2591 The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}. |
2591 The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}. |
2592 |
2592 |
2593 \medskip |
2593 \medskip |
2594 |
2594 |
2595 We end this subsection with some remarks about Morita equivalence of disklike $n$-categories. |
2595 We end this subsection with some remarks about Morita equivalence of disklike $n$-categories. |
2596 Recall that two 1-categories $C$ and $D$ are Morita equivalent if and only if they are equivalent |
2596 Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent |
2597 objects in the 2-category of (linear) 1-categories, bimodules, and intertwinors. |
2597 objects in the 2-category of (linear) 1-categories, bimodules, and intertwinors. |
2598 Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the |
2598 Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the |
2599 $n{+}1$-category of sphere modules. |
2599 $n{+}1$-category of sphere modules. |
2600 |
2600 |
2601 Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in |
2601 Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in |
2602 dimensions 1 and $n+1$ (the middle dimensions come along for free), and this data must satisfy |
2602 dimensions 1 and $n+1$ (the middle dimensions come along for free). |
2603 identities corresponding to Morse cancellations in $n{+}1$-manifolds. |
2603 The $n{+}1$-dimensional part of the data must be invertible and satisfy |
|
2604 identities corresponding to Morse cancellations in $n$-manifolds. |
2604 We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar. |
2605 We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar. |
2605 |
2606 |
2606 Let $C$ and $D$ be (unoriented) disklike 2-categories. |
2607 Let $\cC$ and $\cD$ be (unoriented) disklike 2-categories. |
2607 Let $\cS$ denote the 3-category of 2-category sphere modules. |
2608 Let $\cS$ denote the 3-category of 2-category sphere modules. |
2608 The 1-dimensional part of the data for a Morita equivalence between $C$ and $D$ is a 0-sphere module $M = {}_CM_D$ |
2609 The 1-dimensional part of the data for a Morita equivalence between $\cC$ and $\cD$ is a 0-sphere module $\cM = {}_\cC\cM_\cD$ |
2609 (categorified bimodule) connecting $C$ and $D$. |
2610 (categorified bimodule) connecting $\cC$ and $\cD$. |
2610 Because of the full unoriented symmetry, this can also be thought of as a |
2611 Because of the full unoriented symmetry, this can also be thought of as a |
2611 0-sphere module ${}_DM_C$ connecting $D$ and $C$. |
2612 0-sphere module ${}_\cD\cM_\cC$ connecting $\cD$ and $\cC$. |
2612 |
2613 |
2613 We want $M$ 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 $\cS$ |
2614 between ${}_CM_D \otimes_D {}_DM_C$ and the identity 0-sphere module ${}_CC_C$, and similarly |
2615 between ${}_\cC\cM_\cD \otimes_\cD {}_\cD\cM_\cC$ and the identity 0-sphere module ${}_\cC\cC_\cC$, and similarly |
2615 with the roles of $C$ and $D$ reversed. |
2616 with the roles of $\cC$ and $\cD$ reversed. |
2616 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 |
2617 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 \nn{need figure}. |
2618 |
2619 |
2619 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |
2620 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |
2620 between various compositions of these 2-morphisms and various identity 2-morphisms. |
2621 between various compositions of these 2-morphisms and various identity 2-morphisms. |
2622 to decorated circles. |
2623 to decorated circles. |
2623 Figure \nn{need Figure} shows the intertwinors we need. |
2624 Figure \nn{need Figure} shows the intertwinors we need. |
2624 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
2625 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
2625 on the boundary. |
2626 on the boundary. |
2626 This is the 3-dimensional part of the data for the Morita equivalence. |
2627 This is the 3-dimensional part of the data for the Morita equivalence. |
2627 \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} |
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.) |
2628 |
2629 |
