1114 and modules $\cM_\cC$ and $_\cC\cN$, |
1114 and modules $\cM_\cC$ and $_\cC\cN$, |
1115 \[ |
1115 \[ |
1116 (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
1116 (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
1117 \] |
1117 \] |
1118 |
1118 |
1119 We must now define the things appearing in the above equation. |
1119 In the next few paragraphs define the things appearing in the above equation: |
|
1120 $\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
|
1121 $\hom_\cC$. |
1120 |
1122 |
1121 In the previous subsection we defined a tensor product of $A_\infty$ $n$-cat modules |
1123 In the previous subsection we defined a tensor product of $A_\infty$ $n$-cat modules |
1122 for general $n$. |
1124 for general $n$. |
1123 For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ |
1125 For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ |
1124 and their gluings (antirefinements). |
1126 and their gluings (antirefinements). |
1133 \] |
1135 \] |
1134 (If $D$ denotes the subdivision of $J$, then we denote this complex by $\psi(D)$.) |
1136 (If $D$ denotes the subdivision of $J$, then we denote this complex by $\psi(D)$.) |
1135 To each antirefinement we associate a chain map using the composition law of $\cC$ and the |
1137 To each antirefinement we associate a chain map using the composition law of $\cC$ and the |
1136 module actions of $\cC$ on $\cM$ and $\cN$. |
1138 module actions of $\cC$ on $\cM$ and $\cN$. |
1137 \def\olD{{\overline D}} |
1139 \def\olD{{\overline D}} |
|
1140 \def\cbar{{\bar c}} |
1138 The underlying graded vector space of the homotopy colimit is |
1141 The underlying graded vector space of the homotopy colimit is |
1139 \[ |
1142 \[ |
1140 \bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , |
1143 \bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , |
1141 \] |
1144 \] |
1142 where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ |
1145 where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ |
1143 runs through chains of antirefinements, and $[l]$ denotes a grading shift. |
1146 runs through chains of antirefinements, and $[l]$ denotes a grading shift. |
|
1147 We will denote an element of the summand indexed by $\olD$ by |
|
1148 $\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. |
|
1149 The boundary map is given (ignoring signs) by |
|
1150 \begin{eqnarray*} |
|
1151 \bd(\olD\ot m\ot\cbar\ot n) &=& \olD\ot\bd(m\ot\cbar)\ot n + \olD\ot m\ot\cbar\ot \bd n + \\ |
|
1152 & & \;\; (\bd_+ \olD)\ot m\ot\cbar\ot n + (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) , |
|
1153 \end{eqnarray*} |
|
1154 where $\bd_+ \olD = \sum_{i>0} (D_0, \cdots \widehat{D_i} \cdots , D_l)$ (the part of the simplicial |
|
1155 boundary which retains $D_0$), $\bd_0 \olD = (D_1, \cdots , D_l)$, |
|
1156 and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. |
|
1157 |
|
1158 $(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$: |
|
1159 \[ |
|
1160 \prod_l \prod_{\olD} (\psi(D_0)[l])^* , |
|
1161 \] |
|
1162 where $(\psi(D_0)[l])^*$ denotes the linear dual. |
|
1163 The boundary is given by |
|
1164 \begin{eqnarray*} |
|
1165 (\bd f)(\olD\ot m\ot\cbar\ot n) &=& f(\olD\ot\bd(m\ot\cbar)\ot n) + |
|
1166 f(\olD\ot m\ot\cbar\ot \bd n) + \\ |
|
1167 & & \;\; f((\bd_+ \olD)\ot m\ot\cbar\ot n) + f((\bd_0 \olD)\ot \rho(m\ot\cbar\ot n)) . |
|
1168 \end{eqnarray*} |
|
1169 (Again, we are ignoring signs.) |
|
1170 |
|
1171 Next we define the dual module $(_\cC\cN)^*$. |
|
1172 This will depend on a choice of interval $J$, just as the tensor product did. |
|
1173 Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals |
|
1174 to chain complexes. |
|
1175 Given $J$, we define for each $K\sub J$ which contains the {\it left} endpoint of $J$ |
|
1176 \[ |
|
1177 (_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* , |
|
1178 \] |
|
1179 where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated |
|
1180 to the right-marked interval $J\setmin K$. |
|
1181 This extends to a functor from all left-marked intervals (not just those contained in $J$). |
|
1182 It's easy to verify the remaining module axioms. |
|
1183 |
|
1184 Now re reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
|
1185 as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
1144 |
1186 |
1145 \nn{...} |
1187 \nn{...} |
1146 |
1188 |
1147 |
1189 |
1148 |
1190 |
1155 \subsection{The $n{+}1$-category of sphere modules} |
1197 \subsection{The $n{+}1$-category of sphere modules} |
1156 \label{ssec:spherecat} |
1198 \label{ssec:spherecat} |
1157 |
1199 |
1158 In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1200 In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1159 whose objects correspond to $n$-categories. |
1201 whose objects correspond to $n$-categories. |
1160 This is a version of the familiar algebras-bimodules-intertwiners 2-category. |
1202 When $n=2$ |
|
1203 this is a version of the familiar algebras-bimodules-intertwiners 2-category. |
1161 (Terminology: It is clearly appropriate to call an $S^0$ module a bimodule, |
1204 (Terminology: It is clearly appropriate to call an $S^0$ module a bimodule, |
1162 but this is much less true for higher dimensional spheres, |
1205 but this is much less true for higher dimensional spheres, |
1163 so we prefer the term ``sphere module" for the general case.) |
1206 so we prefer the term ``sphere module" for the general case.) |
1164 |
1207 |
1165 The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1208 The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1166 these first. |
1209 these first. |
1167 The $n{+}1$-dimensional part of $\cS$ consist of intertwiners |
1210 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
1168 (of garden-variety $1$-category modules associated to decorated $n$-balls). |
1211 (of garden-variety $1$-category modules associated to decorated $n$-balls). |
1169 We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
1212 We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
1170 the duality requirements of an $n{+}1$-category, we will have to assume |
1213 the duality requirements of an $n{+}1$-category, we will have to assume |
1171 that our $n$-categories and modules have non-degenerate inner products. |
1214 that our $n$-categories and modules have non-degenerate inner products. |
1172 (In other words, we need to assume some extra duality on the $n$-categories and modules.) |
1215 (In other words, we need to assume some extra duality on the $n$-categories and modules.) |