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Kolimiten

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CDaut 2022-10-15 18:57:33 +02:00 • committed by CDaut
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chapters/Kegel_und_Ko.tex
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@ -341,8 +341,72 @@ aussehen können.
Für dieses $I$ sind die Kolimiten die \textit{Koprodukte}.\\
\underline{Beispiel \cat{Set}:}\\
\begin{center}
\begin{tikzpicture}
\node (L) at (0,0){$M\dot{\cup} N$};
\begin{tikzpicture}[baseline=0mm]
\node[red] (L) at (0,0){$M\dot{\cup} N$};
\node[blue] (T) at (0,-2){$T$};
\node (M) at (-2,2){$M$};
\node (N) at (2,2){$N$};
\draw
(M) edge node[sloped, above]{$m\mapsto m$} (L)
(N) edge node[sloped, above]{$n\mapsto n$}(L)
(L) edge[dotted] node[right]{$\varphi$} (T)
(M) edge[blue, bend right] node[left]{$f_M$} (T)
(N) edge[blue, bend left] node[right]{$f_N$} (T)
;
\end{tikzpicture}
\hspace{20mm}
$\varphi:M\dot{\cup}N\mapsto T$ mit
$x\mapsto
\begin{cases}
f_M(x) & x\in M \\
f_N(x) & x\in N
\end{cases}
$
\end{center}
\end{example}
\underline{Beispiel \cat{K-VR}:}\\
\begin{center}
\begin{tikzpicture}[baseline=0mm]
\node[red] (L) at (0,0){$V\oplus W$};
\node[blue] (T) at (0,-2){$T$};
\node (M) at (-2,2){$V$};
\node (N) at (2,2){$W$};
\draw
(M) edge node[sloped, above]{$v\mapsto (v,0)$} (L)
(N) edge node[sloped, above]{$w\mapsto (0,w)$}(L)
(L) edge[dotted] node[right]{$\varphi$} (T)
(M) edge[blue, bend right] node[left]{$f_V$} (T)
(N) edge[blue, bend left] node[right]{$f_W$} (T)
;
\end{tikzpicture}
\hspace{20mm}
$\varphi:V\oplus W\mapsto T$ mit
$(v,w)\mapsto f_V(v) + f_W(w) $
\end{center}
\underline{Beispiel \cat{Grp}:\\}
\begin{center}
\begin{minipage}{0.3\linewidth}
\begin{tikzpicture}[baseline=0mm]
\node[red] (L) at (0,0){$G*H$};
\node[blue] (T) at (0,-2){$T$};
\node (M) at (-2,2){$G$};
\node (N) at (2,2){$H$};
\draw
(M) edge (L)
(N) edge (L)
(L) edge[dotted] node[right]{$\varphi$} (T)
(M) edge[blue, bend right] node[left]{$f_G$} (T)
(N) edge[blue, bend left] node[right]{$f_H$} (T)
;
\end{tikzpicture}
\end{minipage}
\hspace{25mm}
\begin{minipage}{0.5\linewidth}
$G*H$ bezeichnet dabei die Menge aller Worte in $G$ und $H$.\\
$\varphi: G*H\mapsto T$ ersetzt im Wort $g$ durch $f_G(G)$ und $h$
durch $f_H(h)$.
\end{minipage}
\end{center}
\end{example}