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 \ref{fig:product-regions} shows some examples. |
218 Figure \ref{fig:product-regions} shows some examples. |
219 \begin{figure}[t] |
219 \begin{figure}[t] |
220 $$ |
220 %$$ |
221 \mathfig{0.5}{triangle/triangle2} |
221 %\mathfig{0.5}{triangle/triangle2} |
222 $$ |
222 %$$ |
223 \begin{align*} |
223 \begin{align*} |
224 \begin{tikzpicture}[baseline] |
224 \begin{tikzpicture}[baseline] |
225 \node[draw] (c) at (0,0) [circle through = {(1,0)}] {$f$}; |
225 \node[draw] (c) at (0,0) [circle through = {(1,0)}] {$f$}; |
226 \node (d) at (c.east) [circle through = {(0.25,0)}] {}; |
226 \node (d) at (c.east) [circle through = {(0.25,0)}] {}; |
227 \foreach \n in {1,2} { |
227 \foreach \n in {1,2} { |
761 \end{align*} |
761 \end{align*} |
762 \caption{Horizontal composition of 2-morphisms} |
762 \caption{Horizontal composition of 2-morphisms} |
763 \label{fzo5} |
763 \label{fzo5} |
764 \end{figure} |
764 \end{figure} |
765 \begin{figure}[t] |
765 \begin{figure}[t] |
766 $$ |
766 %$$ |
767 \mathfig{0.6}{triangle/triangle3c} |
767 %\mathfig{0.6}{triangle/triangle3c} |
768 $$ |
768 %$$ |
769 $$ |
769 $$ |
770 \begin{tikzpicture} |
770 \begin{tikzpicture} |
771 \node (fg1) at (0,0) { |
771 \node (fg1) at (0,0) { |
772 \begin{tikzpicture}[baseline=-0.6cm] |
772 \begin{tikzpicture}[baseline=-0.6cm] |
773 \path (0,0) coordinate (f1); |
773 \path (0,0) coordinate (f1); |
781 \draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1); |
781 \draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1); |
782 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
782 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
783 \draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2); |
783 \draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2); |
784 \draw (f2) .. controls +(1,.4) and +(-1,1) .. (g2); |
784 \draw (f2) .. controls +(1,.4) and +(-1,1) .. (g2); |
785 % |
785 % |
786 \draw[blue,->] (-0.8,-1.2) node[below] {$(a \circ d) \times I$} -- (1,-0.5) ; |
786 \draw[blue,->] (-0.8,-1.2) node[below] {$(a \bullet d) \times I$} -- (1,-0.5) ; |
787 \path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2) |
787 \path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2) |
788 .. controls +(1,.4) and +(-1,1) .. (g2) |
788 .. controls +(1,.4) and +(-1,1) .. (g2) |
789 .. controls +(-1,.4) and +(1,.4) .. (g1) |
789 .. controls +(-1,.4) and +(1,.4) .. (g1) |
790 .. controls +(-1,-.4) and +(1,-1) .. (f1); |
790 .. controls +(-1,-.4) and +(1,-1) .. (f1); |
791 \foreach \x in {0,0.1, ..., 6} { |
791 \foreach \x in {0,0.1, ..., 6} { |
840 \draw (f1) .. controls +(1,-.8) and +(-1,-.8) .. (f2); |
840 \draw (f1) .. controls +(1,-.8) and +(-1,-.8) .. (f2); |
841 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
841 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
842 \draw[dashed] (g1) .. controls +(1,.8) and +(-1,.8) .. (g2); |
842 \draw[dashed] (g1) .. controls +(1,.8) and +(-1,.8) .. (g2); |
