From ab54b3ad3373974f5e80b528120f84320cc1479c Mon Sep 17 00:00:00 2001 From: CDaut Date: Sun, 16 Oct 2022 21:49:58 +0200 Subject: [PATCH] =?UTF-8?q?Beispiel=20f=C3=BCr=20Kern?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- chapters/Kegel_und_Ko.tex | 28 ++++++++++++++++++++++++---- main.pdf | Bin 316751 -> 317486 bytes 2 files changed, 24 insertions(+), 4 deletions(-) diff --git a/chapters/Kegel_und_Ko.tex b/chapters/Kegel_und_Ko.tex index 4275a1c..d3fa26d 100644 --- a/chapters/Kegel_und_Ko.tex +++ b/chapters/Kegel_und_Ko.tex @@ -589,8 +589,8 @@ aussehen können. \begin{tikzpicture} \node (L) at (0,0){$\bullet$}; \node (R) at (2,0){$\bullet$}; - \node (K) at (-2,0){$K$}; - \node (KE) at (-1,1){$\bullet$}; + \node[red] (K) at (-2,0){$K$}; + \node[blue] (KE) at (-1,1){$\bullet$}; \draw (L) edge node[above]{$0$} (R) @@ -604,5 +604,25 @@ aussehen können. \caption{Das Kommutative Diagramm für einen Kern} \end{figure}\\ Auch der Kern existiert nicht in jeder Kategorie. Die Kategorien \cat{Ab}, \cat{Grp} und \cat{K-VR} haben - beispielsweise einen Kern wohingegen \cat{Set} keinen Besitzt. -\end{definition} \ No newline at end of file + beispielsweise einen Kern wohingegen \cat{Set} keinen besitzt. +\end{definition} +\begin{example}{Kern der Kategorie \cat{K-VR}\\} + Betrachte folgendes kommutatives Diagramm: + \begin{center} + \begin{tikzpicture} + \node (L) at (0,0){$V$}; + \node (R) at (2,0){$W$}; + \node[red] (K) at (-2,0){$Kern(f)$}; + \node[blue] (KE) at (-1,1.5){$T$}; + + \draw + (L) edge node[above]{$0$} (R) + (L) edge[bend right] node[below]{$f$}(R) + (K) edge[red] node[above]{$\iota$} (L) + (KE) edge[dotted] node[left]{$\varphi$} (K) + (KE) edge[blue] node[right]{$g$} (L) + ; + \end{tikzpicture} + \end{center} + Wobei $Kern(f):=\{v\in V|f(v)=0\}$ und $\varphi(t)=g(t)$ +\end{example} \ No newline at end of file diff --git a/main.pdf b/main.pdf index 6fdfd3b167d1d4d03ee2a5cfc595eb57fd5c78ce..76280708b21b54c6b68793eb29e29d4d419c87ad 100644 GIT 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