677 \draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
677 \draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
678 \draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
678 \draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
679 \draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
679 \draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
680 \draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
680 \draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
681 \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); |
681 \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); |
682 \end{tikzpicture}} \\ |
682 \end{tikzpicture}} \displaybreak[1] \\ |
683 \;&=\; |
683 \;&=\; |
684 \raisebox{-2.1cm}{ |
684 \raisebox{-2.1cm}{ |
685 \begin{tikzpicture} |
685 \begin{tikzpicture} |
686 \draw (0,0) coordinate (p1); |
686 \draw (0,0) coordinate (p1); |
687 \draw (5.8,0) coordinate (p2); |
687 \draw (5.8,0) coordinate (p2); |
703 \draw (p1) .. controls +(1,1) and +(-1,1) .. (pu); |
703 \draw (p1) .. controls +(1,1) and +(-1,1) .. (pu); |
704 \draw (p2) .. controls +(-1,1) and +(1,1) .. (pu); |
704 \draw (p2) .. controls +(-1,1) and +(1,1) .. (pu); |
705 \draw[->, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7); |
705 \draw[->, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7); |
706 \draw[->, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7); |
706 \draw[->, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7); |
707 \draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, below] {$(a\cdot c)\times I$} -- (3.3,-0.5); |
707 \draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, below] {$(a\cdot c)\times I$} -- (3.3,-0.5); |
708 \end{tikzpicture}} \\ |
708 \end{tikzpicture}} \displaybreak[1] \\ |
709 \;&=\; |
709 \;&=\; |
710 \raisebox{-1.9cm}{ |
710 \raisebox{-1.9cm}{ |
711 \begin{tikzpicture}[y=-1cm] |
711 \begin{tikzpicture}[y=-1cm] |
712 \draw (0,0) coordinate (p1); |
712 \draw (0,0) coordinate (p1); |
713 \draw (5.8,0) coordinate (p2); |
713 \draw (5.8,0) coordinate (p2); |
729 \draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
729 \draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
730 \draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
730 \draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
731 \draw[<-, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
731 \draw[<-, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
732 \draw[<-, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
732 \draw[<-, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
733 \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, below] {$(a\cdot d)\times I$} -- (2.5,0); |
733 \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, below] {$(a\cdot d)\times I$} -- (2.5,0); |
734 \end{tikzpicture}} \\ |
734 \end{tikzpicture}} \displaybreak[1] \\ |
735 \;&=\; |
735 \;&=\; |
736 \raisebox{-1.0cm}{ |
736 \raisebox{-1.0cm}{ |
737 \begin{tikzpicture}[y=-1cm] |
737 \begin{tikzpicture}[y=-1cm] |
738 \draw (0,0) coordinate (p1); |
738 \draw (0,0) coordinate (p1); |
739 \draw (5.8,0) coordinate (p2); |
739 \draw (5.8,0) coordinate (p2); |
1023 Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) |
1025 Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) |
1024 as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). |
1026 as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). |
1025 \begin{figure}[t] |
1027 \begin{figure}[t] |
1026 \begin{align*} |
1028 \begin{align*} |
1027 \mathfig{0.4}{triangle/triangle4d} |
1029 \mathfig{0.4}{triangle/triangle4d} |
1028 \mathfig{0.4}{triangle/triangle4e} |
1030 \mathfig{0.4}{triangle/triangle4e} \\ |
|
1031 \end{align*} |
|
1032 \begin{align*} |
|
1033 u \bullet (b \times I) & = |
|
1034 \begin{tikzpicture}[baseline] |
|
