63 |
59 |
64 } %end \noop |
60 } %end \noop |
65 |
61 |
66 |
62 |
67 |
63 |
68 \input{text/intro.tex} |
64 \input{text/intro} |
69 |
65 |
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66 \input{text/definitions} |
70 |
67 |
71 \section{Definitions} |
68 \input{text/basic_properties} |
72 \label{sec:definitions} |
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73 |
69 |
74 \subsection{Systems of fields} |
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75 \label{sec:fields} |
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76 |
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77 Let $\cM_k$ denote the category (groupoid, in fact) with objects |
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78 oriented PL manifolds of dimension |
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79 $k$ and morphisms homeomorphisms. |
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80 (We could equally well work with a different category of manifolds --- |
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81 unoriented, topological, smooth, spin, etc. --- but for definiteness we |
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82 will stick with oriented PL.) |
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83 |
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84 Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. |
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85 |
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86 A $n$-dimensional {\it system of fields} in $\cS$ |
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87 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
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88 together with some additional data and satisfying some additional conditions, all specified below. |
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89 |
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90 \nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris} |
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91 |
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92 Before finishing the definition of fields, we give two motivating examples |
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93 (actually, families of examples) of systems of fields. |
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94 |
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95 The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps |
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96 from X to $B$. |
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97 |
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98 The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be |
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99 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
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100 $j$-morphisms of $C$. |
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101 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
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102 This is described in more detail below. |
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103 |
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104 Now for the rest of the definition of system of fields. |
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105 \begin{enumerate} |
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106 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
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107 and these maps are a natural |
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108 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
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109 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
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110 $\cC(X)$ which restricts to $c$. |
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111 In this context, we will call $c$ a boundary condition. |
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112 \item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. |
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113 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps |
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114 again comprise a natural transformation of functors. |
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115 In addition, the orientation reversal maps are compatible with the boundary restriction maps. |
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116 \item $\cC_k$ is compatible with the symmetric monoidal |
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117 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
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118 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
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119 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
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120 restriction maps. |
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121 \item Gluing without corners. |
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122 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
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123 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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124 Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
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125 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
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126 copies of $Y$ in $\bd X$. |
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127 Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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128 Then (here's the axiom/definition part) there is an injective ``gluing" map |
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129 \[ |
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130 \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
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131 \] |
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132 and this gluing map is compatible with all of the above structure (actions |
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133 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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134 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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135 the gluing map is surjective. |
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136 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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137 gluing surface, we say that fields in the image of the gluing map |
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138 are transverse to $Y$ or cuttable along $Y$. |
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139 \item Gluing with corners. |
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140 Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
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141 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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142 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
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143 (without corners) along two copies of $\bd Y$. |
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144 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a cuttable field on $W\sgl$ and let |
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145 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
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146 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
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147 (This restriction map uses the gluing without corners map above.) |
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148 Using the boundary restriction, gluing without corners, and (in one case) orientation reversal |
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149 maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
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150 copies of $Y$ in $\bd X$. |
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151 Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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152 Then (here's the axiom/definition part) there is an injective ``gluing" map |
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153 \[ |
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154 \Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
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155 \] |
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156 and this gluing map is compatible with all of the above structure (actions |
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157 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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158 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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159 the gluing map is surjective. |
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160 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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161 gluing surface, we say that fields in the image of the gluing map |
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162 are transverse to $Y$ or cuttable along $Y$. |
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163 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
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164 $c \mapsto c\times I$. |
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165 These maps comprise a natural transformation of functors, and commute appropriately |
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166 with all the structure maps above (disjoint union, boundary restriction, etc.). |
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167 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
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168 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
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169 \end{enumerate} |
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170 |
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171 \nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$} |
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172 |
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173 \bigskip |
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174 Using the functoriality and $\bullet\times I$ properties above, together |
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175 with boundary collar homeomorphisms of manifolds, we can define the notion of |
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176 {\it extended isotopy}. |
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177 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
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178 of $\bd M$. |
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179 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$. |
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180 Let $c$ be $x$ restricted to $Y$. |
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181 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
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182 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
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183 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
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184 Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
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185 More generally, we define extended isotopy to be the equivalence relation on fields |
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186 on $M$ generated by isotopy plus all instance of the above construction |
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187 (for all appropriate $Y$ and $x$). |
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188 |
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189 \nn{should also say something about pseudo-isotopy} |
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190 |
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191 %\bigskip |
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192 %\hrule |
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193 %\bigskip |
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194 % |
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195 %\input{text/fields.tex} |
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196 % |
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197 % |
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198 %\bigskip |
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199 %\hrule |
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200 %\bigskip |
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201 |
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202 \nn{note: probably will suppress from notation the distinction |
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203 between fields and their (orientation-reversal) duals} |
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204 |
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205 \nn{remark that if top dimensional fields are not already linear |
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206 then we will soon linearize them(?)} |
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207 |
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208 We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
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209 by $n$-category morphisms. |
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210 |
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211 Given an $n$-category $C$ with the right sort of duality |
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212 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
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213 we can construct a system of fields as follows. |
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214 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
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215 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
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216 We'll spell this out for $n=1,2$ and then describe the general case. |
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217 |
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218 If $X$ has boundary, we require that the cell decompositions are in general |
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219 position with respect to the boundary --- the boundary intersects each cell |
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220 transversely, so cells meeting the boundary are mere half-cells. |
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221 |
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222 Put another way, the cell decompositions we consider are dual to standard cell |
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223 decompositions of $X$. |
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224 |
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225 We will always assume that our $n$-categories have linear $n$-morphisms. |
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226 |
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227 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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228 an object (0-morphism) of the 1-category $C$. |
