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1111 To motivate the definitions which follow, consider algebras $A$ and $B$, right/bi/left modules |
1111 To motivate the definitions which follow, consider algebras $A$ and $B$, right/bi/left modules |
1112 $X_B$, $_BY_A$ and $Z_A$, and the familiar adjunction |
1112 $X_B$, $_BY_A$ and $Z_A$, and the familiar adjunction |
1113 \begin{eqnarray*} |
1113 \begin{eqnarray*} |
1114 \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
1114 \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
1115 f &\mapsto& [x \mapsto f(x\ot -)] \\ |
1115 f &\mapsto& [x \mapsto f(x\ot -)] \\ |
1116 {}[x\ot y \mapsto g(x)(y)] & \leftarrowtail & g . |
1116 {}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g . |
1117 \end{eqnarray*} |
1117 \end{eqnarray*} |
1118 \nn{how to do a left-pointing ``$\mapsto$"?} |
|
1119 If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to |
1118 If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to |
1120 \[ |
1119 \[ |
1121 (X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . |
1120 (X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . |
1122 \] |
1121 \] |
1123 We will establish the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ |
1122 We will establish the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ |