1139 

  1139 Part of the structure of an $n$-cat 0-sphere module is captured my saying it is

  1140 

  1140 a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms)

  1141 of $\cA$ and $\cB$.

  1142 Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior).

  1143 Given a $j$-ball $X$, $0\le j\le n-1$, we define

  1144 \[

  1145 	\cD(X) \deq \cM(X\times J) .

  1146 \]

  1147 The product is pinched over the boundary of $J$.

  1148 $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$

  1149 (see Figure xxxx).

  1150 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$.

  1151 

  1152 More generally, consider an interval with interior marked points, and with the complements

  1153 of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled

  1154 by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$.

  1155 (See Figure xxxx.)

  1156 To this data we can apply to coend construction as in Subsection \ref{moddecss} above

  1157 to obtain an $\cA_0$-$\cA_l$ bimodule and, forgetfully, an $n{-}1$-category.

  1158 This amounts to a definition of taking tensor products of bimodules over $n$-categories.

  1159 

  1160 We could also similarly mark and label a circle, obtaining an $n{-}1$-category

  1161 associated to the marked and labeled circle.

  1162 (See Figure xxxx.)

  1163 If the circle is divided into two intervals, we can think of this $n{-}1$-category

  1164 as the 2-ended tensor product of the two bimodules associated to the two intervals.

  1165 

  1166 \medskip

  1167 

  1168 Next we define $n$-category 1-sphere modules.

  1169 These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled 

  1170 circles (1-spheres) which we just introduced.

  1171 

  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 $S$.

  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.}

  1176 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$.