1109 |
1109 |
1110 We start with 0-sphere modules, which also could reasonably be called (categorified) bimodules. |
1110 We start with 0-sphere modules, which also could reasonably be called (categorified) bimodules. |
1111 (For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
1111 (For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
1112 Define a 0-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard |
1112 Define a 0-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard |
1113 $(B^k, B^{k-1})$, where $B^{k-1}$ is properly embedded in $B^k$. |
1113 $(B^k, B^{k-1})$, where $B^{k-1}$ is properly embedded in $B^k$. |
1114 See Figure xxxx. |
1114 See Figure \ref{feb21a}. |
1115 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1115 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
|
1116 |
|
1117 \begin{figure}[!ht] |
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1118 \begin{equation*} |
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1119 \mathfig{.85}{tempkw/feb21a} |
|
1120 \end{equation*} |
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1121 \caption{0-marked 1-ball and 0-marked 2-ball} |
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1122 \label{feb21a} |
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1123 \end{figure} |
1116 |
1124 |
1117 0-marked balls can be cut into smaller balls in various ways. |
1125 0-marked balls can be cut into smaller balls in various ways. |
1118 These smaller balls could be 0-marked or plain. |
1126 These smaller balls could be 0-marked or plain. |
1119 We can also take the boundary of a 0-marked ball, which is 0-marked sphere. |
1127 We can also take the boundary of a 0-marked ball, which is 0-marked sphere. |
1120 |
1128 |
1144 \[ |
1152 \[ |
1145 \cD(X) \deq \cM(X\times J) . |
1153 \cD(X) \deq \cM(X\times J) . |
1146 \] |
1154 \] |
1147 The product is pinched over the boundary of $J$. |
1155 The product is pinched over the boundary of $J$. |
1148 $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
1156 $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
1149 (see Figure xxxx). |
1157 (see Figure \ref{feb21b}). |
1150 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
1158 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
|
1159 |
|
1160 \begin{figure}[!ht] |
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1161 \begin{equation*} |
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1162 \mathfig{1}{tempkw/feb21b} |
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1163 \end{equation*} |
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1164 \caption{The pinched product $X\times J$} |
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1165 \label{feb21b} |
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1166 \end{figure} |
1151 |
1167 |
1152 More generally, consider an interval with interior marked points, and with the complements |
1168 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 |
1169 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 $\cM_i$. |
1170 by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. |
1155 (See Figure xxxx.) |
1171 (See Figure \ref{feb21c}.) |
1156 To this data we can apply to coend construction as in Subsection \ref{moddecss} above |
1172 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. |
1173 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. |
1174 This amounts to a definition of taking tensor products of bimodules over $n$-categories. |
1159 |
1175 |
|
1176 \begin{figure}[!ht] |
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1177 \begin{equation*} |
|
1178 \mathfig{1}{tempkw/feb21c} |
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1179 \end{equation*} |
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1180 \caption{Marked and labeled 1-manifolds} |
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1181 \label{feb21c} |
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1182 \end{figure} |
|
1183 |
1160 We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1184 We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1161 associated to the marked and labeled circle. |
1185 associated to the marked and labeled circle. |
1162 (See Figure xxxx.) |
1186 (See Figure \ref{feb21c}.) |
1163 If the circle is divided into two intervals, we can think of this $n{-}1$-category |
1187 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. |
1188 as the 2-ended tensor product of the two bimodules associated to the two intervals. |
1165 |
1189 |
1166 \medskip |
1190 \medskip |
1167 |
1191 |
1169 These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
1193 These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
1170 circles (1-spheres) which we just introduced. |
1194 circles (1-spheres) which we just introduced. |
1171 |
1195 |
1172 Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
1196 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 $S$. |
1197 Fix a marked (and labeled) circle $S$. |
1174 Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure xxxx). |
1198 Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
1175 \nn{I need to make up my mind whether marked things are always labeled too. |
1199 \nn{I need to make up my mind whether marked things are always labeled too. |
1176 For the time being, let's say they are.} |
1200 For the time being, let's say they are.} |
1177 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
1201 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. |
1202 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 |
1203 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. |
1204 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. |
1205 We now proceed as in the above module definitions. |
|
1206 |
|
1207 \begin{figure}[!ht] |
|
1208 \begin{equation*} |
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1209 \mathfig{.4}{tempkw/feb21d} |
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1210 \end{equation*} |
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1211 \caption{Cone on a marked circle} |
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1212 \label{feb21d} |
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1213 \end{figure} |
1182 |
1214 |
1183 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1215 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1184 \[ |
1216 \[ |
1185 \cD(X) \deq \cM(X\times C(S)) . |
1217 \cD(X) \deq \cM(X\times C(S)) . |
1186 \] |
1218 \] |