3080 We want $\cM$ to be an equivalence, so we need 2-morphisms in $\cS$ |
3080 We want $\cM$ to be an equivalence, so we need 2-morphisms in $\cS$ |
3081 between ${}_\cC\cM_\cD \otimes_\cD {}_\cD\cM_\cC$ and the identity 0-sphere module ${}_\cC\cC_\cC$, and similarly |
3081 between ${}_\cC\cM_\cD \otimes_\cD {}_\cD\cM_\cC$ and the identity 0-sphere module ${}_\cC\cC_\cC$, and similarly |
3082 with the roles of $\cC$ and $\cD$ reversed. |
3082 with the roles of $\cC$ and $\cD$ reversed. |
3083 These 2-morphisms come for free, in the sense of not requiring additional data, since we can take them to be the labeled |
3083 These 2-morphisms come for free, in the sense of not requiring additional data, since we can take them to be the labeled |
3084 cell complexes (cups and caps) in $B^2$ shown in Figure \ref{morita-fig-1}. |
3084 cell complexes (cups and caps) in $B^2$ shown in Figure \ref{morita-fig-1}. |
|
3085 |
|
3086 \definecolor{C}{named}{orange} |
|
3087 \definecolor{D}{named}{blue} |
|
3088 \definecolor{M}{named}{purple} |
|
3089 |
|
3090 |
3085 \begin{figure}[t] |
3091 \begin{figure}[t] |
|
3092 \todo{Verify that the tikz figure is correct, remove the hand-drawn one.} |
3086 $$\mathfig{.65}{tempkw/morita1}$$ |
3093 $$\mathfig{.65}{tempkw/morita1}$$ |
3087 |
|
3088 |
3094 |
3089 $$ |
3095 $$ |
3090 \begin{tikzpicture} |
3096 \begin{tikzpicture} |
3091 \node(L) at (0,0) {\tikz{ |
3097 \node(L) at (0,0) {\tikz{ |
3092 \draw[orange] (0,0) -- node[below] {$\cC$} (1,0); |
3098 \draw[C] (0,0) -- node[below] {$\cC$} (1,0); |
3093 \draw[blue] (1,0) -- node[below] {$\cD$} (2,0); |
3099 \draw[D] (1,0) -- node[below] {$\cD$} (2,0); |
3094 \draw[orange] (2,0) -- node[below] {$\cC$} (3,0); |
3100 \draw[C] (2,0) -- node[below] {$\cC$} (3,0); |
3095 \node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (1,0) {}; |
3101 \node[M, fill, circle, inner sep=2pt, label=$\cM$] at (1,0) {}; |
3096 \node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (2,0) {}; |
3102 \node[M, fill, circle, inner sep=2pt, label=$\cM$] at (2,0) {}; |
3097 }}; |
3103 }}; |
3098 |
3104 |
3099 \node(R) at (6,0) {\tikz{ |
3105 \node(R) at (6,0) {\tikz{ |
3100 \draw[orange] (0,0) -- node[below] {$\cC$} (3,0); |
3106 \draw[C] (0,0) -- node[below] {$\cC$} (3,0); |
3101 \node[label={\phantom{$\cM$}}] at (1.5,0) {}; |
3107 \node[label={\phantom{$\cM$}}] at (1.5,0) {}; |
3102 }}; |
3108 }}; |
3103 |
3109 |
3104 \node at (-1,-1.5) { $\leftidx{_\cC}{(\cM \tensor_\cD \cM)}{_\cC}$ }; |
3110 \node at (-1,-1.5) { $\leftidx{_\cC}{(\cM \tensor_\cD \cM)}{_\cC}$ }; |
3105 \node at (7,-1.5) { $\leftidx{_\cC}{\cC}{_\cC}$ }; |
3111 \node at (7,-1.5) { $\leftidx{_\cC}{\cC}{_\cC}$ }; |
3106 |
3112 |
3107 \draw[->] (L) to[out=35, in=145] node[below] {$w$} node[above] { \tikz{ |
3113 \draw[->] (L) to[out=35, in=145] node[below] {$w$} node[above] { \tikz{ |
3108 \draw (0,0) circle (16pt); |
3114 \draw (0,0) circle (16pt); |
|
3115 \path[clip] (0,0) circle (16pt); |
|
3116 \draw[fill=C!20] (0,0) circle (16pt); |
|
3117 \draw[M,fill=D!20,line width=2pt] (0,-0.5) circle (16pt); |
3109 }}(R); |
3118 }}(R); |
3110 |
3119 |
3111 \draw[->] (R) to[out=-145, in=-35] node[above] {$x$} node[below] { \tikz{ |
3120 \draw[->] (R) to[out=-145, in=-35] node[above] {$x$} node[below] { \tikz{ |
3112 \draw (0,0) circle (16pt); |
3121 \draw (0,0) circle (16pt); |
|
3122 \path[clip] (0,0) circle (16pt); |
|
3123 \draw[fill=C!20] (0,0) circle (16pt); |
|
3124 \draw[M,fill=D!20,line width=2pt] (0,0.5) circle (16pt); |
3113 }}(L); |
3125 }}(L); |
3114 |
3126 |
3115 |
3127 |
3116 \end{tikzpicture} |
3128 \end{tikzpicture} |
3117 $$ |
3129 $$ |
3118 |
3130 $$ |
|
3131 \begin{tikzpicture} |
|
3132 \node(L) at (0,0) {\tikz{ |
|
3133 \draw[D] (0,0) -- node[below] {$\cD$} (1,0); |
|
3134 \draw[C] (1,0) -- node[below] {$\cC$} (2,0); |
|
3135 \draw[D] (2,0) -- node[below] {$\cD$} (3,0); |
|
3136 \node[M, fill, circle, inner sep=2pt, label=$\cM$] at (1,0) {}; |
|
3137 \node[M, fill, circle, inner sep=2pt, label=$\cM$] at (2,0) {}; |
|
3138 }}; |
|
3139 |
|
3140 \node(R) at (6,0) {\tikz{ |
|
3141 \draw[D] (0,0) -- node[below] {$\cD$} (3,0); |
|
3142 \node[label={\phantom{$\cM$}}] at (1.5,0) {}; |
|
3143 }}; |
|
3144 |
|
3145 \node at (-1,-1.5) { $\leftidx{_\cD}{(\cM \tensor_\cC \cM)}{_\cD}$ }; |
|
3146 \node at (7,-1.5) { $\leftidx{_\cD}{\cD}{_\cD}$ }; |
|
3147 |
|
3148 \draw[->] (L) to[out=35, in=145] node[below] {$y$} node[above] { \tikz{ |
|
3149 \draw (0,0) circle (16pt); |
|
3150 \path[clip] (0,0) circle (16pt); |
|
3151 \draw[fill=D!20] (0,0) circle (16pt); |
|
3152 \draw[M,fill=C!20,line width=2pt] (0,-0.5) circle (16pt); |
|
3153 }}(R); |
|
3154 |
|
3155 \draw[->] (R) to[out=-145, in=-35] node[above] {$z$} node[below] { \tikz{ |
|
3156 \draw (0,0) circle (16pt); |
|
3157 \path[clip] (0,0) circle (16pt); |
|
3158 \draw[fill=D!20] (0,0) circle (16pt); |
|
3159 \draw[M,fill=C!20,line width=2pt] (0,0.5) circle (16pt); |
|
3160 }}(L); |
|
3161 |
|
3162 |
|
3163 \end{tikzpicture} |
|
3164 $$ |
3119 \caption{Cups and caps for free}\label{morita-fig-1} |
3165 \caption{Cups and caps for free}\label{morita-fig-1} |
3120 \end{figure} |
3166 \end{figure} |
3121 |
3167 |
3122 |
3168 |
3123 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |
3169 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |
3125 Recall that the 3-morphisms of $\cS$ are intertwiners between representations of 1-categories associated |
3171 Recall that the 3-morphisms of $\cS$ are intertwiners between representations of 1-categories associated |
3126 to decorated circles. |
3172 to decorated circles. |
3127 Figure \ref{morita-fig-2} |
3173 Figure \ref{morita-fig-2} |
3128 \begin{figure}[t] |
3174 \begin{figure}[t] |
3129 $$\mathfig{.55}{tempkw/morita2}$$ |
3175 $$\mathfig{.55}{tempkw/morita2}$$ |
|
3176 $$ |
|
3177 \begin{tikzpicture} |
|
3178 \node(L) at (0,0) {\tikz{ |
|
3179 \draw[fill=C!20] (0,0) circle (32pt); |
|
3180 \draw[M,fill=D!20,line width=2pt] (0,0) circle (16pt); |
|
3181 }}; |
|
3182 \node(R) at (4,0) {\tikz{ |
|
3183 \draw[fill=C!20] (0,0) circle (32pt); |
|
3184 }}; |
|
3185 \draw[->] (L) to[out=35, in=145] node[below] {$a$} (R); |
|
3186 \draw[->] (R) to[out=-145, in=-35] node[above] {$b$} (L); |
|
3187 \node at (-2,0) {$w \atop x$}; |
|
3188 \node at (6,0) {$1$}; |
|
3189 \end{tikzpicture} |
|
3190 $$ |
|
3191 $$ |
|
3192 \begin{tikzpicture} |
|
3193 \node(L) at (0,0) {\tikz{ |
