--- a/text/appendixes/comparing_defs.tex Sun Dec 11 23:16:19 2011 -0800
+++ b/text/appendixes/comparing_defs.tex Sun Dec 11 23:18:45 2011 -0800
@@ -215,8 +215,85 @@
rely heavily on the extended isotopy invariance of 2-morphisms in $\cC$.
Extended isotopy invariance implies that adding a product collar to a 2-morphism of $\cC$ has no effect,
and by cutting and regluing we can insert (or delete) product regions in the interior of 2-morphisms as well.
-Figure \nn{triangle.pdf 2.a through 2.d} shows some examples.
-
+Figure \ref{fig:product-regions} shows some examples.
+\begin{figure}[t]
+$$
+\mathfig{0.5}{triangle/triangle2}
+$$
+\begin{align*}
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [circle through = {(1,0)}] {$f$};
+\node (d) at (c.east) [circle through = {(0.25,0)}] {};
+\foreach \n in {1,2} {
+ \node (p\n) at (intersection \n of c and d) {};
+ \fill (p\n) circle (2pt);
+}
+\begin{scope}[decoration={brace,amplitude=10,aspect=0.5}]
+ \draw[decorate] (p2.east) -- node[right=2ex] {$a$} (p1.east);
+\end{scope}
+\end{tikzpicture} & =
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [circle through = {(1,0)}] {};
+\begin{scope}
+\path[clip] (c) circle (1);
+\node[draw,dashed] (d) at (c.east) [circle through = {(0.25,0)}] {};
+\foreach \n in {1,2} {
+ \node (p\n) at (intersection \n of c and d) {};
+}
+\node[left] at (c) {$f$};
+\path[clip] (d) circle (0.75);
+\foreach \y in {1,0.86,...,-1} {
+ \draw[green!50!brown] (0,\y)--(1,\y);
+}
+\end{scope}
+\draw[->,blue] (1.5,-1) node[below] {$a \times I$} -- (0.75,0);
+\end{tikzpicture} \\
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};
+\draw[dashed] (c.north) -- (c.south);
+\node[right=6] at (c) {$g$};
+\node[left=6] at (c) {$f$};
+\end{tikzpicture} & =
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};
+\node[right=9] at (c) {$g$};
+\node[left=9] at (c) {$f$};
+\draw[dashed] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north);
+\begin{scope}
+\path[clip] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north);
+\foreach \y in {1,0.86,...,-1} {
+ \draw[green!50!brown] (-1,\y)--(1,\y);
+}
+\end{scope}
+\draw[->,blue] (.75,-1.25) node[below] {$a \times I$} -- (0,-0.25);
+\end{tikzpicture} \\
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};
+\draw[dashed] (c.north) -- (c.south);
+\node[right=18] at (c) {$g$};
+\node[left=10] at (c) {$f$};
+\fill (0,0.4) node (p1) {} circle (2pt);
+\fill (0,-0.4) node (p2) {} circle (2pt);
+\begin{scope}[decoration={brace,amplitude=5,aspect=0.5}]
+ \draw[decorate] (p1.east) -- node[right=0.5ex] {\scriptsize $a$} (p2.east);
+\end{scope}
+\end{tikzpicture} & =
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};
+\node[draw,dashed] (d) at (0,0) [circle, minimum height=1cm,minimum width=1cm] {};
+\draw[dashed] (c.north) -- (d.north) (d.south) -- (c.south);
+\node[right=18] at (c) {$g$};
+\node[left=18] at (c) {$f$};
+\clip (0,0) circle (0.5cm);
+\foreach \y in {1,0.86,...,-1} {
+ \draw[green!50!brown] (-1,\y)--(1,\y);
+}
+\end{tikzpicture}
+\end{align*}
+\todo{fourth case}
+\caption{Examples of inserting or deleting product regions.}
+\label{fig:product-regions}
+\end{figure}
Let $a: y\to x$ be a 1-morphism.
