%!TEX root = ../../blob1.tex

%\section{Morphisms and duals of topological $A_\infty$ modules}

%\label{sec:A-infty-hom-and-duals}%

%

%\begin{defn}

%If $\cM$ and $\cN$ are topological $A_\infty$ left modules over a topological $A_\infty$ category $\cC$, then a morphism $f: \cM \to \cN$ consists of a chain map $f:\cM(J,p;b) \to \cN(J,p;b)$ for each right marked interval $(J,p)$ with a boundary condition $b$, such that  for each interval $J'$ the diagram

%\begin{equation*}

%\xymatrix{

%\cC(J';a,b) \tensor \cM(J,p;b) \ar[r]^{\text{gl}} \ar[d]^{\id \tensor f} & \cM(J' cup J,a) \ar[d]^f \\

%\cC(J';a,b) \tensor \cN(J,p;b) \ar[r]^{\text{gl}}                                & \cN(J' cup J,a) 

%}

%\end{equation*}

%commutes on the nose, and the diagram

%\begin{equation*}

%\xymatrix{

%\CD{(J,p) \to (J',p')} \tensor \cM(J,p;a) \ar[r]^{\text{ev}} \ar[d]^{\id \tensor f} & \cM(J',p';a) \ar[d]^f \\

%\CD{(J,p) \to (J',p')} \tensor \cN(J,p;a) \ar[r]^{\text{ev}}  & \cN(J',p';a) \\

%}

%\end{equation*}

%commutes up to a weakly unique homotopy.

%\end{defn}

%The variations required for right modules and bimodules should be obvious.

%\todo{duals}

%\todo{ the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic}