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.}

  1177 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, 

  1178 where $B^j$ is the standard $j$-ball.

  1179 1-marked $k$-balls can be decomposed in various ways into smaller balls, which are either 

  1180 smaller 1-marked $k$-balls or the product of an unmarked ball with a marked interval.

  1181 We now proceed as in the above module definitions.

  1182 

  1183 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with

  1184 \[

  1185 	\cD(X) \deq \cM(X\times C(S)) .

  1186 \]

  1187 The product is pinched over the boundary of $C(S)$.

  1188 $\cD$ breaks into ``blocks" according to the restriction to the 

  1189 image of $\bd C(S) = S$ in $X\times C(S)$.

  1190 

  1191 More generally, consider a 2-manifold $Y$ 

  1192 (e.g.\ 2-ball or 2-sphere) marked by an embedded 1-complex $K$.

  1193 The components of $Y\setminus K$ are labeled by $n$-categories, 

  1194 the edges of $K$ are labeled by 0-sphere modules, 

  1195 and the 0-cells of $K$ are labeled by 1-sphere modules.

  1196 We can now apply the coend construction and obtain an $n{-}2$-category.

  1197 If $Y$ has boundary then this $n{-}2$-category is a module for the $n{-}1$-manifold

  1198 associated to the (marked, labeled) boundary of $Y$.

  1199 In particular, if $\bd Y$ is a 1-sphere then we get a 1-sphere module as defined above.

  1200 

  1201 \medskip

  1202 

  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,

  1205 and an $m$-sphere module is a representation of such an $n{-}2$-category.

  1206 

  1207 \medskip

  1208 

  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

  1211 these $n$-categories, then choose some collection of 1-sphere modules for the various

  1212 possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on.

  1213 Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen.

  1214 (For convenience, we declare a $(-1)$-sphere module to be an $n$-category.)

  1215 There is a wide range of possibilities.

  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 

  1218 it could contain several.

  1219 

  1220 \nn{...}

  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