2590 

  2591 \medskip

  2592 

  2593 We end this subsection with some remarks about Morita equivalence of disklike $n$-categories.

  2594 Recall that two 1-categories $C$ and $D$ are Morita equivalent if and only if they are equivalent

  2595 objects in the 2-category of (linear) 1-categories, bimodules, and intertwinors.

  2596 Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the

  2597 $n{+}1$-category of sphere modules.

  2598 

  2599 Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in 

  2600 dimensions 1 and $n+1$ (the middle dimensions come along for free), and this data must satisfy

  2601 identities corresponding to Morse cancellations in $n{+}1$-manifolds.

  2602 \noop{ % the following doesn't work; need 2^(k+1) different N's, not 2*(k+1)

  2603 More specifically, the 1-dimensional part of the data is a 0-sphere module $M = {}_CM_D$ 

  2604 (categorified bimodule) connecting $C$ and $D$.

  2605 From $M$ we can construct various $k$-sphere modules $N^k_{j,E}$ for $0 \le k \le n$, $0\le j \le k$, and $E = C$ or $D$.

  2606 $N^k_{j,E}$ can be thought of as the graph of an index $j$ Morse function on the $k$-ball $B^k$

  2607 (so the graph lives in $B^k\times I = B^{k+1}$).

  2608 The positive side of the graph is labeled by $E$, the negative side by $E'$

  2609 (where $C' = D$ and $D' = C$), and the codimension-1 

  2610 submanifold separating the positive and negative regions is labeled by $M$.

  2611 We think of $N^k_{j,E}$ as a $k{+}1$-morphism connecting 

  2612 }

  2613 We plan on treating this in more detail in a future paper.

  2614 \nn{should add a few more details}

  2615