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text/appendixes/comparing_defs.tex
changeset 955 65ef3b339d0a
parent 948 727cefa97b8e
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   213 
   213 
   214 In showing that identity 1-morphisms have the desired properties, we will
 
   214 In showing that identity 1-morphisms have the desired properties, we will
 
   215 rely heavily on the extended isotopy invariance of 2-morphisms in $\cC$.
 
   215 rely heavily on the extended isotopy invariance of 2-morphisms in $\cC$.
 
   216 Extended isotopy invariance implies that adding a product collar to a 2-morphism of $\cC$ has no effect,
 
   216 Extended isotopy invariance implies that adding a product collar to a 2-morphism of $\cC$ has no effect,
 
   217 and by cutting and regluing we can insert (or delete) product regions in the interior of 2-morphisms as well.
 
   217 and by cutting and regluing we can insert (or delete) product regions in the interior of 2-morphisms as well.
 
   218 Figure \nn{triangle.pdf 2.a through 2.d} shows some examples.
 
   218 Figure \ref{fig:product-regions} shows some examples.
 
   219 
   219 \begin{figure}[t]
 
       
   220 $$
 
       
   221 \mathfig{0.5}{triangle/triangle2}
 
       
   222 $$
 
       
   223 \caption{Examples of inserting or deleting product regions.}
 
       
   224 \label{fig:product-regions}
 
       
   225 \end{figure}
 
   220 
   226 
   221 
   227 
   222 Let $a: y\to x$ be a 1-morphism.
 
   228 Let $a: y\to x$ be a 1-morphism.
 
   223 Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$
 
   229 Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$
 
   224 as shown in Figure \ref{fzo2}.
 
   230 as shown in Figure \ref{fzo2}.
 
   655 \label{fzo5}
 
   661 \label{fzo5}
 
   656 \end{figure}
 
   662 \end{figure}
 
   657 \begin{figure}[t]
 
   663 \begin{figure}[t]
 
   658 $$
 
   664 $$
 
   659 \mathfig{0.6}{triangle/triangle3c}
 
   665 \mathfig{0.6}{triangle/triangle3c}
 
       
   666 $$
 
       
   667 $$
 
       
   668 \begin{tikzpicture}
 
       
   669 \node (fg1) at (0,0) {
 
       
   670 \begin{tikzpicture}[baseline=-0.6cm]
 
       
   671 \path (0,0) coordinate (f1);
 
       
   672 \path (3,0) coordinate (f2);
 
       
   673 \path (3,-0.5) coordinate (g1);
 
       
   674 \path (6,-0.5) coordinate (g2);
 
       
   675 \node at (1.5,0.125) {$f$};
 
       
   676 \node at (4.5,-0.625) {$g$};
 
       
   677 \draw (f1) .. controls +(1,.8) and +(-1,.8) .. (f2);
 
       
   678 \draw[dashed] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2);
 
       
   679 \draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1);
 
       
   680 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2);
 
       
   681 \draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2);
 
       
   682 \draw (f2) .. controls +(1,.4) and +(-1,1) .. (g2);
 
       
   683 %
 
       
   684 \draw[blue,->] (-0.8,-1.2) node[below] {$(a \circ d) \times I$} -- (1,-0.5) ;
 
       
   685 \path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2)
 
       
   686                     .. controls +(1,.4) and +(-1,1) .. (g2)
 
       
   687                     .. controls +(-1,.4) and +(1,.4) .. (g1)
 
       
   688                     .. controls +(-1,-.4) and +(1,-1) .. (f1);
 
       
   689 \foreach \x in {0,0.1, ..., 6} {
 
       
   690 	\draw[green!50!brown] (\x,-2) -- + (0,4);
 
       
   691 }
 
       
   692 \end{tikzpicture}
 
       
   693 };
 
       
   694 \node (fg2) at (4,-4) {
 
       
   695 \begin{tikzpicture}[baseline=-0.1cm]
 
       
   696 \path (0,0) coordinate (f1);
 
       
   697 \path (3,0) coordinate (f2);
 
       
   698 \path (3,-0.5) coordinate (g1);
 
       
   699 \path (6,-0.5) coordinate (g2);
 
       
   700 \node at (1.5,0.125) {$f$};
 
       
   701 \node at (4.5,-0.625) {$g$};
 
       
   702 \draw[dashed] (f1) .. controls +(1,.8) and +(-1,.8) .. (f2);
 
       
   703 \draw[dashed] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2);
 
       
   704 \draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1);
 
       
   705 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2);
 
       
   706 \draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2);
 
       
   707 \draw[dashed] (f2) .. controls +(1,.4) and +(-1,1) .. (g2);
 
       
   708 \draw (f1) .. controls +(1,1.5) and +(-1,2)..(g2);
 
       
   709 %
 
       
   710 \begin{scope}
 
       
   711 \path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2)
 
       
   712                     .. controls +(1,.4) and +(-1,1) .. (g2)
 
       
   713                     .. controls +(-1,.4) and +(1,.4) .. (g1)
 
       
   714                     .. controls +(-1,-.4) and +(1,-1) .. (f1);
 
       
   715 \foreach \x in {0,0.1, ..., 6} {
 
       
   716 	\draw[green!50!brown] (\x,-2) -- + (0,4);
 
       
   717 }
 
       
   718 \end{scope}
 
       
   719 \begin{scope}
 
       
   720 \path[clip] (f1) ..  controls +(1,1.5) and +(-1,2).. (g2)
 
       
   721 		      .. controls +(-1,1) and +(1,.4) .. (f2)
 
       
   722 		      .. controls +(-1,.8) and + (1,.8) .. (f1);
 
       
   723 \foreach \x in {0,0.1, ..., 6} {
 
       
   724 	\draw[green!50!brown] (\x,-2) -- + (0,4);
 
       
   725 }
 
       
   726 \end{scope}
 
       
   727 \end{tikzpicture}
 
       
   728 };
 
       
   729 \node (fg3) at (8,0) {
 
       
   730 \begin{tikzpicture}[baseline=-2.45cm]
 
       
   731 \path (0,0) coordinate (f1);
 
       
   732 \path (3,0) coordinate (f2);
 
       
   733 \path (3,0) coordinate (g1);
 
       
   734 \path (6,0) coordinate (g2);
 
       
   735 \node at (1.5,0) {$f$};
 
       
   736 \node at (4.5,0) {$g$};
 
       
   737 \draw[dashed] (f1) .. controls +(1,.8) and +(-1,.8) .. (f2);
 
       
   738 \draw (f1) .. controls +(1,-.8) and +(-1,-.8) .. (f2);
 
       
   739 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2);
 
       
   740 \draw[dashed] (g1) .. controls +(1,.8) and +(-1,.8) .. (g2);
 
       
   741 \draw (f1) .. controls +(1,1.5) and +(-1,1.5)..(g2);
 
       
   742 %
 
       
   743 \draw[blue,->] (4,1.75) node[above] {$(b \circ d) \times I$}-- + (0,-1);
 
       
   744 \begin{scope}
 
       
   745 \path[clip] (f1) ..  controls +(1,1.5) and +(-1,1.5).. (g2)
 
       
   746 		      .. controls +(-1,.8) and +(1,.8) .. (f2)
 
       
   747 		      .. controls +(-1,.8) and + (1,.8) .. (f1);
 
       
   748 \foreach \x in {0,0.1, ..., 6} {
 
       
   749 	\draw[green!50!brown] (\x,-2) -- + (0,4);
 
       
   750 }
 
       
   751 \end{scope}
 
       
   752 \end{tikzpicture}
 
       
   753 };
 
       
   754 \draw[->] ($(fg1.south)+(0,0.5)$) -- node[left=0.5cm] {add $(b \circ d) \times I$} (fg2);
 
       
   755 \draw[->] (fg2) -- node[right=0.5cm] {remove $(a \circ d) \times I$} ($(fg3.south)+(0,1.75)$);
 
       
   756 \path (fg1) -- node {$=$} (fg3);
 
       
   757 \end{tikzpicture}
 
   660 $$
 
   758 $$
 
   661 \caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.}
 
   759 \caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.}
 
   662 \label{fig:horizontal-compositions-equal}
 
   760 \label{fig:horizontal-compositions-equal}
 
   663 \end{figure}
 
   761 \end{figure}
 
   664 
   762 
   754 \caption{Horizontal compositions in the triangle axiom.}
 
   852 \caption{Horizontal compositions in the triangle axiom.}
 
   755 \label{fig:horizontal-composition}
 
   853 \label{fig:horizontal-composition}
 
   756 \end{figure}
 
   854 \end{figure}
 
   757 \begin{figure}[t]
 
   855 \begin{figure}[t]
 
   758 \begin{align*}
 
   856 \begin{align*}
 
   759 \mathfig{0.4}{triangle/triangle4f}
 
   857 \mathfig{0.4}{triangle/triangle4f} \\
 
       
   858 \mathfig{0.4}{triangle/triangle4f_i}
 
   760 \end{align*}
 
   859 \end{align*}
 
   761 \caption{Vertical composition in the triangle axiom.}
 
   860 \caption{Vertical composition in the triangle axiom.}
 
   762 \label{fig:vertical-composition}
 
   861 \label{fig:vertical-composition}
 
   763 \end{figure}
 
   862 \end{figure}
 
   764 \begin{figure}[t]
 
   863 \begin{figure}[t]
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