982 interval with boundary conditions $(J, c_-, c_+)$, with $c_-, c_+ \in A(pt)$, and only ask for gluing maps when the boundary conditions match up: |
982 interval with boundary conditions $(J, c_-, c_+)$, with $c_-, c_+ \in A(pt)$, and only ask for gluing maps when the boundary conditions match up: |
983 \begin{equation*} |
983 \begin{equation*} |
984 \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
984 \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
985 \end{equation*} |
985 \end{equation*} |
986 The action of diffeomorphisms, and $k$-parameter families of diffeomorphisms, ignore the boundary conditions. |
986 The action of diffeomorphisms, and $k$-parameter families of diffeomorphisms, ignore the boundary conditions. |
|
987 \todo{we presumably need to say something about $\Id_c \in A(J, c, c)$.} |
987 |
988 |
988 The definition of a module follows closely the definition of an algebra or category. |
989 The definition of a module follows closely the definition of an algebra or category. |
989 \begin{defn} |
990 \begin{defn} |
990 \label{defn:topological-module}% |
991 \label{defn:topological-module}% |
991 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ consists of the data |
992 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ consists of the data |
1017 Next, we define all the `higher associators' $m_k$ by |
1018 Next, we define all the `higher associators' $m_k$ by |
1018 \todo{} |
1019 \todo{} |
1019 \end{defn} |
1020 \end{defn} |
1020 |
1021 |
1021 Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should |
1022 Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should |
1022 think of this at the generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition |
1023 think of this as a generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition |
1023 in the case the $A$ is actually an associative category. |
1024 in the case the $A$ is actually an associative category. |
1024 \begin{defn} |
1025 \begin{defn} |
1025 \end{defn} |
1026 \end{defn} |
1026 |
1027 |
1027 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
1028 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |