From 8d9228ea70de239780bb0db0ec382813805a66ba Mon Sep 17 00:00:00 2001 From: CDaut Date: Tue, 27 Sep 2022 19:15:02 +0200 Subject: [PATCH] something on functors --- chapters/Grundlagen.tex | 113 ++++++++++++++++++++++++++++++++++++---- main.pdf | Bin 238681 -> 259180 bytes 2 files changed, 103 insertions(+), 10 deletions(-) diff --git a/chapters/Grundlagen.tex b/chapters/Grundlagen.tex index b879b20..a19ce57 100644 --- a/chapters/Grundlagen.tex +++ b/chapters/Grundlagen.tex @@ -19,8 +19,8 @@ Diese Definition ist bewusst recht allgemein gehalten um viele verschiedene Kate 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{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 $H$ @@ -28,7 +28,7 @@ Die folgenden Beispiele erfüllen alle oben genannten Axiome und sind daher Kate \end{itemize} Das ist wohldefiniert und \cat{Grp} ist eine Kategorie. \end{example} -\begin{example}{Mehr Beispiele} +\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 @@ -56,13 +56,106 @@ Die Objekte werden zu Knoten und die Morphismen zu Kanten. \cat{G} würde daher \begin{center} \begin{tikzpicture} - \node[circle,draw,thick](star){$*$}; + \node[circle,draw](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) - + (star) edge[out=321 , in=39 , loop] node[right] {f} (star) + (star) edge[out=51 , in=129 , loop] node[above] {g} (star) + (star) edge[out=141 , in=219 , loop] node[left] {$f\square g$} (star) + (star) edge[out=231 , in=309 , loop] node[below] {$e_G$} (star) ; \end{tikzpicture} -\end{center} \ No newline at end of file +\end{center} + +\begin{example}{Kategorie aus einer partiell geordneten Menge\\} + \label{ex:poset} + Sei $(P,\le)$ eine partiell geordnete Menge ($\le$ ist eine reflexive, transitive und antisymmetrische Relation). + $\cat{P}$ ist definiert als: + \begin{itemize} + \item $\ob\cat{P}:=P$ + \item für $x,y\in P$ definiere $\mor{\cat{P}}{x}{y}= + \begin{cases} + \{*_{xy}\} & x\le y \\ + \varnothing & x\nleq y + \end{cases} + $ + \item Sei $f\in\mor{\cat{P}}{x}{y}, g\in\mor{\cat{P}}{y}{z}$ (also $x=*_{xy}$ und $g=*_{yz}$) + dann $*_{yz}\circ *_{xy}=*_{xz}$ + \item $id_x=*_{xx}$ + \end{itemize} + Diese Kategorie sähe als Graph folgendermaßen aus: + \begin{center} + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node(1) at (1,0){$\cdot$}; + \node(2) at (2,0){$\cdot$}; + \node(3) at (3,0){$\cdot$}; + \node(4) at (4,0){$\cdot$}; + \node(5) at (5,0){$\dots$}; + + + \path + (1) edge (2) + (2) edge (3) + (3) edge (4) + (4) edge (5) + ; + \end{tikzpicture} + \end{center} + Die Identitätsmorphismen und und Kompositionen der Morphismen werden im Graphen der Übersichtlichkeit halber nicht gezeichnet. +\end{example} + + +\begin{definition}{Isomorphismus\\} + $f\in\mor{\mathscr{C}}{A}{B}$ heißt Isomorphismus wenn es $g\in\mor{\mathscr{C}}{B}{A}$ existiert + mit $f\circ g=id_B$ und $g\circ f=id_A$ +\end{definition} +Man kann leicht sehen, dass es in Beispiel \ref{ex:poset} keine nicht trivialen Isomorphismen geben kann. +Dazu wäre es nötig, dass Morphismen der From $*_{yx}$ und $*_{xy}$ existieren. Dazu wäre es aber +nötig, dass $x\le y$ und $y\le x$ gilt. Daraus folgt, dass $x=y$ gilt und somit der einzige Isomorphismus die Identität ist. +\subsection{Funktoren} +Es ist uns bereits Möglich Morphismen innerhalb von Kategorien zu definieren. Um auch zwischen Kategorien abbilden +zu können und diese so in Relation zu setzen definieren wir Funktoren. +\begin{definition}{Funktor (Kovariant)\\} + Seien $\mathscr{C},\mathscr{D}$ Kategorien. Ein Funktor $\mathcal{F}$ von $\mathscr{C}$ nach + $\mathscr{D}$ ist eine Zuordnung + \begin{itemize} + \item $\ob\mathscr{C}\mapsto\ob\mathscr{D}$ $(A\in\ob\mathscr{C}\mapsto\mathcal{F}(A))$ + \item und eine Zuordnung für $A,B\in\ob\mathscr{C}$ mit + $\mor{\mathscr{C}}{A}{B}\mapsto\mor{\mathscr{D}}{\mathcal{F}(A)}{\mathcal{F}(B)}$ + sodass $\mathcal{F}(f\circ g)=\mathcal{F}(f)\circ\mathcal{F}(g)$ + \end{itemize} + \begin{center} + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node(A) at (0,0){$A$}; + \node(C) at (0,-3){$C$}; + \node(B) at (1,-1.5){$B$}; + + \path + (A) edge node[left]{$f\circ g$} (C) + (A) edge node[above]{$f$} (B) + (B) edge node[right]{$g$} (C) + ; + \end{tikzpicture} + \hspace{10mm} + Wird vom Funktor abgebildet auf + \hspace{10mm} + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node(A) at (0,0){$\mathcal{F}(A)$}; + \node(C) at (0,-3){$\mathcal{F}(C)$}; + \node(B) at (1,-1.5){$\mathcal{F}(B)$}; + + \path + (A) edge node[left]{$\mathcal{F}(f\circ g)$} (C) + (A) edge node[right]{$\mathcal{F}(f)$} (B) + (B) edge node[right]{$\mathcal{F}(g)$} (C) + ; + \end{tikzpicture} + \end{center} +\end{definition} +\begin{example}{Vergissfunktor\\} + $?:\cat{K-Vec}\mapsto\cat{Set}$\\ + $(V,+,\cdot)\mapsto V$ bildet einen Vektorraum auf die Menge der Vektoren ab und vergisst dabei alle anderen Informationen + wie z.B. über welchem Körper der Vektorraum definiert war und jegliche Operationen. +\end{example} +\begin{example} + Sei V ein $\mathbb{K}$-Vektorraum +\end{example} \ No newline at end of file diff --git a/main.pdf b/main.pdf index 2357a07310665feacd2c9e26b244035239bfedf0..7d491550e1ef601592340ab3ecf8064b002050b3 100644 GIT binary patch delta 138278 zcmcb4g73`>{tdm1Or|E&CuTEi)}Nhd?|;}p;OO_7F6p-ipS}B|{!EoXUg#X_K@Jx~ALJ zZ*_m!yCUdzd649nb59TRSXiDqd2n*Fsm!jk>#|;9^JqwDh*@%CT&^*=^;e@m)b%fIml)7y^ClekhBPiJCbGBuhW*vYI>e>T#4@@+Gr zeZRvq@B+JC!)+$m6q9v_nr$;XnDr!F@_wcIO;bRsU1_?e4PtU|{uXRe#QU#!ryORPhS~j;25;<$D^{%2Ga^oi_eNq%G|XhR=>{o 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