843 \draw (f1) .. controls +(1,1.5) and +(-1,1.5)..(g2); |
843 \draw (f1) .. controls +(1,1.5) and +(-1,1.5)..(g2); |
844 % |
844 % |
845 \draw[blue,->] (4,1.75) node[above] {$(b \circ d) \times I$}-- + (0,-1); |
845 \draw[blue,->] (4,1.75) node[above] {$(b \bullet d) \times I$}-- + (0,-1); |
846 \begin{scope} |
846 \begin{scope} |
847 \path[clip] (f1) .. controls +(1,1.5) and +(-1,1.5).. (g2) |
847 \path[clip] (f1) .. controls +(1,1.5) and +(-1,1.5).. (g2) |
848 .. controls +(-1,.8) and +(1,.8) .. (f2) |
848 .. controls +(-1,.8) and +(1,.8) .. (f2) |
849 .. controls +(-1,.8) and + (1,.8) .. (f1); |
849 .. controls +(-1,.8) and + (1,.8) .. (f1); |
850 \foreach \x in {0,0.1, ..., 6} { |
850 \foreach \x in {0,0.1, ..., 6} { |
851 \draw[green!50!brown] (\x,-2) -- + (0,4); |
851 \draw[green!50!brown] (\x,-2) -- + (0,4); |
852 } |
852 } |
853 \end{scope} |
853 \end{scope} |
854 \end{tikzpicture} |
854 \end{tikzpicture} |
855 }; |
855 }; |
856 \draw[->] ($(fg1.south)+(0,0.5)$) -- node[left=0.5cm] {add $(b \circ d) \times I$} (fg2); |
856 \draw[->] ($(fg1.south)+(0,0.5)$) -- node[left=0.5cm] {add $(b \bullet d) \times I$} (fg2); |
857 \draw[->] (fg2) -- node[right=0.5cm] {remove $(a \circ d) \times I$} ($(fg3.south)+(0,1.75)$); |
857 \draw[->] (fg2) -- node[right=0.5cm] {remove $(a \bullet d) \times I$} ($(fg3.south)+(0,1.75)$); |
858 \path (fg1) -- node {$=$} (fg3); |
858 \path (fg1) -- node {$=$} (fg3); |
859 \end{tikzpicture} |
859 \end{tikzpicture} |
860 $$ |
860 $$ |
861 \caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.} |
861 \caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.} |
862 \label{fig:horizontal-compositions-equal} |
862 \label{fig:horizontal-compositions-equal} |
864 |
864 |
865 Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ |
865 Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ |
866 as in Figure \ref{fig:associator}. |
866 as in Figure \ref{fig:associator}. |
867 This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$. |
867 This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$. |
868 \begin{figure}[t] |
868 \begin{figure}[t] |
869 $$ |
869 %$$ |
870 \mathfig{0.4}{triangle/triangle4a} |
870 %\mathfig{0.4}{triangle/triangle4a} |
871 $$ |
871 %$$ |
872 $$ |
872 $$ |
873 \begin{tikzpicture} |
873 \begin{tikzpicture} |
874 \node[circle,fill=black,inner sep=1pt] at (1.73,0) {}; |
874 \node[circle,fill=black,inner sep=1pt] at (1.73,0) {}; |
875 \node[circle,fill=black,inner sep=1pt] at (-1.73,0) {}; |
875 \node[circle,fill=black,inner sep=1pt] at (-1.73,0) {}; |
876 \begin{scope}[yshift=-1cm] |
876 \begin{scope}[yshift=-1cm] |
927 (See also Figures \ref{fzo2} and \ref{fig:associator}.) |
927 (See also Figures \ref{fzo2} and \ref{fig:associator}.) |
928 We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$ |
928 We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$ |
929 is equal to the composition of $\alpha$ and $\id_a\bullet v$. |
