blob: comparison text/ncat.tex

equal deleted inserted replaced

-:7d7f9e7c5869
+:3311fa1c93b9
 \begin{figure}[t]
 $$
 \begin{tikzpicture}[baseline=0]
 \begin{scope}
 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4);
-\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);
+\draw[kw-blue-a,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);
 \foreach \x in {0, 0.5, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- (\x,2);
 }
 \end{scope}
-\draw[blue,line width=1.5pt] (0,-3) -- (5.66,-3);
+\draw[kw-blue-a,line width=1.5pt] (0,-3) -- (5.66,-3);
 \draw[->,red,line width=2pt] (2.83,-1.5) -- (2.83,-2.5);
 \end{tikzpicture}
 \qquad \qquad
 \begin{tikzpicture}[baseline=-0.15cm]
 \begin{scope}
 \path[clip] (0,1) arc (90:135:8 and 4)  arc (-135:-90:8 and 4) -- cycle;
-\draw[blue,line width=2pt] (0,1) arc (90:135:8 and 4)  arc (-135:-90:8 and 4) -- cycle;
+\draw[kw-blue-a,line width=2pt] (0,1) arc (90:135:8 and 4)  arc (-135:-90:8 and 4) -- cycle;
 \foreach \x in {-6, -5.5, ..., 0} {
 	\draw[green!50!brown] (\x,-2) -- (\x,2);
 }
 \end{scope}
-\draw[blue,line width=1.5pt] (-5.66,-3.15) -- (0,-3.15);
+\draw[kw-blue-a,line width=1.5pt] (-5.66,-3.15) -- (0,-3.15);
 \draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5);
 \end{tikzpicture}
 $$
 \caption{Examples of pinched products}\label{pinched_prods}
 \end{figure}
 \begin{figure}[t]
 $$
 \begin{tikzpicture}[baseline=0]
 \begin{scope}
 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4);
-\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);
+\draw[kw-blue-a,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);
-\draw[blue] (0,0) -- (5.66,0);
+\draw[kw-blue-a] (0,0) -- (5.66,0);
 \foreach \x in {0, 0.5, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- (\x,2);
 }
 \end{scope}
 \end{tikzpicture}
 \qquad
 \begin{tikzpicture}[baseline=0]
 \begin{scope}
 \path[clip] (0,-1) rectangle (4,1);
-\draw[blue,line width=2pt] (0,-1) rectangle (4,1);
+\draw[kw-blue-a,line width=2pt] (0,-1) rectangle (4,1);
-\draw[blue] (0,0) -- (5,0);
+\draw[kw-blue-a] (0,0) -- (5,0);
 \foreach \x in {0, 0.5, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- (\x,2);
 }
 \end{scope}
 \end{tikzpicture}
 \qquad
 \begin{tikzpicture}[baseline=0]
 \begin{scope}
 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4);
-\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);
+\draw[kw-blue-a,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);
-\draw[blue] (2.83,3) circle (3);
+\draw[kw-blue-a] (2.83,3) circle (3);
 \foreach \x in {0, 0.5, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- (\x,2);
 }
 \end{scope}
 \end{tikzpicture}
 $$
 $$
 \begin{tikzpicture}[baseline=0]
 \begin{scope}
 \path[clip] (0,-1) rectangle (4,1);
-\draw[blue,line width=2pt] (0,-1) rectangle (4,1);
+\draw[kw-blue-a,line width=2pt] (0,-1) rectangle (4,1);
-\draw[blue] (0,-1) -- (4,1);
+\draw[kw-blue-a] (0,-1) -- (4,1);
 \foreach \x in {0, 0.5, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- (\x,2);
 }
 \end{scope}
 \end{tikzpicture}
 \qquad
 \begin{tikzpicture}[baseline=0]
 \begin{scope}
 \path[clip] (0,-1) rectangle (5,1);
-\draw[blue,line width=2pt] (0,-1) rectangle (5,1);
+\draw[kw-blue-a,line width=2pt] (0,-1) rectangle (5,1);
-\draw[blue] (1,-1) .. controls  (2,-1) and (3,1) .. (4,1);
+\draw[kw-blue-a] (1,-1) .. controls  (2,-1) and (3,1) .. (4,1);
 \foreach \x in {0, 0.5, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- (\x,2);
 }
 \end{scope}
 \end{tikzpicture}
 \qquad
 \begin{tikzpicture}[baseline=0]
 \begin{scope}
 \path[clip] (0,0) arc (135:45:4) arc (-45:-135:4);
-\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);
+\draw[kw-blue-a,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);
-\draw[blue] (2.82,-5) -- (2.83,5);
+\draw[kw-blue-a] (2.82,-5) -- (2.83,5);
 \foreach \x in {0, 0.5, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- (\x,2);
 }
 \end{scope}
 \end{tikzpicture}
 \node[right=1mm] at (0.east) {$a$};
 \draw[->] ($(0.east)+(0.75,0)$) -- ($(1.west)+(-0.2,0)$);
 \draw (1-small)  circle (\srad);
 \foreach \theta in {90, 72, ..., -90} {
-	\draw[blue] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$);
+	\draw[kw-blue-a] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$);
 }
 \filldraw[fill=white] (1) circle (\rad);
 \foreach \n in {1,2} {
 	\fill (intersection \n of 1-small and 1) circle (2pt);
 }
 \begin{scope}
 \path[clip] (2) circle (\rad);
 \draw[clip] (2.east) circle (\srad);
 \foreach \y in {1, 0.86, ..., -1} {
-	\draw[blue] ($(2)+(-1,\y) $)-- ($(2)+(1,\y)$);
+	\draw[kw-blue-a] ($(2)+(-1,\y) $)-- ($(2)+(1,\y)$);
 }
 \end{scope}
 \end{tikzpicture}
