2613 |
2613 |
2614 We want $\cM$ to be an equivalence, so we need 2-morphisms in $\cS$ |
2614 We want $\cM$ to be an equivalence, so we need 2-morphisms in $\cS$ |
2615 between ${}_\cC\cM_\cD \otimes_\cD {}_\cD\cM_\cC$ and the identity 0-sphere module ${}_\cC\cC_\cC$, and similarly |
2615 between ${}_\cC\cM_\cD \otimes_\cD {}_\cD\cM_\cC$ and the identity 0-sphere module ${}_\cC\cC_\cC$, and similarly |
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 \nn{need figure}. |
2618 cell complexes (cups and caps) in $B^2$ shown in Figure \ref{morita-fig-1}. |
|
2619 \begin{figure}[t] |
|
2620 $$\mathfig{.65}{tempkw/morita1}$$ |
|
2621 \caption{Cups and caps for free}\label{morita-fig-1} |
|
2622 \end{figure} |
|
2623 |
2619 |
2624 |
2620 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |
2625 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |
2621 between various compositions of these 2-morphisms and various identity 2-morphisms. |
2626 between various compositions of these 2-morphisms and various identity 2-morphisms. |
2622 Recall that the 3-morphisms of $\cS$ are intertwinors between representations of 1-categories associated |
2627 Recall that the 3-morphisms of $\cS$ are intertwinors between representations of 1-categories associated |
2623 to decorated circles. |
2628 to decorated circles. |
2624 Figure \nn{need Figure} shows the intertwinors we need. |
2629 Figure \ref{morita-fig-2} |
|
2630 \begin{figure}[t] |
|
2631 $$\mathfig{.55}{tempkw/morita2}$$ |
|
2632 \caption{Intertwinors for a Morita equivalence}\label{morita-fig-2} |
|
2633 \end{figure} |
|
2634 shows the intertwinors we need. |
2625 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
2635 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
2626 on the boundary. |
2636 on the boundary. |
2627 This is the 3-dimensional part of the data for the Morita equivalence. |
2637 This is the 3-dimensional part of the data for the Morita equivalence. |
2628 (Note that, by symmetry, the $c$ and $d$ arrows of Figure \ref{} are the same (up to rotation), as are the $h$ and $g$ arrows.) |
2638 (Note that, by symmetry, the $c$ and $d$ arrows of Figure \ref{morita-fig-2} |
|
2639 are the same (up to rotation), as are the $h$ and $g$ arrows.) |
2629 |
2640 |
2630 In order for these 3-morphisms to be equivalences, |
2641 In order for these 3-morphisms to be equivalences, |
2631 they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition |
2642 they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition |
2632 they must satisfy identities corresponding to Morse cancellations on 2-manifolds. |
2643 they must satisfy identities corresponding to Morse cancellations on 2-manifolds. |
2633 These are illustrated in Figure \nn{need figure}. |
2644 These are illustrated in Figure \ref{morita-fig-3}. |
|
2645 \begin{figure}[t] |
|
2646 $$\mathfig{.65}{tempkw/morita3}$$ |
|
2647 \caption{Identities for intertwinors}\label{morita-fig-3} |
|
2648 \end{figure} |
2634 Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor. |
2649 Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor. |
2635 |
2650 |
2636 For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional |
2651 For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional |
2637 part of the Morita equivalence. |
2652 part of the Morita equivalence. |
2638 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds |
2653 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds |