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\cdot zu \bullet geändert

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CDaut 2022-10-16 19:55:26 +02:00 • committed by CDaut
parent bce8db5665
commit 62b1f74686
3 changed files with 37 additions and 37 deletions
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16
chapters/Grundlagen.tex
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@ -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)

58
chapters/Kegel_und_Ko.tex
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@ -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]
\begin{center}
\begin{tikzpicture}
\node (X) at (0,0){$\cdot$};
\node (V) at (0,-2){$\cdot$};
\node (W) at (2,0){$\cdot$};
\node (X) at (0,0){$\bullet$};
\node (V) at (0,-2){$\bullet$};
\node (W) at (2,0){$\bullet$};
\node[red] (P) at (2,-2){$P$};
\node[blue] (T) at (3,-3){$T$};

BIN
main.pdf
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