From a765a7fc38444358a9c97b51daf71f05676616a2 Mon Sep 17 00:00:00 2001 From: CDaut Date: Sat, 1 Oct 2022 00:13:53 +0200 Subject: [PATCH] more examples --- chapters/Grundlagen.tex | 68 ++++++++++++++++++++++++++++++++++++++++ main.pdf | Bin 285495 -> 296466 bytes main.tex | 1 + 3 files changed, 69 insertions(+) diff --git a/chapters/Grundlagen.tex b/chapters/Grundlagen.tex index 4634e37..5024e46 100644 --- a/chapters/Grundlagen.tex +++ b/chapters/Grundlagen.tex @@ -261,4 +261,72 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. $\\ Man schränkt den Ring also auf alle invertierbaren Ringelemente ein und lässt nur noch die Abbildungen übrig, die ohnehin schon zwischen invertierbaren Elementen abgebildet haben. +\end{example} +\begin{example}{$\cat{Set}\mapsto\cat{K-VR}$\\} + Definiere\\ + $M\mapsto \text{Abb}_0(M,\mathbb{K})=\{\text{Abb } M\mapsto\mathbb{K} \text{ fast überall } 0\}$ (Freier Vektorraum über M)\\ + $\mathcal{F}(M):=\left\{\displaystyle\sum_{m\in M}\lambda_m\cdot m|\lambda_m=0\text{ ffa } m\in M, \lambda_m\in\mathbb{K}\right\}$\\ + \begin{center}%TDOD: Fix tikz picture + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node(M) at (0,0){$M$}; + \node(N) at (0,-2){$N$}; + \node(FM) at (2,0){$\mathcal{F}(M)$}; + \node(FN) at (2,-2){$\mathcal{F}(N)$}; + \node(Fg) at (4,0){$\sum\lambda_m\cdot m$}; + \node(FFg) at (4,-2){$\sum\lambda_m\cdot f(m)$}; + + \path + (M) edge node[right]{f} (N) + (FM) edge (FN) + (Fg) edge (FFg) + ; + \draw[decorate,decoration={brace, mirror}] (M.north west) -- (N.south west); + \draw[decorate,decoration={brace}] (M.north east) -- (N.south east); + + \end{tikzpicture} + \end{center} +\end{example} +\begin{example}{$\cat{Grp}\mapsto\cat{Ab}$ (Abelisierung)\\} + $G\mapsto G^{ab}=G_{/H}$ mit $H=\langle\langle\{h_1h_2h_1^{-1}h_2^{-1}|h_1,h_2\in G\}\rangle\rangle$ + \begin{center} + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node (G1) at (0,0){$G_1$}; + \node (G2) at (0, -2){$G_2$}; + \node (G1ab) at (2,0){$G_1^{ab}$}; + \node (G2ab) at (2,-2){$G_2^{ab}$}; + + \path + (G1) edge node[right](Lphi){$\phi$} (G2) + (G1ab) edge node[left](Rphi){} node[right]{$\phi$} (G2ab) + (Lphi) edge (Rphi)%TODO: make this edge a mapsto + ; + \end{tikzpicture} + \end{center} +\end{example} +\begin{example}{Graphen\\} + Definiere $\cat{Grph}$ exemplarisch als folgende Kategorie: + \begin{center} + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node (E) at (0,0){E $\cdot$}; + \node (V) at (3,0){$\cdot$ V}; + + \path + (E) edge node[above]{$s,t,u,\dots$} (V) + (E) edge[loop] (E) + (V) edge[loop] (V) + ; + \end{tikzpicture} + \end{center} + Dann gibt es Funktoren $\mathcal{F}: \cat{Grph}\mapsto\cat{Set}$ mit + \begin{center} + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node (E) at (0,0){$\mathcal{F}(E)$ $\cdot$}; + \node (V) at (6,0){$\cdot$ $\mathcal{F}(V)$}; + + \path + (E) edge node[above]{$\mathcal{F}(s),\mathcal{F}(t),\mathcal{F}(u),\dots$} (V) + ; + \end{tikzpicture} + \end{center} + Diese Funktoren bilden Graphen auf ihre Knoten- und Kantenmengendarstellung ab. \end{example} \ No newline at end of file diff --git a/main.pdf b/main.pdf index b7d2499305c42d9ab0d9b547811d0ab36b57775b..ad30f629c6899c468f99b0a0d9ecdc920e9dacdc 100644 GIT binary patch delta 122100 zcmdnKO>k0=&;(^BbMxsRgjqBv`f=9(>5_h{c+9STy+^L9gMq+ChGxY)p@nK1zCZuo zxWO#MJURYn&G};WVu^;s3m^WUb3L|r>vW+2!`EvX-f@H z8m*6=R({h;qn*ETx$o16+H_`VmH70^SKfAO9tH(1SgWvAp7(L4Y2OTY#>X=pS=T-| zv1jck+pY6jKJR=pZC2M}A(v$vC&}Awd)m7#^U{v6Iv&;kKi9@yuVK#2Dl(Yn`s1VbUp|^#82C{Vx3Ox00P_BTHVtTlLxGx5eV zFq<8hvj2ANi06;qCoWmOci)|o>8m}&>nHB$lmGuYwN*K-aHhoB?3LpB#=MU&eEPD; zIdE64@4eGcS6*M?{FgK6iyF)CnRn_8CKoT7!Dn3mB5e2h)p@+DBep#_(wi$lLS59w|JYZ@821-<@hCCyFXGZyq9N7iasb? zbLH3Z`gh8rMa(x!19_}2nJljh77Z2RjC%m30_iK%A5JkiooKs= zG1uRNS+d+x+q36^uiMv&T2Yf@h4d<;KA$aI{p8VahvgNDv(-d=ZXS+^yeyPi{V~y& zZC=uwM?ZU>_H6C^v8Sy~DQG39)Y3080*~(~{F~b-uUT*YY2vJ0i^MJLuWDXy-<^;+ zyG?gv>*S?v-9{|unoOo%YjKIKexaB3+S-Zt{DGzF$HR2C$^KB`IwUUtq4@jte^t9% 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