1063 has the structure of an $n{-}k$-category. |
1063 has the structure of an $n{-}k$-category. |
1064 |
1064 |
1065 \medskip |
1065 \medskip |
1066 |
1066 |
1067 |
1067 |
1068 %\subsection{Tensor products} |
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1069 |
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1070 We will use a simple special case of the above |
1068 We will use a simple special case of the above |
1071 construction to define tensor products |
1069 construction to define tensor products |
1072 of modules. |
1070 of modules. |
1073 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
1071 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
1074 (If $k=1$ and manifolds are oriented, then one should be |
1072 (If $k=1$ and manifolds are oriented, then one should be |
1082 This of course depends (functorially) |
1080 This of course depends (functorially) |
1083 on the choice of 1-ball $J$. |
1081 on the choice of 1-ball $J$. |
1084 |
1082 |
1085 We will define a more general self tensor product (categorified coend) below. |
1083 We will define a more general self tensor product (categorified coend) below. |
1086 |
1084 |
1087 |
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1088 |
|
1089 |
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1090 %\nn{what about self tensor products /coends ?} |
1085 %\nn{what about self tensor products /coends ?} |
1091 |
1086 |
1092 \nn{maybe ``tensor product" is not the best name?} |
1087 \nn{maybe ``tensor product" is not the best name?} |
1093 |
1088 |
1094 %\nn{start with (less general) tensor products; maybe change this later} |
1089 %\nn{start with (less general) tensor products; maybe change this later} |
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1090 |
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1091 |
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1092 |
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1093 |
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1094 \subsection{Morphisms of $A_\infty$ 1-cat modules} |
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1095 |
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1096 In order to state and prove our version of the higher dimensional Deligne conjecture |
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1097 (Section \ref{sec:deligne}), |
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1098 we need to define morphisms of $A_\infty$ 1-cat modules and establish |
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1099 some elementary properties of these. |
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1100 |
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1101 To motivate the definitions which follow, consider algebras $A$ and $B$, right/bi/left modules |
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1102 $X_B$, $_BY_A$ and $Z_A$, and the familiar adjunction |
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1103 \begin{eqnarray*} |
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1104 \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
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1105 f &\mapsto& [x \mapsto f(x\ot -)] \\ |
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1106 {}[x\ot y \mapsto g(x)(y)] & \leftarrowtail & g . |
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1107 \end{eqnarray*} |
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1108 \nn{how to do a left-pointing ``$\mapsto$"?} |
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1109 If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to |
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1110 \[ |
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1111 (X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . |
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1112 \] |
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1113 We will establish the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ |
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1114 and modules $\cM_\cC$ and $_\cC\cN$, |
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1115 \[ |
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1116 (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
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1117 \] |
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1118 |
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1119 We must now define the things appearing in the above equation. |
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1120 |
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1121 In the previous subsection we defined a tensor product of $A_\infty$ $n$-cat modules |
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1122 for general $n$. |
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1123 For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ |
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1124 and their gluings (antirefinements). |
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1125 (The tensor product will depend (functorially) on the choice of $J$.) |
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1126 To a subdivision |
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1127 \[ |
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1128 J = I_1\cup \cdots\cup I_m |
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1129 \] |
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1130 we associate the chain complex |
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1131 \[ |
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1132 \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) . |
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1133 \] |
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1134 (If $D$ denotes the subdivision of $J$, then we denote this complex by $\psi(D)$.) |
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1135 To each antirefinement we associate a chain map using the composition law of $\cC$ and the |
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1136 module actions of $\cC$ on $\cM$ and $\cN$. |
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1137 \def\olD{{\overline D}} |
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1138 The underlying graded vector space of the homotopy colimit is |
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1139 \[ |
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1140 \bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , |
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1141 \] |
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1142 where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ |
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1143 runs through chains of antirefinements, and $[l]$ denotes a grading shift. |
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1144 |
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1145 \nn{...} |
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1146 |
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1147 |
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1148 |
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1149 |
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1150 |
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1151 |
1095 |
1152 |
1096 |
1153 |
1097 |
1154 |
1098 \subsection{The $n{+}1$-category of sphere modules} |
1155 \subsection{The $n{+}1$-category of sphere modules} |
1099 \label{ssec:spherecat} |
1156 \label{ssec:spherecat} |