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229 A field on a 1-manifold $S$ consists of |
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230 \begin{itemize} |
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231 \item A cell decomposition of $S$ (equivalently, a finite collection |
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232 of points in the interior of $S$); |
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233 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
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234 by an object (0-morphism) of $C$; |
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235 \item a transverse orientation of each 0-cell, thought of as a choice of |
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236 ``domain" and ``range" for the two adjacent 1-cells; and |
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237 \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
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238 domain and range determined by the transverse orientation and the labelings of the 1-cells. |
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239 \end{itemize} |
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240 |
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241 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
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242 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
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243 interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
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244 of the algebra. |
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245 |
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246 \medskip |
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247 |
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248 For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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249 that are common in the literature. |
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250 We describe these carefully here. |
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251 |
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252 A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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253 an object of the 2-category $C$. |
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254 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
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255 A field on a 2-manifold $Y$ consists of |
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256 \begin{itemize} |
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257 \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
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258 that each component of the complement is homeomorphic to a disk); |
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259 \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
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260 by a 0-morphism of $C$; |
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261 \item a transverse orientation of each 1-cell, thought of as a choice of |
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262 ``domain" and ``range" for the two adjacent 2-cells; |
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263 \item a labeling of each 1-cell by a 1-morphism of $C$, with |
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264 domain and range determined by the transverse orientation of the 1-cell |
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265 and the labelings of the 2-cells; |
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266 \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
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267 of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
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268 to $\pm 1 \in S^1$; and |
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269 \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
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270 determined by the labelings of the 1-cells and the parameterizations of the previous |
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271 bullet. |
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272 \end{itemize} |
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273 \nn{need to say this better; don't try to fit everything into the bulleted list} |
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274 |
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275 For general $n$, a field on a $k$-manifold $X^k$ consists of |
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276 \begin{itemize} |
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277 \item A cell decomposition of $X$; |
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278 \item an explicit general position homeomorphism from the link of each $j$-cell |
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279 to the boundary of the standard $(k-j)$-dimensional bihedron; and |
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280 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
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281 domain and range determined by the labelings of the link of $j$-cell. |
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282 \end{itemize} |
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283 |
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284 %\nn{next definition might need some work; I think linearity relations should |
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285 %be treated differently (segregated) from other local relations, but I'm not sure |
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286 %the next definition is the best way to do it} |
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287 |
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288 \medskip |
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289 |
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290 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
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291 in the linearized space of fields. |
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292 By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
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293 the vector space of finite |
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294 linear combinations of fields on $X$. |
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295 If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
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296 Thus the restriction (to boundary) maps are well defined because we never |
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297 take linear combinations of fields with differing boundary conditions. |
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298 |
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299 In some cases we don't linearize the default way; instead we take the |
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300 spaces $\lf(X; a)$ to be part of the data for the system of fields. |
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301 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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302 Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
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303 obvious relations on 0-cell labels. |
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304 More specifically, let $L$ be a cell decomposition of $X$ |
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305 and let $p$ be a 0-cell of $L$. |
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306 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
