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1 \section{Gluing - needs to be rewritten/replaced} |
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2 \label{sec:gluing}% |
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3 |
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4 We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction |
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5 \begin{itemize} |
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6 %\mbox{}% <-- gets the indenting right |
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7 \item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
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8 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
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9 |
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10 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
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11 $A_\infty$ module for $\bc_*(Y \times I)$. |
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12 |
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13 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
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14 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
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15 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
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16 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
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17 \begin{equation*} |
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18 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
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19 \end{equation*} |
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20 \end{itemize} |
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21 |
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22 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 |
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23 definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions, |
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24 and then prove the gluing formula in the topological langauge. Section \ref{sec:topological-A-infty} below explains these definitions, and establishes |
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25 the desired equivalence. This is quite involved, and in particular requires us to generalise the definition of blob homology to allow $A_\infty$ algebras |
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26 as inputs, and to re-establish many of the properties of blob homology in this generality. Many readers may prefer to read the |
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27 Definitions \ref{defn:topological-algebra} and \ref{defn:topological-module} of `topological' $A_\infty$-categories, and Definition \ref{???} of the |
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28 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 |
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29 of the gluing formula in the topological context. |
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30 |
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31 \subsection{`Topological' $A_\infty$ $n$-categories} |
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32 \label{sec:topological-A-infty}% |
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33 |
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34 This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}. |
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35 The main result of this section is |
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36 |
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37 \begin{thm} |
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38 Topological $A_\infty$-$1$-categories are equivalent to the usual notion of |
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39 $A_\infty$-$1$-categories. |
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40 \end{thm} |
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41 |
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42 Before proving this theorem, we embark upon a long string of definitions. |
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43 For expository purposes, we begin with the $n=1$ special cases,\scott{Why are we treating the $n>1$ cases at all?} and define |
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44 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
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45 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
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46 \nn{Something about duals?} |
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47 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
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48 \kevin{probably we should say something about the relation |
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49 to [framed] $E_\infty$ algebras |
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50 } |
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51 |
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52 \todo{} |
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53 Various citations we might want to make: |
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54 \begin{itemize} |
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55 \item \cite{MR2061854} McClure and Smith's review article |
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56 \item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) |
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57 \item \cite{MR0236922,MR0420609} Boardman and Vogt |
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58 \item \cite{MR1256989} definition of framed little-discs operad |
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59 \end{itemize} |
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60 |
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61 \begin{defn} |
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62 \label{defn:topological-algebra}% |
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63 A ``topological $A_\infty$-algebra'' $A$ consists of the following data. |
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64 \begin{enumerate} |
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65 \item For each $1$-manifold $J$ diffeomorphic to the standard interval |
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66 $I=\left[0,1\right]$, a complex of vector spaces $A(J)$. |
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67 % either roll functoriality into the evaluation map |
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68 \item For each pair of intervals $J,J'$ an `evaluation' chain map |
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69 $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$. |
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70 \item For each decomposition of intervals $J = J'\cup J''$, |
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71 a gluing map $\gl_{J',J''} : A(J') \tensor A(J'') \to A(J)$. |
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72 % or do it as two separate pieces of data |
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73 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
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74 %\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, |
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75 %\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
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76 \end{enumerate} |
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77 This data is required to satisfy the following conditions. |
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78 \begin{itemize} |
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79 \item The evaluation chain map is associative, in that the diagram |
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80 \begin{equation*} |
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81 \xymatrix{ |
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82 & \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} & \\ |
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83 \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''}} \\ |
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84 & A(J'') & |
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85 } |
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86 \end{equation*} |
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87 commutes up to homotopy. |
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88 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. |
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89 %% or the version for separate pieces of data: |
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90 %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
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91 %\item The evaluation chain map is associative, in that the diagram |
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92 %\begin{equation*} |
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93 %\xymatrix{ |
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94 %\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\id \tensor \ev_J} \ar[d]_{\compose \tensor \id} & |
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95 %\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ |
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96 %\CD{J} \tensor A(J) \ar[r]_{\ev_J} & |
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97 %A(J) |
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98 %} |