929 is equal to the composition of $\alpha$ and $\id_a\bullet v$. |
930 (Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.) |
930 (Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.) |
931 \begin{figure}[t] |
931 \begin{figure}[t] |
932 \begin{align*} |
932 %\begin{align*} |
933 \mathfig{0.4}{triangle/triangle4a} \\ |
933 %\mathfig{0.4}{triangle/triangle4a} \\ |
934 \mathfig{0.4}{triangle/triangle4b} \\ |
934 %\mathfig{0.4}{triangle/triangle4b} \\ |
935 \mathfig{0.4}{triangle/triangle4c} |
935 %\mathfig{0.4}{triangle/triangle4c} |
936 \end{align*} |
936 %\end{align*} |
937 \begin{align*} |
937 \begin{align*} |
938 \alpha & = |
938 \alpha & = |
939 \begin{tikzpicture}[baseline] |
939 \begin{tikzpicture}[baseline] |
940 \node[circle,fill=black,inner sep=1pt] at (1.73,0) {}; |
940 \node[circle,fill=black,inner sep=1pt] at (1.73,0) {}; |
941 \node[circle,fill=black,inner sep=1pt] at (-1.73,0) {}; |
941 \node[circle,fill=black,inner sep=1pt] at (-1.73,0) {}; |
1010 \path (R) to[out=0,in=-140] node[pos=\x/10] (RQ\x) {} (Q); |
1010 \path (R) to[out=0,in=-140] node[pos=\x/10] (RQ\x) {} (Q); |
1011 \draw[brown] (PQ\x.center) -- (RQ\x.center); |
1011 \draw[brown] (PQ\x.center) -- (RQ\x.center); |
1012 } |
1012 } |
1013 \end{tikzpicture} \\ |
1013 \end{tikzpicture} \\ |
1014 \end{align*} |
1014 \end{align*} |
1015 \nn{remember to change `assoc' to $\alpha$} |
|
1016 \caption{Ingredients for the triangle axiom.} |
1015 \caption{Ingredients for the triangle axiom.} |
1017 \label{fig:ingredients-triangle-axiom} |
1016 \label{fig:ingredients-triangle-axiom} |
1018 \end{figure} |
1017 \end{figure} |
1019 |
1018 |
1020 The horizontal compositions $u\bullet\id_b$ and $\id_a\bullet v$ are shown in Figure \ref{fig:horizontal-composition} |
1019 The horizontal compositions $u\bullet\id_b$ and $\id_a\bullet v$ are shown in Figure \ref{fig:horizontal-composition} |
1023 Figure \ref{fig:adding-a-collar} shows that we can add a collar to $u\bullet\id_b$ so that the result differs from |
1022 Figure \ref{fig:adding-a-collar} shows that we can add a collar to $u\bullet\id_b$ so that the result differs from |
1024 Figure \ref{fig:vertical-composition} by an isotopy rel boundary. |
1023 Figure \ref{fig:vertical-composition} by an isotopy rel boundary. |
1025 Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) |
1024 Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) |
1026 as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). |
1025 as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). |
1027 \begin{figure}[t] |
1026 \begin{figure}[t] |
|
1027 %\begin{align*} |
|
1028 %\mathfig{0.4}{triangle/triangle4d} |
|
1029 %\mathfig{0.4}{triangle/triangle4e} \\ |
|
1030 %\end{align*} |
1028 \begin{align*} |
1031 \begin{align*} |
1029 \mathfig{0.4}{triangle/triangle4d} |
1032 u *_h (b \times I) & = |
1030 \mathfig{0.4}{triangle/triangle4e} \\ |
|
1031 \end{align*} |
|
1032 \begin{align*} |
|
1033 u \bullet (b \times I) & = |
|
1034 \begin{tikzpicture}[baseline] |
