From f00a95b0cc20b624b3d838556b24c24861f030b0 Mon Sep 17 00:00:00 2001 From: CDaut Date: Thu, 13 Oct 2022 11:30:46 +0200 Subject: [PATCH] limiten --- chapters/Kegel_und_Ko.tex | 23 +++++++++++++++++++++++ main.pdf | Bin 349238 -> 351284 bytes 2 files changed, 23 insertions(+) diff --git a/chapters/Kegel_und_Ko.tex b/chapters/Kegel_und_Ko.tex index 72df334..0cd8280 100644 --- a/chapters/Kegel_und_Ko.tex +++ b/chapters/Kegel_und_Ko.tex @@ -46,3 +46,26 @@ $\mathcal{F}(f)\circ f_i=f_j$ für alle $f\in\mor{I}{i}{j}$.\\ Der Kegel ist hier der blau markierte Teil. Das Diagramm ist der Funktor $\mathcal{F}$, der $I$ in $\mathscr{C}$ \glqq einbettet''. +\begin{definition}{Limes\\} + Ein Limes über einem Diagramm $\mathcal{F}$ ist ein universeller Kegel L: + \begin{center} + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node (Fi) at (0,0) {$\mathcal{F}(i)$}; + \node (Fj) at (3,0) {$\mathcal{F}(j)$}; + \node [red] (L) at (1.5,1.5) {$L$}; + \node [blue] (K) at (1.5,3) {$K$}; + + \draw + (L) edge[red] node[left]{$g_i$} (Fi) + (L) edge[red] node[right]{$g_j$} (Fj) + (K) edge[blue, bend right] node[left]{$f_i$} (Fi) + (K) edge[blue, bend left] node[right]{$f_j$} (Fj) + (K) edge[dotted] (L) + (Fi) edge node[below]{$\mathcal{F}(f)$} (Fj) + ; + \end{tikzpicture}\\ + \end{center} + Ein Kegel, sodass für jeden Kegel $K$ über $\mathcal{F}$ ein eindeutiger + Morphismus $K\mapsto L$ existiert, sodass $f_i=g_i\circ (K\mapsto L)$. + Der Limes ist hier rot dargestellt und der Beispielkegel blau. +\end{definition} \ No newline at end of file diff --git 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