1 %!TEX root = ../../blob1.tex

     2 

     3 

     4 

     5 %\section{Morphisms and duals of topological $A_\infty$ modules}

     6 %\label{sec:A-infty-hom-and-duals}%

     7 %

     8 %\begin{defn}

     9 %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

    10 %\begin{equation*}

    11 %\xymatrix{

    12 %\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 \\

    13 %\cC(J';a,b) \tensor \cN(J,p;b) \ar[r]^{\text{gl}}                                & \cN(J' cup J,a) 

    14 %}

    15 %\end{equation*}

    16 %commutes on the nose, and the diagram

    17 %\begin{equation*}

    18 %\xymatrix{

    19 %\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 \\

    20 %\CD{(J,p) \to (J',p')} \tensor \cN(J,p;a) \ar[r]^{\text{ev}}  & \cN(J',p';a) \\

    21 %}

    22 %\end{equation*}

    23 %commutes up to a weakly unique homotopy.

    24 %\end{defn}

    25 

    26 %The variations required for right modules and bimodules should be obvious.

    27 

    28 %\todo{duals}

    29 %\todo{ the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic}

    30