 \end{equation*}
 \begin{equation*}
 $(B^k, B^{k-1})$.
 See Figure \ref{feb21a}.
 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$.
 \begin{figure}[t]
-$$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue][fill=blue!30!white] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$
+$$\tikz[baseline,line width=2pt]{\draw[kw-blue-a] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[kw-blue-a][fill=kw-blue-a!30!white] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$
 \caption{0-marked 1-ball and 0-marked 2-ball}
 \label{feb21a}
 \end{figure}
 The $0$-marked balls can be cut into smaller balls in various ways.
 The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$
 (see Figure \ref{feb21b}).
 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$.
 \begin{figure}[t] \centering
-\begin{tikzpicture}[blue,line width=2pt]
+\begin{tikzpicture}[kw-blue-a,line width=2pt]
 \draw (0,1) -- (0,-1) node[below] {$X$};
 \draw (2,0) -- (4,0) node[below] {$J$};
 \fill[red] (3,0) circle (0.1);
-\draw[fill=blue!30!white] (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4);
+\draw[fill=kw-blue-a!30!white] (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4);
 \draw[red] (top.center) -- (bottom.center);
 \fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$};
 \fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$};
 \path (bottom) node[below]{$X \times J$};
 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category.
 This amounts to a definition of taking tensor products of $0$-sphere modules over $n$-categories.
 \begin{figure}[t] \centering
 \begin{tikzpicture}[baseline,line width = 2pt]
-\draw[blue] (0,0) -- (6,0);
+\draw[kw-blue-a] (0,0) -- (6,0);
 \foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} {
 	\path (\x,0)  node[below] {\color{green!50!brown}$\cA_{\n}$};
 }
 \foreach \x/\n in {1/0,2/1,4/2,5/3} {
 	\fill[red] (\x,0) circle (0.1) node[above] {\color{green!50!brown}$\cM_{\n}$};
 }
 \end{tikzpicture}
 \qquad
 \qquad
 \begin{tikzpicture}[baseline,line width = 2pt]
-\draw[blue] (0,0) circle (2);
+\draw[kw-blue-a] (0,0) circle (2);
 \foreach \q/\n in {-45/0,90/1,180/2} {
 	\path (\q:2.4)  node {\color{green!50!brown}$\cA_{\n}$};
 }
 \foreach \q/\n in {60/0,120/1,-120/2} {
 	\fill[red] (\q:2) circle (0.1);
 (See Figure \ref{subdividing1marked}.)
 We now proceed as in the above module definitions.
 \begin{figure}[t] \centering
 \begin{tikzpicture}[baseline,line width = 2pt]
-\draw[blue][fill=blue!15!white] (0,0) circle (2);
+\draw[kw-blue-a][fill=kw-blue-a!15!white] (0,0) circle (2);
 \fill[red] (0,0) circle (0.1);
 \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} {
 	\draw[red] (0,0) -- (\qm:2);
 	\path (\qa:1) node {\color{green!50!brown} $\cA_\n$};
 	\path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$};
 \label{feb21d}
 \end{figure}
 \begin{figure}[t] \centering
 \begin{tikzpicture}[baseline,line width = 2pt]
-\draw[blue][fill=blue!15!white] (0,0) circle (2);
+\draw[kw-blue-a][fill=kw-blue-a!15!white] (0,0) circle (2);
 \fill[red] (0,0) circle (0.1);
 \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} {
 	\draw[red] (0,0) -- (\qm:2);
 %	\path (\qa:1) node {\color{green!50!brown} $\cA_\n$};
 %	\path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$};
 \definecolor{D}{named}{blue}
 \definecolor{M}{named}{purple}
 \begin{figure}[t]
-\todo{Verify that the tikz figure is correct, remove the hand-drawn one.}
+%$$\mathfig{.65}{tempkw/morita1}$$
-$$\mathfig{.65}{tempkw/morita1}$$
 $$
 \begin{tikzpicture}
 \node(L) at (0,0) {\tikz{
 	\draw[C] (0,0) -- node[below] {$\cC$} (1,0);
 between various compositions of these 2-morphisms and various identity 2-morphisms.
 Recall that the 3-morphisms of $\cS$ are intertwiners between representations of 1-categories associated
 to decorated circles.
 Figure \ref{morita-fig-2}
 \begin{figure}[t]
-$$\mathfig{.55}{tempkw/morita2}$$
+%$$\mathfig{.55}{tempkw/morita2}$$
 $$
 \begin{tikzpicture}
 \node(L) at (0,0) {\tikz{
 	\draw[fill=C!20] (0,0) circle (32pt);
 	\draw[M,fill=D!20,line width=2pt] (0,0) circle (16pt);
 In order for these 3-morphisms to be equivalences,
 they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition
 they must satisfy identities corresponding to Morse cancellations on 2-manifolds.
 These are illustrated in Figure \ref{morita-fig-3}.
 \begin{figure}[t]
-$$\mathfig{.65}{tempkw/morita3}$$
+%$$\mathfig{.65}{tempkw/morita3}$$
 $$
 \begin{tikzpicture}
 \node(L) at (0,0) {\tikz{
 \draw[fill=C!20] (0,0) circle (32pt);
 \path[clip] (0,0) circle (32pt);

changeset 931	3311fa1c93b9
parent 930	7d7f9e7c5869
child 945	341c2a09f9a8