From c9bc50332119d9a00c59cdd720b4fc801acbf3bf Mon Sep 17 00:00:00 2001 From: CDaut Date: Tue, 27 Sep 2022 14:06:35 +0200 Subject: [PATCH] one more example --- chapters/Grundlagen.tex | 37 +++++++++++++++++++++++++++++++------ main.pdf | Bin 227043 -> 238681 bytes main.tex | 1 + 3 files changed, 32 insertions(+), 6 deletions(-) diff --git a/chapters/Grundlagen.tex b/chapters/Grundlagen.tex index daf8c78..b879b20 100644 --- a/chapters/Grundlagen.tex +++ b/chapters/Grundlagen.tex @@ -6,13 +6,13 @@ Zunächst müssen wir definieren was eine Kategorie ist: Eine Kategorie $\mathscr{C}$ ist: \begin{itemize} \item eine Klasse von Objekten $\ob\mathscr{C}$ - \item für $A,B\in\ob\mathscr{C}$ eine Menge $\mor{\mathscr{C}}{A}{B}$ eine Menge - von Morphismen zwischen den Objekten - \item für $A,B,C\in\ob\mathscr{C}$ eine Abbildung $\circ: \mor{\mathscr{C}}{B}{C}\times\mor{\mathscr{C}}{A}{B}\mapsto\mor{\mathscr{C}}{A}{C}$ also - Verkettung von Morphismen + \item für $A,B\in\ob\mathscr{C}$ eine Menge $\mor{\mathscr{C}}{A}{B}$ eine Menge + von Morphismen zwischen den Objekten + \item für $A,B,C\in\ob\mathscr{C}$ eine Abbildung $\circ: \mor{\mathscr{C}}{B}{C}\times\mor{\mathscr{C}}{A}{B}\mapsto\mor{\mathscr{C}}{A}{C}$ also + Verkettung von Morphismen \item Assoziativität der Morphismen bezüglich $\circ$ % TODO:Definition \item Für jedes $A\in\ob\mathscr{C}$ ein $id_A\in\mor{\mathscr{C}}{A}{A}$ mit - $id_A \circ f=f$ und $g\circ id_A=g$ für alle $f\in\mor{\mathscr{C}}{B}{A}, g\in\mor{\mathscr{C}}{A}{B}$ + $id_A \circ f=f$ und $g\circ id_A=g$ für alle $f\in\mor{\mathscr{C}}{B}{A}, g\in\mor{\mathscr{C}}{A}{B}$ \end{itemize} \end{definition} Diese Definition ist bewusst recht allgemein gehalten um viele verschiedene Kategorien zu erlauben. @@ -30,7 +30,7 @@ Die folgenden Beispiele erfüllen alle oben genannten Axiome und sind daher Kate \end{example} \begin{example}{Mehr Beispiele} \begin{itemize} - \item \cat{$\mathbb{K}$-VR} die Kategorie der $\mathbb{K}$-Vektorräume und Vektorraumhomomorphismen + \item \cat{$\mathbb{K}$-VR} die Kategorie der $\mathbb{K}$-Vektorräume und Vektorraumhomomorphismen \item \cat{Ringe} Die Kategorie der Ringe und Ringhomomorphismen \item \cat{Körper} Die Kategorie der Körper und Körperhomomorphismen \item \cat{Set} Die Kategorie der Mengen und aller Abbildungen @@ -41,3 +41,28 @@ Die folgenden Beispiele erfüllen alle oben genannten Axiome und sind daher Kate \item \cat{Set$^*$} Kategorie der punktierten Mengen ($(M,m)$, $M$ Menge, $m\in M$ $(M,m)\xmapsto{f}(N,n)$ Abbildung mit $f(m)=n$) \end{itemize} \end{example} +\begin{example}{Eine Kategorie in der die Morphismen keine Abbildungen sind\\} + Sei $(G,\square)$ eine Gruppe. $\cat{G}$ ist definiert als: + \begin{itemize} + \item $\ob G :=\{*\}$ + \item $\mor{\cat{G}}{*}{*}=G$ + \item für $f,g\in\mor{\cat{G}}{*}{*}$ definiere $f\circ g:=f\square g$ + \item $id_*=e_G$ + \end{itemize} + Die Morphismen von \cat{G} sind hier also keine Abbildungen, sondern die Elemente der Gruppe. +\end{example} +Oft werden Kategorien mit Hilfe von Graphen dargestellt. +Die Objekte werden zu Knoten und die Morphismen zu Kanten. \cat{G} würde daher folgendermaßen dargestellt: + +\begin{center} + \begin{tikzpicture} + \node[circle,draw,thick](star){$*$}; + \path + (star) edge[loop,thick] node {f} (star) + (star) edge[loop,thick] node {g} (star) + (star) edge[loop,thick] node {$f\square g$}(star) + (star) edge[loop,thick] node {$e_G$}(star) + + ; + \end{tikzpicture} +\end{center} \ No newline at end of file diff --git a/main.pdf b/main.pdf index 2851725b3a094c1a1308cf800303e00c40655bad..2357a07310665feacd2c9e26b244035239bfedf0 100644 GIT binary patch delta 89169 zcmaF-miOifJ{8pf7dKr)JyR~#3s;Wb;4)J%P{>cp;Y&C+H}$~?lvqgObYr&P=Q>G$Nv3W|mse#HHLckj91 zGXsY9zit18-|s$q*U!u0=$g$;yM#HGOKca?@k-m;xUN0CXvYupOx@fiStl!_R#klQ zmUw2;>T!Q}d(ILDOJlZD!>p3i0Xt<>opo1VEUWKQG%pUnqFKGl@b064c~M43XT4g! z^_e&WZ{)FnFQ+akh)h29;QIwD<){pvbgyJr+u!|{Ow+UzJx@F+o~nPP&}g~p&qJbp 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