2369 For each pair of $n$-categories in $L_0$, $L_1$ could contain no 0-sphere modules at all or |
2369 For each pair of $n$-categories in $L_0$, $L_1$ could contain no 0-sphere modules at all or |
2370 it could contain several. |
2370 it could contain several. |
2371 The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
2371 The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
2372 constructed out of labels taken from $L_j$ for $j<k$. |
2372 constructed out of labels taken from $L_j$ for $j<k$. |
2373 |
2373 |
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2374 We remind the reader again that $\cS = \cS_{\{L_i\}, \{z_Y\}}$ depends on |
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2375 the choice of $L_i$ above as well as the choice of |
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2376 families of inner products below. |
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2377 |
2374 We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all |
2378 We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all |
2375 cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
2379 cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
2376 by elements of $L_j$. |
2380 by elements of $L_j$. |
2377 As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
2381 As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
2378 for the $n{-}k{+}1$-category associated to its decorated boundary. |
2382 for the $n{-}k{+}1$-category associated to its decorated boundary. |
2394 making arbitrary choices of incoming and outgoing boundaries for each $n$-ball. |
2398 making arbitrary choices of incoming and outgoing boundaries for each $n$-ball. |
2395 The additional duality assumptions are needed to prove independence of our definition from these choices. |
2399 The additional duality assumptions are needed to prove independence of our definition from these choices. |
2396 |
2400 |
2397 Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
2401 Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
2398 by a cell complex labeled by 0- through $n$-morphisms, as above. |
2402 by a cell complex labeled by 0- through $n$-morphisms, as above. |
2399 Choose an $n{-}1$-sphere $E\sub \bd X$ which divides |
2403 Choose an $n{-}1$-sphere $E\sub \bd X$, transverse to $c$, which divides |
2400 $\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |
2404 $\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |
2401 Let $E_c$ denote $E$ decorated by the restriction of $c$ to $E$. |
2405 Let $E_c$ denote $E$ decorated by the restriction of $c$ to $E$. |
2402 Recall from above the associated 1-category $\cS(E_c)$. |
2406 Recall from above the associated 1-category $\cS(E_c)$. |
2403 We can also have $\cS(E_c)$ modules $\cS(\bd_-X_c)$ and $\cS(\bd_+X_c)$. |
2407 We can also have $\cS(E_c)$ modules $\cS(\bd_-X_c)$ and $\cS(\bd_+X_c)$. |
2404 Define |
2408 Define |