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1279 \begin{figure}[t] |
1279 \begin{figure}[t] |
1280 $$\mathfig{.55}{ncat/boundary-collar}$$ |
1280 $$\mathfig{.55}{ncat/boundary-collar}$$ |
1281 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1281 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1282 |
1282 |
1283 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
1283 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
1284 Call such a thing a {marked $k{-}1$-hemisphere}. |
1284 Call such a thing a {\it marked $k{-}1$-hemisphere}. |
|
1285 (A marked $k{-}1$-hemisphere is, of course, just a $k{-}1$-ball with its entire boundary marked. |
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1286 We call it a hemisphere instead of a ball because it plays a role analogous |
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1287 to the $k{-}1$-spheres in the $n$-category definition.) |
1285 |
1288 |
1286 \begin{lem} |
1289 \begin{lem} |
1287 \label{lem:hemispheres} |
1290 \label{lem:hemispheres} |
1288 {For each $0 \le k \le n-1$, we have a functor $\cl\cM_k$ from |
1291 {For each $0 \le k \le n-1$, we have a functor $\cl\cM_k$ from |
1289 the category of marked $k$-hemispheres and |
1292 the category of marked $k$-hemispheres and |