blob: comparison text/appendixes/comparing

equal deleted inserted replaced

-:8372e04e4b7d
+:2232d94b90b8
 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 \ref{fig:product-regions} shows some examples.
 \begin{figure}[t]
-$$
+%$$
-\mathfig{0.5}{triangle/triangle2}
+%\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} {
 \end{align*}
 \caption{Horizontal composition of 2-morphisms}
 \label{fzo5}
 \end{figure}
 \begin{figure}[t]
-$$
+%$$
-\mathfig{0.6}{triangle/triangle3c}
+%\mathfig{0.6}{triangle/triangle3c}
-$$
+%$$
 $$
 \begin{tikzpicture}
 \node (fg1) at (0,0) {
 \begin{tikzpicture}[baseline=-0.6cm]
 \path (0,0) coordinate (f1);
 \draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1);
 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2);
 \draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2);
 \draw (f2) .. controls +(1,.4) and +(-1,1) .. (g2);
 %
-\draw[blue,->] (-0.8,-1.2) node[below] {$(a \circ d) \times I$} -- (1,-0.5) ;
+\draw[blue,->] (-0.8,-1.2) node[below] {$(a \bullet d) \times I$} -- (1,-0.5) ;
 \path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2)
 .. controls +(1,.4) and +(-1,1) .. (g2)
 .. controls +(-1,.4) and +(1,.4) .. (g1)
 .. controls +(-1,-.4) and +(1,-1) .. (f1);
 \foreach \x in {0,0.1, ..., 6} {
 \draw (f1) .. controls +(1,-.8) and +(-1,-.8) .. (f2);
 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2);
 \draw[dashed] (g1) .. controls +(1,.8) and +(-1,.8) .. (g2);
 \draw (f1) .. controls +(1,1.5) and +(-1,1.5)..(g2);
 %
-\draw[blue,->] (4,1.75) node[above] {$(b \circ d) \times I$}-- + (0,-1);
+\draw[blue,->] (4,1.75) node[above] {$(b \bullet d) \times I$}-- + (0,-1);
 \begin{scope}
 \path[clip] (f1) ..  controls +(1,1.5) and +(-1,1.5).. (g2)
 		      .. controls +(-1,.8) and +(1,.8) .. (f2)
 		      .. controls +(-1,.8) and + (1,.8) .. (f1);
 \foreach \x in {0,0.1, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- + (0,4);
 }
 \end{scope}
 \end{tikzpicture}
 };
-\draw[->] ($(fg1.south)+(0,0.5)$) -- node[left=0.5cm] {add $(b \circ d) \times I$} (fg2);
+\draw[->] ($(fg1.south)+(0,0.5)$) -- node[left=0.5cm] {add $(b \bullet d) \times I$} (fg2);
-\draw[->] (fg2) -- node[right=0.5cm] {remove $(a \circ d) \times I$} ($(fg3.south)+(0,1.75)$);
+\draw[->] (fg2) -- node[right=0.5cm] {remove $(a \bullet d) \times I$} ($(fg3.south)+(0,1.75)$);
 \path (fg1) -- node {$=$} (fg3);
 \end{tikzpicture}
 $$
 \caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.}
 \label{fig:horizontal-compositions-equal}
 Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$
 as in Figure \ref{fig:associator}.
 This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$.
 \begin{figure}[t]
-$$
+%$$
-\mathfig{0.4}{triangle/triangle4a}
+%\mathfig{0.4}{triangle/triangle4a}
-$$
+%$$
 $$
 \begin{tikzpicture}
 \node[circle,fill=black,inner sep=1pt] at (1.73,0) {};
 \node[circle,fill=black,inner sep=1pt] at (-1.73,0) {};
 \begin{scope}[yshift=-1cm]
 (See also Figures \ref{fzo2} and \ref{fig:associator}.)
 We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$
 is equal to the composition of $\alpha$ and $\id_a\bullet v$.
 (Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.)
 \begin{figure}[t]
-\begin{align*}
+%\begin{align*}
-\mathfig{0.4}{triangle/triangle4a} \\
+%\mathfig{0.4}{triangle/triangle4a} \\
-\mathfig{0.4}{triangle/triangle4b} \\
+%\mathfig{0.4}{triangle/triangle4b} \\
-\mathfig{0.4}{triangle/triangle4c}
+%\mathfig{0.4}{triangle/triangle4c}
-\end{align*}
+%\end{align*}
 \begin{align*}
 \alpha & =
 \begin{tikzpicture}[baseline]
 \node[circle,fill=black,inner sep=1pt] at (1.73,0) {};
 \node[circle,fill=black,inner sep=1pt] at (-1.73,0) {};
 	\path (R) to[out=0,in=-140] node[pos=\x/10] (RQ\x) {} (Q);
 	\draw[brown] (PQ\x.center) -- (RQ\x.center);
 }
 \end{tikzpicture} \\
 \end{align*}
-\nn{remember to change `assoc' to $\alpha$}
 \caption{Ingredients for the triangle axiom.}
 \label{fig:ingredients-triangle-axiom}
 \end{figure}
 The horizontal compositions $u\bullet\id_b$ and $\id_a\bullet v$ are shown in Figure \ref{fig:horizontal-composition}
 Figure \ref{fig:adding-a-collar} shows that we can add a collar to $u\bullet\id_b$ so that the result differs from
 Figure  \ref{fig:vertical-composition} by an isotopy rel boundary.
 Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3)
 as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1).
 \begin{figure}[t]
+%\begin{align*}
+%\mathfig{0.4}{triangle/triangle4d}
+%\mathfig{0.4}{triangle/triangle4e} \\
+%\end{align*}
 \begin{align*}
-\mathfig{0.4}{triangle/triangle4d}
+u *_h (b \times I) & =
-\mathfig{0.4}{triangle/triangle4e} \\
-\end{align*}
-\begin{align*}
-u \bullet (b \times I) & =
 \begin{tikzpicture}[baseline]
 \coordinate (L) at (0,0);
 \coordinate (R) at (3,0);
 \coordinate (T) at (1.5,0.8);
 \coordinate (M) at (1.5,-0.4);
 	\path (M) to[out=-20,in=-150] node[coordinate,pos=\n/8] (MR\n) {} (R);
 	\path (T) to[out=0,in=140] node[coordinate,pos=\n/8] (TR\n) {} (R);
 	\draw[brown] (MR\n) -- (TR\n);
 }
 \end{tikzpicture} \\
-(a \times I) \bullet v & =
+(a \times I) *_h v & =
 \begin{tikzpicture}[baseline]
 \coordinate (L) at (0,0);
 \coordinate (R) at (3,0);
 \coordinate (T) at (1.5,0.8);
 \coordinate (M) at (1.5,-0.4);
 \caption{Horizontal compositions in the triangle axiom.}
 \label{fig:horizontal-composition}
 \end{figure}
 \begin{figure}[t]
 \begin{align*}
-\mathfig{0.4}{triangle/triangle4f} \\
+%\mathfig{0.4}{triangle/triangle4f} \\
 \begin{tikzpicture}
 \node[circle,fill=black,inner sep=1pt] (A) at (1.73,0) {};
 \node[circle,fill=black,inner sep=1pt] (B) at (-1.73,0) {};
 \draw[dashed] (A) -- (B);
 \node[circle,fill=black,inner sep=1pt] (C) at (0,0) {};
 \end{align*}
 \caption{Vertical composition in the triangle axiom.}
 \label{fig:vertical-composition}
 \end{figure}
 \begin{figure}[t]
-\begin{align*}
+%\begin{align*}
-\mathfig{0.4}{triangle/triangle5}
+%\mathfig{0.4}{triangle/triangle5}
-\end{align*}
+%\end{align*}
 \begin{align*}
 \begin{tikzpicture}[baseline]
 \coordinate (L) at (0,0);
 \coordinate (R) at (3,0);
 \coordinate (T) at (1.5,0.8);

changeset 969	2232d94b90b8
parent 967	6e4f0ed47e0e
child 970	7f47bf84b0f1