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1 %!TEX root = ../../blob1.tex |
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2 |
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3 |
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4 |
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5 %\section{Morphisms and duals of topological $A_\infty$ modules} |
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6 %\label{sec:A-infty-hom-and-duals}% |
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7 % |
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8 %\begin{defn} |
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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 |
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10 %\begin{equation*} |
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11 %\xymatrix{ |
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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 \\ |
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13 %\cC(J';a,b) \tensor \cN(J,p;b) \ar[r]^{\text{gl}} & \cN(J' cup J,a) |
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14 %} |
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15 %\end{equation*} |
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16 %commutes on the nose, and the diagram |
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17 %\begin{equation*} |
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18 %\xymatrix{ |
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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 \\ |
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20 %\CD{(J,p) \to (J',p')} \tensor \cN(J,p;a) \ar[r]^{\text{ev}} & \cN(J',p';a) \\ |
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21 %} |
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22 %\end{equation*} |
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23 %commutes up to a weakly unique homotopy. |
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24 %\end{defn} |
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25 |
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26 %The variations required for right modules and bimodules should be obvious. |
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27 |
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28 %\todo{duals} |
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29 %\todo{ the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic} |
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30 |
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