2629 In order for these 3-morphisms to be equivalences, they must satisfy identities corresponding to Morse cancellations |
2630 In order for these 3-morphisms to be equivalences, |
2630 on 3-manifolds. |
2631 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. |
2631 These are illustrated in Figure \nn{need figure}. |
2633 These are illustrated in Figure \nn{need figure}. |
2632 Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor. |
2634 Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor. |
2633 |
2635 |
2634 For general $n$, we start with an $n$-category 0-sphere module $M$ which is the data for the 1-dimensional |
2636 For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional |
2635 part of the Morita equivalence. |
2637 part of the Morita equivalence. |
2636 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds |
2638 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds |
2637 labeled by $C$, $D$ and $M$; no additional data is needed for these parts. |
2639 labeled by $\cC$, $\cD$ and $\cM$; no additional data is needed for these parts. |
2638 The $n{+}1$-dimensional part of the equivalence is given by certain intertwinors, and these intertwinors must satisfy |
2640 The $n{+}1$-dimensional part of the equivalence is given by certain intertwinors, and these intertwinors must |
2639 identities corresponding to Morse cancellations in $n{+}1$-manifolds. |
2641 be invertible and satisfy |
2640 |
2642 identities corresponding to Morse cancellations in $n$-manifolds. |
2641 |
2643 |
|
2644 \noop{ |
|
2645 One way of thinking of these conditions is as follows. |
|
2646 Given a decorated $n{+}1$-manifold, with a codimension 1 submanifold labeled by $\cM$ and |
|
2647 codimension 0 submanifolds labeled by $\cC$ and $\cD$, we can make any local modification we like without |
|
2648 changing |
|
2649 } |
|
2650 |
|
2651 If $\cC$ and $\cD$ are Morita equivalent $n$-categories, then it is easy to show that for any $n-j$-manifold |
|
2652 $Y$ the $j$-categories $\cC(Y)$ and $\cD(Y)$ are Morita equivalent. |
|
2653 When $j=0$ this means that the TQFT Hilbert spaces $\cC(Y)$ and $\cD(Y)$ are isomorphic |
|
2654 (if we are enriching over vector spaces). |
2642 |
2655 |
2643 |
2656 |
2644 |
2657 |
2645 |
2658 |
2646 \noop{ % the following doesn't work; need 2^(k+1) different N's, not 2*(k+1) |
2659 \noop{ % the following doesn't work; need 2^(k+1) different N's, not 2*(k+1) |
2647 More specifically, the 1-dimensional part of the data is a 0-sphere module $M = {}_CM_D$ |
2660 More specifically, the 1-dimensional part of the data is a 0-sphere module $\cM = {}_\cCM_\cD$ |
2648 (categorified bimodule) connecting $C$ and $D$. |
2661 (categorified bimodule) connecting $\cC$ and $\cD$. |
2649 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 $D$. |
2662 From $\cM$ we can construct various $k$-sphere modules $N^k_{j,E}$ for $0 \le k \le n$, $0\le j \le k$, and $E = \cC$ or $\cD$. |
2650 $N^k_{j,E}$ can be thought of as the graph of an index $j$ Morse function on the $k$-ball $B^k$ |
2663 $N^k_{j,E}$ can be thought of as the graph of an index $j$ Morse function on the $k$-ball $B^k$ |
2651 (so the graph lives in $B^k\times I = B^{k+1}$). |
2664 (so the graph lives in $B^k\times I = B^{k+1}$). |
2652 The positive side of the graph is labeled by $E$, the negative side by $E'$ |
2665 The positive side of the graph is labeled by $E$, the negative side by $E'$ |
2653 (where $C' = D$ and $D' = C$), and the codimension-1 |
2666 (where $\cC' = \cD$ and $\cD' = \cC$), and the codimension-1 |
2654 submanifold separating the positive and negative regions is labeled by $M$. |
2667 submanifold separating the positive and negative regions is labeled by $\cM$. |
2655 We think of $N^k_{j,E}$ as a $k{+}1$-morphism connecting |
2668 We think of $N^k_{j,E}$ as a $k{+}1$-morphism connecting |
2656 We plan on treating this in more detail in a future paper. |
2669 We plan on treating this in more detail in a future paper. |
2657 \nn{should add a few more details} |
2670 \nn{should add a few more details} |
2658 } |
2671 } |
2659 |
2672 |