From 54220ccac9ecd5fa0d1a9186ecc82fa951975403 Mon Sep 17 00:00:00 2001 From: CDaut Date: Sat, 1 Oct 2022 00:40:14 +0200 Subject: [PATCH] contravariant Hom and pointed sets --- chapters/Grundlagen.tex | 51 ++++++++++++++++++++++++++++++++++++---- main.pdf | Bin 296466 -> 296292 bytes main.tex | 2 +- 3 files changed, 48 insertions(+), 5 deletions(-) diff --git a/chapters/Grundlagen.tex b/chapters/Grundlagen.tex index 5024e46..ff8d5ff 100644 --- a/chapters/Grundlagen.tex +++ b/chapters/Grundlagen.tex @@ -259,7 +259,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \end{aligned} \right] $\\ - Man schränkt den Ring also auf alle invertierbaren Ringelemente ein und + Man schränkt den Ring also auf alle invertierbaren Ringelemente ein und lässt nur noch die Abbildungen übrig, die ohnehin schon zwischen invertierbaren Elementen abgebildet haben. \end{example} \begin{example}{$\cat{Set}\mapsto\cat{K-VR}$\\} @@ -282,7 +282,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. ; \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} @@ -309,7 +309,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \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) @@ -322,11 +322,54 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \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} +\begin{example}{Der Hom-Funktor (Kontravariant)\\} + Es sei $\mathscr{C}$ eine Kategorie, $X\in\ob\mathscr{C}$\\ + Definiere den Funktor $\mor{\mathscr{C}}{\_}{X}: \mathscr{C}\mapsto\cat{Set}$ + \begin{itemize} + \item $Y\mapsto\mor{\mathscr{C}}{Y}{X}$\\ + Ein $Y\in\mathscr{C}$ wird also auf die Morphismenmenge $\mor{\mathscr{C}}{Y}{X}$, die Morphismen, die in $\mathscr{C}$ von Y auf X existieren, abgebildet. + \item $[Y\xrightarrow{f}Z]\mapsto\mor{\mathscr{C}}{X}{f}:= + \left[ + \begin{aligned} + \mor{\mathscr{C}}{Z}{X}\mapsto\mor{\mathscr{C}}{Y}{X} \\ + [g:Z\mapsto X]\mapsto[g\circ f:Y\mapsto X] + \end{aligned} + \right] + $\\ + Der Funktor bildet also jeden Morphismus $g$ von $Z$ nach $X$ auf die oben definierte Funktion ab. + \end{itemize} +\end{example} +\begin{example}{Punktierung von Mengen\\} + Definitions und Bildkategorie: $\cat{Set}\mapsto\cat{Set}^*$\\ + Objekte: $M\mapsto(M\cup \{\star_m\}, \star_M)$\\ + Morphismen: + \begin{center} + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node (M) at (0,0){$M$}; + \node (N) at (0,-2){$N$}; + + \path + (M) edge node[left]{$f$} (N) + ; + \end{tikzpicture} + $\mapsto$%TODO: THis needs to go higher... + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node (M) at (0,0){$M\cup\{\star_M\}$}; + \node (N) at (0,-2){$N\cup\{\star_N\}$}; + + \path + (M) edge node[right]{$f\cup id_\star:m\mapsto f(m) \text{ und } \star_M\mapsto\star_N$} (N) + ; + \end{tikzpicture} + \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 diff --git a/main.pdf b/main.pdf index ad30f629c6899c468f99b0a0d9ecdc920e9dacdc..92a80be525cabd8a4fb2300a0320f2595f9ec201 100644 GIT binary patch delta 108022 zcmbQ#BlM(8Xv0P}CNtydiMlKrlW((e)oYlq(mKSXFz@E6#5qSEr<{7a>dh4Xn>U$d 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a/main.tex +++ b/main.tex @@ -16,7 +16,7 @@ \theoremstyle{definition} \newtheorem{definition}{Definition}[section] -\newtheorem{example}{Beispiel}[section] +\newtheorem{example}{Beispiel}[subsection] \newcommand{\ob}{\mathsf{ob}\text{ }} \newcommand{\mor}[3]{\mathsf{Hom}_#1(#2,#3)}