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11 of the corresponding traditional $n$-category to be $\cC(B^k)$, where |
11 of the corresponding traditional $n$-category to be $\cC(B^k)$, where |
12 $B^k$ is the {\it standard} $k$-ball. |
12 $B^k$ is the {\it standard} $k$-ball. |
13 One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms. |
13 One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms. |
14 One should also show that composing the two arrows (between traditional and disk-like $n$-categories) |
14 One should also show that composing the two arrows (between traditional and disk-like $n$-categories) |
15 yields the appropriate sort of equivalence on each side. |
15 yields the appropriate sort of equivalence on each side. |
16 Since we haven't given a definition for functors between disk-like $n$-categories |
16 Since we haven't given a definition for functors between disk-like $n$-categories, we do not pursue this here. |
17 (the paper is already too long!), we do not pursue this here. |
|
18 |
17 |
19 We emphasize that we are just sketching some of the main ideas in this appendix --- |
18 We emphasize that we are just sketching some of the main ideas in this appendix --- |
20 it falls well short of proving the definitions are equivalent. |
19 it falls well short of proving the definitions are equivalent. |
21 |
20 |
22 %\nn{cases to cover: (a) ordinary $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
21 %\nn{cases to cover: (a) ordinary $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |