From 0d57fb92cf993c8323e9397408479e4455a6f6f6 Mon Sep 17 00:00:00 2001 From: CDaut Date: Thu, 13 Oct 2022 17:26:16 +0200 Subject: [PATCH] Limiten sind eindeutig --- chapters/Kegel_und_Ko.tex | 56 +++++++++++++++++++++++++++++++++++++- main.pdf | Bin 358104 -> 360256 bytes 2 files changed, 55 insertions(+), 1 deletion(-) diff --git a/chapters/Kegel_und_Ko.tex b/chapters/Kegel_und_Ko.tex index a01ade0..e034394 100644 --- a/chapters/Kegel_und_Ko.tex +++ b/chapters/Kegel_und_Ko.tex @@ -206,4 +206,58 @@ aussehen können. \end{center} Wobei T hier ein Testobjekt aus \cat{set}, $\Pi_M((m,n))=m$ und der Morphismus $T\mapsto M\times N$ definiert ist als $t\mapsto (f_M(t),f_N(t))$. Analog funktionieren die Limiten für \cat{Grp} und \cat{K-VR}. -\end{example} \ No newline at end of file +\end{example} +\begin{lemma}{Limiten sind eindeutig bis auf eindeutige Isomorphismen.\\} + Sei $D:\mathcal{I}\mapsto\mathscr{C}$ ein Diagramm, $(L,f_*),(\tilde{L},\tilde{f_*})$ Limiten über $D$. + \begin{center} + \begin{figure}[h] + \centering + \begin{tikzpicture} + \node (L1) at (0,3) {$L$}; + \node (L2) at (3,3) {$\tilde{L}$}; + \node (P1) at (0,0) {}; + \node (P2) at (1.5,0) {}; + \node (P3) at (3,0) {}; + + \draw + (L1) edge[dotted] (P1) + (L1) edge node[left]{$f_i$} (P2) + (L2) edge node[right]{$\tilde{f_i}$} (P2) + (L2) edge[dotted] (P3) + (L1) edge[dotted, bend right] node[below]{$\tilde{\varphi}$}(L2) + (L2) edge[dotted, bend right] node[above]{$\varphi$} (L1) + ; + \draw (P2) ellipse (2 and 0.5); + \end{tikzpicture} + \caption{Zwei Limiten über einem Diagramm und zugehörige Morphismen} + \end{figure} + \end{center} + $L$ Limes $\Rightarrow$ Es existiert ein $\varphi:\tilde{L}\mapsto L$ mit + $f_i\circ\varphi=\tilde{f_i}$ für alle $i\in I$.\\ + $\tilde{L}$ Limes $\Rightarrow$ Es existiert ein $\tilde{\varphi}: L\mapsto\tilde{L}$ mit + $\tilde{f_i}\circ\tilde{\varphi}=f_i$\\ + \textbf{Beh}: $\varphi=\tilde{\varphi}^{-1}$\\ + \begin{center} + \begin{tikzpicture} + \node (L1) at (0,2) {$L$}; + \node (L2) at (2,2) {$L$}; + \node (P1) at (0,0) {}; + \node (P2) at (2,0) {}; + \node (center) at (1,0){}; + + \draw + (L1) edge node[right]{$f_i$} (P1) + (L2) edge (P1) + (L1) edge (P2) + (L2) edge node[left]{$f_i$} (P2) + (L2) edge node[above]{$\varphi\circ\tilde{\varphi}$}(L1) + ; + \draw (center) ellipse (2 and 0.5); + \end{tikzpicture} + \end{center} + $f_i\circ (\varphi\circ\tilde{\varphi})=\tilde{f_i}\circ\tilde{\varphi}=f_i$ + Also $f_i\circ(id_L)=f_i$\\ + $\Rightarrow\varphi\circ\tilde{\varphi}=id_L$\\ + Analog: $\tilde{\varphi}\circ\varphi=id_{\tilde{L}}$ + \qed +\end{lemma} \ No newline at end of file diff --git a/main.pdf b/main.pdf index 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