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2145 that our $n$-categories and modules have non-degenerate inner products. |
2145 that our $n$-categories and modules have non-degenerate inner products. |
2146 (In other words, we need to assume some extra duality on the $n$-categories and modules.) |
2146 (In other words, we need to assume some extra duality on the $n$-categories and modules.) |
2147 |
2147 |
2148 \medskip |
2148 \medskip |
2149 |
2149 |
2150 Our first task is to define an $n$-category $m$-sphere modules, for $0\le m \le n-1$. |
2150 Our first task is to define an $n$-category $m$-sphere module, for $0\le m \le n-1$. |
2151 These will be defined in terms of certain classes of marked balls, very similarly |
2151 These will be defined in terms of certain classes of marked balls, very similarly |
2152 to the definition of $n$-category modules above. |
2152 to the definition of $n$-category modules above. |
2153 (This, in turn, is very similar to our definition of $n$-category.) |
2153 (This, in turn, is very similar to our definition of $n$-category.) |
2154 Because of this similarity, we only sketch the definitions below. |
2154 Because of this similarity, we only sketch the definitions below. |
2155 |
2155 |
2390 The construction of the 0- through $n$-morphisms was easy and tautological, but the |
2390 The construction of the 0- through $n$-morphisms was easy and tautological, but the |
2391 $n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional |
2391 $n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional |
2392 duality assumptions on the lower morphisms. |
2392 duality assumptions on the lower morphisms. |
2393 These are required because we define the spaces of $n{+}1$-morphisms by |
2393 These are required because we define the spaces of $n{+}1$-morphisms by |
2394 making arbitrary choices of incoming and outgoing boundaries for each $n$-ball. |
2394 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 form these choices. |
2395 The additional duality assumptions are needed to prove independence of our definition from these choices. |
2396 |
2396 |
2397 Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
2397 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. |
2398 by a cell complex labeled by 0- through $n$-morphisms, as above. |
2399 Choose an $n{-}1$-sphere $E\sub \bd X$ which divides |
2399 Choose an $n{-}1$-sphere $E\sub \bd X$ which divides |
2400 $\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |
2400 $\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |