981 commutes. |
981 commutes. |
982 \end{itemize} |
982 \end{itemize} |
983 \end{defn} |
983 \end{defn} |
984 |
984 |
985 \begin{rem} |
985 \begin{rem} |
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 |
986 We can restrict the evaluation map to $0$-chains, and see that $J \mapsto A(J)$ and $(\phi:J \to J') \mapsto \ev_{J \to J'}(\phi, \bullet)$ together |
987 constitute a functor from the category of 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. |
988 Further, once this functor has been specified, we only need to know how the evaluation map acts when $J = J'$. |
988 Further, once this functor has been specified, we only need to know how the evaluation map acts when $J = J'$. |
989 \end{rem} |
989 \end{rem} |
990 |
990 |
991 %% if we do things separately, we should say this: |
991 %% if we do things separately, we should say this: |
1008 The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions. |
1008 The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions. |
1009 \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 |
1010 |
1011 At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$. |
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} |
1012 \begin{defn} |
1013 Define the topological $A_\infty$ category $C_*(\Maps(- \to M))$ by |
1013 Define the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ by |
1014 \begin{enumerate} |
1014 \begin{enumerate} |
1015 \item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$, |
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?!}, |
1016 \item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition |
|
1017 \begin{align*} |
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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)), |
|
1019 \end{align*} |
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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?!}, |
1017 \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
1021 \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
1018 \end{enumerate} |
1022 \end{enumerate} |
1019 The associativity conditions are trivially satisfied. |
1023 The associativity conditions are trivially satisfied. |
1020 \end{defn} |
1024 \end{defn} |
1021 |
1025 |
1026 The definition of a module follows closely the definition of an algebra or category. |
1030 The definition of a module follows closely the definition of an algebra or category. |
1027 \begin{defn} |
1031 \begin{defn} |
1028 \label{defn:topological-module}% |
1032 \label{defn:topological-module}% |
1029 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ consists of the data |
1033 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ consists of the data |
1030 \begin{enumerate} |
1034 \begin{enumerate} |
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, |
1035 \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, |
1032 \item along with an `evaluation' map $\ev_K : \CD{K} \tensor M(K) \to M(K)$ |
1036 \item along with an `evaluation' map $\ev_K : \CD{K} \tensor M(K) \to M(K)$ |
1033 \item and for each interval $J$ and interval $K$ a marked point on the right boundary, a gluing map |
1037 \item and for each interval $J$ and interval $K$ a marked point on the upper boundary, a gluing map |
1034 $\gl_{J,K} : A(J) \tensor M(K) \to M(J \cup K)$ |
1038 $\gl_{J,K} : A(J) \tensor M(K) \to M(J \cup K)$ |
1035 \end{enumerate} |
1039 \end{enumerate} |
1036 satisfying the obvious conditions analogous to those in Definition \ref{defn:topological-algebra}. |
1040 satisfying the obvious conditions analogous to those in Definition \ref{defn:topological-algebra}. |
1037 \end{defn} |
1041 \end{defn} |
|
1042 |
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1043 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 |
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1044 $\bc_*(Y)$, the topological $A_\infty$ category described above. For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
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1045 (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 |
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1046 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
1038 |
1047 |
1039 \todo{Bimodules, and gluing} |
1048 \todo{Bimodules, and gluing} |
1040 |
1049 |
1041 \todo{the motivating example $C_*(\maps(X, M))$} |
1050 \todo{the motivating example $C_*(\maps(X, M))$} |
1042 |
1051 |
1057 \end{defn} |
1066 \end{defn} |
1058 |
1067 |
1059 Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should |
1068 Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should |
1060 think of this as a generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition |
1069 think of this as a generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition |
1061 in the case the $A$ is actually an associative category. |
1070 in the case the $A$ is actually an associative category. |
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1071 |
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1072 We'll first define $\cT_{k,n}$ to be the set of planar forests consisting of $n-k$ trees, with a total of $n$ leaves. Thus |
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1073 \todo{$\cT_{0,n}$ has 1 element, with $n$ vertical lines, $\cT_{1,n}$ has $n-1$ elements, each with a single trivalent vertex, $\cT_{2,n}$ etc...} |
|
1074 \begin{align*} |
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1075 \end{align*} |
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1076 |
1062 \begin{defn} |
1077 \begin{defn} |
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1078 The topological $A_\infty$ category $\bc_*^A$ is doubly graded, by `blob degree' and `internal degree'. We'll write $\bc_k^A$ for the blob degree $k$ piece. |
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1079 The homological degree of an element $a \in \bc_*^A(J)$ |
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1080 is the sum of the blob degree and the internal degree. |
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1081 |
|
1082 We first define $\bc_0^A(J)$ as a vector space by |
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1083 \begin{equation*} |
