From 34e67174f5a241261a49cde8f761b5705f889cb2 Mon Sep 17 00:00:00 2001 From: CDaut Date: Sat, 8 Oct 2022 17:01:49 +0200 Subject: [PATCH] Co vs Kontravariant --- chapters/Grundlagen.tex | 11 ++++++++++- main.pdf | Bin 296292 -> 297339 bytes main.tex | 1 + 3 files changed, 11 insertions(+), 1 deletion(-) diff --git a/chapters/Grundlagen.tex b/chapters/Grundlagen.tex index ff8d5ff..8bfe5f5 100644 --- a/chapters/Grundlagen.tex +++ b/chapters/Grundlagen.tex @@ -372,4 +372,13 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \end{center} Man fügt als ein Element hinzu, das jetzt das punktierte Element ist und definiert die Morphismen so dass sie auf die Elemente der punktierten Menge wie vorher angewandt werden und das punktierte Element in M auf das punktierte Element in N abgebildet wird. -\end{example} \ No newline at end of file +\end{example} +\begin{definition}{Opposite Kategorien\\} + Sei $\mathscr{C}$ eine Kategorie. Definiere $\mathscr{C}^{op}$ wie folgt:\\ + \begin{itemize} + \item $\ob\mathscr{C}^{op}:=\ob \mathscr{C}$ + \item Für $A,B\in\ob\mathscr{C^{op}}$ definiere $\mor{\mathscr{C}^{op}}{A}{B}:=\mor{\mathscr{C}}{B}{A}$ + \end{itemize} + Wir drehen also alle \glqq Morphismenpfeile'' um und nutzen die gleichen Objekte.\\ +\end{definition} +Man sieht, dass ein kontravarianter Funktor $\mathscr{C}\mapsto\mathscr{D}$ einem kovarianten Funktor $\mathscr{C}^{op}\mapsto\mathscr{D}^{op}$ entspricht. \ No newline at end of file diff --git a/main.pdf b/main.pdf index 92a80be525cabd8a4fb2300a0320f2595f9ec201..fa78e1b755aa7ec1aef6ea776d4dce766f2f7ba5 100644 GIT binary patch delta 33977 zcmaFTCG>le&;~a~>6m33C&}Awd)m7#^U{v6I@SL_*T!D2Va{eKGT)rYIMtrT$i%>G zdSMicM*Y>O%=|mYB>%08Ke#1cliSy)&b&Bj#-$5y{;KL} zD{Va`Vq9@-^Y_nhoa>HdRQ~$-xBqVY{e5DK*LQ?Tt(W<+U#DKjp@PM!^O9`GuVaTj zqz)D&?Z{00w=b@0!KzathO0^pdW3X+!XC5-^8YaBKXyi7y3!V*n+&hktU2Wn;j&x7 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