1037 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
1037 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
1038 |
1038 |
1039 The definition of a module follows closely the definition of an algebra or category. |
1039 The definition of a module follows closely the definition of an algebra or category. |
1040 \begin{defn} |
1040 \begin{defn} |
1041 \label{defn:topological-module}% |
1041 \label{defn:topological-module}% |
1042 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ consists of the data |
1042 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ |
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1043 consists of the following data. |
1043 \begin{enumerate} |
1044 \begin{enumerate} |
1044 \item a functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a upper boundary, to complexes of vector spaces, |
1045 \item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a upper boundary, to complexes of vector spaces. |
1045 \item along with an `evaluation' map $\ev_K : \CD{K} \tensor M(K) \to M(K)$ |
1046 \item For each pair of such marked intervals, |
1046 \item and for each interval $J$ and interval $K$ a marked point on the upper boundary, a gluing map |
1047 an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$. |
1047 $\gl_{J,K} : A(J) \tensor M(K) \to M(J \cup K)$ |
1048 \item For each decomposition $K = J\cup K'$ of the marked interval |
|
1049 $K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map |
|
1050 $\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$. |
1048 \end{enumerate} |
1051 \end{enumerate} |
1049 satisfying the obvious conditions analogous to those in Definition \ref{defn:topological-algebra}. |
1052 The above data is required to satisfy |
|
1053 conditions analogous to those in Definition \ref{defn:topological-algebra}. |
1050 \end{defn} |
1054 \end{defn} |
1051 |
1055 |
1052 Any manifold $X$ with $\bdy X = Y$ (or indeed just with $Y$ a codimension $0$-submanifold of $\bdy X$) becomes a topological $A_\infty$ module over |
1056 Any manifold $X$ with $\bdy X = Y$ (or indeed just with $Y$ a codimension $0$-submanifold of $\bdy X$) becomes a topological $A_\infty$ module over |
1053 $\bc_*(Y)$, the topological $A_\infty$ category described above. For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
1057 $\bc_*(Y)$, the topological $A_\infty$ category described above. For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
1054 (Here we glue $Y \times pt$ to $X$, where $pt$ is the marked point of $K$.) Again, the evaluation and gluing maps come directly from Properties |
1058 (Here we glue $Y \times pt$ to $X$, where $pt$ is the marked point of $K$.) Again, the evaluation and gluing maps come directly from Properties |