1134 Corresponding to this decomposition we have an action and/or composition map |
1134 Corresponding to this decomposition we have an action and/or composition map |
1135 from the product of these various sets into $\cM(X)$. |
1135 from the product of these various sets into $\cM(X)$. |
1136 |
1136 |
1137 \medskip |
1137 \medskip |
1138 |
1138 |
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) |
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1141 of $\cA$ and $\cB$. |
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1142 Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
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1143 Given a $j$-ball $X$, $0\le j\le n-1$, we define |
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1144 \[ |
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1145 \cD(X) \deq \cM(X\times J) . |
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1146 \] |
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1147 The product is pinched over the boundary of $J$. |
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1148 $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
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1149 (see Figure xxxx). |
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1150 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
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1151 |
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1152 More generally, consider an interval with interior marked points, and with the complements |
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1153 of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
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1154 by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. |
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1155 (See Figure xxxx.) |
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1156 To this data we can apply to coend construction as in Subsection \ref{moddecss} above |
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1157 to obtain an $\cA_0$-$\cA_l$ bimodule and, forgetfully, an $n{-}1$-category. |
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1158 This amounts to a definition of taking tensor products of bimodules over $n$-categories. |
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1159 |
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1160 We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
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1161 associated to the marked and labeled circle. |
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1162 (See Figure xxxx.) |
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1163 If the circle is divided into two intervals, we can think of this $n{-}1$-category |
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1164 as the 2-ended tensor product of the two bimodules associated to the two intervals. |
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1165 |
|
1166 \medskip |
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1167 |
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1168 Next we define $n$-category 1-sphere modules. |
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1169 These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
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1170 circles (1-spheres) which we just introduced. |
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1171 |
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1172 Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
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1173 Fix a marked (and labeled) circle $S$. |
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1174 Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure xxxx). |
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1175 \nn{I need to make up my mind whether marked things are always labeled too.} |
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1176 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$. |
1141 |
1177 |
1142 \medskip |
1178 \medskip |
1143 \hrule |
1179 \hrule |
1144 \medskip |
1180 \medskip |
1145 |
1181 |