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307 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
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308 Then the subspace $K$ is generated by things of the form |
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309 $\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
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310 to infer the meaning of $\alpha_{\lambda c + d}$. |
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311 Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
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312 |
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313 \nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
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314 will do something similar below; in general, whenever a label lives in a linear |
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315 space we do something like this; ? say something about tensor |
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316 product of all the linear label spaces? Yes:} |
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317 |
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318 For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
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319 Define an ``almost-field" to be a field without labels on the 0-cells. |
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320 (Recall that 0-cells are labeled by $n$-morphisms.) |
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321 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
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322 space determined by the labeling of the link of the 0-cell. |
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323 (If the 0-cell were labeled, the label would live in this space.) |
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324 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
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325 We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
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326 above tensor products. |
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327 |
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328 |
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329 |
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330 \subsection{Local relations} |
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331 \label{sec:local-relations} |
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332 |
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333 |
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334 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
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335 for all $n$-manifolds $B$ which are |
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336 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
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337 satisfying the following properties. |
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338 \begin{enumerate} |
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339 \item functoriality: |
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340 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
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341 \item local relations imply extended isotopy: |
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342 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
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343 to $y$, then $x-y \in U(B; c)$. |
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344 \item ideal with respect to gluing: |
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345 if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ |
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346 \end{enumerate} |
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347 See \cite{kw:tqft} for details. |
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348 |
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349 |
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350 For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, |
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351 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
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352 |
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353 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
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354 $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
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355 domain and range. |
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356 |
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357 \nn{maybe examples of local relations before general def?} |
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358 |
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359 Given a system of fields and local relations, we define the skein space |
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360 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
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361 the $n$-manifold $Y$ modulo local relations. |
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362 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
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363 is defined to be the dual of $A(Y; c)$. |
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364 (See \cite{kw:tqft} or xxxx for details.) |
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365 |
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366 \nn{should expand above paragraph} |
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367 |
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368 The blob complex is in some sense the derived version of $A(Y; c)$. |
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369 |
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370 |
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371 |
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372 \subsection{The blob complex} |
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373 \label{sec:blob-definition} |
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374 |
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375 Let $X$ be an $n$-manifold. |
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376 Assume a fixed system of fields and local relations. |
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377 In this section we will usually suppress boundary conditions on $X$ from the notation |
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378 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
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379 |
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380 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
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381 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
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382 $\overline{X \setmin Y}$. |
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383 |
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384 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
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385 |
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386 Define $\bc_0(X) = \lf(X)$. |
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387 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
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388 We'll omit this sort of detail in the rest of this section.) |
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389 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
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390 |
