# HG changeset patch # User Scott Morrison # Date 1322706713 28800 # Node ID 50af564d0e043e2c8772eaa0c486e49366354541 # Parent 54503d88c9690d96ba2fee6ac92188a73d5546db preliminary tikz diagrams for all the morita equivalences diff -r 54503d88c969 -r 50af564d0e04 text/ncat.tex --- a/text/ncat.tex Wed Nov 30 16:29:23 2011 -0800 +++ b/text/ncat.tex Wed Nov 30 18:31:53 2011 -0800 @@ -3082,22 +3082,28 @@ with the roles of $\cC$ and $\cD$ reversed. These 2-morphisms come for free, in the sense of not requiring additional data, since we can take them to be the labeled cell complexes (cups and caps) in $B^2$ shown in Figure \ref{morita-fig-1}. + +\definecolor{C}{named}{orange} +\definecolor{D}{named}{blue} +\definecolor{M}{named}{purple} + + \begin{figure}[t] +\todo{Verify that the tikz figure is correct, remove the hand-drawn one.} $$\mathfig{.65}{tempkw/morita1}$$ - $$ \begin{tikzpicture} \node(L) at (0,0) {\tikz{ - \draw[orange] (0,0) -- node[below] {$\cC$} (1,0); - \draw[blue] (1,0) -- node[below] {$\cD$} (2,0); - \draw[orange] (2,0) -- node[below] {$\cC$} (3,0); - \node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (1,0) {}; - \node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (2,0) {}; + \draw[C] (0,0) -- node[below] {$\cC$} (1,0); + \draw[D] (1,0) -- node[below] {$\cD$} (2,0); + \draw[C] (2,0) -- node[below] {$\cC$} (3,0); + \node[M, fill, circle, inner sep=2pt, label=$\cM$] at (1,0) {}; + \node[M, fill, circle, inner sep=2pt, label=$\cM$] at (2,0) {}; }}; \node(R) at (6,0) {\tikz{ - \draw[orange] (0,0) -- node[below] {$\cC$} (3,0); + \draw[C] (0,0) -- node[below] {$\cC$} (3,0); \node[label={\phantom{$\cM$}}] at (1.5,0) {}; }}; @@ -3106,16 +3112,56 @@ \draw[->] (L) to[out=35, in=145] node[below] {$w$} node[above] { \tikz{ \draw (0,0) circle (16pt); + \path[clip] (0,0) circle (16pt); + \draw[fill=C!20] (0,0) circle (16pt); + \draw[M,fill=D!20,line width=2pt] (0,-0.5) circle (16pt); }}(R); \draw[->] (R) to[out=-145, in=-35] node[above] {$x$} node[below] { \tikz{ \draw (0,0) circle (16pt); + \path[clip] (0,0) circle (16pt); + \draw[fill=C!20] (0,0) circle (16pt); + \draw[M,fill=D!20,line width=2pt] (0,0.5) circle (16pt); }}(L); \end{tikzpicture} $$ - +$$ +\begin{tikzpicture} +\node(L) at (0,0) {\tikz{ + \draw[D] (0,0) -- node[below] {$\cD$} (1,0); + \draw[C] (1,0) -- node[below] {$\cC$} (2,0); + \draw[D] (2,0) -- node[below] {$\cD$} (3,0); + \node[M, fill, circle, inner sep=2pt, label=$\cM$] at (1,0) {}; + \node[M, fill, circle, inner sep=2pt, label=$\cM$] at (2,0) {}; +}}; + +\node(R) at (6,0) {\tikz{ + \draw[D] (0,0) -- node[below] {$\cD$} (3,0); + \node[label={\phantom{$\cM$}}] at (1.5,0) {}; +}}; + +\node at (-1,-1.5) { $\leftidx{_\cD}{(\cM \tensor_\cC \cM)}{_\cD}$ }; +\node at (7,-1.5) { $\leftidx{_\cD}{\cD}{_\cD}$ }; + +\draw[->] (L) to[out=35, in=145] node[below] {$y$} node[above] { \tikz{ + \draw (0,0) circle (16pt); + \path[clip] (0,0) circle (16pt); + \draw[fill=D!20] (0,0) circle (16pt); + \draw[M,fill=C!20,line width=2pt] (0,-0.5) circle (16pt); +}}(R); + +\draw[->] (R) to[out=-145, in=-35] node[above] {$z$} node[below] { \tikz{ + \draw (0,0) circle (16pt); + \path[clip] (0,0) circle (16pt); + \draw[fill=D!20] (0,0) circle (16pt); + \draw[M,fill=C!20,line width=2pt] (0,0.5) circle (16pt); +}}(L); + + +\end{tikzpicture} +$$ \caption{Cups and caps for free}\label{morita-fig-1} \end{figure} @@ -3127,6 +3173,77 @@ Figure \ref{morita-fig-2} \begin{figure}[t] $$\mathfig{.55}{tempkw/morita2}$$ +$$ +\begin{tikzpicture} +\node(L) at (0,0) {\tikz{ + \draw[fill=C!20] (0,0) circle (32pt); + \draw[M,fill=D!20,line width=2pt] (0,0) circle (16pt); +}}; +\node(R) at (4,0) {\tikz{ + \draw[fill=C!20] (0,0) circle (32pt); +}}; +\draw[->] (L) to[out=35, in=145] node[below] {$a$} (R); +\draw[->] (R) to[out=-145, in=-35] node[above] {$b$} (L); +\node at (-2,0) {$w \atop x$}; +\node at (6,0) {$1$}; +\end{tikzpicture} +$$ +$$ +\begin{tikzpicture} +\node(L) at (0,0) {\tikz{ + \draw[fill=C!20] (0,0) circle (32pt); + \path[clip] (0,0) circle (32pt); + \draw[M,fill=D!20,line width=2pt] (0,1) circle (16pt); + \draw[M,fill=D!20,line width=2pt] (0,-1) circle (16pt); +}}; +\node(R) at (4,0) {\tikz{ + \draw[fill=D!20] (0,0) circle (32pt); + \path[clip] (0,0) circle (32pt); + \draw[M,fill=C!20,line width=2pt] (5,0) circle (130pt); + \draw[M,fill=C!20,line width=2pt] (-5,0) circle (130pt); +}}; +\draw[->] (L) to[out=35, in=145] node[below] {$c$} (R); +\draw[->] (R) to[out=-145, in=-35] node[above] {$d$} (L); +\node at (-2,0) {$x \atop w$}; +\node at (6,0) {$1$}; +\end{tikzpicture} +$$ +$$ +\begin{tikzpicture} +\node(L) at (0,0) {\tikz{ + \draw[fill=D!20] (0,0) circle (32pt); + \draw[M,fill=C!20,line width=2pt] (0,0) circle (16pt); +}}; +\node(R) at (4,0) {\tikz{ + \draw[fill=D!20] (0,0) circle (32pt); +}}; +\draw[->] (L) to[out=35, in=145] node[below] {$e$} (R); +\draw[->] (R) to[out=-145, in=-35] node[above] {$f$} (L); +\node at (-2,0) {$y \atop z$}; +\node at (6,0) {$1$}; +\end{tikzpicture} +$$ +$$ +\begin{tikzpicture} +\node(L) at (0,0) {\tikz{ + \draw[fill=D!20] (0,0) circle (32pt); + \path[clip] (0,0) circle (32pt); + \draw[M,fill=C!20,line width=2pt] (0,1) circle (16pt); + \draw[M,fill=C!20,line width=2pt] (0,-1) circle (16pt); +}}; +\node(R) at (4,0) {\tikz{ + \draw[fill=C!20] (0,0) circle (32pt); + \path[clip] (0,0) circle (32pt); + \draw[M,fill=D!20,line width=2pt] (5,0) circle (130pt); + \draw[M,fill=D!20,line width=2pt] (-5,0) circle (130pt); +}}; +\draw[->] (L) to[out=35, in=145] node[below] {$g$} (R); +\draw[->] (R) to[out=-145, in=-35] node[above] {$h$} (L); +\node at (-2,0) {$z \atop y$}; +\node at (6,0) {$1$}; +\end{tikzpicture} +$$ + \caption{intertwiners for a Morita equivalence}\label{morita-fig-2} \end{figure} shows the intertwiners we need. @@ -3142,6 +3259,58 @@ These are illustrated in Figure \ref{morita-fig-3}. \begin{figure}[t] $$\mathfig{.65}{tempkw/morita3}$$ +$$ +\begin{tikzpicture} +\node(L) at (0,0) {\tikz{ +\draw[fill=C!20] (0,0) circle (32pt); +\path[clip] (0,0) circle (32pt); +\draw[M,fill=D!20,line width=2pt] (-5,0) circle (130pt); +}}; +\node(C) at (4,0) {\tikz{ +\draw[fill=C!20] (0,0) circle (32pt); +\path[clip] (0,0) circle (32pt); +\draw[M,fill=D!20,line width=2pt] (-5,0) circle (130pt); +\draw[M,fill=D!20,line width=2pt] (0.25,0) circle (6pt); +}}; +\node(R) at (8,0) {\tikz{ +\draw[fill=C!20] (0,0) circle (32pt); +\path[clip] (0,0) circle (32pt); +\draw[M,line width=4pt] (-0.75,2) .. controls +(0,-2) and +(0,0.5) .. (0.2,0) .. controls +(0,-0.5) and +(0,2) .. (-0.75,-2) -- (-5,-2) -- (-5,2) -- cycle; +\path[clip] (-0.75,2) .. controls +(0,-2) and +(0,0.5) .. (0.2,0) .. controls +(0,-0.5) and +(0,2) .. (-0.75,-2) -- (-5,-2) -- (-5,2) -- cycle; +\path[fill=D!20] (-5,-2) rectangle (5,2); +}}; +\draw[<-] (L) to[out=35, in=145] node[above] {$a$} (C); +\draw[<-] (C) to[out=35, in=145] node[above] {$d$} (R); +\draw[<-] (R) to[out=-145, in=-35] node[below] {$c$} (C); +\draw[<-] (C) to[out=-145, in=-35] node[below] {$b$} (L); +\end{tikzpicture} +$$ +$$ +\begin{tikzpicture} +\node(L) at (0,0) {\tikz{ +\draw[fill=D!20] (0,0) circle (32pt); +\path[clip] (0,0) circle (32pt); +\draw[M,fill=C!20,line width=2pt] (-5,0) circle (130pt); +}}; +\node(C) at (4,0) {\tikz{ +\draw[fill=D!20] (0,0) circle (32pt); +\path[clip] (0,0) circle (32pt); +\draw[M,fill=C!20,line width=2pt] (-5,0) circle (130pt); +\draw[M,fill=C!20,line width=2pt] (0.25,0) circle (6pt); +}}; +\node(R) at (8,0) {\tikz{ +\draw[fill=D!20] (0,0) circle (32pt); +\path[clip] (0,0) circle (32pt); +\draw[M,line width=4pt] (-0.75,2) .. controls +(0,-2) and +(0,0.5) .. (0.2,0) .. controls +(0,-0.5) and +(0,2) .. (-0.75,-2) -- (-5,-2) -- (-5,2) -- cycle; +\path[clip] (-0.75,2) .. controls +(0,-2) and +(0,0.5) .. (0.2,0) .. controls +(0,-0.5) and +(0,2) .. (-0.75,-2) -- (-5,-2) -- (-5,2) -- cycle; +\path[fill=C!20] (-5,-2) rectangle (5,2); +}}; +\draw[<-] (L) to[out=35, in=145] node[above] {$e$} (C); +\draw[<-] (C) to[out=35, in=145] node[above] {$c$} (R); +\draw[<-] (R) to[out=-145, in=-35] node[below] {$d$} (C); +\draw[<-] (C) to[out=-145, in=-35] node[below] {$f$} (L); +\end{tikzpicture} +$$ \caption{Identities for intertwiners}\label{morita-fig-3} \end{figure} Each line shows a composition of two intertwiners which we require to be equal to the identity intertwiner.