898 \draw[dashed] (L1) -- (U1); |
898 \draw[dashed] (L1) -- (U1); |
899 \draw[dashed] (L2) -- (U2); |
899 \draw[dashed] (L2) -- (U2); |
900 \begin{scope} |
900 \begin{scope} |
901 \path[clip] (0,1) circle (2); |
901 \path[clip] (0,1) circle (2); |
902 \path[clip] (0,-1) circle (2); |
902 \path[clip] (0,-1) circle (2); |
903 \foreach \t in {-2.5,-2.4,...,-1.2} { |
903 \foreach \t in {-2.5,-2.3,...,-1.2} { |
904 \draw[green!50!brown] (\t,-1) -- +(1.19,2); |
904 \draw[green!50!brown] (\t,-1) -- +(1.19,2); |
905 } |
905 } |
906 \foreach \t in {-1.1,-1.0,...,0} { |
906 \foreach \t in {-1.1,-0.9,...,0} { |
907 \draw[blue!50!brown] (\t,-1) -- +(1.19,2); |
907 \draw[blue!50!brown] (\t,-1) -- +(1.19,2); |
908 } |
908 } |
909 \foreach \t in {0.1,0.2,...,2.5} { |
909 \foreach \t in {0.1,0.3,...,2.5} { |
910 \draw[red!50!brown] (\t,-1) -- +(1.19,2); |
910 \draw[red!50!brown] (\t,-1) -- +(1.19,2); |
911 } |
911 } |
912 \end{scope} |
912 \end{scope} |
913 \draw[->, thick, blue!75!yellow] (-2,0.5) node[black, left] {$a\times I$} -- (-1,0.5); |
913 \draw[->, thick, blue!75!yellow] (-2,0.5) node[black, left] {$a\times I$} -- (-1,0.5); |
914 \draw[->, thick, blue!75!yellow] (-0.25,-1.5) node[black, below] {$b\times I$} -- (-0.25,-0.5); |
914 \draw[->, thick, blue!75!yellow] (-0.25,-1.5) node[black, below] {$b\times I$} -- (-0.25,-0.5); |
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} \\ |
|
936 \alpha & = |
|
937 \begin{tikzpicture}[baseline] |
|
938 \node[circle,fill=black,inner sep=1pt] at (1.73,0) {}; |
|
939 \node[circle,fill=black,inner sep=1pt] at (-1.73,0) {}; |
|
940 \begin{scope}[yshift=-1cm] |
|
941 \path[clip] (0,0) circle (2); |
|
942 \begin{scope}[yshift=2cm] |
|
943 \draw (0,0) circle (2); |
|
944 \node[circle,fill=black,inner sep=1pt] (L2) at (-90:2) {}; |
|
945 \node[circle,fill=black,inner sep=1pt] (L1) at (-120:2) {}; |
|
946 \node at (-60:2.25) {$b$}; |
|
947 \node at (-105:2.25) {\scriptsize $y \times I$}; |
|
948 \node at (-135:2.25) {$a$}; |
|
949 \end{scope} |
|
950 \end{scope} |
|
951 \begin{scope}[yshift=1cm] |
|
952 \path[clip] (0,0) circle (2); |
|
953 \begin{scope}[yshift=-2cm] |
|
954 \node at (120:2.25) {$a$}; |
|
955 \node at (75:2.25) {\scriptsize $y \times I$}; |
|
956 \node at (45:2.25) {$b$}; |
|
957 \draw (0,0) circle (2); |
|
958 \node[circle,fill=black,inner sep=1pt] (U1) at (90:2) {}; |
|
959 \node[circle,fill=black,inner sep=1pt] (U2) at (60:2) {}; |
|
960 \end{scope} |
|
961 \end{scope} |
|
962 \draw[dashed] (L1) -- (U1); |
|
963 \draw[dashed] (L2) -- (U2); |
|
964 \begin{scope} |
|
965 \path[clip] (0,1) circle (2); |
|
966 \path[clip] (0,-1) circle (2); |
|