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1084 \bc_0^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \left(\{J_i\}, \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A) \right). |
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1085 \end{equation*} |
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1086 (That is, for each division of $J$ into finitely many subintervals, |
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1087 we have the tensor product of chains of diffeomorphisms from each subinterval to the standard interval, |
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1088 and a copy of $A$ for each subinterval.) |
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1089 The internal degree of an element $(f_1 \tensor a_1, \ldots, f_n \tensor a_n)$ is the sum of the dimensions of the singular chains |
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1090 plus the sum of the homological degrees of the elements of $A$. |
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1091 The differential is defined just by the graded Leibniz rule and the differentials on $\CD{J_i \to I}$ and on $A$. |
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1092 |
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1093 Next, |
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1094 \begin{equation*} |
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1095 \bc_1^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \DirectSum_{T \in \cT_{1,n}} \left(\{J_i\}, T, \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A) \right). |
|
1096 \end{equation*} |
1063 \end{defn} |
1097 \end{defn} |
|
1098 |
|
1099 \newcommand{\tm}{\widetilde{m}} |
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1100 \newcommand{\ttm}{\widetilde{\widetilde{m}}} |
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1101 |
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1102 Define $\ttm_k$ by |
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1103 \begin{align*} |
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1104 \ttm_k(a_1 \tensor \cdots \tensor a_k) & = m_k(a_1 \tensor \cdots \tensor a_k) \\ |
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1105 \ttm_k(a_1 \tensor \cdots \tensor a_{k-1} \tensor z) & = z \tensor \tm_{k-1}(a_1 \tensor \cdots \tensor a_{k-1}) \\ |
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1106 \intertext{and} |
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1107 \ttm_k(a_1 \tensor \cdots \tensor a_{k-2} \tensor z \tensor a_k) & = z \tensor \tm_{k-2}(a_1 \tensor \cdots \tensor a_{k-2}) \tensor a_k. |
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1108 \end{align*} |
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1109 |
|
1110 Let $\tm_1(a) = a$. |
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1111 |
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1112 Then define |
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1113 \begin{align*} |
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1114 \bdy(\tm_k(a_1 \tensor \cdots \tensor a_k)) & = \sum_{j=1}^{k} \tm_k(a_1 \tensor \cdots \tensor \bdy a_j \tensor \cdots \tensor a_k) + \\ |
|
1115 & z \perp \sum_{q=2}^{k-1} \sum_{p=1}^{k-q+2} \ttm_{k-q+1}(a_1 \tensor \cdots a_{p-1} \tensor \ttm_q(a_p \tensor \cdots \tensor a_{p+q-1}) \tensor a_{p+q} \tensor \cdots \tensor a_{k+1}). |
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1116 \end{align*} |
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1117 where here $a_{k+1}$ is just notation for $z$. |
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1118 \todo{err... here I mean $z \perp z \tensor x = x$...} |
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1119 \todo{actually, if you let $q$ start from 1 you don't need the first term} |
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1120 |
|
1121 \begin{align*} |
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1122 \bdy(\tm_2(a \tensor b)) & = (\tm_2(\bdy a \tensor b) + \tm_2(a \tensor \bdy b)) + \\ |
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1123 & \qquad + a \tensor b + \\ |
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1124 & \qquad + m_2(a \tensor b) \\ |
|
1125 \bdy(\tm_3(a \tensor b \tensor c)) & = (\tm_3(\bdy a \tensor b \tensor c) + \tm_3(a \tensor \bdy b \tensor c) + \tm_3(a \tensor b \tensor \bdy c)) + \\ |
|
1126 & + (\tm_2(a \tensor b) \tensor c + a \tensor \tm_2(b \tensor c)) + \\ |
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1127 & + (\tm_2(m_2(a \tensor b) \tensor c) + \tm_2(a, m_2(b \tensor c)) + m_3(a \tensor b \tensor c)) \\ |
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1128 \bdy(\tm_4(a \tensor b \tensor c \tensor d)) & = (\tm_4(\bdy a \tensor b \tensor c \tensor d) + \cdots + \tm_4(a \tensor b \tensor c \tensor \bdy d)) + \\ |
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1129 & + (\tm_3(a \tensor b \tensor c) \tensor d + \tm_2(a \tensor b) \tensor \tm_2(c \tensor d) + a \tensor \tm_3(b \tensor c \tensor d)) + \\ |
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1130 & + (\tm_3(m_2(a \tensor b) \tensor c \tensor d) + \tm_3(a \tensor m_2(b \tensor c) \tensor d) + \tm_3(a \tensor b \tensor m_2(c \tensor d)) + \\ |
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1131 & + \tm_2(m_3(a \tensor b \tensor c) \tensor d) + \tm_2(a \tensor m_3(b \tensor c \tensor d)) + m_4(a \tensor b \tensor c \tensor d)) \\ |
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1132 %d(\tm_k(x_1 \tensor \cdots \tensor x_k)) & = \sum_{i=1}^k (-1)^{\sum_{j=1}^{i-1} \deg(x_j)} \tm_k(x_1 \tensor \cdots \tensor d x_i \tensor \cdots \tensor x_k) + \\ |
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1133 % & \qquad + + \\ |
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1134 % & \qquad + |
|
1135 \end{align*} |
1064 |
1136 |
1065 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
1137 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
1066 $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
1138 $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
1067 easy, I think, so maybe it should be done earlier??} |
1139 easy, I think, so maybe it should be done earlier??} |
1068 |
1140 |