1199 \nn{this completes proof} |
1199 \nn{this completes proof} |
1200 |
1200 |
1201 \input{text/explicit.tex} |
1201 \input{text/explicit.tex} |
1202 |
1202 |
1203 \section{Comparing definitions of $A_\infty$ algebras} |
1203 \section{Comparing definitions of $A_\infty$ algebras} |
|
1204 \label{sec:comparing-A-infty} |
1204 In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}. |
1205 In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}. |
1205 |
1206 |
1206 We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, which we can alternatively characterise as: |
1207 We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, which we can alternatively characterise as: |
1207 \begin{defn} |
1208 \begin{defn} |
1208 A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
1209 A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
1260 \end{align*} |
1261 \end{align*} |
1261 as required (c.f. \cite[p. 6]{MR1854636}). |
1262 as required (c.f. \cite[p. 6]{MR1854636}). |
1262 \todo{then the general case.} |
1263 \todo{then the general case.} |
1263 We won't describe a reverse construction (producing a topological $A_\infty$ category from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts. |
1264 We won't describe a reverse construction (producing a topological $A_\infty$ category from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts. |
1264 |
1265 |
|
1266 \section{Morphisms and duals of topological $A_\infty$ modules} |
|
1267 \label{sec:A-infty-hom-and-duals}% |
|
1268 |
|
1269 \begin{defn} |
|
1270 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 |
|
1271 \begin{equation*} |
|
1272 \xymatrix{ |
|
1273 \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 \\ |
|
1274 \cC(J';a,b) \tensor \cN(J,p;b) \ar[r]^{\text{gl}} & \cN(J' cup J,a) |
|
1275 } |
|
1276 \end{equation*} |
|
1277 commutes on the nose, and the diagram |
|
1278 \begin{equation*} |
|
1279 \xymatrix{ |
|
1280 \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 \\ |
|
1281 \CD{(J,p) \to (J',p')} \tensor \cN(J,p;a) \ar[r]^{\text{ev}} & \cN(J',p';a) \\ |
|
1282 } |
|
1283 \end{equation*} |
|
1284 commutes up to a weakly unique homotopy. |
|
1285 \end{defn} |
|
1286 |
|
1287 The variations required for right modules and bimodules should be obvious. |
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1288 |
|
1289 \todo{duals} |
|
1290 \todo{ the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic} |
1265 |
1291 |
1266 |
1292 |
1267 \input{text/obsolete.tex} |
1293 \input{text/obsolete.tex} |
1268 |
1294 |
1269 % ---------------------------------------------------------------- |
1295 % ---------------------------------------------------------------- |