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. |
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 For expository purposes, we begin with the $n=1$ special cases, and define |
932 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
931 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
933 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
932 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
934 \nn{Something about duals?} |
933 \nn{Something about duals?} |
935 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
934 \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 |
935 \kevin{probably we should say something about the relation |
937 to [framed] $E_\infty$ algebras} |
936 to [framed] $E_\infty$ algebras |
|
937 } |
|
938 |
|
939 \todo{} |
|
940 Various citations we might want to make: |
|
941 \begin{itemize} |
|
942 \item \cite{MR2061854} McClure and Smith's review article |
|
943 \item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) |
|
944 \item \cite{MR0236922,MR0420609} Boardman and Vogt |
|
945 \item \cite{MR1256989} definition of framed little-discs operad |
|
946 \end{itemize} |
938 |
947 |
939 \begin{defn} |
948 \begin{defn} |
940 \label{defn:topological-algebra}% |
949 \label{defn:topological-algebra}% |
941 A ``topological $A_\infty$-algebra'' $A$ consists of the following data. |
950 A ``topological $A_\infty$-algebra'' $A$ consists of the following data. |
942 \begin{enumerate} |
951 \begin{enumerate} |
943 \item For each $1$-manifold $J$ diffeomorphic to the standard interval |
952 \item For each $1$-manifold $J$ diffeomorphic to the standard interval |
944 $I=\left[0,1\right]$, a complex of vector spaces $A(J)$. |
953 $I=\left[0,1\right]$, a complex of vector spaces $A(J)$. |
945 % either roll functoriality into the evaluation map |
954 % either roll functoriality into the evaluation map |
946 \item For each pair of intervals $J,J'$ an `evaluation' chain map |
955 \item For each pair of intervals $J,J'$ an `evaluation' chain map |
947 $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$. |
956 $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$. |
948 \item For each decomposition of intervals $J = J'\cup J''$, |
957 \item For each decomposition of intervals $J = J'\cup J''$, |
949 a gluing map $\gl_{J,J'} : A(J') \tensor A(J'') \to A(J)$. |
958 a gluing map $\gl_{J,J'} : A(J') \tensor A(J'') \to A(J)$. |
950 % or do it as two separate pieces of data |
959 % or do it as two separate pieces of data |
951 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
960 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
961 \CD{J' \to J''} \tensor A(J') \ar[d]^{\ev_{J' \to J''}} \\ |
970 \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''}} & |
971 \CD{J \to J''} \tensor A(J) \ar[r]_{\ev_{J \to J''}} & |
963 A(J'') |
972 A(J'') |
964 } |
973 } |
965 \end{equation*} |
974 \end{equation*} |
966 commutes. |
975 commutes. |
967 \kevin{commutes up to homotopy? in the blob case the evaluation map is ambiguous up to homotopy} |
976 \kevin{commutes up to homotopy? in the blob case the evaluation map is ambiguous up to homotopy} |
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.) |
977 (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.) |
969 %% or the version for separate pieces of data: |
978 %% or the version for separate pieces of data: |
970 %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
979 %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
971 %\item The evaluation chain map is associative, in that the diagram |
980 %\item The evaluation chain map is associative, in that the diagram |
1041 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
1050 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
1042 |
1051 |
1043 The definition of a module follows closely the definition of an algebra or category. |
1052 The definition of a module follows closely the definition of an algebra or category. |
1044 \begin{defn} |
1053 \begin{defn} |
1045 \label{defn:topological-module}% |
1054 \label{defn:topological-module}% |
1046 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ |
1055 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ |
1047 consists of the following data. |
1056 consists of the following data. |
1048 \begin{enumerate} |
1057 \begin{enumerate} |
1049 \item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a upper boundary, to complexes of vector spaces. |
1058 \item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a upper boundary, to complexes of vector spaces. |
1050 \item For each pair of such marked intervals, |
1059 \item For each pair of such marked intervals, |
1051 an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$. |
1060 an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$. |
1052 \item For each decomposition $K = J\cup K'$ of the marked interval |
1061 \item For each decomposition $K = J\cup K'$ of the marked interval |
1053 $K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map |
1062 $K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map |
1054 $\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$. |
1063 $\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$. |
1055 \end{enumerate} |
