2602 We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar.

  2603 

  2604 Let $C$ and $D$ be (unoriented) disklike 2-categories.

  2605 Let $\cS$ denote the 3-category of 2-category sphere modules.

  2606 The 1-dimensional part of the data for a Morita equivalence between $C$ and $D$ is a 0-sphere module $M = {}_CM_D$ 

  2607 (categorified bimodule) connecting $C$ and $D$.

  2608 Because of the full unoriented symmetry, this can also be thought of as a 

  2609 0-sphere module ${}_DM_C$ connecting $D$ and $C$.

  2610 

  2611 We want $M$ to be an equivalence, so we need 2-morphisms in $\cS$ 

  2612 between ${}_CM_D \otimes_D {}_DM_C$ and the identity 0-sphere module ${}_CC_C$, and similarly

  2613 with the roles of $C$ and $D$ reversed.

  2614 These 2-morphisms come for free, in the sense of not requiring additional data, since we can take them to be the labeled 

  2615 cell complexes (cups and caps) in $B^2$ shown in Figure \nn{need figure}.

  2616 

  2617 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms

  2618 between various compositions of these 2-morphisms and various identity 2-morphisms.

  2619 Recall that the 3-morphisms of $\cS$ are intertwinors between representations of 1-categories associated

  2620 to decorated circles.

  2621 Figure \nn{need Figure} shows the intertwinors we need.

  2622 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle

  2623 on the boundary.

  2624 This is the 3-dimensional part of the data for the Morita equivalence.

  2625 \nn{?? note that, by symmetry, the x and y arrows of Fig xxxx are the same (up to rotation), as are the z and w arrows}

  2626 

  2627 In order for these 3-morphisms to be equivalences, they must satisfy identities corresponding to Morse cancellations

  2628 on 3-manifolds.

  2629 These are illustrated in Figure \nn{need figure}.

  2630 Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor.

  2631 

  2632 For general $n$, we start with an $n$-category 0-sphere module $M$ which is the data for the 1-dimensional

  2633 part of the Morita equivalence.

  2634 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds

  2635 labeled by $C$, $D$ and $M$; no additional data is needed for these parts.

  2636 The $n{+}1$-dimensional part of the equivalence is given by certain intertwinors, and these intertwinors must satisfy

  2637 identities corresponding to Morse cancellations in $n{+}1$-manifolds. 

  2638 

  2639 

  2640 

  2641 

  2642 

  2643