From aa571440405916a0d9bba1e883d2591bd1597cd7 Mon Sep 17 00:00:00 2001 From: CDaut Date: Tue, 27 Sep 2022 03:17:44 +0200 Subject: [PATCH] basics --- chapters/Grundlagen.tex | 43 ++++++++++++++++++++++++++++++++++++++++ main.pdf | Bin 95118 -> 227043 bytes main.tex | 15 ++++++++++++++ 3 files changed, 58 insertions(+) create mode 100644 chapters/Grundlagen.tex diff --git a/chapters/Grundlagen.tex b/chapters/Grundlagen.tex new file mode 100644 index 0000000..daf8c78 --- /dev/null +++ b/chapters/Grundlagen.tex @@ -0,0 +1,43 @@ +\section{Grundlagen} +Oft betrachtet man in der Kategorientheorie Kategorien und sogenannte Funktoren, die quasi als Abbildungen zwischen Kategorien aufgefasst werden können. +\subsection{Kategorien} +Zunächst müssen wir definieren was eine Kategorie ist: +\begin{definition}[Kategorie] + 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 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 $f\in\mor{\mathscr{C}}{B}{A}, g\in\mor{\mathscr{C}}{A}{B}$ + \end{itemize} +\end{definition} +Diese Definition ist bewusst recht allgemein gehalten um viele verschiedene Kategorien zu erlauben. +Die Morphismen müssen dabei nicht unbedingt Abbildungen sein, sondern lediglich die oben genannte Definition erfüllen. +Eine Klasse kann hier für alle Zwecke als Menge betrachtet werden.\\ +Die folgenden Beispiele erfüllen alle oben genannten Axiome und sind daher Kategorien: +\begin{example}{\cat{Grp}} + Ist die Kategorie der Gruppen\\ + \begin{itemize} + \item $\ob\cat{Grp}=$ alle Gruppen + \item $\mor{\cat{Grp}}{G}{H}=$ Gruppenhompmorphismen von $G$ nach $H$ + \item $\circ:$ Komposition von Gruppenhomomorphismen + \end{itemize} + Das ist wohldefiniert und \cat{Grp} ist eine Kategorie. +\end{example} +\begin{example}{Mehr Beispiele} + \begin{itemize} + \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 + \item \cat{Ab} Abelsche Gruppen und Gruppenhomomorphismen + \item \cat{Top} Topologische Räume und stetige Abbildungen + \item \cat{MRäume} Metrische Räume + \item \cat{Euklid} Euklidische Vektorräume und Subisometrien + \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} diff --git a/main.pdf b/main.pdf index 841ec9fc4472e185136fb4400c6a5314678ca407..2851725b3a094c1a1308cf800303e00c40655bad 100644 GIT binary patch delta 184510 zcmeBs&ieQ*?}l1NW+Mah$qkH}^?N65>^o$@)Arr2^>~WMxw1pdDCltgd4%zHIZ zK`3kKwvq=gIYs79$V=ap^80t)&C?MzYhoEbUHf@JfPaI)=|cetS--L;*KKt!&APmx{*j)vmcp6WPUXx^58q5SIJ~6SVAc9OpUHfFSCd~w<;t`r zZaQ;V=ZEsDnGa^(_%gk$eqI0B75~Wbh>C<vhk?j1vx8QjowEb8<#q4OK5dh3ZDD^92E^}eO2z1t}Bl-}#2d26m*GbkX`gxqtPWXI~c0XT7~-^`_J; zKkjEw`nTlVuwy;Arm1d98()z`m8n*4$e!<6kC)7=TkzM;Lq2W0%(J5_KV%t3Tb=Q9 z?l?ZD`2CKg_0u_7CcjzyQaRt{#IcDftFJCGTEli(zalJj%HL?-ho@F0viIbkJ-E`+ zqt1_esiE$!mzm2BhqX1#dRPD3UGE5!XV*Lyx5?7{oLy1R6tYcMEZ?$9BB8mf$Ao{H z+}A4MKdU?c`JeV^H}~>)U)y$iR+6{gLhYRi=l6M7)ND}xbEk0qazm}B4eKh8-rN1O zqD7g-__y|9`xBqybAL_mlD=r~n7s6GNABUce6y+>P7Av~zIB}W)gyPKd200LdQOQ& zzqPCL=dhj>YWTc8w}feF!}ST>v!2u_RW_{QozPg{^2TjeIz#xLyc>L)tXm38dLta1 zCkB_wzF=RapLu$w)tPIS_a6sNtYPH8_-r$`sgtzG-uWdVr-~izK3Hb)J~5Qz$p1R2 zcK-|G=d2CQEyhthe~MkdU+KUSa98*ob7Z|^(u4|$@bzp?5kWuqSzbD+x<~HKy#Qer zHJ@qym67Xg4IURx*ukVQb<=^Ms0@bc8Dz@c&KNzb%uj4!+|&anGvv!olvnn|8!5KIo>sKlXHAdssukCg%lbT$5BvX_D?ru5SfB@{<%kC|+c7QZg}g5M~ysKePG5r8l<@ev`VkFZ|Db z_X`IruQ(L<^R3w#dB{`t@%M&Tt#9{yUR-zJ{bHFv8Vw(euccJ09~XSaFD7BTXrp^` z`b@Se<+xxz&EL1mb;SG2e8df}U4U2eW z^wy8b{JVCSzux_yVbvWSi^ZUPiJC)=3=|9$Kv~tshD+ZkH7~s+L&3aqelJN zNbkwF&4l*-4%aYWrRDTN<-n3}27ZqUjh=dimYr%(DOxanN?Ug49932SQ~T?7Gk##? 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+\theoremstyle{definition} +\newtheorem{definition}{Definition}[section] +\newtheorem{example}{Beispiel}[section] + +\newcommand{\ob}{\mathsf{ob}\text{ }} +\newcommand{\mor}[3]{\mathsf{Hom}_#1(#2,#3)} +\newcommand{\cat}[1]{\textnormal{\textsf{\textbf{#1}}}} + \begin{document} \include{./chapters/deckblatt.tex} \include{./chapters/motivation.tex} +\include{./chapters/Grundlagen.tex} \end{document} \ No newline at end of file