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99 %\end{equation*} |
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100 %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 $\Diff(J)$.) |
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101 \item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram |
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102 \begin{equation*} |
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103 \xymatrix{ |
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104 A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \id} \ar[d]_{\id \tensor \gl_{J',J''}} && |
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105 A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ |
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106 A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && |
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107 A(J \cup J' \cup J'') |
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108 } |
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109 \end{equation*} |
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110 commutes. |
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111 \item The gluing and evaluation maps are compatible. |
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112 \nn{give diagram, or just say ``in the obvious way", or refer to diagram in blob eval map section?} |
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113 \end{itemize} |
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114 \end{defn} |
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115 |
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116 \begin{rem} |
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117 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 |
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118 constitute a functor from the category of intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
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119 Further, once this functor has been specified, we only need to know how the evaluation map acts when $J = J'$. |
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120 \end{rem} |
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121 |
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122 %% if we do things separately, we should say this: |
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123 %\begin{rem} |
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124 %Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of |
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125 %intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
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126 %Further, one can combine the second and third pieces of data, asking instead for a map |
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127 %\begin{equation*} |
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128 %\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J'). |
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129 %\end{equation*} |
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130 %(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 |
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131 %diffeomorphisms in $\CD{J'}$.) |
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132 %\end{rem} |
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133 |
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134 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 |
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135 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: |
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136 \begin{equation*} |
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137 \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
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138 \end{equation*} |
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139 The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions. |
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140 \todo{we presumably need to say something about $\id_c \in A(J, c, c)$.} |
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141 |
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142 At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$. |
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143 \begin{defn} |
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144 Define the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ by |
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145 \begin{enumerate} |
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146 \item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$, |
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147 \item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition |
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148 \begin{align*} |
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149 \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)), |
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150 \end{align*} |
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151 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism, |
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152 \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
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153 \end{enumerate} |
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154 The associativity conditions are trivially satisfied. |
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155 \end{defn} |
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156 |
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157 The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$. |
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158 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 |
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159 \ref{property:evaluation} and \ref{property:gluing-map} respectively. We'll often write $bc_*(Y)$ for this algebra. |
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160 |
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161 The definition of a module follows closely the definition of an algebra or category. |
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162 \begin{defn} |
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163 \label{defn:topological-module}% |
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164 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ |
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165 consists of the following data. |
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166 \begin{enumerate} |
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167 \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. |
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168 \item For each pair of such marked intervals, |
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169 an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$. |
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170 \item For each decomposition $K = J\cup K'$ of the marked interval |
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171 $K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map |
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172 $\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$. |
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173 \end{enumerate} |
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174 The above data is required to satisfy |
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175 conditions analogous to those in Definition \ref{defn:topological-algebra}. |
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176 \end{defn} |
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177 |
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178 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 |
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179 a topological $A_\infty$ module over $\bc_*(Y)$, the topological $A_\infty$ category described above. |
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180 For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
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181 (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 |
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182 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
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183 |
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184 The definition of a bimodule is like the definition of a module, |
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185 except that we have two disjoint marked intervals $K$ and $L$, one with a marked point |
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186 on the upper boundary and the other with a marked point on the lower boundary. |
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187 There are evaluation maps corresponding to gluing unmarked intervals |
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188 to the unmarked ends of $K$ and $L$. |
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189 |
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190 Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a |
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191 codimension-0 submanifold of $\bdy X$. |
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192 Then the the assignment $K,L \mapsto \bc_*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the |
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193 structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$. |
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194 |
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195 Next we define the coend |
