11 @@ This means that once we choose weak equivalences and fibrations for a category $
\Lemma{model-cats-pushouts}{ \Lemma{model-cats-pushouts}{
Let $\cat{C}$ be a model category. Consider the following two diagrams where $P$ is the pushout and pullback respectively. Let $\cat{C}$ be a model category. Consider the following two diagrams where $P$ is the pushout and pullback respectively.
\cimage{Model_Cats_Pushouts} \cdiagram{Model_Cats_Pushouts}
\begin{itemize} \begin{itemize}
\item If $i$ is a (trivial) cofibrations, so is $j$. \item If $i$ is a (trivial) cofibration, so is $j$.
\item If $p$ is a (trivial) fibrations, so is $q$. \item If $p$ is a (trivial) fibration, so is $q$.
\end{itemize} \end{itemize}
} }
@ -146,8 +110,7 @@ In this thesis we often restrict to $1$-connected spaces. The full subcategory $
\Lemma{topr-no-colimit}{ \Lemma{topr-no-colimit}{
Let $r > 0$ and $\Top_r$ be the full subcategory of $r$-connected spaces. The diagrams Let $r > 0$ and $\Top_r$ be the full subcategory of $r$-connected spaces. The diagrams
\cimage[scale=0.5]{Topr_No_Coequalizer} \cdiagram{Topr_No_Coequalizer}
\cimage[scale=0.5]{Topr_No_Equalizer}
have no coequalizer and respectively no equalizer in $\Top_r$. have no coequalizer and respectively no equalizer in $\Top_r$.
} }

23
thesis/preamble.tex

@ -18,10 +18,14 @@
% for multiple cites % for multiple cites
\usepackage{cite} \usepackage{cite}
% fancy diagrams
\usepackage{tikz} \usepackage{tikz}
\usetikzlibrary{matrix, arrows, decorations} \usetikzlibrary{matrix, arrows, decorations}
\tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto} \tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto}
% simple diagrams
\usepackage[all,cmtip]{xy}
\usepackage{graphicx} \usepackage{graphicx}
\graphicspath{ {./images/} } \graphicspath{ {./images/} }
\usepackage{caption} \usepackage{caption}
@ -73,6 +77,12 @@
\newcommand{\fib}{\twoheadrightarrow} % fibration \newcommand{\fib}{\twoheadrightarrow} % fibration
\newcommand{\we}{\tot{\simeq}} % weak equivalence \newcommand{\we}{\tot{\simeq}} % weak equivalence
% for use in xy diagrams
\newcommand{\arcof}{\ar@{^{(}->}}
\newcommand{\artcof}{\ar@{^{(}->}|\simeq}
\newcommand{\arfib}{\ar@{->>}}
\newcommand{\artfib}{\ar@{->>}|\simeq}
%\newcommand{\leftadj}{\ooalign{\hss\rightleftarrows\hss\cr\bot}} %\newcommand{\leftadj}{\ooalign{\hss\rightleftarrows\hss\cr\bot}}
\newcommand{\leftadj}{\rightleftarrows} \newcommand{\leftadj}{\rightleftarrows}
@ -167,4 +177,15 @@
\begin{center} \begin{center}
\includegraphics[#1]{#2} \includegraphics[#1]{#2}
\end{center} \end{center}
} }
% simple way to center a diagram
\newcommand{\cdiagrambase}[1]{
\begin{displaymath}
\input{#1}
\end{displaymath}
}
\newcommand{\cdiagram}[1]{
\cdiagrambase{diagrams/#1}
}

8
thesis/test_diagram.sh

@ -0,0 +1,8 @@
#!/bin/bash
set -e
file=$1
sed "s|__INPUT__|$file|" test_diagram.tex | xelatex -file-line-error -output-directory=build
mv build/texput.pdf test_diagram.pdf
exo-open test_diagram.pdf
rm build/texput.*

14
thesis/test_diagram.tex

@ -0,0 +1,14 @@
\documentclass[a4paper, 12pt]{amsart}
\input{style}
\input{preamble}
\begin{document}
Your diagram:
\cdiagrambase{__INPUT__}
For reference, some default latex stuff:
$$ A \cof B \we X \fib Y $$
\end{document}