984 A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && |
984 A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && |
985 A(J \cup J' \cup J'') |
985 A(J \cup J' \cup J'') |
986 } |
986 } |
987 \end{equation*} |
987 \end{equation*} |
988 commutes. |
988 commutes. |
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989 \item The gluing and evaluation maps are compatible. |
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990 \nn{give diagram, or just say ``in the obvious way", or refer to diagram in blob eval map section?} |
989 \end{itemize} |
991 \end{itemize} |
990 \end{defn} |
992 \end{defn} |
991 |
993 |
992 \begin{rem} |
994 \begin{rem} |
993 We can restrict the evaluation map to $0$-chains, and see that $J \mapsto A(J)$ and $(\phi:J \to J') \mapsto \ev_{J \to J'}(\phi, \bullet)$ together |
995 We can restrict the evaluation map to $0$-chains, and see that $J \mapsto A(J)$ and $(\phi:J \to J') \mapsto \ev_{J \to J'}(\phi, \bullet)$ together |
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 |
1058 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 |
1057 $\bc_*(Y)$, the topological $A_\infty$ category described above. For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
1059 $\bc_*(Y)$, the topological $A_\infty$ category described above. For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
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 |
1060 (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 |
1059 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
1061 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
1060 |
1062 |
1061 \todo{Bimodules, and gluing} |
1063 The definition of a bimodule is like the definition of a module, |
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1064 except that we have two disjoint marked intervals $K$ and $L$, one with a marked point |
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1065 on the upper boundary and the other with a marked point on the lower boundary. |
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1066 There are evaluation maps corresponding to gluing unmarked intervals |
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1067 to the unmarked ends of $K$ and $L$. |
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1068 |
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1069 Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a |
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1070 codimension-0 submanifold of $\bdy X$. |
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1071 Then the the assignment $K,L \mapsto \bc*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the |
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1072 structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$. |
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1073 |
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1074 Next we define the coend |
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1075 (or gluing or tensor product or self tensor product, depending on the context) |
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1076 $\gl(M)$ of a topological $A_\infty$ bimodule $M$. |
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1077 $\gl(M)$ is defined to be the universal thing with the following structure. |
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1078 |
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1079 \nn{...} |
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1080 |
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1081 |
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1082 |
1062 |
1083 |
1063 \todo{the motivating example $C_*(\maps(X, M))$} |
1084 \todo{the motivating example $C_*(\maps(X, M))$} |
1064 |
1085 |
1065 \todo{higher $n$} |
1086 \todo{higher $n$} |
1066 |
1087 |