1033 \begin{tikzpicture}[baseline] |
1035 \coordinate (L) at (0,0); |
1034 \coordinate (L) at (0,0); |
1036 \coordinate (R) at (3,0); |
1035 \coordinate (R) at (3,0); |
1037 \coordinate (T) at (1.5,0.8); |
1036 \coordinate (T) at (1.5,0.8); |
1038 \coordinate (M) at (1.5,-0.4); |
1037 \coordinate (M) at (1.5,-0.4); |
1056 \path (M) to[out=-20,in=-150] node[coordinate,pos=\n/8] (MR\n) {} (R); |
1055 \path (M) to[out=-20,in=-150] node[coordinate,pos=\n/8] (MR\n) {} (R); |
1057 \path (T) to[out=0,in=140] node[coordinate,pos=\n/8] (TR\n) {} (R); |
1056 \path (T) to[out=0,in=140] node[coordinate,pos=\n/8] (TR\n) {} (R); |
1058 \draw[brown] (MR\n) -- (TR\n); |
1057 \draw[brown] (MR\n) -- (TR\n); |
1059 } |
1058 } |
1060 \end{tikzpicture} \\ |
1059 \end{tikzpicture} \\ |
1061 (a \times I) \bullet v & = |
1060 (a \times I) *_h v & = |
1062 \begin{tikzpicture}[baseline] |
1061 \begin{tikzpicture}[baseline] |
1063 \coordinate (L) at (0,0); |
1062 \coordinate (L) at (0,0); |
1064 \coordinate (R) at (3,0); |
1063 \coordinate (R) at (3,0); |
1065 \coordinate (T) at (1.5,0.8); |
1064 \coordinate (T) at (1.5,0.8); |
1066 \coordinate (M) at (1.5,-0.4); |
1065 \coordinate (M) at (1.5,-0.4); |
1090 \caption{Horizontal compositions in the triangle axiom.} |
1089 \caption{Horizontal compositions in the triangle axiom.} |
1091 \label{fig:horizontal-composition} |
1090 \label{fig:horizontal-composition} |
1092 \end{figure} |
1091 \end{figure} |
1093 \begin{figure}[t] |
1092 \begin{figure}[t] |
1094 \begin{align*} |
1093 \begin{align*} |
1095 \mathfig{0.4}{triangle/triangle4f} \\ |
1094 %\mathfig{0.4}{triangle/triangle4f} \\ |
1096 \begin{tikzpicture} |
1095 \begin{tikzpicture} |
1097 \node[circle,fill=black,inner sep=1pt] (A) at (1.73,0) {}; |
1096 \node[circle,fill=black,inner sep=1pt] (A) at (1.73,0) {}; |
1098 \node[circle,fill=black,inner sep=1pt] (B) at (-1.73,0) {}; |
1097 \node[circle,fill=black,inner sep=1pt] (B) at (-1.73,0) {}; |
1099 \draw[dashed] (A) -- (B); |
1098 \draw[dashed] (A) -- (B); |
1100 \node[circle,fill=black,inner sep=1pt] (C) at (0,0) {}; |
1099 \node[circle,fill=black,inner sep=1pt] (C) at (0,0) {}; |
1145 \end{align*} |
1144 \end{align*} |
1146 \caption{Vertical composition in the triangle axiom.} |
1145 \caption{Vertical composition in the triangle axiom.} |
1147 \label{fig:vertical-composition} |
1146 \label{fig:vertical-composition} |
1148 \end{figure} |
1147 \end{figure} |
1149 \begin{figure}[t] |
1148 \begin{figure}[t] |
1150 \begin{align*} |
1149 %\begin{align*} |
1151 \mathfig{0.4}{triangle/triangle5} |
1150 %\mathfig{0.4}{triangle/triangle5} |
1152 \end{align*} |
1151 %\end{align*} |
1153 \begin{align*} |
1152 \begin{align*} |
1154 \begin{tikzpicture}[baseline] |
1153 \begin{tikzpicture}[baseline] |
1155 \coordinate (L) at (0,0); |
1154 \coordinate (L) at (0,0); |
1156 \coordinate (R) at (3,0); |
1155 \coordinate (R) at (3,0); |
1157 \coordinate (T) at (1.5,0.8); |
1156 \coordinate (T) at (1.5,0.8); |