From 1c7ffd57053551e2ade4ae1308897070707da2d3 Mon Sep 17 00:00:00 2001 From: CDaut Date: Thu, 13 Oct 2022 12:59:24 +0200 Subject: [PATCH] Beispiele Kegel & Limiten --- chapters/Grundlagen.tex | 28 +++---- chapters/Kegel_und_Ko.tex | 151 +++++++++++++++++++++++++++++--------- main.pdf | Bin 351284 -> 354693 bytes main.tex | 4 + 4 files changed, 134 insertions(+), 49 deletions(-) diff --git a/chapters/Grundlagen.tex b/chapters/Grundlagen.tex index 974d144..10a9688 100644 --- a/chapters/Grundlagen.tex +++ b/chapters/Grundlagen.tex @@ -84,7 +84,7 @@ Die Objekte werden zu Knoten und die Morphismen zu Kanten. \cat{G} würde daher \end{itemize} Diese Kategorie sähe als Graph folgendermaßen aus: \begin{center} - \begin{tikzpicture}[every edge/.style = {draw, -to}] + \begin{tikzpicture} \node(1) at (1,0){$\cdot$}; \node(2) at (2,0){$\cdot$}; \node(3) at (3,0){$\cdot$}; @@ -124,7 +124,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. sodass $\mathcal{F}(f\circ g)=\mathcal{F}(f)\circ\mathcal{F}(g)$ \end{itemize} \begin{center} - \begin{tikzpicture}[every edge/.style = {draw, -to}] + \begin{tikzpicture} \node(A) at (0,0){$A$}; \node(C) at (0,-3){$C$}; \node(B) at (1,-1.5){$B$}; @@ -138,7 +138,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \hspace{10mm} Wird vom Funktor abgebildet auf \hspace{10mm} - \begin{tikzpicture}[every edge/.style = {draw, -to}] + \begin{tikzpicture} \node(A) at (0,0){$\mathcal{F}(A)$}; \node(C) at (0,-3){$\mathcal{F}(C)$}; \node(B) at (1,-1.5){$\mathcal{F}(B)$}; @@ -159,7 +159,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \begin{example}{Ein Endofunktor\\} Sei V ein $\mathbb{K}$-Vektorraum. $\mathcal{F}:\cat{K-Vec}\mapsto\cat{K-Vec}$ mit $W\mapsto V\times W$. Funktoren mit gleicher Definitions und Bildkategorie nennt man Endofunktoren. \begin{center} - \begin{tikzpicture}[every edge/.style = {draw, -to}] + \begin{tikzpicture} \node(W) at (0,0){$W$}; \node(X) at (0,-1.5){$X$}; @@ -182,7 +182,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. aus der Bildkategorie abgebildet. (Das Diagramm muss nicht unbedingt kommutieren)\\ Bleibt noch zu zeigen, dass $\mathcal{F}(g\circ f)=\mathcal{F}(g)\circ\mathcal{F}(f)$ gilt. \begin{center} - \begin{tikzpicture}[every edge/.style = {draw, -to}] + \begin{tikzpicture} \node(W1) at (0,0){$W_1$}; \node(W2) at (2, 0){$W_2$}; \node(W3) at (4,0){$W_3$}; @@ -195,7 +195,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \end{center} Also: \begin{center} - \begin{tikzpicture}[every edge/.style = {draw, -to}] + \begin{tikzpicture} \node(VW1) at (0,0){$V\times W_1$}; \node(VW3) at (4, 0){$V\times W_3$}; \node(vw) at (0,-1){$(v,w)$}; @@ -210,7 +210,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \end{center} und \begin{center} - \begin{tikzpicture}[every edge/.style = {draw, -to}] + \begin{tikzpicture} \node(VW1) at (0,0){$V\times W_1$}; \node(VW2) at (2,0){$V\times W_2$}; \node(VW3) at (4, 0){$V\times W_3$}; @@ -271,7 +271,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. $M\mapsto \text{Abb}_0(M,\mathbb{K})=\{\text{Abb } M\mapsto\mathbb{K} \text{ fast überall } 0\}$ (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}] + \begin{tikzpicture} \node(M) at (0,0){$M$}; \node(N) at (0,-2){$N$}; \node(FM) at (2,0){$\mathcal{F}(M)$}; @@ -293,7 +293,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \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}] + \begin{tikzpicture} \node (G1) at (0,0){$G_1$}; \node (G2) at (0, -2){$G_2$}; \node (G1ab) at (2,0){$G_1^{ab}$}; @@ -310,7 +310,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \begin{example}{Graphen\\} Definiere $\cat{Grph}$ exemplarisch als folgende Kategorie: \begin{center} - \begin{tikzpicture}[every edge/.style = {draw, -to}] + \begin{tikzpicture} \node (E) at (0,0){E $\cdot$}; \node (V) at (3,0){$\cdot$ V}; @@ -323,7 +323,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \end{center} Dann gibt es Funktoren $\mathcal{F}: \cat{Grph}\mapsto\cat{Set}$ mit \begin{center} - \begin{tikzpicture}[every edge/.style = {draw, -to}] + \begin{tikzpicture} \node (E) at (0,0){$\mathcal{F}(E)$ $\cdot$}; \node (V) at (6,0){$\cdot$ $\mathcal{F}(V)$}; @@ -356,7 +356,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. Objekte: $M\mapsto(M\cup \{\star_m\}, \star_M)$\\ Morphismen: \begin{center} - \begin{tikzpicture}[every edge/.style = {draw, -to}] + \begin{tikzpicture}[baseline=-10mm] \node (M) at (0,0){$M$}; \node (N) at (0,-2){$N$}; @@ -364,8 +364,8 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. (M) edge node[left]{$f$} (N) ; \end{tikzpicture} - $\mapsto$%TODO: THis needs to go higher... - \begin{tikzpicture}[every edge/.style = {draw, -to}] + $\mapsto$ + \begin{tikzpicture}[baseline=-10mm] \node (M) at (0,0){$M\cup\{\star_M\}$}; \node (N) at (0,-2){$N\cup\{\star_N\}$}; diff --git a/chapters/Kegel_und_Ko.tex b/chapters/Kegel_und_Ko.tex index 0cd8280..710db90 100644 --- a/chapters/Kegel_und_Ko.tex +++ b/chapters/Kegel_und_Ko.tex @@ -1,5 +1,5 @@ \section{Kegel und Ko} -\subsection{Nötige Definitionen} +\subsection{Kegel und Limiten} \begin{definition}{Diagramm\\} Ein Diagramm in einer Kategorie $\mathscr{C}$ ist ein Funktor\\ $\mathcal{F}:I\mapsto\mathscr{C}$ von einer kleinen\footnote{Eine Kategorie heißt klein, wenn die Klasse ihrer Morphismen eine Menge ist. Nicht kleine Kategorien werden hier nicht betrachtet.} Kategorie I. Ein Diagramm bettet also quasi eine kleine Kategorie @@ -11,45 +11,48 @@ alle Diagramme kommutieren. \end{definition} \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 [blue] (K) at (1.5,2) {$K$}; - \node (ul) at (-1, 3){}; - \node (lr) at (4, -1){}; - \node (C) at (-0.5, 2.5) {$\mathscr{C}$}; - \draw - (K) edge[blue] (Fi) - (K) edge[blue] (Fj) - (Fi) edge node[below]{$\mathcal{F}(f)$} (Fj) - (Fi) edge[loop, out=90+45, in=180, looseness=3] (Fi) - (Fj) edge[loop, out=90-45, in=0, looseness=3] (Fj) + \begin{figure}[h] + \centering + \begin{tikzpicture} + \node (Fi) at (0,0) {$\mathcal{F}(i)$}; + \node (Fj) at (3,0) {$\mathcal{F}(j)$}; + \node [blue] (K) at (1.5,2) {$K$}; + \node (ul) at (-1, 3){}; + \node (lr) at (4, -1){}; + \node (C) at (-0.5, 2.5) {$\mathscr{C}$}; + \draw + (K) edge[blue] (Fi) + (K) edge[blue] (Fj) + (Fi) edge node[below]{$\mathcal{F}(f)$} (Fj) + (Fi) edge[loop, out=90+45, in=180, looseness=3] (Fi) + (Fj) edge[loop, out=90-45, in=0, looseness=3] (Fj) - (ul) rectangle (lr); - ; - \end{tikzpicture}\\ - \begin{tikzpicture}[every edge/.style = {draw, -to}] - \node (i) at (0,0) {$i$}; - \node (j) at (3,0){$j$}; - \node (ul) at (-1,1.5){}; - \node (lr) at (4,-1){}; - \node (I) at (-0.5, 1){$I$}; + (ul) rectangle (lr); + ; + \end{tikzpicture}\\ + \begin{tikzpicture} + \node (i) at (0,0) {$i$}; + \node (j) at (3,0){$j$}; + \node (ul) at (-1,1.5){}; + \node (lr) at (4,-1){}; + \node (I) at (-0.5, 1){$I$}; - \draw - (i) edge node[below]{$f$} (j) - (i) edge[loop, out=90, in=180, looseness=4] (i) - (j) edge[loop, out=90, in=0, looseness=4] (j) - (ul) rectangle (lr) - ; - \end{tikzpicture} + \draw + (i) edge node[below]{$f$} (j) + (i) edge[loop, out=90, in=180, looseness=4] (i) + (j) edge[loop, out=90, in=0, looseness=4] (j) + (ul) rectangle (lr) + ; + \end{tikzpicture} + \caption{Eine Kategorie $\mathscr{C}$ und eine Kleine Kategorie $I$, die vom Diagramm + $\mathcal{F}$ in $\mathscr{C}$ abgebildet wird. Außerdem der Kegel $K$ in blau. } + \end{figure} \end{center} -$\mathcal{F}(f)\circ f_i=f_j$ 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''. +$\mathcal{F}(f)\circ f_i=f_j$ für alle $f\in\mor{I}{i}{j}$. \begin{definition}{Limes\\} Ein Limes über einem Diagramm $\mathcal{F}$ ist ein universeller Kegel L: \begin{center} - \begin{tikzpicture}[every edge/.style = {draw, -to}] + \begin{tikzpicture} \node (Fi) at (0,0) {$\mathcal{F}(i)$}; \node (Fj) at (3,0) {$\mathcal{F}(j)$}; \node [red] (L) at (1.5,1.5) {$L$}; @@ -68,4 +71,82 @@ der $I$ in $\mathscr{C}$ \glqq einbettet''. 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 +\end{definition} +Nachdem nun definiert wurde was Diagramme, Kegel und Limiten sind, sollen nun einige Beispiele +angeführt werden, wie eben solche Kegel und Limiten über bestimmten Kategorien +aussehen können. + +\begin{example}{Leere Kategorie\\} + Es sei $I$ die leere Kategorie. Also $\ob I=\varnothing=\mor{I}{\cdot}{\cdot}$.\\ + \begin{itemize} + \item Allgemein: $lim(I) =$ terminales Objekt + \item Für $\cat{set}:\text{Kegel}_I =\ob\cat{set}$ und $lim(I)=\{*\}$ + \item Für $\cat{grp}=\{e\}$ + \item Für $\cat{K-VR}=\{0\}$ + \end{itemize} + Dieses Beispiel ist relativ uninteressant, da $I$ keine Objekte enthält und somit + die Anforderungen um Kegel bzw. Limiten zu finden nicht besonders hoch sind. +\end{example} +\newpage +\begin{example}{Einelementige Kategorie\\} + Sei $I=$ + \begin{tikzpicture}[baseline=2mm] + \node (i) at (0,0) {i}; + \draw + (i) edge[loop, looseness=5] (i) + ; + \end{tikzpicture} + die Kategorie mit einem Element und dem Identitätsmorphismus. Betrachten + wir nun allgemein eine Kategorie $\mathscr{C}$ und ein Diagramm $\mathcal{F}: \mathscr{C}\mapsto I$. + Dann gilt $\text{Kegel}_{\mathscr{C}}(I)=$ + \begin{tikzpicture}[baseline=4mm] + \node (1)[blue] at (0,1) {$\cdot$}; + \node (2) at (0,0) {$\cdot$}; + \draw + (1) edge[blue] (2) + ; + \draw[decorate,decoration={brace}] (1.north east) -- (2.south east); + \draw[decorate,decoration={brace, mirror}] (1.north west) -- (2.south west); + \end{tikzpicture} + $=$Alle Objekte in $\mathscr{C}$, die einen Morphismus nach $\mathcal{F}(i)$ haben.\\ + $\text{lim}(\mathcal{F})=\mathcal{F}(i)$:\\ + \begin{center} + \begin{figure}[h] + \centering + \begin{minipage}{0.3\linewidth} + \centering + \begin{tikzpicture} + \node (Fi) at (0,0) {$\mathcal{F}(i)$}; + \node (L)[red] at (0,2){$\cdot$}; + \node (K)[blue] at (1,1){$\cdot$}; + + \draw + (L) edge[red] (Fi) + (K) edge[dotted] (L) + (K) edge[blue] (Fi) + ; + \end{tikzpicture} + \caption{Ein allgemeines Diagramm für Kegel (blau) und Limes (rot) über dieser Kategorie} + \end{minipage} + \hspace{10mm} + \begin{minipage}{0.3\linewidth} + \centering + \begin{tikzpicture} + \node (Fi) at (0,0) {$\mathcal{F}(i)$}; + \node (L)[red] at (0,2){$\mathcal{F}(i)$}; + \node (K)[blue] at (2,2){$\cdot$}; + + \draw + (L) edge[red] (Fi) + (K) edge[dotted] (L) + (K) edge[blue] (Fi) + ; + \end{tikzpicture} + \caption{Das selbe Diagramm erhält man, wenn man $\text{lim }\mathcal{F}=\mathcal{F}(i)$ setzt.} + \end{minipage} + \end{figure} + \end{center} +\end{example} +\begin{example}{Kategorie mit zwei Objekten\\} + +\end{example} \ No newline at end of file diff --git a/main.pdf b/main.pdf index 8f6c59afe4d30649f1d1225f7f8ab90da1598903..84488313c4092e455d9a3ac2875ffa42672aced4 100644 GIT binary patch delta 77000 zcmdn8L$vj>=!Q;4M$^fP%$oIAgKhl}8wlL}t!;Cw`RrZGvPYtwVF5Qbh4Kh;u5|TW zV)FLD`r`)|eraZG3p1Eoy7PP7pLdcGXB*C{8C-i^Q)GH|TH>OTq|T@3b&9)WHcpvS zv>-BW6YKumX-8MZ=eN4WmF7>a7n1t1>0|cC8TVP+rYN=)wlgyucs8ED$>F9bThCaZ zQN3R@?DL}Uxynn}vVDF?aFlhJC}}9A-Cd>`?iyR`6Wnk~gZ+0=hUn50Dp{#Q=0+

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a/main.tex b/main.tex index 7e09729..495c87d 100644 --- a/main.tex +++ b/main.tex @@ -12,6 +12,10 @@ \usepackage{tikz} \usetikzlibrary{positioning,decorations.pathreplacing} +\tikzset{ +every edge/.style = {draw, -to} +} + \biolinum \theoremstyle{definition}