2585 We define product $n{+}1$-morphisms to be identity maps of modules. |
2585 We define product $n{+}1$-morphisms to be identity maps of modules. |
2586 |
2586 |
2587 To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator |
2587 To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator |
2588 then compose the module maps. |
2588 then compose the module maps. |
2589 The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}. |
2589 The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}. |
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2590 |
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2591 \medskip |
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2592 |
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2593 We end this subsection with some remarks about Morita equivalence of disklike $n$-categories. |
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2594 Recall that two 1-categories $C$ and $D$ are Morita equivalent if and only if they are equivalent |
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2595 objects in the 2-category of (linear) 1-categories, bimodules, and intertwinors. |
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2596 Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the |
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2597 $n{+}1$-category of sphere modules. |
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2598 |
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2599 Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in |
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2600 dimensions 1 and $n+1$ (the middle dimensions come along for free), and this data must satisfy |
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2601 identities corresponding to Morse cancellations in $n{+}1$-manifolds. |
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2602 \noop{ % the following doesn't work; need 2^(k+1) different N's, not 2*(k+1) |
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2603 More specifically, the 1-dimensional part of the data is a 0-sphere module $M = {}_CM_D$ |
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2604 (categorified bimodule) connecting $C$ and $D$. |
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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 $D$. |
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2606 $N^k_{j,E}$ can be thought of as the graph of an index $j$ Morse function on the $k$-ball $B^k$ |
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2607 (so the graph lives in $B^k\times I = B^{k+1}$). |
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2608 The positive side of the graph is labeled by $E$, the negative side by $E'$ |
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2609 (where $C' = D$ and $D' = C$), and the codimension-1 |
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2610 submanifold separating the positive and negative regions is labeled by $M$. |
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2611 We think of $N^k_{j,E}$ as a $k{+}1$-morphism connecting |
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2612 } |
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2613 We plan on treating this in more detail in a future paper. |
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2614 \nn{should add a few more details} |
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2615 |