From b784784eaea3e9740403232ec3c778a313b33ae0 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Mon, 27 Oct 2014 13:08:39 +0100 Subject: [PATCH] Adds xymatrix diagrams --- thesis/Makefile | 16 ++++---- thesis/diagrams/Model_Cats_Pushouts.tex | 9 +++++ thesis/diagrams/Model_Liftproblem.tex | 4 ++ thesis/diagrams/Model_Replacements.tex | 9 +++++ thesis/diagrams/Topr_No_Coequalizer.tex | 11 ++++++ thesis/images/Model_Cats_Pushouts.png | Bin 30538 -> 0 bytes thesis/images/Topr_No_Coequalizer.png | Bin 9997 -> 0 bytes thesis/images/Topr_No_Equalizer.png | Bin 18104 -> 0 bytes thesis/notes/Model_Categories.tex | 49 +++--------------------- thesis/preamble.tex | 23 ++++++++++- thesis/test_diagram.sh | 8 ++++ thesis/test_diagram.tex | 14 +++++++ 12 files changed, 92 insertions(+), 51 deletions(-) create mode 100644 thesis/diagrams/Model_Cats_Pushouts.tex create mode 100644 thesis/diagrams/Model_Liftproblem.tex create mode 100644 thesis/diagrams/Model_Replacements.tex create mode 100644 thesis/diagrams/Topr_No_Coequalizer.tex delete mode 100644 thesis/images/Model_Cats_Pushouts.png delete mode 100644 thesis/images/Topr_No_Coequalizer.png delete mode 100644 thesis/images/Topr_No_Equalizer.png create mode 100755 thesis/test_diagram.sh create mode 100644 thesis/test_diagram.tex diff --git a/thesis/Makefile b/thesis/Makefile index 37751f6..0dff3f3 100644 --- a/thesis/Makefile +++ b/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 # 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 +thesis: dirs cp references.bib build/ xelatex -file-line-error -output-directory=build thesis.tex cd build; bibtex thesis @@ -12,15 +11,18 @@ thesis: xelatex -file-line-error -output-directory=build thesis.tex cp build/thesis.pdf ./ -fast: - mkdir -p build +fast: dirs xelatex -file-line-error -output-directory=build thesis.tex cp build/thesis.pdf ./ -images: - mkdir -p build +images: dirs 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 ssh moerman@stitch.science.ru.nl 'pdf2svg wvlt_images.pdf wvlt_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 diff --git a/thesis/diagrams/Model_Cats_Pushouts.tex b/thesis/diagrams/Model_Cats_Pushouts.tex new file mode 100644 index 0000000..beb496c --- /dev/null +++ b/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 +} \ No newline at end of file diff --git a/thesis/diagrams/Model_Liftproblem.tex b/thesis/diagrams/Model_Liftproblem.tex new file mode 100644 index 0000000..1707236 --- /dev/null +++ b/thesis/diagrams/Model_Liftproblem.tex @@ -0,0 +1,4 @@ +\xymatrix{ + A \ar[d]^i \ar[r] & X \ar[d]^p \\ + B \ar[r] & Y +} diff --git a/thesis/diagrams/Model_Replacements.tex b/thesis/diagrams/Model_Replacements.tex new file mode 100644 index 0000000..2fd9341 --- /dev/null +++ b/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] & +} diff --git a/thesis/diagrams/Topr_No_Coequalizer.tex b/thesis/diagrams/Topr_No_Coequalizer.tex new file mode 100644 index 0000000..a5747b1 --- /dev/null +++ b/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 +} diff --git a/thesis/images/Model_Cats_Pushouts.png b/thesis/images/Model_Cats_Pushouts.png deleted file mode 100644 index 19c50a8ac9a137bbdf49a26164d144fa1bab4928..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 30538 zcmce-WmH>F)IN&41&UK7xRc=S!M#Nar4S$#iU&)floksRptwtMD^7tzi@RHa;+jHn zC{FqFd*A0hMi``C!@t)-`qzd}Sn$V;zA z_lNqiiO%VnxwopjtE+>X_oIk`VQ4I<2NDz!Wi^r(yP5xCgMq<{p{1t$%>UPZHX@7Z zc}sil-r<61x-OS0D?b~d8Vx&ZytuPHRcf9;uJShDM_Sr8KC?F3YJR@Ay|_&*{_4kk zAEStnA8jU-u($*u1d$d57K0L6T!|I|WveVECh+pPJqK8MYkM5aVAU*fw_@8)635Hx?9Lo6EZ9UE_I^TojAA99v@yO=JV_HLl zqdC<>DeQ*6n46?1~Ed#O|^D7K~(Ck$rE`}%ya*3bi=Z*L1OPhbiKS^cK91m)5CJjWsg$^k_Ofley9E#hPP|glH(>E_o2|#;y9iQ0QzRh=n z=I!0JS1AsS#em4d#^EG${;Wcyo|4TMv&NV|N~P7FNd#>-EjVcC=ms zN51BUE=F(QctE@EM96eq3NYjwXtRbHG&_5dE7Bx>&6KS!(#n_Hi?=z@TR+`o_%%B-Ej>3yljg#_mlyl2mtE?e3IayAv;)}&{imtXuOcnGz? z?;uMt?sioli07u);Mc(0s3xkRv%p^(W+~m)%lf@PdRgV@Q)U_k;)V!gw{>VVM~NbM 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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[MC4] In any commuting square as follows where $i \in \Cof$ and $p \in \Fib$, - \begin{center} - \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} + \cdiagram{Model_Liftproblem} there exist a lift $h: B \to Y$ if either \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: -\begin{center} -\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} +\cdiagram{Model_Replacements} 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}{ 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} - \item If $i$ is a (trivial) cofibrations, so is $j$. - \item If $p$ is a (trivial) fibrations, so is $q$. + \item If $i$ is a (trivial) cofibration, so is $j$. + \item If $p$ is a (trivial) fibration, so is $q$. \end{itemize} } @@ -146,8 +110,7 @@ In this thesis we often restrict to $1$-connected spaces. The full subcategory $ \Lemma{topr-no-colimit}{ Let $r > 0$ and $\Top_r$ be the full subcategory of $r$-connected spaces. The diagrams - \cimage[scale=0.5]{Topr_No_Coequalizer} - \cimage[scale=0.5]{Topr_No_Equalizer} + \cdiagram{Topr_No_Coequalizer} have no coequalizer and respectively no equalizer in $\Top_r$. } diff --git a/thesis/preamble.tex b/thesis/preamble.tex index 66b0598..6209c5b 100644 --- a/thesis/preamble.tex +++ b/thesis/preamble.tex @@ -18,10 +18,14 @@ % for multiple cites \usepackage{cite} +% fancy diagrams \usepackage{tikz} \usetikzlibrary{matrix, arrows, decorations} \tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto} +% simple diagrams +\usepackage[all,cmtip]{xy} + \usepackage{graphicx} \graphicspath{ {./images/} } \usepackage{caption} @@ -73,6 +77,12 @@ \newcommand{\fib}{\twoheadrightarrow} % fibration \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}{\rightleftarrows} @@ -167,4 +177,15 @@ \begin{center} \includegraphics[#1]{#2} \end{center} -} \ No newline at end of file +} + +% simple way to center a diagram +\newcommand{\cdiagrambase}[1]{ + \begin{displaymath} + \input{#1} + \end{displaymath} +} +\newcommand{\cdiagram}[1]{ + \cdiagrambase{diagrams/#1} +} + diff --git a/thesis/test_diagram.sh b/thesis/test_diagram.sh new file mode 100755 index 0000000..45a183c --- /dev/null +++ b/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.* diff --git a/thesis/test_diagram.tex b/thesis/test_diagram.tex new file mode 100644 index 0000000..5932e9b --- /dev/null +++ b/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}