例0.9 位相空間の普遍性
eg0.9.tikz(tex)\usetikzlibrary {arrows.meta}
\begin{document}
\begin{tikzpicture}[auto,arrows={Hooks[left]->}]
\node (UV) at (2,0) {$U\cap V$};
\node (V) at (0,0) {$V$};
\node (U) at (2,2) {$U$};
\node (X) at (0,2) {$\underbar X$};
\draw (UV) -- node {$j$} (V);
\draw (UV) -- node[swap] {$i$} (U);
\draw (U) -- node {$j'$} (X);
\draw (V) -- node {$i'$} (X);
\end{tikzpicture}
\end{document}
eg0.9-2.tikz(tex)\usetikzlibrary {arrows.meta}
\begin{document}
\begin{tikzpicture}[auto]
\tikzset{hook/.style={arrows={Hooks[left]->}}}
\node (UV) at (2,0) {$U\cap V$};
\node (V) at (0,0) {$V$};
\node (U) at (2,2) {$U$};
\node (X) at (0,2) {$\underbar X$};
\node (Y) at (-2,4) {$\forall Y$};
\draw[hook] (UV) -- node {$j$} (V);
\draw[hook] (UV) -- node[swap] {$i$} (U);
\draw[hook] (U) -- node {$j'$} (X);
\draw[hook] (V) -- node {$i'$} (X);
\draw[dashed] (X) -- node[anchor=center] {$\exists!h$} (Y);
\draw (U) to[bend right] node {$\forall f$} (Y);
\draw (V) to[bend left] node {$\forall g$} (Y);
\end{tikzpicture}
\end{document}>都合のよい条件下においては、誘導された基本群の間の図式
eg0.9-3.tikz(tex)\begin{document}
\begin{tikzpicture}[auto,->]
\node (UV) at (2,0) {$\pi_1(U\cap V)$};
\node (V) at (0,0) {$\pi_1(V)$};
\node (U) at (2,2) {$\pi_1(U)$};
\node (X) at (0,2) {$\pi_1(\underbar X)$};
\draw (UV) -- node {$j_*$} (V);
\draw (UV) -- node[swap] {$i_*$} (U);
\draw (U) -- node {$j'_*$} (X);
\draw (V) -- node {$i'_*$} (X);
\end{tikzpicture}
\end{document}>も、群と群準同型の世界で同じ普遍性をもつ。これがvan Kampenの定理である
この例は特に説明がないのでパス