967 \foreach \t in {-2.5,-2.3,...,-1.2} { |
|
968 \draw[green!50!brown] (\t,-1) -- +(1.19,2); |
|
969 } |
|
970 \foreach \t in {-1.1,-0.9,...,0} { |
|
971 \draw[blue!50!brown] (\t,-1) -- +(1.19,2); |
|
972 } |
|
973 \foreach \t in {0.1,0.3,...,2.5} { |
|
974 \draw[red!50!brown] (\t,-1) -- +(1.19,2); |
|
975 } |
|
976 \foreach \x in {0,1,...,10} { |
|
977 \path (L1) to[out=60,in=-120] node[pos=\x/10] (p1\x) {}(U1); |
|
978 \path (L2) to[out=60,in=-120] node[pos=\x/10] (p2\x) {} (U2); |
|
979 \draw[blue!50!brown] (p1\x.center) -- (p2\x.center); |
|
980 } |
|
981 \end{scope} |
|
982 \draw[->, thick, blue!75!yellow] (-2,0.5) node[black, left] {$a\times I$} -- (-1,0.5); |
|
983 \draw[->, thick, blue!75!yellow] (-0.1,-1.5) node[black, below] {$y\times I \times I$} -- (-0.25,-0.5); |
|
984 \draw[->, thick, blue!75!yellow] (2,-0.5) node[black, right] {$b\times I$} -- (1,-0.5); |
|
985 \end{tikzpicture} \\ |
935 u & = |
986 u & = |
936 \mathfig{0.4}{triangle/triangle4c} |
987 \begin{tikzpicture}[baseline] |
|
988 \coordinate (P) at (0,0); |
|
989 \coordinate (Q) at (3,0); |
|
990 \coordinate (R) at (1.5,-0.5); |
|
991 \draw (P) \vertex to[out=40,in=140] node[above] {$a$} (Q) \vertex -- node[below] {\scriptsize $y \times I$} (R) \vertex to[out=-180,in=-40] node[below] {$a$} (P); |
|
992 \clip (P) to[out=40,in=140] (Q) -- (R) to[out=-180,in=-40] (P); |
|
993 \foreach \x in {1,...,9} { |
|
994 \path (P) to[out=40,in=140] node[pos=\x/10] (PQ\x) {} (Q); |
|
995 \path (P) to[out=-40,in=-180] node[pos=\x/10] (PR\x) {} (R); |
|
996 \draw[green!50!brown] (PQ\x.center) -- (PR\x.center); |
|
997 } |
|
998 \end{tikzpicture} \\ |
|
999 v & = |
|
1000 \begin{tikzpicture}[baseline] |
|
1001 \coordinate (P) at (0,0); |
|
1002 \coordinate (Q) at (3,0); |
|
1003 \coordinate (R) at (1.5,-0.5); |
|
1004 \draw (P) \vertex to[out=40,in=140] node[above] {$b$} (Q) \vertex to[out=-140,in=0] node[below] {$b$} (R) -- node[below] {\scriptsize $y \times I$} (P); |
|
1005 \clip (P) to[out=40,in=140] (Q) to[out=-140,in=0] (R) -- (P); |
|
1006 \foreach \x in {1,...,9} { |
|
1007 \path (P) to[out=40,in=140] node[pos=\x/10] (PQ\x) {} (Q); |
|
1008 \path (R) to[out=0,in=-140] node[pos=\x/10] (RQ\x) {} (Q); |
|
1009 \draw[brown] (PQ\x.center) -- (RQ\x.center); |
|
1010 } |
|
1011 \end{tikzpicture} \\ |
937 \end{align*} |
1012 \end{align*} |
938 \nn{remember to change `assoc' to $\alpha$} |
1013 \nn{remember to change `assoc' to $\alpha$} |
939 \caption{Ingredients for the triangle axiom.} |
1014 \caption{Ingredients for the triangle axiom.} |
940 \label{fig:ingredients-triangle-axiom} |
1015 \label{fig:ingredients-triangle-axiom} |
941 \end{figure} |
1016 \end{figure} |