1363 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
1363 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
1364 |
1364 |
1365 Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$. |
1365 Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$. |
1366 We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
1366 We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
1367 |
1367 |
|
1368 \noop{ %%%%%%% |
1368 \begin{lem}[Module to category restrictions] |
1369 \begin{lem}[Module to category restrictions] |
1369 {For each marked $k$-hemisphere $H$ there is a restriction map |
1370 {For each marked $k$-hemisphere $H$ there is a restriction map |
1370 $\cl\cM(H)\to \cC(H)$. |
1371 $\cl\cM(H)\to \cC(H)$. |
1371 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1372 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1372 These maps comprise a natural transformation of functors.} |
1373 These maps comprise a natural transformation of functors.} |
1373 \end{lem} |
1374 \end{lem} |
1374 |
1375 } %%%%%%% end \noop |
|
1376 |
|
1377 It follows from the definition of the colimit $\cl\cM(H)$ that |
|
1378 given any (unmarked) $k{-}1$-ball $Y$ in the interior of $H$ there is a restriction map |
|
1379 from a subset $\cl\cM(H)_{\trans{\bdy Y}}$ of $\cl\cM(H)$ to $\cC(Y)$. |
|
1380 Combining this with the boundary map $\cM(B,N) \to \cl\cM(\bd(B,N))$, we also have a restriction |
|
1381 map from a subset $\cM(B,N)_{\trans{\bdy Y}}$ of $\cM(B,N)$ to $\cC(Y)$ whenever $Y$ is in the interior of $\bd B \setmin N$. |
|
1382 This fact will be used below. |
|
1383 |
|
1384 \noop{ %%%% |
1375 Note that combining the various boundary and restriction maps above |
1385 Note that combining the various boundary and restriction maps above |
1376 (for both modules and $n$-categories) |
1386 (for both modules and $n$-categories) |
1377 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
1387 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
1378 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
1388 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
1379 This subset $\cM(B,N)\trans{\bdy Y}$ is the subset of morphisms which are appropriately splittable (transverse to the |
1389 This subset $\cM(B,N)\trans{\bdy Y}$ is the subset of morphisms which are appropriately splittable (transverse to the |
1380 cutting submanifolds). |
1390 cutting submanifolds). |
1381 This fact will be used below. |
1391 This fact will be used below. |
|
1392 } %%%%% end \noop |
1382 |
1393 |
1383 In our example, the various restriction and gluing maps above come from |
1394 In our example, the various restriction and gluing maps above come from |
1384 restricting and gluing maps into $T$. |
1395 restricting and gluing maps into $T$. |
1385 |
1396 |
1386 We require two sorts of composition (gluing) for modules, corresponding to two ways |
1397 We require two sorts of composition (gluing) for modules, corresponding to two ways |