1170 circles (1-spheres) which we just introduced. |
1170 circles (1-spheres) which we just introduced. |
1171 |
1171 |
1172 Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
1172 Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
1173 Fix a marked (and labeled) circle $S$. |
1173 Fix a marked (and labeled) circle $S$. |
1174 Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure xxxx). |
1174 Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure xxxx). |
1175 \nn{I need to make up my mind whether marked things are always labeled too.} |
1175 \nn{I need to make up my mind whether marked things are always labeled too. |
1176 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$. |
1176 For the time being, let's say they are.} |
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1177 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
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1178 where $B^j$ is the standard $j$-ball. |
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1179 1-marked $k$-balls can be decomposed in various ways into smaller balls, which are either |
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1180 smaller 1-marked $k$-balls or the product of an unmarked ball with a marked interval. |
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1181 We now proceed as in the above module definitions. |
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1182 |
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1183 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
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1184 \[ |
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1185 \cD(X) \deq \cM(X\times C(S)) . |
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1186 \] |
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1187 The product is pinched over the boundary of $C(S)$. |
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1188 $\cD$ breaks into ``blocks" according to the restriction to the |
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1189 image of $\bd C(S) = S$ in $X\times C(S)$. |
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1190 |
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1191 More generally, consider a 2-manifold $Y$ |
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1192 (e.g.\ 2-ball or 2-sphere) marked by an embedded 1-complex $K$. |
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1193 The components of $Y\setminus K$ are labeled by $n$-categories, |
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1194 the edges of $K$ are labeled by 0-sphere modules, |
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1195 and the 0-cells of $K$ are labeled by 1-sphere modules. |
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1196 We can now apply the coend construction and obtain an $n{-}2$-category. |
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1197 If $Y$ has boundary then this $n{-}2$-category is a module for the $n{-}1$-manifold |
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1198 associated to the (marked, labeled) boundary of $Y$. |
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1199 In particular, if $\bd Y$ is a 1-sphere then we get a 1-sphere module as defined above. |
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1200 |
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1201 \medskip |
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1202 |
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1203 It should now be clear how to define $n$-category $m$-sphere modules for $0\le m \le n-1$. |
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1204 For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
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1205 and an $m$-sphere module is a representation of such an $n{-}2$-category. |
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1206 |
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1207 \medskip |
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1208 |
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1209 We can now define the $n$- or less dimensional part of our $n{+}1$-category $\cS$. |
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1210 Choose some collection of $n$-categories, then choose some collections of bimodules for |
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1211 these $n$-categories, then choose some collection of 1-sphere modules for the various |
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1212 possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. |
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1213 Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
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1214 (For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
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1215 There is a wide range of possibilities. |
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1216 $L_0$ could contain infinitely many $n$-categories or just one. |
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1217 For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or |
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1218 it could contain several. |
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1219 |
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1220 \nn{...} |
1177 |
1221 |
1178 \medskip |
1222 \medskip |
1179 \hrule |
1223 \hrule |
1180 \medskip |
1224 \medskip |
1181 |
1225 |
1185 |
1229 |
1186 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
1230 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
1187 a separate paper): |
1231 a separate paper): |
1188 \begin{itemize} |
1232 \begin{itemize} |
1189 \item spell out what difference (if any) Top vs PL vs Smooth makes |
1233 \item spell out what difference (if any) Top vs PL vs Smooth makes |
1190 \item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
1234 \item discuss Morita equivalence |
1191 a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
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1192 \item morphisms of modules; show that it's adjoint to tensor product |
1235 \item morphisms of modules; show that it's adjoint to tensor product |
1193 (need to define dual module for this) |
1236 (need to define dual module for this) |
1194 \item functors |
1237 \item functors |
1195 \end{itemize} |
1238 \end{itemize} |
1196 |
1239 |