1035 \coordinate (L) at (0,0); |
|
1036 \coordinate (R) at (3,0); |
|
1037 \coordinate (T) at (1.5,0.8); |
|
1038 \coordinate (M) at (1.5,-0.4); |
|
1039 \coordinate (B) at (0.75,-0.8); |
|
1040 \path (L) \vertex (T) \vertex (R) \vertex (M) \vertex (B) \vertex; |
|
1041 \draw (L) to[out=40,in=180] node[above] {$a$} (T) |
|
1042 to[out=0,in=140] node[above] {$b$} (R) |
|
1043 to[out=-150,in=-20] node[below] {$b$} (M) |
|
1044 -- node[below] {\tiny $y \times I$} (B) |
|
1045 to[out=150,in=-45] node[below] {$a$} (L); |
|
1046 \draw[dashed] (L) to[out=0,in=150] (M) |
|
1047 to[out=30,in=150] (R); |
|
1048 \draw[dashed] (T) -- (M); |
|
1049 \foreach \n in {0,...,7} { |
|
1050 \path (L) to[out=-45,in=150] node[coordinate,pos=\n/8] (LB\n) {} (B); |
|
1051 \path (L) to[out=0,in=150] node[coordinate,pos=\n/8] (LM\n) {} (M); |
|
1052 \path (L) to[out=40,in=180] node[coordinate,pos=\n/8] (LT\n) {} (T); |
|
1053 \draw[green!50!brown] (LB\n) -- (LM\n) -- (LT\n); |
|
1054 } |
|
1055 \foreach \n in {1,...,7} { |
|
1056 \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); |
|
1058 \draw[brown] (MR\n) -- (TR\n); |
|
1059 } |
|
1060 \end{tikzpicture} \\ |
|
1061 (a \times I) \bullet v & = |
|
1062 \begin{tikzpicture}[baseline] |
|
1063 \coordinate (L) at (0,0); |
|
1064 \coordinate (R) at (3,0); |
|
1065 \coordinate (T) at (1.5,0.8); |
|
1066 \coordinate (M) at (1.5,-0.4); |
|
1067 \coordinate (B) at (2.25,-0.8); |
|
1068 \path (L) \vertex (T) \vertex (R) \vertex (M) \vertex (B) \vertex; |
|
1069 \draw (L) to[out=40,in=180] node[above] {$a$} (T) |
|
1070 to[out=0,in=140] node[above] {$b$} (R) |
|
1071 to[out=-150,in=45] node[below] {$b$} (B) |
|
1072 -- node[below=2] {\tiny $y \times I$} (M) |
|
1073 to[out=-160,in=-30] node[below] {$a$} (L); |
|
1074 \draw[dashed] (L) to[out=30,in=150] (M) |
|
1075 to[out=30,in=180] (R); |
|
1076 \draw[dashed] (T) -- (M); |
|
1077 \foreach \n in {1,...,7} { |
|
1078 \path (B) to[out=45,in=-150] node[coordinate,pos=\n/8] (BR\n) {} (R); |
|
1079 \path (M) to[out=30,in=180] node[coordinate,pos=\n/8] (MR\n) {} (R); |
|
1080 \path (T) to[out=0,in=140] node[coordinate,pos=\n/8] (TR\n) {} (R); |
|
1081 \draw[brown] (BR\n) -- (MR\n) -- (TR\n); |
|
1082 } |
|
1083 \foreach \n in {1,...,7} { |
|
1084 \path (L) to[out=-20,in=-150] node[coordinate,pos=\n/8] (LM\n) {} (M); |
|
1085 \path (L) to[out=40,in=180] node[coordinate,pos=\n/8] (LT\n) {} (T); |
|
1086 \draw[green!50!brown] (LM\n) -- (LT\n); |
|
1087 } |
|
1088 \end{tikzpicture} |
1029 \end{align*} |
1089 \end{align*} |
1030 \caption{Horizontal compositions in the triangle axiom.} |
1090 \caption{Horizontal compositions in the triangle axiom.} |
1031 \label{fig:horizontal-composition} |
1091 \label{fig:horizontal-composition} |
1032 \end{figure} |
1092 \end{figure} |
1033 \begin{figure}[t] |
1093 \begin{figure}[t] |
1087 \label{fig:vertical-composition} |
1147 \label{fig:vertical-composition} |
1088 \end{figure} |
1148 \end{figure} |
1089 \begin{figure}[t] |
1149 \begin{figure}[t] |
1090 \begin{align*} |
1150 \begin{align*} |
1091 \mathfig{0.4}{triangle/triangle5} |
1151 \mathfig{0.4}{triangle/triangle5} |
|
1152 \end{align*} |
|
1153 \begin{align*} |
|
1154 \begin{tikzpicture}[baseline] |
|
1155 \coordinate (L) at (0,0); |
|
1156 \coordinate (R) at (3,0); |
|
1157 \coordinate (T) at (1.5,0.8); |
|
1158 \coordinate (M) at (1.5,-0.4); |
|
1159 \coordinate (B) at (0.75,-0.8); |
|
1160 \path (L) \vertex (T) \vertex (R) \vertex (M) \vertex (B) \vertex; |
|
1161 \draw (L) to[out=40,in=180] (T) |
|
1162 to[out=0,in=140] (R) |
|
1163 to[out=-150,in=-20] (M) |
|
1164 -- (B) |
|
1165 to[out=150,in=-45] (L); |
|
1166 \draw[dashed] (T) -- (M); |
|
1167 \foreach \n in {0,...,7} { |
|
1168 \path (L) to[out=-45,in=150] node[coordinate,pos=\n/8] (LB\n) {} (B); |
|
