例0.9 位相空間の普遍性
2つの開集合$ U,Vで被覆された位相空間$ X:=(U\cup V,{\cal O})を考える https://scrapbox.io/files/65c702e8aa434f0025fab763.svg
code:eg0.9.tikz(tex)
\usetikzlibrary {arrows.meta}
\begin{document}
\begin{tikzpicture}[auto,arrows={Hooksleft->}] \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) -- nodeswap {$i$} (U); \draw (U) -- node {$j'$} (X);
\draw (V) -- node {$i'$} (X);
\end{tikzpicture}
\end{document}
https://scrapbox.io/files/65c702f724a71e002410f9c7.svg
code:eg0.9-2.tikz(tex)
\usetikzlibrary {arrows.meta}
\begin{document}
\tikzset{hook/.style={arrows={Hooksleft->}}} \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) {$\forall Y$};
\drawhook (UV) -- node {$j$} (V); \drawhook (U) -- node {$j'$} (X); \drawhook (V) -- node {$i'$} (X); \draw (V) tobend 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
code:eg0.9-3.tikz(tex)
\begin{document}
\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) -- nodeswap {$i_*$} (U); \draw (U) -- node {$j'_*$} (X);
\draw (V) -- node {$i'_*$} (X);
\end{tikzpicture}
\end{document}
この例は特に説明がないのでパス