935 \begin{defn} |
935 \begin{defn} |
936 \label{defn:topological-algebra}% |
936 \label{defn:topological-algebra}% |
937 A ``topological $A_\infty$-algebra'' $A$ consists of the data |
937 A ``topological $A_\infty$-algebra'' $A$ consists of the data |
938 \begin{enumerate} |
938 \begin{enumerate} |
939 \item for each $1$-manifold $J$ diffeomorphic to the standard interval $I=\left[0,1\right]$, a complex of vector spaces $A(J)$, |
939 \item for each $1$-manifold $J$ diffeomorphic to the standard interval $I=\left[0,1\right]$, a complex of vector spaces $A(J)$, |
940 \item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
940 % either roll functoriality into the evaluation map |
941 \item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, |
941 \item and for each pair of intervals $J,J'$ an `evaluation' chain map $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$, |
942 \item and whenever $\bdy J \cap \bdy J'$ is a single point, a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
942 \item and a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
|
943 % or do it as two separate pieces of data |
|
944 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
|
945 %\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 $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
943 \end{enumerate} |
947 \end{enumerate} |
944 satisfying the following conditions. |
948 satisfying the following conditions. |
945 \begin{itemize} |
949 \begin{itemize} |
946 \item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
|
947 \item The evaluation chain map is associative, in that the diagram |
950 \item The evaluation chain map is associative, in that the diagram |
948 \begin{equation*} |
951 \begin{equation*} |
949 \xymatrix{ |
952 \xymatrix{ |
950 \CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\Id \tensor \ev_J} \ar[d]_{\compose \tensor \Id} & |
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} & |
951 \CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ |
954 \CD{J' \to J''} \tensor A(J') \ar[d]^{\ev_{J' \to J''}} \\ |
952 \CD{J} \tensor A(J) \ar[r]_{\ev_J} & |
955 \CD{J \to J''} \tensor A(J) \ar[r]_{\ev_{J \to J''}} & |
953 A(J) |
956 A(J'') |
954 } |
957 } |
955 \end{equation*} |
958 \end{equation*} |
956 commutes. (Here the map $\compose : \CD{J} \tensor \CD{J} \to \CD{J}$ is a composition: take products of singular chains first, then use the group multiplication in $\Diff(J)$.) |
959 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.) |
|
960 %% or the version for separate pieces of data: |
|
961 %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
|
962 %\item The evaluation chain map is associative, in that the diagram |
|
963 %\begin{equation*} |
|
964 %\xymatrix{ |
|
965 %\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\Id \tensor \ev_J} \ar[d]_{\compose \tensor \Id} & |
|
966 %\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ |
|
967 %\CD{J} \tensor A(J) \ar[r]_{\ev_J} & |
|
968 %A(J) |
|
969 %} |
|
970 %\end{equation*} |
|
971 %commutes. (Here the map $\compose : \CD{J} \tensor \CD{J} \to \CD{J}$ is a composition: take products of singular chains first, then use the group multiplication in $\Diff(J)$.) |
957 \item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram |
972 \item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram |
958 \begin{equation*} |
973 \begin{equation*} |
959 \xymatrix{ |
974 \xymatrix{ |
960 A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \Id} \ar[d]_{\Id \tensor \gl_{J',J''}} && |
975 A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \Id} \ar[d]_{\Id \tensor \gl_{J',J''}} && |
961 A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ |
976 A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ |
966 commutes. |
981 commutes. |
967 \end{itemize} |
982 \end{itemize} |
968 \end{defn} |
983 \end{defn} |
969 |
984 |
970 \begin{rem} |
985 \begin{rem} |
971 Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of |
986 We can restrict the evaluation map to $0$-chains, and see that $J \mapsto A(J)$ and $(\phi:J \to J') \mapsto \ev_{J \to J'}(\phi, -)$ together |
972 intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
987 constitute a functor from the category of intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
973 Further, one can combine the second and third pieces of data, asking instead for a map |
988 Further, once this functor has been specified, we only need to know how the evaluation map acts when $J = J'$. |
974 \begin{equation*} |
|
975 \ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J'). |
|
976 \end{equation*} |
|