1169 \path (L) to[out=0,in=150] node[coordinate,pos=\n/8] (LM\n) {} (M); |
|
1170 \path (L) to[out=40,in=180] node[coordinate,pos=\n/8] (LT\n) {} (T); |
|
1171 \draw[green!50!brown] (LB\n) -- (LM\n) -- (LT\n); |
|
1172 } |
|
1173 \foreach \n in {1,...,7} { |
|
1174 \path (M) to[out=-20,in=-150] node[coordinate,pos=\n/8] (MR\n) {} (R); |
|
1175 \path (T) to[out=0,in=140] node[coordinate,pos=\n/8] (TR\n) {} (R); |
|
1176 \draw[brown] (MR\n) -- (TR\n); |
|
1177 } |
|
1178 \coordinate (B') at ($(B)+(0.125,-0.25)$); |
|
1179 \coordinate (M') at ($(M)+(0.125,-0.25)$); |
|
1180 \coordinate (R') at ($(R)+(0.125,-0.25)$); |
|
1181 \coordinate (X) at ($(M')+(0,-0.6)$); |
|
1182 \path (X) \vertex; |
|
1183 \draw[dashed] (M') -- (X); |
|
1184 \draw[clip] (B') to[out=-30,in=180] (X) to[out=0,in=-135] (R') to[out=-150,in=-20] (M') -- cycle; |
|
1185 \foreach \n in {1,...,7} { |
|
1186 \path (M') to[out=-20,in=-150] node[coordinate,pos=\n/8] (M'R'\n) {} (R'); |
|
1187 \draw[brown] (M'R'\n) -- +(0,-1); |
|
1188 } |
|
1189 \foreach \n in {1,...,5} { |
|
1190 \path (B') to[out=-30,in=180] node[coordinate,pos=\n/6] (B'X\n) {} (X); |
|
1191 \path (M') to[out=-90,in=90] node[coordinate,pos=\n/6] (M'X\n) {} (X); |
|
1192 \path (B') to[out=30,in=-150] node[coordinate,pos=\n/6] (B'M'\n) {} (M'); |
|
1193 \draw[blue] (B'X\n) -- (M'X\n); |
|
1194 \draw[blue] (B'M'\n) -- (B'X\n); |
|
1195 } |
|
1196 \end{tikzpicture} |
|
1197 \xrightarrow{\text{collar}} |
|
1198 \begin{tikzpicture}[baseline] |
|
1199 \coordinate (L) at (0,0); |
|
1200 \coordinate (R) at (3,0); |
|
1201 \coordinate (T) at (1.5,0.8); |
|
1202 \coordinate (M) at (1.5,-0.4); |
|
1203 \coordinate (B) at (0.75,-0.8); |
|
1204 \path (L) \vertex (T) \vertex (R) \vertex (M) \vertex (B) \vertex; |
|
1205 \draw (L) to[out=40,in=180] (T) |
|
1206 to[out=0,in=140] (R); |
|
1207 \draw[dashed] (R) to[out=-150,in=-20] (M) |
|
1208 -- (B); |
|
1209 \draw (B) to[out=150,in=-45] (L); |
|
1210 \draw[dashed] (T) -- (M); |
|
1211 \foreach \n in {0,...,7} { |
|
1212 \path (L) to[out=-45,in=150] node[coordinate,pos=\n/8] (LB\n) {} (B); |
|
1213 \path (L) to[out=0,in=150] node[coordinate,pos=\n/8] (LM\n) {} (M); |
|
1214 \path (L) to[out=40,in=180] node[coordinate,pos=\n/8] (LT\n) {} (T); |
|
1215 \draw[green!50!brown] (LB\n) -- (LM\n) -- (LT\n); |
|
1216 } |
|
1217 \foreach \n in {1,...,7} { |
|
1218 \path (M) to[out=-20,in=-150] node[coordinate,pos=\n/8] (MR\n) {} (R); |
|
1219 \path (T) to[out=0,in=140] node[coordinate,pos=\n/8] (TR\n) {} (R); |
|
1220 \draw[brown] (MR\n) -- (TR\n); |
|
1221 } |
|
1222 \coordinate (B') at (B); |
|
1223 \coordinate (M') at (M); |
|
1224 \coordinate (R') at (R); |
|
1225 \coordinate (X) at ($(M')+(0,-0.6)$); |
|
1226 \path (X) \vertex; |
|
1227 \draw[dashed] (M') -- (X); |
|
1228 \draw (B') to[out=-30,in=180] (X) to[out=0,in=-135] (R'); |
|
1229 \path[clip] (B') to[out=-30,in=180] (X) to[out=0,in=-135] (R') to[out=-150,in=-20] (M') -- cycle; |
|
1230 \foreach \n in {1,...,7} { |
|
1231 \path (M') to[out=-20,in=-150] node[coordinate,pos=\n/8] (M'R'\n) {} (R'); |
|
1232 \draw[brown] (M'R'\n) -- +(0,-1); |
|
1233 } |
|
1234 \foreach \n in {1,...,5} { |
|
1235 \path (B') to[out=-30,in=180] node[coordinate,pos=\n/6] (B'X\n) {} (X); |
|
1236 \path (M') to[out=-90,in=90] node[coordinate,pos=\n/6] (M'X\n) {} (X); |
|
1237 \path (B') to[out=30,in=-150] node[coordinate,pos=\n/6] (B'M'\n) {} (M'); |
|
1238 \draw[blue] (B'X\n) -- (M'X\n); |
|
1239 \draw[blue] (B'M'\n) -- (B'X\n); |
|
1240 } |
|
1241 \end{tikzpicture} |
1092 \end{align*} |
1242 \end{align*} |
1093 \caption{Adding a collar in the proof of the triangle axiom.} |
1243 \caption{Adding a collar in the proof of the triangle axiom.} |
1094 \label{fig:adding-a-collar} |
1244 \label{fig:adding-a-collar} |
1095 \end{figure} |
1245 \end{figure} |
1096 |
1246 |