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196 (or gluing or tensor product or self tensor product, depending on the context) |
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197 $\gl(M)$ of a topological $A_\infty$ bimodule $M$. This will be an `initial' or `universal' object satisfying various properties. |
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198 \begin{defn} |
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199 We define a category $\cG(M)$. Objects consist of the following data. |
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200 \begin{itemize} |
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201 \item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N). |
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202 \item For each pair of intervals $N,N'$ an evaluation chain map |
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203 $\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$. |
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204 \item For each decomposition of intervals $N = K\cup L$, |
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205 a gluing map $\gl_{K,L} : M(K,L) \to C(N)$. |
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206 \end{itemize} |
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207 This data must satisfy the following conditions. |
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208 \begin{itemize} |
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209 \item The evaluation maps are associative. |
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210 \nn{up to homotopy?} |
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211 \item Gluing is strictly associative. |
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212 That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to |
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213 $K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$ |
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214 agree. |
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215 \item the gluing and evaluation maps are compatible. |
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216 \end{itemize} |
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217 |
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218 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, |
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219 satisfying the following conditions. |
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220 \begin{itemize} |
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221 \item For each pair of intervals $N,N'$, the diagram |
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222 \begin{equation*} |
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223 \xymatrix{ |
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224 \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} \\ |
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225 C(N) \ar[r]_{f_N} & C'(N) |
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226 } |
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227 \end{equation*} |
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228 commutes. |
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229 \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 |
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230 $$M(K,L) \xto{\gl_{K,L}} C(N) \xto{f_N} C'(N).$$ |
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231 \end{itemize} |
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232 \end{defn} |
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233 |
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234 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 $\cG{M}$, |
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235 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 $M(K,L) \to C'(N)$ |
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236 factor through the gluing maps for $\gl(M)$. |
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237 |
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238 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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239 is essentially $C_*(\Maps(S^1 \to M))$. \todo{} |
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240 |
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241 For the second example, given $X$ and $Y\du -Y \sub \bdy X$, the assignment |
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242 $$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y))$$ clearly gives an object in $\cG{M}$. |
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243 Showing that it is an initial object is the content of the gluing theorem proved below. |
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244 |
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245 The definitions for a topological $A_\infty$-$n$-category are very similar to the above |
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246 $n=1$ case. |
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247 One replaces intervals with manifolds diffeomorphic to the ball $B^n$. |
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248 Marked points are replaced by copies of $B^{n-1}$ in $\bdy B^n$. |
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249 |
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250 \nn{give examples: $A(J^n) = \bc_*(Z\times J)$ and $A(J^n) = C_*(\Maps(J \to M))$.} |
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251 |
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252 \todo{the motivating example $C_*(\maps(X, M))$} |
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253 |
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254 |
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255 |
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256 \newcommand{\skel}[1]{\operatorname{skeleton}(#1)} |
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257 |
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258 Given a topological $A_\infty$-category $\cC$, we can construct an `algebraic' $A_\infty$ category $\skel{\cC}$. First, pick your |
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259 favorite diffeomorphism $\phi: I \cup I \to I$. |
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260 \begin{defn} |
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261 We'll write $\skel{\cC} = (A, m_k)$. Define $A = \cC(I)$, and $m_2 : A \tensor A \to A$ by |
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262 \begin{equation*} |
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263 m_2 \cC(I) \tensor \cC(I) \xrightarrow{\gl_{I,I}} \cC(I \cup I) \xrightarrow{\cC(\phi)} \cC(I). |
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264 \end{equation*} |
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265 Next, we define all the `higher associators' $m_k$ by |
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266 \todo{} |
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267 \end{defn} |
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268 |
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269 Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should |
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270 think of this as a generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition |
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271 in the case the $A$ is actually an associative category. |
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272 |
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273 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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274 \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...} |
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275 \begin{align*} |
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276 \end{align*} |
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277 |
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278 \begin{defn} |
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279 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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280 The homological degree of an element $a \in \bc_*^A(J)$ |
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281 is the sum of the blob degree and the internal degree. |
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282 |
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283 We first define $\bc_0^A(J)$ as a vector space by |
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284 \begin{equation*} |
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285 \bc_0^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A). |
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286 \end{equation*} |
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287 (That is, for each division of $J$ into finitely many subintervals, |
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288 we have the tensor product of chains of diffeomorphisms from each subinterval to the standard interval, |
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289 and a copy of $A$ for each subinterval.) |
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290 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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291 plus the sum of the homological degrees of the elements of $A$. |
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292 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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293 |
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294 Next, |
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295 \begin{equation*} |