|
3194 \draw[fill=C!20] (0,0) circle (32pt); |
|
3195 \path[clip] (0,0) circle (32pt); |
|
3196 \draw[M,fill=D!20,line width=2pt] (0,1) circle (16pt); |
|
3197 \draw[M,fill=D!20,line width=2pt] (0,-1) circle (16pt); |
|
3198 }}; |
|
3199 \node(R) at (4,0) {\tikz{ |
|
3200 \draw[fill=D!20] (0,0) circle (32pt); |
|
3201 \path[clip] (0,0) circle (32pt); |
|
3202 \draw[M,fill=C!20,line width=2pt] (5,0) circle (130pt); |
|
3203 \draw[M,fill=C!20,line width=2pt] (-5,0) circle (130pt); |
|
3204 }}; |
|
3205 \draw[->] (L) to[out=35, in=145] node[below] {$c$} (R); |
|
3206 \draw[->] (R) to[out=-145, in=-35] node[above] {$d$} (L); |
|
3207 \node at (-2,0) {$x \atop w$}; |
|
3208 \node at (6,0) {$1$}; |
|
3209 \end{tikzpicture} |
|
3210 $$ |
|
3211 $$ |
|
3212 \begin{tikzpicture} |
|
3213 \node(L) at (0,0) {\tikz{ |
|
3214 \draw[fill=D!20] (0,0) circle (32pt); |
|
3215 \draw[M,fill=C!20,line width=2pt] (0,0) circle (16pt); |
|
3216 }}; |
|
3217 \node(R) at (4,0) {\tikz{ |
|
3218 \draw[fill=D!20] (0,0) circle (32pt); |
|
3219 }}; |
|
3220 \draw[->] (L) to[out=35, in=145] node[below] {$e$} (R); |
|
3221 \draw[->] (R) to[out=-145, in=-35] node[above] {$f$} (L); |
|
3222 \node at (-2,0) {$y \atop z$}; |
|
3223 \node at (6,0) {$1$}; |
|
3224 \end{tikzpicture} |
|
3225 $$ |
|
3226 $$ |
|
3227 \begin{tikzpicture} |
|
3228 \node(L) at (0,0) {\tikz{ |
|
3229 \draw[fill=D!20] (0,0) circle (32pt); |
|
3230 \path[clip] (0,0) circle (32pt); |
|
3231 \draw[M,fill=C!20,line width=2pt] (0,1) circle (16pt); |
|
3232 \draw[M,fill=C!20,line width=2pt] (0,-1) circle (16pt); |
|
3233 }}; |
|
3234 \node(R) at (4,0) {\tikz{ |
|
3235 \draw[fill=C!20] (0,0) circle (32pt); |
|
3236 \path[clip] (0,0) circle (32pt); |
|
3237 \draw[M,fill=D!20,line width=2pt] (5,0) circle (130pt); |
|
3238 \draw[M,fill=D!20,line width=2pt] (-5,0) circle (130pt); |
|
3239 }}; |
|
3240 \draw[->] (L) to[out=35, in=145] node[below] {$g$} (R); |
|
3241 \draw[->] (R) to[out=-145, in=-35] node[above] {$h$} (L); |
|
3242 \node at (-2,0) {$z \atop y$}; |
|
3243 \node at (6,0) {$1$}; |
|
3244 \end{tikzpicture} |
|
3245 $$ |
|
3246 |
3130 \caption{intertwiners for a Morita equivalence}\label{morita-fig-2} |
3247 \caption{intertwiners for a Morita equivalence}\label{morita-fig-2} |
3131 \end{figure} |
3248 \end{figure} |
3132 shows the intertwiners we need. |
3249 shows the intertwiners we need. |
3133 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
3250 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
3134 on the boundary. |
3251 on the boundary. |
3140 they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition |
3257 they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition |
3141 they must satisfy identities corresponding to Morse cancellations on 2-manifolds. |
3258 they must satisfy identities corresponding to Morse cancellations on 2-manifolds. |
3142 These are illustrated in Figure \ref{morita-fig-3}. |
3259 These are illustrated in Figure \ref{morita-fig-3}. |
3143 \begin{figure}[t] |
3260 \begin{figure}[t] |
3144 $$\mathfig{.65}{tempkw/morita3}$$ |
3261 $$\mathfig{.65}{tempkw/morita3}$$ |
|
3262 $$ |
|
3263 \begin{tikzpicture} |
|
3264 \node(L) at (0,0) {\tikz{ |
|
3265 \draw[fill=C!20] (0,0) circle (32pt); |
|
3266 \path[clip] (0,0) circle (32pt); |
|
3267 \draw[M,fill=D!20,line width=2pt] (-5,0) circle (130pt); |
|
3268 }}; |
|
3269 \node(C) at (4,0) {\tikz{ |
|
3270 \draw[fill=C!20] (0,0) circle (32pt); |
|
3271 \path[clip] (0,0) circle (32pt); |
|
3272 \draw[M,fill=D!20,line width=2pt] (-5,0) circle (130pt); |
|
3273 \draw[M,fill=D!20,line width=2pt] (0.25,0) circle (6pt); |
|
3274 }}; |
|
3275 \node(R) at (8,0) {\tikz{ |
|
3276 \draw[fill=C!20] (0,0) circle (32pt); |
|
3277 \path[clip] (0,0) circle (32pt); |
|
3278 \draw[M,line width=4pt] (-0.75,2) .. controls +(0,-2) and +(0,0.5) .. (0.2,0) .. controls +(0,-0.5) and +(0,2) .. (-0.75,-2) -- (-5,-2) -- (-5,2) -- cycle; |
|
3279 \path[clip] (-0.75,2) .. controls +(0,-2) and +(0,0.5) .. (0.2,0) .. controls +(0,-0.5) and +(0,2) .. (-0.75,-2) -- (-5,-2) -- (-5,2) -- cycle; |
|
3280 \path[fill=D!20] (-5,-2) rectangle (5,2); |
|
3281 }}; |
|
3282 \draw[<-] (L) to[out=35, in=145] node[above] {$a$} (C); |
|
3283 \draw[<-] (C) to[out=35, in=145] node[above] {$d$} (R); |
|
3284 \draw[<-] (R) to[out=-145, in=-35] node[below] {$c$} (C); |
|
3285 \draw[<-] (C) to[out=-145, in=-35] node[below] {$b$} (L); |
|
3286 \end{tikzpicture} |
|
3287 $$ |
|
3288 $$ |
|
3289 \begin{tikzpicture} |
|
3290 \node(L) at (0,0) {\tikz{ |
|
3291 \draw[fill=D!20] (0,0) circle (32pt); |
|
3292 \path[clip] (0,0) circle (32pt); |
|
3293 \draw[M,fill=C!20,line width=2pt] (-5,0) circle (130pt); |
|
3294 }}; |
|
3295 \node(C) at (4,0) {\tikz{ |
|
3296 \draw[fill=D!20] (0,0) circle (32pt); |
|
3297 \path[clip] (0,0) circle (32pt); |
|
3298 \draw[M,fill=C!20,line width=2pt] (-5,0) circle (130pt); |
|
3299 \draw[M,fill=C!20,line width=2pt] (0.25,0) circle (6pt); |
|
3300 }}; |
|
3301 \node(R) at (8,0) {\tikz{ |
|
3302 \draw[fill=D!20] (0,0) circle (32pt); |
|
3303 \path[clip] (0,0) circle (32pt); |
|
3304 \draw[M,line width=4pt] (-0.75,2) .. controls +(0,-2) and +(0,0.5) .. (0.2,0) .. controls +(0,-0.5) and +(0,2) .. (-0.75,-2) -- (-5,-2) -- (-5,2) -- cycle; |
|
3305 \path[clip] (-0.75,2) .. controls +(0,-2) and +(0,0.5) .. (0.2,0) .. controls +(0,-0.5) and +(0,2) .. (-0.75,-2) -- (-5,-2) -- (-5,2) -- cycle; |
|
3306 \path[fill=C!20] (-5,-2) rectangle (5,2); |
|
3307 }}; |
|
3308 \draw[<-] (L) to[out=35, in=145] node[above] {$e$} (C); |
|
3309 \draw[<-] (C) to[out=35, in=145] node[above] {$c$} (R); |
|
3310 \draw[<-] (R) to[out=-145, in=-35] node[below] {$d$} (C); |
|
3311 \draw[<-] (C) to[out=-145, in=-35] node[below] {$f$} (L); |
|
3312 \end{tikzpicture} |
|
3313 $$ |
3145 \caption{Identities for intertwiners}\label{morita-fig-3} |
3314 \caption{Identities for intertwiners}\label{morita-fig-3} |
3146 \end{figure} |
3315 \end{figure} |
3147 Each line shows a composition of two intertwiners which we require to be equal to the identity intertwiner. |
3316 Each line shows a composition of two intertwiners which we require to be equal to the identity intertwiner. |
3148 The modules corresponding leftmost and rightmost disks in the figure can be identified via the obvious isotopy. |
3317 The modules corresponding leftmost and rightmost disks in the figure can be identified via the obvious isotopy. |
3149 |
3318 |