From 419e02adc2366f88d6b55ead5b010170188abc7d Mon Sep 17 00:00:00 2001 From: CDaut Date: Sat, 15 Oct 2022 11:41:00 +0200 Subject: [PATCH] Beispiel Kolimiten --- chapters/Kegel_und_Ko.tex | 72 +++++++++++++++++++++++++++++++++++--- main.pdf | Bin 364561 -> 386254 bytes 2 files changed, 67 insertions(+), 5 deletions(-) diff --git a/chapters/Kegel_und_Ko.tex b/chapters/Kegel_und_Ko.tex index 531b97a..38ac0da 100644 --- a/chapters/Kegel_und_Ko.tex +++ b/chapters/Kegel_und_Ko.tex @@ -261,12 +261,12 @@ aussehen können. Analog: $\tilde{\varphi}\circ\varphi=id_{\tilde{L}}$ \qed \end{lemma} -\subsection{Kokegel, Kolimes und Koprodukt} +\subsection{Kokegel und Kolimes} \begin{definition}{Kokegel \& Kolimes\\} Sei $D:I\mapsto\mathscr{C}$ ein Diagramm.\\ Kokegel: \begin{tikzpicture}[baseline=5mm] - \node[blue] (K) at (0, 0){$K$}; + \node[blue] (K) at (0, 0){$K_{co}$}; \node (P1) at (-1, 1){$\cdot$}; \node (P2) at (1, 1){$\cdot$}; \node (center) at (0,1){}; @@ -280,7 +280,69 @@ aussehen können. \end{tikzpicture} Ein Objekt $K$ aus $\mathscr{C}$, sodass von jedem Objekt aus $\mathscr{C}$ Morphismen auf $K$ existieren und alle Dreiecke kommutieren.\\ - Ein Kolimes ist ein universeller Kokegel: - + Ein Kolimes ist ein universeller Kokegel:\\ + \begin{center} + \begin{tikzpicture} + \node[red] (L) at (0, 0){$L_{co}$}; + \node (P1) at (-2, 2){$\cdot$}; + \node (P2) at (2, 2){$\cdot$}; + \node (center) at (0,2){}; + \node[blue] (K) at (0, -1.5){$K_{co}$}; -\end{definition} \ No newline at end of file + \draw + (P1) edge (P2) + (P1) edge[red] (L) + (P2) edge[red] (L) + (P1) edge[blue, bend right] (K) + (P2) edge[blue, bend left] (K) + (L) edge[dotted] (K) + ; + \draw (center) ellipse (3 and 0.5); + \end{tikzpicture} + \end{center} + Ein Kokegel, sodass auf jeden anderen Kokegel ein Morphismus existiert und alle + Dreiecke kommutieren. +\end{definition} +\begin{example}{Kolimiten für $I=\varnothing$\\} + \begin{figure}[h] + \begin{center} + \begin{tikzpicture} + \node (center) at (0, 2){}; + \node[red] (L) at (0,1){$L_{co}$}; + \node[blue] (K) at (0,0){$K_{co}$}; + \draw + (L) edge[dotted] (K) + (center) ellipse (1 and 0.25) + ; + \end{tikzpicture} + \end{center} + \caption{Die leere Kategorie als Bild des Diagramms und wie der zu findende + Kolimes im Verhältnis zu dieser stehen muss.} + \end{figure}\\ + Diesen Kolimes nennt man allgemein auch \textit{Initiales Objekt}. Folgende Objekte sind initiale Objekte der jeweiligen Kategorie: + \begin{itemize} + \item \cat{set} $\varnothing$ + \item \cat{Grp} $\{e\}$ + \item \cat{K-VR} $\{0\}$ + \item \cat{Top} $\varnothing$ + \item \cat{R1ng} $\mathbb{Z}$ + \end{itemize} + Dies lässt sich leicht verifizieren, indem man prüft, dass die jeweiligen initialen Objekte die Kolimeseigenschaften der respektiven Kategorie erfüllen. +\end{example} +\begin{example} + {$I=$ + \begin{tikzpicture}[baseline=-1mm] + \node (P1) at (0,0){$\cdot$}; + \node (P2) at (0.5,0){$\cdot$}; + \node (center) at (0.25,0){}; + \draw (center) ellipse (0.5 and 0.25); + \end{tikzpicture} + \\} + Für dieses $I$ sind die Kolimiten die \textit{Koprodukte}.\\ + \underline{Beispiel \cat{Set}:}\\ + \begin{center} + \begin{tikzpicture} + \node (L) at (0,0){$M\dot{\cup} N$}; + \end{tikzpicture} + \end{center} +\end{example} \ No newline at end of file diff --git a/main.pdf b/main.pdf index 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