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Pushout über KVR

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CDaut 2022-10-16 00:38:27 +02:00 • committed by CDaut
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chapters/Kegel_und_Ko.tex
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@ -535,9 +535,35 @@ aussehen können.
(W) edge[blue, bend left] (T)
(P) edge[dotted] (T)
;
\end{tikzpicture}
\end{center}
\caption{Das kommutative Diagramm für einen Pushout.}
\end{figure}
\end{definition}
\end{definition}
\begin{example}{Pushout über \cat{K-VR}\\}
Betrachte folgendes kommutatives Diagramm:
\begin{center}
\begin{tikzpicture}
\node (X) at (0,0){$X$};
\node (V) at (0,-2){$V$};
\node (W) at (2,0){$W$};
\node[red] (P) at (2,-2){$V\coprod_X W$};
\node[blue] (T) at (3,-3){$T$};
\draw
(X) edge node[left]{$f_V$} (V)
(X) edge node[above]{$f_W$}(W)
(V) edge[red] (P)
(W) edge[red] (P)
(V) edge[blue, bend right] node[below]{$g_V$} (T)
(W) edge[blue, bend left] node[right]{$g_W$}(T)
(P) edge[dotted] node[above]{$\varphi$} (T)
;
\end{tikzpicture}
\end{center}
Wobei $V\coprod_X W:=\{(v,w)\in V\times W\}_{/U}$ mit $U=\{(-f_V(x),f_W(x))|x\in X\}$\\
$V\mapsto V\coprod_X W$ ist definiert als $v\mapsto [(v,0)]$ und\\
$\varphi: V\coprod_X W\mapsto T$ ist definiert als $[(v,w)]\mapsto g_V(v)+g_W(w)$
\end{example}