1200 |
1200 |
1201 \medskip |
1201 \medskip |
1202 |
1202 |
1203 It should now be clear how to define $n$-category $m$-sphere modules for $0\le m \le n-1$. |
1203 It should now be clear how to define $n$-category $m$-sphere modules for $0\le m \le n-1$. |
1204 For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
1204 For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
1205 and an $m$-sphere module is a representation of such an $n{-}2$-category. |
1205 and a 2-sphere module is a representation of such an $n{-}2$-category. |
1206 |
1206 |
1207 \medskip |
1207 \medskip |
1208 |
1208 |
1209 We can now define the $n$- or less dimensional part of our $n{+}1$-category $\cS$. |
1209 We can now define the $n$- or less dimensional part of our $n{+}1$-category $\cS$. |
1210 Choose some collection of $n$-categories, then choose some collections of bimodules for |
1210 Choose some collection of $n$-categories, then choose some collections of bimodules for |
1214 (For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
1214 (For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
1215 There is a wide range of possibilities. |
1215 There is a wide range of possibilities. |
1216 $L_0$ could contain infinitely many $n$-categories or just one. |
1216 $L_0$ could contain infinitely many $n$-categories or just one. |
1217 For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or |
1217 For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or |
1218 it could contain several. |
1218 it could contain several. |
|
1219 The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
|
1220 constructed out of labels taken from $L_j$ for $j<k$. |
|
1221 |
|
1222 We now define $\cS(X)$, for $X$ of dimension at most $n$, to be the set of all |
|
1223 cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
|
1224 by elements of $L_j$. |
|
1225 As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
|
1226 for the $n{-}k{+}1$-category associated to its decorated boundary. |
|
1227 Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
|
1228 of as $n$-category $k{-}1$-sphere modules |
|
1229 (generalizations of bimodules). |
|
1230 On the other hand, we can equally think of the $k$-morphisms as decorations on $k$-balls, |
|
1231 and from this (official) point of view it is clear that they satisfy all of the axioms of an |
|
1232 $n{+}1$-category. |
|
1233 (All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
|
1234 |
|
1235 \medskip |
|
1236 |
|
1237 Next we define the $n{+}1$-morphisms of $\cS$. |
|
1238 |
|
1239 |
|
1240 |
|
1241 |
|
1242 |
|
1243 |
1219 |
1244 |
1220 \nn{...} |
1245 \nn{...} |
1221 |
1246 |
1222 \medskip |
1247 \medskip |
1223 \hrule |
1248 \hrule |
1224 \medskip |
1249 \medskip |
1225 |
1250 |
1226 \nn{to be continued...} |
1251 \nn{to be continued...} |
1227 \medskip |
1252 \medskip |
|
1253 |
|
1254 |
|
1255 |
|
1256 |
1228 |
1257 |
1229 |
1258 |
1230 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
1259 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
1231 a separate paper): |
1260 a separate paper): |
1232 \begin{itemize} |
1261 \begin{itemize} |