2616 with the roles of $\cC$ and $\cD$ reversed. |
2616 with the roles of $\cC$ and $\cD$ reversed. |
2617 These 2-morphisms come for free, in the sense of not requiring additional data, since we can take them to be the labeled |
2617 These 2-morphisms come for free, in the sense of not requiring additional data, since we can take them to be the labeled |
2618 cell complexes (cups and caps) in $B^2$ shown in Figure \ref{morita-fig-1}. |
2618 cell complexes (cups and caps) in $B^2$ shown in Figure \ref{morita-fig-1}. |
2619 \begin{figure}[t] |
2619 \begin{figure}[t] |
2620 $$\mathfig{.65}{tempkw/morita1}$$ |
2620 $$\mathfig{.65}{tempkw/morita1}$$ |
|
2621 |
|
2622 |
|
2623 $$ |
|
2624 \begin{tikzpicture} |
|
2625 \node(L) at (0,0) {\tikz{ |
|
2626 \draw[orange] (0,0) -- node[below] {$\cC$} (1,0); |
|
2627 \draw[blue] (1,0) -- node[below] {$\cD$} (2,0); |
|
2628 \draw[orange] (2,0) -- node[below] {$\cC$} (3,0); |
|
2629 \node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (1,0) {}; |
|
2630 \node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (2,0) {}; |
|
2631 }}; |
|
2632 |
|
2633 \node(R) at (6,0) {\tikz{ |
|
2634 \draw[orange] (0,0) -- node[below] {$\cC$} (3,0); |
|
2635 \node[label={\phantom{$\cM$}}] at (1.5,0) {}; |
|
2636 }}; |
|
2637 |
|
2638 \node at (-1,-1.5) { $\leftidx{_\cC}{(\cM \tensor_\cD \cM)}{_\cC}$ }; |
|
2639 \node at (7,-1.5) { $\leftidx{_\cC}{\cC}{_\cC}$ }; |
|
2640 |
|
2641 \draw[->] (L) to[out=35, in=145] node[below] {$w$} node[above] { \tikz{ |
|
2642 \draw (0,0) circle (16pt); |
|
2643 }}(R); |
|
2644 |
|
2645 \draw[->] (R) to[out=-145, in=-35] node[above] {$x$} node[below] { \tikz{ |
|
2646 \draw (0,0) circle (16pt); |
|
2647 }}(L); |
|
2648 |
|
2649 |
|
2650 \end{tikzpicture} |
|
2651 $$ |
|
2652 |
2621 \caption{Cups and caps for free}\label{morita-fig-1} |
2653 \caption{Cups and caps for free}\label{morita-fig-1} |
2622 \end{figure} |
2654 \end{figure} |
2623 |
2655 |
2624 |
2656 |
2625 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |
2657 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |