924 \begin{thm} |
924 \begin{thm} |
925 Topological $A_\infty$-$1$-categories are equivalent to `standard' |
925 Topological $A_\infty$-$1$-categories are equivalent to `standard' |
926 $A_\infty$-$1$-categories. |
926 $A_\infty$-$1$-categories. |
927 \end{thm} |
927 \end{thm} |
928 |
928 |
929 Before proving this theorem, we embark upon a long string of definitions. For expository purposes, we begin with the $n=1$ special cases, and define |
929 Before proving this theorem, we embark upon a long string of definitions. |
|
930 \kevin{the \\kevin macro seems to be truncating text of the left side of the page} |
|
931 For expository purposes, we begin with the $n=1$ special cases, and define |
930 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
932 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
931 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
933 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
932 \nn{Something about duals?} |
934 \nn{Something about duals?} |
933 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
935 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
|
936 \kevin{probably we should say something about the relation |
|
937 to [framed] $E_\infty$ algebras} |
934 |
938 |
935 \begin{defn} |
939 \begin{defn} |
936 \label{defn:topological-algebra}% |
940 \label{defn:topological-algebra}% |
937 A ``topological $A_\infty$-algebra'' $A$ consists of the data |
941 A ``topological $A_\infty$-algebra'' $A$ consists of the following data. |
938 \begin{enumerate} |
942 \begin{enumerate} |
939 \item for each $1$-manifold $J$ diffeomorphic to the standard interval $I=\left[0,1\right]$, a complex of vector spaces $A(J)$, |
943 \item For each $1$-manifold $J$ diffeomorphic to the standard interval |
|
944 $I=\left[0,1\right]$, a complex of vector spaces $A(J)$. |
940 % either roll functoriality into the evaluation map |
945 % either roll functoriality into the evaluation map |
941 \item and for each pair of intervals $J,J'$ an `evaluation' chain map $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$, |
946 \item For each pair of intervals $J,J'$ an `evaluation' chain map |
942 \item and a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
947 $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$. |
|
948 \item For each decomposition of intervals $J = J'\cup J''$, |
|
949 a gluing map $\gl_{J,J'} : A(J') \tensor A(J'') \to A(J)$. |
943 % or do it as two separate pieces of data |
950 % or do it as two separate pieces of data |
944 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
951 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
945 %\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, |
952 %\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, |
946 %\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
953 %\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
947 \end{enumerate} |
954 \end{enumerate} |
948 satisfying the following conditions. |
955 This data is required to satisfy the following conditions. |
949 \begin{itemize} |
956 \begin{itemize} |
950 \item The evaluation chain map is associative, in that the diagram |
957 \item The evaluation chain map is associative, in that the diagram |
951 \begin{equation*} |
958 \begin{equation*} |
952 \xymatrix{ |
959 \xymatrix{ |
953 \CD{J' \to J''} \tensor \CD{J \to J'} \tensor A(J) \ar[r]^{\Id \tensor \ev_{J \to J'}} \ar[d]_{\compose \tensor \Id} & |
960 \CD{J' \to J''} \tensor \CD{J \to J'} \tensor A(J) \ar[r]^{\Id \tensor \ev_{J \to J'}} \ar[d]_{\compose \tensor \Id} & |
1016 \item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition |
1023 \item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition |
1017 \begin{align*} |
1024 \begin{align*} |
1018 \CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
1025 \CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
1019 \end{align*} |
1026 \end{align*} |
1020 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism\todo{inverse, really?!}, |
1027 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism\todo{inverse, really?!}, |
|
1028 \kevin{I think that's fine. If we recoil at taking inverses, |
|
1029 we should use smooth maps instead of diffeos} |
1021 \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
1030 \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
1022 \end{enumerate} |
1031 \end{enumerate} |
1023 The associativity conditions are trivially satisfied. |
1032 The associativity conditions are trivially satisfied. |
1024 \end{defn} |
1033 \end{defn} |
1025 |
1034 |