author | Scott Morrison <scott@tqft.net> |
Sat, 29 May 2010 23:13:20 -0700 | |
changeset 303 | 2252c53bd449 |
parent 194 | 8d3f0bc6a76e |
permissions | -rw-r--r-- |
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%!TEX root = ../../blob1.tex |
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%\section{Morphisms and duals of topological $A_\infty$ modules} |
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%\label{sec:A-infty-hom-and-duals}% |
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% |
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%\begin{defn} |
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%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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%\begin{equation*} |
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%\xymatrix{ |
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%\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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%\cC(J';a,b) \tensor \cN(J,p;b) \ar[r]^{\text{gl}} & \cN(J' cup J,a) |
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%} |
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%\end{equation*} |
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%commutes on the nose, and the diagram |
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%\begin{equation*} |
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%\xymatrix{ |
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%\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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%\CD{(J,p) \to (J',p')} \tensor \cN(J,p;a) \ar[r]^{\text{ev}} & \cN(J',p';a) \\ |
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%} |
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%\end{equation*} |
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%commutes up to a weakly unique homotopy. |
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%\end{defn} |
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%The variations required for right modules and bimodules should be obvious. |
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%\todo{duals} |
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%\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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