913 \nn{say something about associativity here} |
919 \nn{say something about associativity here} |
914 |
920 |
915 \section{Gluing} |
921 \section{Gluing} |
916 \label{sec:gluing}% |
922 \label{sec:gluing}% |
917 |
923 |
|
924 We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction |
|
925 \begin{itemize} |
|
926 %\mbox{}% <-- gets the indenting right |
|
927 \item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
|
928 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
|
929 |
|
930 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
|
931 $A_\infty$ module for $\bc_*(Y \times I)$. |
|
932 |
|
933 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
|
934 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
|
935 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
|
936 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
|
937 \begin{equation*} |
|
938 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} |
|
939 \end{equation*} |
|
940 \todo{How do you write self tensor product?} |
|
941 \end{itemize} |
|
942 |
|
943 Although this gluing formula is stated in terms of $A_\infty$ categories and their (bi-)modules, it will be more natural for us to give alternative |
|
944 definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions, |
|
945 and then prove the gluing formula in the topological langauge. Section \ref{sec:topological-A-infty} below explains these definitions, and establishes |
|
946 the desired equivalence. This is quite involved, and in particular requires us to generalise the definition of blob homology to allow $A_\infty$ algebras |
|
947 as inputs, and to re-establish many of the properties of blob homology in this generality. Many readers may prefer to read the |
|
948 Definitions \ref{defn:topological-algebra} and \ref{defn:topological-module} of `topological' $A_\infty$-categories, and Definition \ref{???} of the |
|
949 self-tensor product of a `topological' $A_\infty$-bimodule, then skip to \S \ref{sec:boundary-action} and \S \ref{sec:gluing-formula} for the proofs |
|
950 of the gluing formula in the topological context. |
|
951 |
918 \subsection{`Topological' $A_\infty$ $n$-categories} |
952 \subsection{`Topological' $A_\infty$ $n$-categories} |
919 \label{sec:topological-A-infty}% |
953 \label{sec:topological-A-infty}% |
920 |
954 |
921 This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}. |
955 This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}. |
922 The main result of this section is |
956 The main result of this section is |
923 |
957 |
924 \begin{thm} |
958 \begin{thm} |
925 Topological $A_\infty$-$1$-categories are equivalent to `standard' |
959 Topological $A_\infty$-$1$-categories are equivalent to the usual notion of |
926 $A_\infty$-$1$-categories. |
960 $A_\infty$-$1$-categories. |
927 \end{thm} |
961 \end{thm} |
928 |
962 |
929 Before proving this theorem, we embark upon a long string of definitions. |
963 Before proving this theorem, we embark upon a long string of definitions. |
930 For expository purposes, we begin with the $n=1$ special cases, and define |
964 For expository purposes, we begin with the $n=1$ special cases,\scott{Why are we treating the $n>1$ cases at all?} and define |
931 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
965 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
932 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
966 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
933 \nn{Something about duals?} |
967 \nn{Something about duals?} |
934 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
968 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
935 \kevin{probably we should say something about the relation |
969 \kevin{probably we should say something about the relation |
953 $I=\left[0,1\right]$, a complex of vector spaces $A(J)$. |
987 $I=\left[0,1\right]$, a complex of vector spaces $A(J)$. |
954 % either roll functoriality into the evaluation map |
988 % either roll functoriality into the evaluation map |
955 \item For each pair of intervals $J,J'$ an `evaluation' chain map |
989 \item For each pair of intervals $J,J'$ an `evaluation' chain map |
956 $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$. |
990 $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$. |
957 \item For each decomposition of intervals $J = J'\cup J''$, |
991 \item For each decomposition of intervals $J = J'\cup J''$, |
958 a gluing map $\gl_{J,J'} : A(J') \tensor A(J'') \to A(J)$. |
992 a gluing map $\gl_{J',J''} : A(J') \tensor A(J'') \to A(J)$. |
959 % or do it as two separate pieces of data |
993 % or do it as two separate pieces of data |
960 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
994 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
961 %\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, |
995 %\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, |
962 %\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