@@ -658,6 +735,98 @@
$$
\mathfig{0.6}{triangle/triangle3c}
$$
+$$
+\begin{tikzpicture}
+\node (fg1) at (0,0) {
+\begin{tikzpicture}[baseline=-0.6cm]
+\path (0,0) coordinate (f1);
+\path (3,0) coordinate (f2);
+\path (3,-0.5) coordinate (g1);
+\path (6,-0.5) coordinate (g2);
+\node at (1.5,0.125) {$f$};
+\node at (4.5,-0.625) {$g$};
+\draw (f1) .. controls +(1,.8) and +(-1,.8) .. (f2);
+\draw[dashed] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2);
+\draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1);
+\draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2);
+\draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2);
+\draw (f2) .. controls +(1,.4) and +(-1,1) .. (g2);
+%
+\draw[blue,->] (-0.8,-1.2) node[below] {$(a \circ d) \times I$} -- (1,-0.5) ;
+\path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2)
+ .. controls +(1,.4) and +(-1,1) .. (g2)
+ .. controls +(-1,.4) and +(1,.4) .. (g1)
+ .. controls +(-1,-.4) and +(1,-1) .. (f1);
+\foreach \x in {0,0.1, ..., 6} {
+ \draw[green!50!brown] (\x,-2) -- + (0,4);
+}
+\end{tikzpicture}
+};
+\node (fg2) at (4,-4) {
+\begin{tikzpicture}[baseline=-0.1cm]
+\path (0,0) coordinate (f1);
+\path (3,0) coordinate (f2);
+\path (3,-0.5) coordinate (g1);
+\path (6,-0.5) coordinate (g2);
+\node at (1.5,0.125) {$f$};
+\node at (4.5,-0.625) {$g$};
+\draw[dashed] (f1) .. controls +(1,.8) and +(-1,.8) .. (f2);
+\draw[dashed] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2);
+\draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1);
+\draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2);
+\draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2);
+\draw[dashed] (f2) .. controls +(1,.4) and +(-1,1) .. (g2);
+\draw (f1) .. controls +(1,1.5) and +(-1,2)..(g2);
+%
+\begin{scope}
+\path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2)
+ .. controls +(1,.4) and +(-1,1) .. (g2)
+ .. controls +(-1,.4) and +(1,.4) .. (g1)
+ .. controls +(-1,-.4) and +(1,-1) .. (f1);
+\foreach \x in {0,0.1, ..., 6} {
+ \draw[green!50!brown] (\x,-2) -- + (0,4);
+}
+\end{scope}
+\begin{scope}
+\path[clip] (f1) .. controls +(1,1.5) and +(-1,2).. (g2)
+ .. controls +(-1,1) and +(1,.4) .. (f2)
+ .. controls +(-1,.8) and + (1,.8) .. (f1);
+\foreach \x in {0,0.1, ..., 6} {
+ \draw[green!50!brown] (\x,-2) -- + (0,4);
+}
+\end{scope}
+\end{tikzpicture}
+};
+\node (fg3) at (8,0) {
+\begin{tikzpicture}[baseline=-2.45cm]
+\path (0,0) coordinate (f1);
+\path (3,0) coordinate (f2);
+\path (3,0) coordinate (g1);
+\path (6,0) coordinate (g2);
+\node at (1.5,0) {$f$};
+\node at (4.5,0) {$g$};
+\draw[dashed] (f1) .. controls +(1,.8) and +(-1,.8) .. (f2);
+\draw (f1) .. controls +(1,-.8) and +(-1,-.8) .. (f2);
+\draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2);
+\draw[dashed] (g1) .. controls +(1,.8) and +(-1,.8) .. (g2);
+\draw (f1) .. controls +(1,1.5) and +(-1,1.5)..(g2);
+%
+\draw[blue,->] (4,1.75) node[above] {$(b \circ d) \times I$}-- + (0,-1);
+\begin{scope}
+\path[clip] (f1) .. controls +(1,1.5) and +(-1,1.5).. (g2)
+ .. controls +(-1,.8) and +(1,.8) .. (f2)
+ .. controls +(-1,.8) and + (1,.8) .. (f1);
+\foreach \x in {0,0.1, ..., 6} {
+ \draw[green!50!brown] (\x,-2) -- + (0,4);
+}
+\end{scope}
+\end{tikzpicture}
+};
+\draw[->] ($(fg1.south)+(0,0.5)$) -- node[left=0.5cm] {add $(b \circ d) \times I$} (fg2);
+\draw[->] (fg2) -- node[right=0.5cm] {remove $(a \circ d) \times I$} ($(fg3.south)+(0,1.75)$);
+\path (fg1) -- node {$=$} (fg3);
+\end{tikzpicture}
+$$
\caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.}
\label{fig:horizontal-compositions-equal}
\end{figure}
@@ -756,7 +925,8 @@
\end{figure}
\begin{figure}[t]
\begin{align*}
-\mathfig{0.4}{triangle/triangle4f}
+\mathfig{0.4}{triangle/triangle4f} \\
+\mathfig{0.4}{triangle/triangle4f_i}
\end{align*}
\caption{Vertical composition in the triangle axiom.}
\label{fig:vertical-composition}