1090 

  1091 

  1092 

  1093 

  1094 \subsection{Morphisms of $A_\infty$ 1-cat modules}

  1095 

  1096 In order to state and prove our version of the higher dimensional Deligne conjecture

  1097 (Section \ref{sec:deligne}),

  1098 we need to define morphisms of $A_\infty$ 1-cat modules and establish

  1099 some elementary properties of these.

  1100 

  1101 To motivate the definitions which follow, consider algebras $A$ and $B$, right/bi/left modules

  1102 $X_B$, $_BY_A$ and $Z_A$, and the familiar adjunction

  1103 \begin{eqnarray*}

  1104 	\hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\

  1105 	f &\mapsto& [x \mapsto f(x\ot -)] \\

  1106 	{}[x\ot y \mapsto g(x)(y)] & \leftarrowtail & g .

  1107 \end{eqnarray*}

  1108 \nn{how to do a left-pointing ``$\mapsto$"?}

  1109 If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to

  1110 \[

  1111 	(X_B\ot {_BY})^* \cong  \hom_B(X_B \to (_BY)^*) .

  1112 \]

  1113 We will establish the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$

  1114 and modules $\cM_\cC$ and $_\cC\cN$,

  1115 \[

  1116 	(\cM_\cC\ot {_\cC\cN})^* \cong  \hom_\cC(\cM_\cC \to (_\cC\cN)^*) .

  1117 \]

  1118 

  1119 We must now define the things appearing in the above equation.

  1120 

  1121 In the previous subsection we defined a tensor product of $A_\infty$ $n$-cat modules

  1122 for general $n$.

  1123 For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$

  1124 and their gluings (antirefinements).

  1125 (The tensor product will depend (functorially) on the choice of $J$.)

  1126 To a subdivision 

  1127 \[

  1128 	J = I_1\cup \cdots\cup I_m

  1129 \]

  1130 we associate the chain complex

  1131 \[

  1132 	\cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) .

  1133 \]

  1134 (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 

  1136 module actions of $\cC$ on $\cM$ and $\cN$.

  1137 \def\olD{{\overline D}}

  1138 The underlying graded vector space of the homotopy colimit is

  1139 \[

  1140 	\bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] ,

  1141 \]

  1142 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.

  1144 

  1145 \nn{...}

  1146 

  1147 

  1148 

  1149 

  1150 

  1151