1064 \end{enumerate} |
1056 The above data is required to satisfy |
1065 The above data is required to satisfy |
1057 conditions analogous to those in Definition \ref{defn:topological-algebra}. |
1066 conditions analogous to those in Definition \ref{defn:topological-algebra}. |
1058 \end{defn} |
1067 \end{defn} |
1059 |
1068 |
1060 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 |
1069 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 |
1061 $\bc_*(Y)$, the topological $A_\infty$ category described above. For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
1070 $\bc_*(Y)$, the topological $A_\infty$ category described above. For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
1066 except that we have two disjoint marked intervals $K$ and $L$, one with a marked point |
1075 except that we have two disjoint marked intervals $K$ and $L$, one with a marked point |
1067 on the upper boundary and the other with a marked point on the lower boundary. |
1076 on the upper boundary and the other with a marked point on the lower boundary. |
1068 There are evaluation maps corresponding to gluing unmarked intervals |
1077 There are evaluation maps corresponding to gluing unmarked intervals |
1069 to the unmarked ends of $K$ and $L$. |
1078 to the unmarked ends of $K$ and $L$. |
1070 |
1079 |
1071 Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a |
1080 Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a |
1072 codimension-0 submanifold of $\bdy X$. |
1081 codimension-0 submanifold of $\bdy X$. |
1073 Then the the assignment $K,L \mapsto \bc*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the |
1082 Then the the assignment $K,L \mapsto \bc*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the |
1074 structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$. |
1083 structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$. |
1075 |
1084 |
1076 Next we define the coend |
1085 Next we define the coend |
1077 (or gluing or tensor product or self tensor product, depending on the context) |
1086 (or gluing or tensor product or self tensor product, depending on the context) |
1078 $\gl(M)$ of a topological $A_\infty$ bimodule $M$. |
1087 $\gl(M)$ of a topological $A_\infty$ bimodule $M$. |
1079 $\gl(M)$ is defined to be the universal thing with the following structure. |
1088 $\gl(M)$ is defined to be the universal thing with the following structure. |
1080 |
1089 |
1081 \begin{itemize} |
1090 \begin{itemize} |
1082 \item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N). |
1091 \item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N). |
1083 \item For each pair of intervals $N,N'$ an evaluation chain map |
1092 \item For each pair of intervals $N,N'$ an evaluation chain map |
1084 $\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$. |
1093 $\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$. |
1085 \item For each decomposition of intervals $N = K\cup L$, |
1094 \item For each decomposition of intervals $N = K\cup L$, |
1086 a gluing map $\gl_{K,L} : M(K,L) \to C(N)$. |
1095 a gluing map $\gl_{K,L} : M(K,L) \to C(N)$. |
1087 \item The evaluation maps are associative. |
1096 \item The evaluation maps are associative. |
1088 \nn{up to homotopy?} |
1097 \nn{up to homotopy?} |
1089 \item Gluing is strictly associative. |
1098 \item Gluing is strictly associative. |
1090 That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to |
1099 That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to |
1091 $K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$ |
1100 $K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$ |
1092 agree. |
1101 agree. |
1093 \item the gluing and evaluation maps are compatible. |
1102 \item the gluing and evaluation maps are compatible. |
1094 \end{itemize} |
1103 \end{itemize} |
1095 |
1104 |
1096 Bu universal we mean that given any other collection of chain complexes, evaluation maps |
1105 Bu universal we mean that given any other collection of chain complexes, evaluation maps |
1097 and gluing maps, they factor through the universal thing. |
1106 and gluing maps, they factor through the universal thing. |
1098 \nn{need to say this in more detail, in particular give the properties of the factoring map} |
1107 \nn{need to say this in more detail, in particular give the properties of the factoring map} |
1099 |
1108 |
1100 Given $X$ and $Y\du -Y \sub \bdy X$ as above, the assignment |
1109 Given $X$ and $Y\du -Y \sub \bdy X$ as above, the assignment |
1101 $N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y)$ clearly has the structure described |
1110 $N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y)$ clearly has the structure described |
1102 in the above bullet points. |
1111 in the above bullet points. |
1103 Showing that it is the universal such thing is the content of the gluing theorem proved below. |
1112 Showing that it is the universal such thing is the content of the gluing theorem proved below. |
1104 |
1113 |
1105 The definitions for a topological $A_\infty$-$n$-category are very similar to the above |
1114 The definitions for a topological $A_\infty$-$n$-category are very similar to the above |
1106 $n=1$ case. |
1115 $n=1$ case. |