2597 $n{+}1$-category of sphere modules. |
2597 $n{+}1$-category of sphere modules. |
2598 |
2598 |
2599 Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in |
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 |
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. |
2601 identities corresponding to Morse cancellations in $n{+}1$-manifolds. |
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2602 We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar. |
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2603 |
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2604 Let $C$ and $D$ be (unoriented) disklike 2-categories. |
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2605 Let $\cS$ denote the 3-category of 2-category sphere modules. |
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2606 The 1-dimensional part of the data for a Morita equivalence between $C$ and $D$ is a 0-sphere module $M = {}_CM_D$ |
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2607 (categorified bimodule) connecting $C$ and $D$. |
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2608 Because of the full unoriented symmetry, this can also be thought of as a |
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2609 0-sphere module ${}_DM_C$ connecting $D$ and $C$. |
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2610 |
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2611 We want $M$ to be an equivalence, so we need 2-morphisms in $\cS$ |
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2612 between ${}_CM_D \otimes_D {}_DM_C$ and the identity 0-sphere module ${}_CC_C$, and similarly |
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2613 with the roles of $C$ and $D$ reversed. |
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2614 These 2-morphisms come for free, in the sense of not requiring additional data, since we can take them to be the labeled |
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2615 cell complexes (cups and caps) in $B^2$ shown in Figure \nn{need figure}. |
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2616 |
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2617 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |
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2618 between various compositions of these 2-morphisms and various identity 2-morphisms. |
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2619 Recall that the 3-morphisms of $\cS$ are intertwinors between representations of 1-categories associated |
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2620 to decorated circles. |
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2621 Figure \nn{need Figure} shows the intertwinors we need. |
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2622 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
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2623 on the boundary. |
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2624 This is the 3-dimensional part of the data for the Morita equivalence. |
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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} |
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2626 |
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2627 In order for these 3-morphisms to be equivalences, they must satisfy identities corresponding to Morse cancellations |
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2628 on 3-manifolds. |
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2629 These are illustrated in Figure \nn{need figure}. |
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2630 Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor. |
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2631 |
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2632 For general $n$, we start with an $n$-category 0-sphere module $M$ which is the data for the 1-dimensional |
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2633 part of the Morita equivalence. |
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2634 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds |
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2635 labeled by $C$, $D$ and $M$; no additional data is needed for these parts. |
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2636 The $n{+}1$-dimensional part of the equivalence is given by certain intertwinors, and these intertwinors must satisfy |
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2637 identities corresponding to Morse cancellations in $n{+}1$-manifolds. |
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2638 |
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2639 |
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2640 |
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2641 |
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2642 |
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2643 |
2602 \noop{ % the following doesn't work; need 2^(k+1) different N's, not 2*(k+1) |
2644 \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 $M = {}_CM_D$ |
2645 More specifically, the 1-dimensional part of the data is a 0-sphere module $M = {}_CM_D$ |
2604 (categorified bimodule) connecting $C$ and $D$. |
2646 (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 $D$. |
2647 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 $B^k$ |
2648 $N^k_{j,E}$ 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 $B^k\times I = B^{k+1}$). |
2649 (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 $E'$ |
2650 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 |
2651 (where $C' = D$ and $D' = C$), and the codimension-1 |
2610 submanifold separating the positive and negative regions is labeled by $M$. |
2652 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 |
2653 We think of $N^k_{j,E}$ as a $k{+}1$-morphism connecting |
2612 } |
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2613 We plan on treating this in more detail in a future paper. |
2654 We plan on treating this in more detail in a future paper. |
2614 \nn{should add a few more details} |
2655 \nn{should add a few more details} |
2615 |
2656 } |
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2657 |