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391 $\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
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392 Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
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393 combinations of 1-blob diagrams, where a 1-blob diagram to consists of |
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394 \begin{itemize} |
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395 \item An embedded closed ball (``blob") $B \sub X$. |
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396 \item A field $r \in \cC(X \setmin B; c)$ |
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397 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
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398 \item A local relation field $u \in U(B; c)$ |
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399 (same $c$ as previous bullet). |
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400 \end{itemize} |
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401 In order to get the linear structure correct, we (officially) define |
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402 \[ |
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403 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
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404 \] |
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405 The first direct sum is indexed by all blobs $B\subset X$, and the second |
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406 by all boundary conditions $c \in \cC(\bd B)$. |
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407 Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
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408 |
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409 Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
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410 \[ |
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411 (B, u, r) \mapsto u\bullet r, |
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412 \] |
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413 where $u\bullet r$ denotes the linear |
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414 combination of fields on $X$ obtained by gluing $u$ to $r$. |
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415 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
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416 just erasing the blob from the picture |
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417 (but keeping the blob label $u$). |
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418 |
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419 Note that the skein space $A(X)$ |
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420 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
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421 |
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422 $\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the |
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423 local relations encoded in $\bc_1(X)$. |
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424 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
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425 2-blob diagrams, of which there are two types, disjoint and nested. |
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426 |
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427 A disjoint 2-blob diagram consists of |
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428 \begin{itemize} |
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429 \item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
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430 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
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431 (where $c_i \in \cC(\bd B_i)$). |
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432 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
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433 \end{itemize} |
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434 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
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435 reversing the order of the blobs changes the sign. |
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436 Define $\bd(B_0, B_1, u_0, u_1, r) = |
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437 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
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438 In other words, the boundary of a disjoint 2-blob diagram |
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439 is the sum (with alternating signs) |
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440 of the two ways of erasing one of the blobs. |
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441 It's easy to check that $\bd^2 = 0$. |
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442 |
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443 A nested 2-blob diagram consists of |
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444 \begin{itemize} |
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445 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
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446 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
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447 (for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$. |
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448 \item A local relation field $u_0 \in U(B_0; c_0)$. |
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449 \end{itemize} |
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450 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
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451 (for some $c_1 \in \cC(B_1)$) and |
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452 $r' \in \cC(X \setmin B_1; c_1)$. |
|
453 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
|
454 Note that the requirement that |
|
455 local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
|
456 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
|
457 sum of the two ways of erasing one of the blobs. |
|
458 If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
|
459 It is again easy to check that $\bd^2 = 0$. |
|
460 |
|
461 \nn{should draw figures for 1, 2 and $k$-blob diagrams} |
|
462 |
|
463 As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
|
464 (officially) |
|
465 \begin{eqnarray*} |
|
466 \bc_2(X) & \deq & |
|
467 \left( |
|
468 \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} |
|
469 U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) |
|
470 \right) \\ |
|
471 && \bigoplus \left( |
|
472 \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
|
473 U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
|
474 \right) . |
|
475 \end{eqnarray*} |
|
476 The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$, |
|
477 but we didn't feel like introducing a notation for that. |
|
478 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
|
479 (rather than a new, linearly independent 2-blob diagram). |
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480 |
|
481 Now for the general case. |
|
482 A $k$-blob diagram consists of |
|
483 \begin{itemize} |
|
484 \item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
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485 For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or |
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486 $B_i \sub B_j$ or $B_j \sub B_i$. |
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487 (The case $B_i = B_j$ is allowed. |
|
488 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
|
489 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
|
490 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
|
491 (These are implied by the data in the next bullets, so we usually |
|
492 suppress them from the notation.) |
|
493 $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
|
494 if the latter space is not empty. |
|
495 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
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496 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
|
497 is determined by the $c_i$'s. |
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498 $r$ is required to be cuttable along the boundaries of all blobs, twigs or not. |
|
499 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
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500 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
|
501 If $B_i = B_j$ then $u_i = u_j$. |
|
502 \end{itemize} |
|
503 |
|
504 If two blob diagrams $D_1$ and $D_2$ |
|
505 differ only by a reordering of the blobs, then we identify |
|
506 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
|
507 |
|
508 $\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. |
|
509 As before, the official definition is in terms of direct sums |
|
510 of tensor products: |
|
511 \[ |
|
512 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
|
513 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
|
514 \] |
|
515 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
|
516 $\overline{c}$ runs over all boundary conditions, again as described above. |
|
517 $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are cuttable along all of the blobs in $\overline{B}$. |
|
518 |
|
519 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
|
520 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
|
521 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
|
522 If $B_j$ is not a twig blob, this involves only decrementing |
|
523 the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
|
524 If $B_j$ is a twig blob, we have to assign new local relation labels |
|
525 if removing $B_j$ creates new twig blobs. |
|
526 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
|
527 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
|
528 Finally, define |
|
529 \eq{ |
|
530 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
|
531 } |
|
532 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
|
533 Thus we have a chain complex. |
|
534 |
|
535 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
|
536 |
|
537 \nn{?? remark about dendroidal sets} |
|
538 |
|
539 |
|
540 |
|
541 \section{Basic properties of the blob complex} |
|
542 \label{sec:basic-properties} |
|
543 |
|
544 \begin{prop} \label{disjunion} |
|
545 There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
|
546 \end{prop} |
|
547 \begin{proof} |
|
548 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
|
549 (putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
|
550 blob diagram $(b_1, b_2)$ on $X \du Y$. |
|
551 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
|
552 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
|
553 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
|
554 a pair of blob diagrams on $X$ and $Y$. |
|
555 These two maps are compatible with our sign conventions. |
|
556 The two maps are inverses of each other. |
|
557 \nn{should probably say something about sign conventions for the differential |
|
558 in a tensor product of chain complexes; ask Scott} |
|
559 \end{proof} |
|
560 |
|
561 For the next proposition we will temporarily restore $n$-manifold boundary |
|
562 conditions to the notation. |
|
563 |
|
564 Suppose that for all $c \in \cC(\bd B^n)$ |
|
565 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
|
566 of the quotient map |
|
567 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
|
568 For example, this is always the case if you coefficient ring is a field. |
|
569 Then |
|
570 \begin{prop} \label{bcontract} |
|
571 For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
|
572 is a chain homotopy equivalence |
|
573 with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
|
574 Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0. |
|
575 \end{prop} |
|
576 \begin{proof} |
|
577 By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map |
|
578 $h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$. |
|
579 For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
|
580 an $(i{+}1)$-st blob equal to all of $B^n$. |
|
581 In other words, add a new outermost blob which encloses all of the others. |
|
582 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
|
583 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
|
584 \end{proof} |
|
585 |
|
586 Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
|
587 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
|
588 |
|
589 For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
|
590 where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
|
591 |
|
592 \medskip |
|
593 |
|
594 \nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction. |
|
595 But I think it's worth saying that the Diff actions will be enhanced later. |
|
596 Maybe put that in the intro too.} |
|
597 |
|
598 As we noted above, |
|
599 \begin{prop} |
|
600 There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
|
601 \qed |
|
602 \end{prop} |
|
603 |
|
604 |
|
605 \begin{prop} |
|
606 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
|
607 of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
|
608 (chain map) isomorphisms. |
|
609 \qed |
|
610 \end{prop} |
|
611 |
|
612 In particular, |
|
613 \begin{prop} \label{diff0prop} |
|
614 There is an action of $\Diff(X)$ on $\bc_*(X)$. |
|
615 \qed |
|
616 \end{prop} |
|
617 |
|
618 The above will be greatly strengthened in Section \ref{sec:evaluation}. |
|
619 |
|
620 \medskip |
|
621 |
|
622 For the next proposition we will temporarily restore $n$-manifold boundary |
|
623 conditions to the notation. |
|
624 |
|
625 Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
|
626 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
|
627 with boundary $Z\sgl$. |
|
628 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
|
629 we have the blob complex $\bc_*(X; a, b, c)$. |
|
630 If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
|
631 $X$ to get blob diagrams on $X\sgl$: |
|
632 |
|
633 \begin{prop} |
|
634 There is a natural chain map |
|
635 \eq{ |
|
636 \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
|
637 } |
|
638 The sum is over all fields $a$ on $Y$ compatible at their |
|
639 ($n{-}2$-dimensional) boundaries with $c$. |
|
640 `Natural' means natural with respect to the actions of diffeomorphisms. |
|
641 \qed |
|
642 \end{prop} |
|
643 |
|
644 The above map is very far from being an isomorphism, even on homology. |
|
645 This will be fixed in Section \ref{sec:gluing} below. |
|
646 |
|
647 \nn{Next para not need, since we already use bullet = gluing notation above(?)} |
|
648 |
|
649 An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
|
650 and $X\sgl = X_1 \cup_Y X_2$. |
|
651 (Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
|
652 For $x_i \in \bc_*(X_i)$, we introduce the notation |
|
653 \eq{ |
|
654 x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
|
655 } |
|
656 Note that we have resumed our habit of omitting boundary labels from the notation. |
|
657 |
|
658 |
|
659 |
|
660 |
|
661 |
|
662 \section{Hochschild homology when $n=1$} |
|
663 \label{sec:hochschild} |
|
664 \input{text/hochschild} |
70 \input{text/hochschild} |
665 |
71 |
666 |
|
667 |
|
668 |
|
669 \section{Action of $\CD{X}$} |
|
670 \label{sec:evaluation} |
|
671 \input{text/evmap} |
72 \input{text/evmap} |
672 |
73 |
|
74 \input{text/ncat} |
673 |
75 |
|
76 \input{text/A-infty} |
674 |
77 |
675 \input{text/ncat.tex} |
78 \input{text/gluing} |
676 |
79 |
677 \input{text/A-infty.tex} |
80 \input{text/comm_alg} |
678 |
|
679 \input{text/gluing.tex} |
|
680 |
|
681 |
|
682 |
|
683 \section{Commutative algebras as $n$-categories} |
|
684 |