996 %\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
963 \end{enumerate} |
997 \end{enumerate} |
964 This data is required to satisfy the following conditions. |
998 This data is required to satisfy the following conditions. |
965 \begin{itemize} |
999 \begin{itemize} |
966 \item The evaluation chain map is associative, in that the diagram |
1000 \item The evaluation chain map is associative, in that the diagram |
967 \begin{equation*} |
1001 \begin{equation*} |
968 \xymatrix{ |
1002 \xymatrix{ |
969 \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} & |
1003 & \quad \mathclap{\CD{J' \to J''} \tensor \CD{J \to J'} \tensor A(J)} \quad \ar[dr]^{\id \tensor \ev_{J \to J'}} \ar[dl]_{\compose \tensor \id} & \\ |
970 \CD{J' \to J''} \tensor A(J') \ar[d]^{\ev_{J' \to J''}} \\ |
1004 \CD{J' \to J''} \tensor A(J') \ar[dr]^{\ev_{J' \to J''}} & & \CD{J \to J''} \tensor A(J) \ar[dl]_{\ev_{J \to J''}} \\ |
971 \CD{J \to J''} \tensor A(J) \ar[r]_{\ev_{J \to J''}} & |
1005 & A(J'') & |
972 A(J'') |
1006 } |
973 } |
1007 \end{equation*} |
974 \end{equation*} |
1008 commutes up to homotopy. |
975 commutes. |
1009 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. |
976 \kevin{commutes up to homotopy? in the blob case the evaluation map is ambiguous up to homotopy} |
|
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.) |
|
978 %% or the version for separate pieces of data: |
1010 %% or the version for separate pieces of data: |
979 %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
1011 %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
980 %\item The evaluation chain map is associative, in that the diagram |
1012 %\item The evaluation chain map is associative, in that the diagram |
981 %\begin{equation*} |
1013 %\begin{equation*} |
982 %\xymatrix{ |
1014 %\xymatrix{ |
983 %\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\Id \tensor \ev_J} \ar[d]_{\compose \tensor \Id} & |
1015 %\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\id \tensor \ev_J} \ar[d]_{\compose \tensor \id} & |
984 %\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ |
1016 %\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ |
985 %\CD{J} \tensor A(J) \ar[r]_{\ev_J} & |
1017 %\CD{J} \tensor A(J) \ar[r]_{\ev_J} & |
986 %A(J) |
1018 %A(J) |
987 %} |
1019 %} |
988 %\end{equation*} |
1020 %\end{equation*} |
989 %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 $\Diff(J)$.) |
1021 %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 $\Diff(J)$.) |
990 \item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram |
1022 \item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram |
991 \begin{equation*} |
1023 \begin{equation*} |
992 \xymatrix{ |
1024 \xymatrix{ |
993 A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \Id} \ar[d]_{\Id \tensor \gl_{J',J''}} && |
1025 A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \id} \ar[d]_{\id \tensor \gl_{J',J''}} && |
994 A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ |
1026 A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ |
995 A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && |
1027 A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && |
996 A(J \cup J' \cup J'') |
1028 A(J \cup J' \cup J'') |
997 } |
1029 } |
998 \end{equation*} |
1030 \end{equation*} |
1024 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: |
1056 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: |
1025 \begin{equation*} |
1057 \begin{equation*} |
1026 \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
1058 \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
1027 \end{equation*} |
1059 \end{equation*} |
1028 The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions. |
1060 The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions. |
1029 \todo{we presumably need to say something about $\Id_c \in A(J, c, c)$.} |
1061 \todo{we presumably need to say something about $\id_c \in A(J, c, c)$.} |
1030 |
1062 |
1031 At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$. |
1063 At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$. |
1032 \begin{defn} |
1064 \begin{defn} |
1033 Define the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ by |
1065 Define the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ by |
1034 \begin{enumerate} |
1066 \begin{enumerate} |
1035 \item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$, |
1067 \item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$, |
1036 \item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition |
1068 \item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition |
1037 \begin{align*} |
1069 \begin{align*} |
1038 \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)), |
1070 \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)), |
1039 \end{align*} |
1071 \end{align*} |
1040 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism\todo{inverse, really?!}, |
1072 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism, |
1041 \kevin{I think that's fine. If we recoil at taking inverses, |
|
1042 we should use smooth maps instead of diffeos} |
|
1043 \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
1073 \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
1044 \end{enumerate} |
1074 \end{enumerate} |
1045 The associativity conditions are trivially satisfied. |
1075 The associativity conditions are trivially satisfied. |
1046 \end{defn} |
1076 \end{defn} |
1047 |
1077 |
1048 The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$. |
1078 The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$. |
1049 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 |
1079 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 |
1050 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
1080 \ref{property:evaluation} and \ref{property:gluing-map} respectively. We'll often write $bc_*(Y)$ for this algebra. |
1051 |
1081 |
1052 The definition of a module follows closely the definition of an algebra or category. |
1082 The definition of a module follows closely the definition of an algebra or category. |
1053 \begin{defn} |
1083 \begin{defn} |
1054 \label{defn:topological-module}% |
1084 \label{defn:topological-module}% |
1055 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ |
1085 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ |
1056 consists of the following data. |
1086 consists of the following data. |
1057 \begin{enumerate} |
1087 \begin{enumerate} |
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. |
1088 \item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with the upper boundary point `marked', to complexes of vector spaces. |
1059 \item For each pair of such marked intervals, |
1089 \item For each pair of such marked intervals, |
1060 an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$. |
1090 an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$. |
1061 \item For each decomposition $K = J\cup K'$ of the marked interval |
1091 \item For each decomposition $K = J\cup K'$ of the marked interval |
1062 $K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map |
1092 $K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map |
1063 $\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$. |
1093 $\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$. |
1064 \end{enumerate} |
1094 \end{enumerate} |
1065 The above data is required to satisfy |
1095 The above data is required to satisfy |
1066 conditions analogous to those in Definition \ref{defn:topological-algebra}. |
1096 conditions analogous to those in Definition \ref{defn:topological-algebra}. |
1067 \end{defn} |
1097 \end{defn} |
1068 |
1098 |
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 |
1099 For any manifold $X$ with $\bdy X = Y$ (or indeed just with $Y$ a codimension $0$-submanifold of $\bdy X$) we can think of $\bc_*(X)$ as |
1070 $\bc_*(Y)$, the topological $A_\infty$ category described above. For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
1100 a topological $A_\infty$ module over $\bc_*(Y)$, the topological $A_\infty$ category described above. |
1071 (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 |
1101 For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
|
1102 (Here we glue $Y \times pt$ to $Y \subset \bdy X$, where $pt$ is the marked point of $K$.) Again, the evaluation and gluing maps come directly from Properties |
1072 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
1103 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
1073 |
1104 |
1074 The definition of a bimodule is like the definition of a module, |
1105 The definition of a bimodule is like the definition of a module, |
1075 except that we have two disjoint marked intervals $K$ and $L$, one with a marked point |
1106 except that we have two disjoint marked intervals $K$ and $L$, one with a marked point |
1076 on the upper boundary and the other with a marked point on the lower boundary. |
1107 on the upper boundary and the other with a marked point on the lower boundary. |
1077 There are evaluation maps corresponding to gluing unmarked intervals |
1108 There are evaluation maps corresponding to gluing unmarked intervals |
1078 to the unmarked ends of $K$ and $L$. |
1109 to the unmarked ends of $K$ and $L$. |
1079 |
1110 |
1080 Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a |
1111 Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a |
1081 codimension-0 submanifold of $\bdy X$. |
1112 codimension-0 submanifold of $\bdy X$. |
1082 Then the the assignment $K,L \mapsto \bc*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the |
1113 Then the the assignment $K,L \mapsto \bc_*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the |
1083 structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$. |
1114 structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$. |
1084 |
1115 |
1085 Next we define the coend |
1116 Next we define the coend |
1086 (or gluing or tensor product or self tensor product, depending on the context) |
1117 (or gluing or tensor product or self tensor product, depending on the context) |
1087 $\gl(M)$ of a topological $A_\infty$ bimodule $M$. |
1118 $\gl(M)$ of a topological $A_\infty$ bimodule $M$. This will be an `initial' or `universal' object satisfying various properties. |
1088 $\gl(M)$ is defined to be the universal thing with the following structure. |
1119 \begin{defn} |
1089 |
1120 We define a category $\cG(M)$. Objects consist of the following data. |
1090 \begin{itemize} |
1121 \begin{itemize} |
1091 \item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N). |
1122 \item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N). |
1092 \item For each pair of intervals $N,N'$ an evaluation chain map |
1123 \item For each pair of intervals $N,N'$ an evaluation chain map |
1093 $\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$. |
1124 $\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$. |
1094 \item For each decomposition of intervals $N = K\cup L$, |
1125 \item For each decomposition of intervals $N = K\cup L$, |
1095 a gluing map $\gl_{K,L} : M(K,L) \to C(N)$. |
1126 a gluing map $\gl_{K,L} : M(K,L) \to C(N)$. |
|
1127 \end{itemize} |
|
1128 This data must satisfy the following conditions. |
|
1129 \begin{itemize} |
1096 \item The evaluation maps are associative. |
1130 \item The evaluation maps are associative. |
1097 \nn{up to homotopy?} |
1131 \nn{up to homotopy?} |
1098 \item Gluing is strictly associative. |
1132 \item Gluing is strictly associative. |
1099 That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to |
1133 That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to |
1100 $K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$ |
1134 $K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$ |
1101 agree. |
1135 agree. |
1102 \item the gluing and evaluation maps are compatible. |
1136 \item the gluing and evaluation maps are compatible. |
1103 \end{itemize} |
1137 \end{itemize} |
1104 |
1138 |
1105 Bu universal we mean that given any other collection of chain complexes, evaluation maps |
1139 A morphism $f$ between such objects $C$ and $C'$ is a chain map $f_N : C(N) \to C'(N)$ for each interval $N$ with both endpoints marked, |
1106 and gluing maps, they factor through the universal thing. |
1140 satisfying the following conditions. |
1107 \nn{need to say this in more detail, in particular give the properties of the factoring map} |
1141 \begin{itemize} |
1108 |
1142 \item For each pair of intervals $N,N'$, the diagram |
1109 Given $X$ and $Y\du -Y \sub \bdy X$ as above, the assignment |
1143 \begin{equation*} |
1110 $N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y)$ clearly has the structure described |
1144 \xymatrix{ |
1111 in the above bullet points. |
1145 \CD{N \to N'} \tensor C(N) \ar[d]_{\ev} \ar[r]^{\id \tensor f_N} & \CD{N \to N'} \tensor C'(N) \ar[d]^{\ev} \\ |
1112 Showing that it is the universal such thing is the content of the gluing theorem proved below. |
1146 C(N) \ar[r]_{f_N} & C'(N) |
|
1147 } |
|
1148 \end{equation*} |
|
1149 commutes. |
|
1150 \item For each decomposition of intervals $N = K \cup L$, the gluing map for $C'$, $\gl'_{K,L} : M(K,L) \to C'(N)$ is the composition |
|
1151 $$M(K,L) \xto{\gl_{K,L}} C(N) \xto{f_N} C'(N).$$ |
|
1152 \end{itemize} |
|
1153 \end{defn} |
|
1154 |
|
1155 We now define $\gl(M)$ to be an initial object in the category $\cG{M}$. This just says that for any other object $C'$ in $\cG{M}$, |
|
1156 there are chain maps $f_N: \gl(M)(N) \to C'(N)$, compatible with the action of families of diffeomorphisms, so that the gluing maps $M(K,L) \to C'(N)$ |
|
1157 factor through the gluing maps for $\gl(M)$. |
|
1158 |
|
1159 We return to our two favourite examples. First, the coend of the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ as a bimodule over itself |
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1160 is essentially $C_*(\Maps(S^1 \to M))$. \todo{} |
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1161 |
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1162 For the second example, given $X$ and $Y\du -Y \sub \bdy X$, the assignment |
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1163 $$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y))$$ clearly gives an object in $\cG{M}$. |
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1164 Showing that it is an initial object is the content of the gluing theorem proved below. |
1113 |
1165 |
1114 The definitions for a topological $A_\infty$-$n$-category are very similar to the above |
1166 The definitions for a topological $A_\infty$-$n$-category are very similar to the above |
1115 $n=1$ case. |
1167 $n=1$ case. |
1116 One replaces intervals with manifolds diffeomorphic to the ball $B^n$. |
1168 One replaces intervals with manifolds diffeomorphic to the ball $B^n$. |
1117 Marked points are replaced by copies of $B^{n-1}$ in $\bdy B^n$. |
1169 Marked points are replaced by copies of $B^{n-1}$ in $\bdy B^n$. |