961 \CD{J' \to J''} \tensor A(J') \ar[d]^{\ev_{J' \to J''}} \\ |
961 \CD{J' \to J''} \tensor A(J') \ar[d]^{\ev_{J' \to J''}} \\ |
962 \CD{J \to J''} \tensor A(J) \ar[r]_{\ev_{J \to J''}} & |
962 \CD{J \to J''} \tensor A(J) \ar[r]_{\ev_{J \to J''}} & |
963 A(J'') |
963 A(J'') |
964 } |
964 } |
965 \end{equation*} |
965 \end{equation*} |
966 commutes. (Here the map $\compose : \CD{J' \to J''} \tensor \CD{J \to J'} \to \CD{J \to J''}$ is a composition: take products of singular chains first, then compose diffeomorphisms.) |
966 commutes. |
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967 \kevin{commutes up to homotopy? in the blob case the evaluation map is ambiguous up to homotopy} |
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968 (Here the map $\compose : \CD{J' \to J''} \tensor \CD{J \to J'} \to \CD{J \to J''}$ is a composition: take products of singular chains first, then compose diffeomorphisms.) |
967 %% or the version for separate pieces of data: |
969 %% or the version for separate pieces of data: |
968 %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
970 %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
969 %\item The evaluation chain map is associative, in that the diagram |
971 %\item The evaluation chain map is associative, in that the diagram |
970 %\begin{equation*} |
972 %\begin{equation*} |
971 %\xymatrix{ |
973 %\xymatrix{ |
1074 Next we define the coend |
1076 Next we define the coend |
1075 (or gluing or tensor product or self tensor product, depending on the context) |
1077 (or gluing or tensor product or self tensor product, depending on the context) |
1076 $\gl(M)$ of a topological $A_\infty$ bimodule $M$. |
1078 $\gl(M)$ of a topological $A_\infty$ bimodule $M$. |
1077 $\gl(M)$ is defined to be the universal thing with the following structure. |
1079 $\gl(M)$ is defined to be the universal thing with the following structure. |
1078 |
1080 |
1079 \nn{...} |
1081 \begin{itemize} |
1080 |
1082 \item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N). |
1081 |
1083 \item For each pair of intervals $N,N'$ an evaluation chain map |
1082 |
1084 $\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$. |
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1085 \item For each decomposition of intervals $N = K\cup L$, |
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1086 a gluing map $\gl_{K,L} : M(K,L) \to C(N)$. |
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1087 \item The evaluation maps are associative. |
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1088 \nn{up to homotopy?} |
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1089 \item Gluing is strictly associative. |
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1090 That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to |
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1091 $K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$ |
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1092 agree. |
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1093 \item the gluing and evaluation maps are compatible. |
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1094 \end{itemize} |
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1095 |
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1096 Bu universal we mean that given any other collection of chain complexes, evaluation maps |
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1097 and gluing maps, they factor through the universal thing. |
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1098 \nn{need to say this in more detail, in particular give the properties of the factoring map} |
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1099 |
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1100 Given $X$ and $Y\du -Y \sub \bdy X$ as above, the assignment |
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1101 $N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y)$ clearly has the structure described |
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1102 in the above bullet points. |
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1103 Showing that it is the universal such thing is the content of the gluing theorem proved below. |
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1104 |
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1105 The definitions for a topological $A_\infty$-$n$-category are very similar to the above |
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1106 $n=1$ case. |
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1107 One replaces intervals with manifolds diffeomorphic to the ball $B^n$. |
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1108 Marked points are replaced by copies of $B^{n-1}$ in $\bdy B^n$. |
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1109 |
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1110 \nn{give examples: $A(J^n) = \bc_*(Z\times J)$ and $A(J^n) = C_*(\Maps(J \to M))$.} |
1083 |
1111 |
1084 \todo{the motivating example $C_*(\maps(X, M))$} |
1112 \todo{the motivating example $C_*(\maps(X, M))$} |
1085 |
1113 |
1086 \todo{higher $n$} |
|
1087 |
1114 |
1088 |
1115 |
1089 \newcommand{\skel}[1]{\operatorname{skeleton}(#1)} |
1116 \newcommand{\skel}[1]{\operatorname{skeleton}(#1)} |
1090 |
1117 |
1091 Given a topological $A_\infty$-category $\cC$, we can construct an `algebraic' $A_\infty$ category $\skel{\cC}$. First, pick your |
1118 Given a topological $A_\infty$-category $\cC$, we can construct an `algebraic' $A_\infty$ category $\skel{\cC}$. First, pick your |