From dbc4d45820f19d2972c1753bd3743d9fab31492c Mon Sep 17 00:00:00 2001 From: CDaut Date: Sat, 15 Oct 2022 18:57:33 +0200 Subject: [PATCH] Kolimiten --- chapters/Kegel_und_Ko.tex | 70 ++++++++++++++++++++++++++++++++++++-- main.pdf | Bin 386254 -> 391303 bytes 2 files changed, 67 insertions(+), 3 deletions(-) diff --git a/chapters/Kegel_und_Ko.tex b/chapters/Kegel_und_Ko.tex index 38ac0da..ec90da6 100644 --- a/chapters/Kegel_und_Ko.tex +++ b/chapters/Kegel_und_Ko.tex @@ -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} \ No newline at end of file + \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} diff --git a/main.pdf b/main.pdf index 20a025163353851783a69cda8ddbfa96908cde28..2bfcfca8ede137f45b3dbfa9949c9bbc8ec33b9a 100644 GIT binary patch delta 83468 zcmX?iQoQ}6_=a{yW@9sx>5h|FHS6zAvGqS}Akg;xA6Nf3fydU?etVQIaxq13>rz~# zxOJ9V(5$it*NYWBU$T1`1TbEXOut`!t~h^9#6gkSzE=d+Up{#?B$~}FD|zD5^Ezch zvYR;OmUwM0Ir!|)wy3n!-8VUTZSv!a{;Oo0OEw?sbI|~$6YHMt;IeBZ(3hDcu3t07B1++>9dWxT)>~8fgRnPmqaC0R?Tfl?Du(?V{ z7$1flnU%jfG}>JDs6k8o-<9?io&6fK7QcD9U!cC{oz0i+@BYqx;u<$;YqG+w?z)K_ zy?fMFUuv9jJpSwMy6?Xgyo_#pF~om)UA_NXuK8Y#lKO+|@4u_FlX~;9HmgAQ>j|HP zBd@0(n9={`*hGO{OHN;38vlIuJGN)>_B`jL_>S`*&fMdvzCMe``2<&LUP^IEQEFl? zh?Sp|#bq=-uz*!~vnkVYW^rQ$0|k(njSZK+PikIzNrr-vftm5-hmIQcXCu8Q-!>E4 z_d8s}e3jNACWU!7PbJPd@;K$x(^YS#@ZY@2EE_1bPPJ&-6dTR|zeDy8JxTRSKSv*@Gb>BXFDB_TAd`4d9(wC3Q_Wme8 zG+q0chVQ!9$vf|*-rf`Q@A)_N2v70)9;TV8J0|Ic$$a0Kck%Yw#GPI{eedaq-%com(w- 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