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296 \bc_1^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \DirectSum_{T \in \cT_{1,n}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A). |
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297 \end{equation*} |
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298 \end{defn} |
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299 |
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300 \begin{figure}[!ht] |
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301 \begin{equation*} |
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302 \mathfig{0.7}{associahedron/A4-vertices} |
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303 \end{equation*} |
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304 \caption{The vertices of the $k$-dimensional associahedron are indexed by binary trees on $k+2$ leaves.} |
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305 \label{fig:A4-vertices} |
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306 \end{figure} |
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307 |
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308 \begin{figure}[!ht] |
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309 \begin{equation*} |
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310 \mathfig{0.7}{associahedron/A4-faces} |
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311 \end{equation*} |
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312 \caption{The faces of the $k$-dimensional associahedron are indexed by trees with $2$ vertices on $k+2$ leaves.} |
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313 \label{fig:A4-vertices} |
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314 \end{figure} |
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315 |
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316 \newcommand{\tm}{\widetilde{m}} |
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317 |
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318 Let $\tm_1(a) = a$. |
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319 |
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320 We now define $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$, first giving an opaque formula, then explaining the combinatorics behind it. |
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321 \begin{align} |
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322 \notag \bdy(\tm_k(a_1 & \tensor \cdots \tensor a_k)) = \\ |
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323 \label{eq:bdy-tm-k-1} & \phantom{+} \sum_{\ell'=0}^{k-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_k(a_1 \tensor \cdots \tensor \bdy a_{\ell'+1} \tensor \cdots \tensor a_k) + \\ |
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324 \label{eq:bdy-tm-k-2} & + \sum_{\ell=1}^{k-1} \tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k) + \\ |
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325 \label{eq:bdy-tm-k-3} & + \sum_{\ell=1}^{k-1} \sum_{\ell'=0}^{l-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell + 1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell + 1}) \tensor \cdots \tensor a_k) |
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326 \end{align} |
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327 The first set of terms in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ just have $\bdy$ acting on each argument $a_i$. |
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328 The terms appearing in \eqref{eq:bdy-tm-k-2} and \eqref{eq:bdy-tm-k-3} are indexed by trees with $2$ vertices on $k+1$ leaves. |
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329 Note here that we have one more leaf than there arguments of $\tm_k$. |
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330 (See Figure \ref{fig:A4-vertices}, in which the rightmost branches are helpfully drawn in red.) |
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331 We will treat the vertices which involve a rightmost (red) branch differently from the vertices which only involve the first $k$ leaves. |
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332 The terms in \eqref{eq:bdy-tm-k-2} arise in the cases in which both |
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333 vertices are rightmost, and the corresponding term in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ is a tensor product of the form |
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334 $$\tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k)$$ |
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335 where $\ell + 1$ and $k - \ell + 1$ are the number of branches entering the vertices. |
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336 If only one vertex is rightmost, we get the term $$\tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell+1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell}) \tensor \cdots \tensor a_k)$$ |
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337 in \eqref{eq:bdy-tm-k-3}, |
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338 where again $\ell + 1$ is the number of branches entering the rightmost vertex, $k-\ell+1$ is the number of branches entering the other vertex, and $\ell'$ is the number of edges meeting the rightmost vertex which start to the left of the other vertex. |
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339 For example, we have |
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340 \begin{align*} |
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341 \bdy(\tm_2(a \tensor b)) & = \left(\tm_2(\bdy a \tensor b) + (-1)^{\abs{a}} \tm_2(a \tensor \bdy b)\right) + \\ |
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342 & \qquad - a \tensor b + m_2(a \tensor b) \\ |
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343 \bdy(\tm_3(a \tensor b \tensor c)) & = \left(- \tm_3(\bdy a \tensor b \tensor c) + (-1)^{\abs{a} + 1} \tm_3(a \tensor \bdy b \tensor c) + (-1)^{\abs{a} + \abs{b} + 1} \tm_3(a \tensor b \tensor \bdy c)\right) + \\ |
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344 & \qquad + \left(- \tm_2(a \tensor b) \tensor c + a \tensor \tm_2(b \tensor c)\right) + \\ |
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345 & \qquad + \left(- \tm_2(m_2(a \tensor b) \tensor c) + \tm_2(a, m_2(b \tensor c)) + m_3(a \tensor b \tensor c)\right) |
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346 \end{align*} |
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347 \begin{align*} |
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348 \bdy(& \tm_4(a \tensor b \tensor c \tensor d)) = \left(\tm_4(\bdy a \tensor b \tensor c \tensor d) + \cdots + \tm_4(a \tensor b \tensor c \tensor \bdy d)\right) + \\ |
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349 & + \left(\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)\right) + \\ |
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350 & + \left(\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))\right. + \\ |
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351 & + \left.\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)\right) \\ |
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352 \end{align*} |
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353 See Figure \ref{fig:A4-terms}, comparing it against Figure \ref{fig:A4-faces}, to see this illustrated in the case $k=4$. There the $3$ faces closest |
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354 to the top of the diagram have two rightmost vertices, while the other $6$ faces have only one. |
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355 |
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356 \begin{figure}[!ht] |
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357 \begin{equation*} |
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358 \mathfig{1.0}{associahedron/A4-terms} |
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359 \end{equation*} |
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360 \caption{The terms of $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ correspond to the faces of the $k-1$ dimensional associahedron.} |
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361 \label{fig:A4-terms} |
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362 \end{figure} |
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363 |
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364 \begin{lem} |
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365 This definition actually results in a chain complex, that is $\bdy^2 = 0$. |
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366 \end{lem} |
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367 \begin{proof} |
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368 \newcommand{\T}{\text{---}} |
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369 \newcommand{\ssum}[1]{{\sum}^{(#1)}} |
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370 For the duration of this proof, inside a summation over variables $l_1, \ldots, l_m$, an expression with $m$ dashes will be interpreted |
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371 by replacing each dash with contiguous factors from $a_1 \tensor \cdots \tensor a_k$, so the first dash takes the first $l_1$ factors, the second |
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372 takes the next $l_2$ factors, and so on. Further, we'll write $\ssum{m}$ for $\sum_{\sum_{i=1}^m l_i = k}$. |
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373 In this notation, the formula for the differential becomes |
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374 \begin{align} |
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375 \notag |
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376 \bdy \tm(\T) & = \ssum{2} \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} + \ssum{3} \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\ |
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377 \intertext{and we calculate} |
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378 \notag |
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379 \bdy^2 \tm(\T) & = \ssum{2} \bdy \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} \\ |
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380 \notag & \qquad + \ssum{2} \tm(\T) \tensor \bdy \tm(\T) \times \sigma_{0;l_1,l_2} \\ |
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381 \notag & \qquad + \ssum{3} \bdy \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\ |
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382 \label{eq:d21} & = \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2,l_3} \sigma_{0;l_1,l_2} \\ |
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383 \label{eq:d22} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2+l_3,l_4} \tau_{0;l_1,l_2,l_3} \\ |
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384 \label{eq:d23} & \qquad + \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2+l_3} \sigma_{l_1;l_2,l_3} \\ |
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385 \label{eq:d24} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \sigma_{0;l_1,l_2+l_3+l_4} \tau_{l_1;l_2,l_3,l_4} \\ |
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386 \label{eq:d25} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \tau_{0;l_1,l_2,l_3+l_4} ??? \\ |
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387 \label{eq:d26} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2,l_3,l_4} \sigma_{0;l_1,l_2} \\ |
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388 \label{eq:d27} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2+l_3,l_4,l_5} \tau_{0;l_1,l_2,l_3} \\ |
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389 \label{eq:d28} & \qquad + \ssum{5} \tm(\T \tensor m(\T \tensor m(\T) \tensor \T) \tensor \T) \times \tau_{0;l_1,l_2+l_3+l_4,l_5} ??? \\ |
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390 \label{eq:d29} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3+l_4+l_5} ??? |
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391 \end{align} |
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392 Now, we see the the expressions on the right hand side of line \eqref{eq:d21} and those on \eqref{eq:d23} cancel. Similarly, line \eqref{eq:d22} cancels |
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393 with \eqref{eq:d25}, \eqref{eq:d24} with \eqref{eq:d26}, and \eqref{eq:d27} with \eqref{eq:d29}. Finally, we need to see that \eqref{eq:d28} gives $0$, |
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394 by the usual relations between the $m_k$ in an $A_\infty$ algebra. |
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395 \end{proof} |
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396 |
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397 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
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398 $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
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399 easy, I think, so maybe it should be done earlier??} |
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400 |
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401 \bigskip |
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402 |
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403 Outline: |
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404 \begin{itemize} |
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405 \item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product. |
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406 use graphical/tree point of view, rather than following Keller exactly |
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407 \item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration? |
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408 \item topological $A_\infty$ cat def (maybe this should go first); also modules gluing |
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409 \item motivating example: $C_*(\maps(X, M))$ |
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410 \item maybe incorporate dual point of view (for $n=1$), where points get |
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411 object labels and intervals get 1-morphism labels |
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412 \end{itemize} |
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413 |
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414 |
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415 \subsection{$A_\infty$ action on the boundary} |
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416 \label{sec:boundary-action}% |
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417 Let $Y$ be an $n{-}1$-manifold. |
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418 The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
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419 conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
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420 of an $A_\infty$ category. |
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421 |
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422 Composition of morphisms (multiplication) depends of a choice of homeomorphism |
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423 $I\cup I \cong I$. Given this choice, gluing gives a map |
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424 \eq{ |
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425 \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
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426 \cong \bc_*(Y\times I; a, c) |
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427 } |
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428 Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
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429 higher associators of the $A_\infty$ structure, more or less canonically. |
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430 |
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431 \nn{is this obvious? does more need to be said?} |
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432 |
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433 Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
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434 |
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435 Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
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436 $(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
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437 (variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the |
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438 $A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
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439 Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
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440 of $Y$ in $X$. |
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441 |
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442 In the next section we use the above $A_\infty$ actions to state and prove |
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443 a gluing theorem for the blob complexes of $n$-manifolds. |
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444 |
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445 |
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446 \subsection{The gluing formula} |
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447 \label{sec:gluing-formula}% |
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448 Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
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449 of $Y \du -Y$ contained in its boundary. |
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450 Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
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451 We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
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452 of $X$. |
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453 More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
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454 where $c\sgl \in \cC(\bd X\sgl)$, |
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455 in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
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456 of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
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457 |
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458 \begin{thm} |
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459 $\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
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460 of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
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461 \end{thm} |
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462 |
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463 The proof will occupy the remainder of this section. |
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464 |
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465 \nn{...} |
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466 |
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467 \bigskip |
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468 |
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469 \nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
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470 |