977 (Any $k$-parameter family of diffeomorphisms in $C_k(\Diff(J \to J'))$ factors into a single diffeomorphism $J \to J'$ and a $k$-parameter family of |
|
978 diffeomorphisms in $\CD{J'}$.) |
|
979 \end{rem} |
989 \end{rem} |
|
990 |
|
991 %% if we do things separately, we should say this: |
|
992 %\begin{rem} |
|
993 %Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of |
|
994 %intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
|
995 %Further, one can combine the second and third pieces of data, asking instead for a map |
|
996 %\begin{equation*} |
|
997 %\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J'). |
|
998 %\end{equation*} |
|
999 %(Any $k$-parameter family of diffeomorphisms in $C_k(\Diff(J \to J'))$ factors into a single diffeomorphism $J \to J'$ and a $k$-parameter family of |
|
1000 %diffeomorphisms in $\CD{J'}$.) |
|
1001 %\end{rem} |
980 |
1002 |
981 To generalise the definition to that of a category, we simply introduce a set of objects which we call $A(pt)$. Now we associate complexes to each |
1003 To generalise the definition to that of a category, we simply introduce a set of objects which we call $A(pt)$. Now we associate complexes to each |
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: |
1004 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*} |
1005 \begin{equation*} |
984 \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
1006 \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
985 \end{equation*} |
1007 \end{equation*} |
986 The action of diffeomorphisms, and $k$-parameter families of diffeomorphisms, ignore the boundary conditions. |
1008 The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions. |
987 \todo{we presumably need to say something about $\Id_c \in A(J, c, c)$.} |
1009 \todo{we presumably need to say something about $\Id_c \in A(J, c, c)$.} |
|
1010 |
|
1011 At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$. |
|
1012 \begin{defn} |
|
1013 Define the topological $A_\infty$ category $C_*(\Maps(- \to M))$ by |
|
1014 \begin{enumerate} |
|
1015 \item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$, |
|
1016 \item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition $\CD{J \to J'} \tensor C_*(\Maps(J \to M)) \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \to C_*(\Maps(J' \to M))$, where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism\todo{inverse, really?!}, |
|
1017 \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
|
1018 \end{enumerate} |
|
1019 The associativity conditions are trivially satisfied. |
|
1020 \end{defn} |
|
1021 |
|
1022 The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$. |
|
1023 Observe $\Diff(J \to J')$ embeds into $\Diff(Y \times J \to Y \times J')$. The evaluation and gluing maps then come directly from Properties |
|
1024 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
988 |
1025 |
989 The definition of a module follows closely the definition of an algebra or category. |
1026 The definition of a module follows closely the definition of an algebra or category. |
990 \begin{defn} |
1027 \begin{defn} |
991 \label{defn:topological-module}% |
1028 \label{defn:topological-module}% |
992 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ consists of the data |
1029 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ consists of the data |
993 \begin{enumerate} |
1030 \begin{enumerate} |
994 \item a functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a boundary, to complexes of vector spaces, |
1031 \item a functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a boundary, to complexes of vector spaces, |
995 \item along with an `evaluation' map $\ev_K : \CD{K} \tensor M(K) \to M(K)$ |
1032 \item along with an `evaluation' map $\ev_K : \CD{K} \tensor M(K) \to M(K)$ |
996 \item whenever $\bdy J \cap K$ is a single point, and isn't the marked point of $K$ \todo{ugh, that's so gross}, a gluing map |
1033 \item and for each interval $J$ and interval $K$ a marked point on the right boundary, a gluing map |
997 $\gl_{J,K} : A(J) \tensor M(K) \to M(J \cup K)$ |
1034 $\gl_{J,K} : A(J) \tensor M(K) \to M(J \cup K)$ |
998 \end{enumerate} |
1035 \end{enumerate} |
999 satisfying the obvious analogous conditions as in Definition \ref{defn:topological-algebra}. |
1036 satisfying the obvious conditions analogous to those in Definition \ref{defn:topological-algebra}. |
1000 \end{defn} |
1037 \end{defn} |
1001 |
1038 |
1002 \todo{Bimodules, and gluing} |
1039 \todo{Bimodules, and gluing} |
1003 |
1040 |
1004 \todo{the motivating example $C_*(\maps(X, M))$} |
1041 \todo{the motivating example $C_*(\maps(X, M))$} |