From bce8db56657ac7a160fd42fb80248cd8938d454c Mon Sep 17 00:00:00 2001 From: CDaut Date: Sun, 16 Oct 2022 19:50:28 +0200 Subject: [PATCH] Kern --- chapters/Kegel_und_Ko.tex | 40 +++++++++++++++++++++++++++++++++++++- chapters/nat_trafo.tex | 1 + main.pdf | Bin 314136 -> 316746 bytes main.tex | 3 ++- 4 files changed, 42 insertions(+), 2 deletions(-) create mode 100644 chapters/nat_trafo.tex diff --git a/chapters/Kegel_und_Ko.tex b/chapters/Kegel_und_Ko.tex index 504457e..9cbb22a 100644 --- a/chapters/Kegel_und_Ko.tex +++ b/chapters/Kegel_und_Ko.tex @@ -567,4 +567,42 @@ aussehen können. $V\mapsto V\coprod_X W$ 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} -\subsection{Kern und Kokern} \ No newline at end of file +\subsection{Kern und Kokern} +\begin{definition}{Kern\\} + Der Kern ist der Limes über dem Diagramm + \begin{tikzpicture}[baseline=-1mm] + \node (L) at (0,0){$\bullet$}; + \node (R) at (1,0){$\bullet$}; + + \draw + (L) edge node[above]{$0$} (R) + (L) edge[bend right] (R) + ; + \end{tikzpicture}. + $0_{A,B}\in\mor{\mathscr{C}}{A}{B}$ ist dabei so definiert, dass für jeden Morphismus + $f\in\mor{\mathscr{C}}{A}{B}$ $0_{A,B}+f=f=f+0_{A,B}$ gilt. Nicht in jeder Kategorie + existiert solch eine Abbildung.\\ + Gesucht ist also ein Kegel $K$ und eine Abbildung $\iota$, sodass folgendes Diagramm + kommutiert und $0\circ\iota=f\circ\iota$ gilt:\\ + \begin{figure}[h] + \begin{center} + \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){$\bullet$}; + + \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] (K) + (KE) edge[blue] (L) + ; + \end{tikzpicture} + \end{center} + \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 diff --git a/chapters/nat_trafo.tex b/chapters/nat_trafo.tex new file mode 100644 index 0000000..468ef77 --- /dev/null +++ b/chapters/nat_trafo.tex @@ -0,0 +1 @@ +\section{Natürliche Transformationen} \ No newline at end of file diff --git a/main.pdf b/main.pdf index 1a704ba7afc0c69477777c6754e5e105678ca875..df06af473562ceb26bf35b71426b9695684d759e 100644 GIT binary patch delta 103270 zcmbO+P59I*;SHsX%qE70(+k(JXx6`-W}Dw^Akg;x4_ANMfv@J@qE>d^+7imPZ40w6 z$3_K(j}sT}-^b?cy=X~by4BpLyFYuLzgum@#(5yY@bKQf{ORSzYK#v%3OyZnFncMl z(&Sj-CS@#;cTmdi*>8>2s~0a%h^af4w)wp8)4lEfdXl$2d~c=j)SMC&KlN1b{B^HG z_0t!b9baMQ`tE0sfZN)`YV+NL#rJB=u9@xXdB^Q>gXZgkrA>vJN`jMRezYhm{$ih? z>(Ozb;H4nrVk^I{OIzM^=qhevd)z6_{b~2lgtx+LR~-Ib+$|ftj(x?JcB}kPTl7wq zap(lOUbVlJ?eX*E+nrv263dQXK7Z$nzHL|HoFg;qp9i15<8@GW*Y8E4EPMtb+eHjZ 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-\include{chapters/Kegel_und_Ko.tex} +\include{./chapters/Kegel_und_Ko.tex} +\include{./chapters/nat_trafo.tex} \end{document} \ No newline at end of file