From 62b1f746868b0572b61752cc5b15afc94f60702a Mon Sep 17 00:00:00 2001 From: CDaut Date: Sun, 16 Oct 2022 19:55:26 +0200 Subject: [PATCH] =?UTF-8?q?\cdot=20zu=20\bullet=20ge=C3=A4ndert?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- chapters/Grundlagen.tex | 16 +++++------ chapters/Kegel_und_Ko.tex | 58 +++++++++++++++++++------------------- main.pdf | Bin 316746 -> 316751 bytes 3 files changed, 37 insertions(+), 37 deletions(-) diff --git a/chapters/Grundlagen.tex b/chapters/Grundlagen.tex index a4c1874..abdaeaa 100644 --- a/chapters/Grundlagen.tex +++ b/chapters/Grundlagen.tex @@ -85,10 +85,10 @@ Die Objekte werden zu Knoten und die Morphismen zu Kanten. \cat{G} würde daher Diese Kategorie sähe als Graph folgendermaßen aus: \begin{center} \begin{tikzpicture} - \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(1) at (1,0){$\bullet$}; + \node(2) at (2,0){$\bullet$}; + \node(3) at (3,0){$\bullet$}; + \node(4) at (4,0){$\bullet$}; \node(5) at (5,0){$\dots$}; @@ -311,8 +311,8 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. Definiere $\cat{Grph}$ exemplarisch als folgende Kategorie: \begin{center} \begin{tikzpicture} - \node (E) at (0,0){E $\cdot$}; - \node (V) at (3,0){$\cdot$ V}; + \node (E) at (0,0){E}; + \node (V) at (3,0){V}; \path (E) edge node[above]{$s,t,u,\dots$} (V) @@ -324,8 +324,8 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. Dann gibt es Funktoren $\mathcal{F}: \cat{Grph}\mapsto\cat{Set}$ mit \begin{center} \begin{tikzpicture} - \node (E) at (0,0){$\mathcal{F}(E)$ $\cdot$}; - \node (V) at (6,0){$\cdot$ $\mathcal{F}(V)$}; + \node (E) at (0,0){$\mathcal{F}(E)$}; + \node (V) at (6,0){$\mathcal{F}(V)$}; \path (E) edge node[above]{$\mathcal{F}(s),\mathcal{F}(t),\mathcal{F}(u),\dots$} (V) diff --git a/chapters/Kegel_und_Ko.tex b/chapters/Kegel_und_Ko.tex index 9cbb22a..4275a1c 100644 --- a/chapters/Kegel_und_Ko.tex +++ b/chapters/Kegel_und_Ko.tex @@ -100,8 +100,8 @@ aussehen können. wir nun allgemein eine Kategorie $\mathscr{C}$ und ein Diagramm $\mathcal{F}: \mathscr{C}\mapsto I$. Dann gilt $\text{Kegel}_{\mathscr{C}}(I)=$ \begin{tikzpicture}[baseline=4mm] - \node (1)[blue] at (0,1) {$\cdot$}; - \node (2) at (0,0) {$\cdot$}; + \node (1)[blue] at (0,1) {$\bullet$}; + \node (2) at (0,0) {$\bullet$}; \draw (1) edge[blue] (2) ; @@ -117,8 +117,8 @@ aussehen können. \centering \begin{tikzpicture} \node (Fi) at (0,0) {$\mathcal{F}(i)$}; - \node (L)[red] at (0,2){$\cdot$}; - \node (K)[blue] at (1,1){$\cdot$}; + \node (L)[red] at (0,2){$\bullet$}; + \node (K)[blue] at (1,1){$\bullet$}; \draw (L) edge[red] (Fi) @@ -134,7 +134,7 @@ aussehen können. \begin{tikzpicture} \node (Fi) at (0,0) {$\mathcal{F}(i)$}; \node (L)[red] at (0,2){$\mathcal{F}(i)$}; - \node (K)[blue] at (2,2){$\cdot$}; + \node (K)[blue] at (2,2){$\bullet$}; \draw (L) edge[red] (Fi) @@ -150,8 +150,8 @@ aussehen können. \begin{example}{Kategorie mit zwei Objekten\\} Es sei I= \begin{tikzpicture}[baseline=-1mm] - \node(A) at (0,0){$\cdot$}; - \node(B) at (0.5,0){$\cdot$}; + \node(A) at (0,0){$\bullet$}; + \node(B) at (0.5,0){$\bullet$}; \node (center) at (0.25, 0){}; \draw (center) ellipse (0.5 and 0.25); @@ -162,8 +162,8 @@ aussehen können. \begin{tikzpicture} - \node (Fi) at (0,0) {$\cdot$}; - \node (Fj) at (3,0) {$\cdot$}; + \node (Fi) at (0,0) {$\bullet$}; + \node (Fj) at (3,0) {$\bullet$}; \node [red] (L) at (1.5,1.5) {$L$}; \node [blue] (K) at (1.5,3) {$K$}; @@ -179,8 +179,8 @@ aussehen können. Für \cat{set}:\\ gegeben $M,N$. Dann ist $\text{lim}\left( \begin{tikzpicture}[baseline=-1mm] - \node(A) at (0,0){$\cdot$}; - \node(B) at (0.5,0){$\cdot$}; + \node(A) at (0,0){$\bullet$}; + \node(B) at (0.5,0){$\bullet$}; \node (center) at (0.25, 0){}; \draw (center) ellipse (0.5 and 0.25); @@ -267,8 +267,8 @@ aussehen können. Kokegel: \begin{tikzpicture}[baseline=5mm] \node[blue] (K) at (0, 0){$K_{co}$}; - \node (P1) at (-1, 1){$\cdot$}; - \node (P2) at (1, 1){$\cdot$}; + \node (P1) at (-1, 1){$\bullet$}; + \node (P2) at (1, 1){$\bullet$}; \node (center) at (0,1){}; \draw @@ -284,8 +284,8 @@ aussehen können. \begin{center} \begin{tikzpicture} \node[red] (L) at (0, 0){$L_{co}$}; - \node (P1) at (-2, 2){$\cdot$}; - \node (P2) at (2, 2){$\cdot$}; + \node (P1) at (-2, 2){$\bullet$}; + \node (P2) at (2, 2){$\bullet$}; \node (center) at (0,2){}; \node[blue] (K) at (0, -1.5){$K_{co}$}; @@ -332,8 +332,8 @@ aussehen können. \begin{example} {$I=$ \begin{tikzpicture}[baseline=-1mm] - \node (P1) at (0,0){$\cdot$}; - \node (P2) at (0.5,0){$\cdot$}; + \node (P1) at (0,0){$\bullet$}; + \node (P2) at (0.5,0){$\bullet$}; \node (center) at (0.25,0){}; \draw (center) ellipse (0.5 and 0.25); \end{tikzpicture} @@ -414,9 +414,9 @@ aussehen können. \begin{definition}{Pullback\\} Ein Pullback ist der Limes des Diagramms \begin{tikzpicture}[baseline=0mm] - \node (A) at (0,0){$\cdot$}; - \node (B) at (1,0){$\cdot$}; - \node (C) at (1,1){$\cdot$}; + \node (A) at (0,0){$\bullet$}; + \node (B) at (1,0){$\bullet$}; + \node (C) at (1,1){$\bullet$}; \draw (A) edge (B) @@ -427,9 +427,9 @@ aussehen können. \begin{figure}[h] \begin{center} \begin{tikzpicture} - \node (A) at (0,0){$\cdot$}; - \node (B) at (2,0){$\cdot$}; - \node (C) at (2,2){$\cdot$}; + \node (A) at (0,0){$\bullet$}; + \node (B) at (2,0){$\bullet$}; + \node (C) at (2,2){$\bullet$}; \node[red] (L) at (0,2){$P$}; \node[blue] (T) at (-1,3){$T$}; @@ -507,9 +507,9 @@ aussehen können. \begin{definition}{Pushout\\} Ein Pushout ist der Kolimes des Diagramms \begin{tikzpicture}[baseline=-5mm] - \node (X) at (0,0){$\cdot$}; - \node (V) at (0,-1){$\cdot$}; - \node (W) at (1,0){$\cdot$}; + \node (X) at (0,0){$\bullet$}; + \node (V) at (0,-1){$\bullet$}; + \node (W) at (1,0){$\bullet$}; \draw (X) edge (V) @@ -520,9 +520,9 @@ aussehen können. \begin{figure}[h] 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