|
685 \nn{this should probably not be a section by itself. i'm just trying to write down the outline |
|
686 while it's still fresh in my mind.} |
|
687 |
|
688 If $C$ is a commutative algebra it |
|
689 can (and will) also be thought of as an $n$-category whose $j$-morphisms are trivial for |
|
690 $j<n$ and whose $n$-morphisms are $C$. |
|
691 The goal of this \nn{subsection?} is to compute |
|
692 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
|
693 |
|
694 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
|
695 |
|
696 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
|
697 unlabeled points in $M$. |
|
698 Note that $\Sigma^0(M)$ is a point. |
|
699 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
|
700 |
|
701 Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$. |
|
702 |
|
703 \begin{prop} \label{sympowerprop} |
|
704 $\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
|
705 \end{prop} |
|
706 |
|
707 \begin{proof} |
|
708 To define the chain maps between the two complexes we will use the following lemma: |
|
709 |
|
710 \begin{lemma} |
|
711 Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
|
712 a basis (e.g.\ blob diagrams or singular simplices). |
|
713 For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
|
714 such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$. |
|
715 Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
|
716 $f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
|
717 \end{lemma} |
|
718 |
|
719 \begin{proof} |
|
720 \nn{easy, but should probably write the details eventually} |
|
721 \end{proof} |
|
722 |
|
723 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
|
724 satisfying the conditions of the above lemma. |
|
725 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
|
726 finite unordered collection of points of $M$ with multiplicities, which is |
|
727 a point in $\Sigma^\infty(M)$. |
|
728 Define $R(b)_*$ to be the singular chain complex of this point. |
|
729 If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). |
|
730 The path components of $\Sigma^\infty(D)$ are contractible, and these components are indexed |
|
731 by the numbers of points in each component of $D$. |
|
732 We may assume that the blob labels $u$ have homogeneous $t$ degree in $k[t]$, and so |
|
733 $u$ picks out a component $X \sub \Sigma^\infty(D)$. |
|
734 The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, |
|
735 and using this point we can embed $X$ in $\Sigma^\infty(M)$. |
|
736 Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a |
|
737 subspace of $\Sigma^\infty(M)$. |
|
738 It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from the above lemma. |
|
739 Thus we have defined (up to homotopy) a map from |
|
740 $\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$. |
|
741 |
|
742 Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace |
|
743 $R(c)_* \sub \bc_*(M^n, k[t])$. |
|
744 If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and |
|
745 $\Sigma^\infty(M)$ described above. |
|
746 Now let $c$ be an $i$-simplex of $\Sigma^j(M)$. |
|
747 Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. |
|
748 We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ |
|
749 is homotopy equivalent to the subcomplex of small simplices. |
|
750 How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. |
|
751 Let $T\sub M$ be the ``track" of $c$ in $M$. |
|
752 \nn{do we need to define this precisely?} |
|
753 Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. |
|
754 \nn{need to say more precisely how small} |
|
755 Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$. |
|
756 This is contractible by \ref{bcontract}. |
|
757 We can arrange that the boundary/inclusion condition is satisfied if we start with |
|
758 low-dimensional simplices and work our way up. |
|
759 \nn{need to be more precise} |
|
760 |
|
761 \nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
|
762 (for this, might need a lemma that says we can assume that blob diameters are small)} |
|
763 \end{proof} |
|
764 |
|
765 |
|
766 \begin{prop} \label{ktcdprop} |
|
767 The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
|
768 \end{prop} |
|
769 |
|
770 \begin{proof} |
|
771 The actions agree in degree 0, and both are compatible with gluing. |
|
772 (cf. uniqueness statement in \ref{CDprop}.) |
|
773 \end{proof} |
|
774 |
|
775 \medskip |
|
776 |
|
777 In view of \ref{hochthm}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
|
778 and that the cyclic homology of $k[t]$ is related to the action of rotations |
|
779 on $C_*(\Sigma^\infty(S^1), k)$. |
|
780 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
|
781 Let us check this directly. |
|
782 |
|
783 According to \nn{Loday, 3.2.2}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$. |
|
784 \nn{say something about $t$-degree? is this in [Loday]?} |
|
785 |
|
786 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
|
787 The fixed points of this flow are the equally spaced configurations. |
|
788 This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
|
789 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
|
790 and the holonomy of the $\Delta^{j-1}$ bundle |
|
791 over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices. |
|
792 |
|
793 In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
|
794 of course $\Sigma^0(S^1)$ is a point. |
|
795 Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
|
796 and is zero for $i\ge 2$. |
|
797 \nn{say something about $t$-degrees also matching up?} |
|
798 |
|
799 By xxxx and \ref{ktcdprop}, |
|
800 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
|
801 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
|
802 If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree |
|
803 0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. |
|
804 The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |
|
805 degrees and 0 in odd degrees. |
|
806 This agrees with the calculation in \nn{Loday, 3.1.7}. |
|
807 |
|
808 \medskip |
|
809 |
|
810 Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. |
|
811 Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations |
|
812 of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable. |
|
813 The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers |
|
814 corresponding to the number of points of each color of a configuration. |
|
815 A proof similar to that of \ref{sympowerprop} shows that |
|
816 |
|
817 \begin{prop} |
|
818 $\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
|
819 \end{prop} |
|
820 |
|
821 According to \nn{Loday, 3.2.2}, |
|
822 \[ |
|
823 HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
|
824 \] |
|
825 Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
|
826 We will content ourselves with the case $k = \z$. |
|
827 One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the same color repel each other and points of different colors do not interact. |
|
828 This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent |
|
829 to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple |
|
830 corresponding to $X$. |
|
831 The homology calculation we desire follows easily from this. |
|
832 |
|
833 \nn{say something about cyclic homology in this case? probably not necessary.} |
|
834 |
|
835 \medskip |
|
836 |
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837 Next we consider the case $C = k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. |
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838 Define $\Delta_l \sub \Sigma^\infty(M)$ to be those configurations of points with $l$ or |
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839 more points coinciding. |
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840 |
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841 \begin{prop} |
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842 $\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ |
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843 (relative singular chains with coefficients in $k$). |
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844 \end{prop} |
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845 |
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846 \begin{proof} |
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847 \nn{...} |
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848 \end{proof} |
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849 |
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850 \nn{...} |
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851 |
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852 |
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853 |
81 |
854 |
82 |
855 \appendix |
83 \appendix |
856 |
84 |
857 \input{text/famodiff.tex} |
85 \input{text/famodiff} |
858 |
86 |
859 \section{Comparing definitions of $A_\infty$ algebras} |
87 \input{text/misc_appendices} |
860 \label{sec:comparing-A-infty} |
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861 In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}. |
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862 |
88 |
863 We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, which we can alternatively characterise as: |
89 \input{text/obsolete} |
864 \begin{defn} |
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865 A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
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866 \begin{itemize} |
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867 \item an action of the operad of $\Obj(\cC)$-labeled cell decompositions |
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868 \item and a compatible action of $\CD{[0,1]}$. |
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869 \end{itemize} |
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870 \end{defn} |
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871 Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. In the $X$-labeled case, we insist that the appropriate labels match up. Saying we have an action of this operad means that for each labeled cell decomposition $0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these chain maps compose exactly as the cell decompositions. |
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872 An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which is supported on the subintervals determined by $\pi$, then the two possible operations (glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy). |
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873 |
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874 Translating between these definitions is straightforward. To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels) |
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875 $$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$ |
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876 where the factors of the first map are induced by the linear isometries $[0,1] \to [x_i, x_{i+1}]$, and the second map is just gluing. The action of $\CD{[0,1]}$ carries across, and is automatically compatible. Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism $\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map $\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. You can readily check that this gluing map is associative on the nose. \todo{really?} |
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877 |
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878 %First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$. Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$. |
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879 |
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880 %\begin{defn} |
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881 %A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'. |
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882 |
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883 %A \emph{composition map} $f$ is a family of chain maps, one for each decomposition of the interval, $f_\cJ : A^{\tensor k} \to A$, making $\cC$ into a category over the coloured little intervals operad, with labels $\cL = \Obj(\cC)$. Thus the chain maps satisfy the identity |
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884 %\begin{equation*} |
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885 %f_{\cJ^{(1)} \circ_m \cJ^{(2)}} = f_{\cJ^{(1)}} \circ (\id^{\tensor m-1} \tensor f_{\cJ^{(2)}} \tensor \id^{\tensor k^{(1)} - m}). |
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886 %\end{equation*} |
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887 |
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888 %An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that |
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889 %\begin{enumerate} |
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890 %\item The diagram |
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891 %\begin{equation*} |
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892 %\xymatrix{ |
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893 %\CD{[0,1]} \tensor \CD{[0,1]} \tensor A \ar[r]^{\id \tensor ev} \ar[d]^{\circ \tensor \id} & \CD{[0,1]} \tensor A \ar[d]^{ev} \\ |
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894 %\CD{[0,1]} \tensor A \ar[r]^{ev} & A |
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895 %} |
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896 %\end{equation*} |
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897 %commutes up to weakly unique homotopy. |
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898 %\item If $\phi \in \Diff([0,1])$ and $\cJ$ is a decomposition of the interval, we obtain a new decomposition $\phi(\cJ)$ and a collection $\phi_m \in \Diff([0,1])$ of diffeomorphisms obtained by taking the restrictions $\restrict{\phi}{[a_m,b_m]} : [a_m,b_m] \to [\phi(a_m),\phi(b_m)]$ and pre- and post-composing these with the linear diffeomorphisms $[0,1] \to [a_m,b_m]$ and $[\phi(a_m),\phi(b_m)] \to [0,1]$. We require that |
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899 %\begin{equation*} |
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900 %\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)). |
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901 %\end{equation*} |
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902 %\end{enumerate} |
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903 %\end{defn} |
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904 |
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905 From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' $A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. We'll just describe the algebra case (that is, a category with only one object), as the modifications required to deal with multiple objects are trivial. Define $A = \cC$ as a chain complex (so $m_1 = d$). Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms of $[0,1]$ that interpolates linearly between the identity and the piecewise linear diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define |
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906 \begin{equation*} |
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907 m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)). |
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908 \end{equation*} |
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909 |
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910 It's then easy to calculate that |
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911 \begin{align*} |
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912 d(m_3(a,b,c)) & = ev(d \phi_3, m_2(m_2(a,b),c)) - ev(\phi_3 d m_2(m_2(a,b), c)) \\ |
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913 & = ev( \phi_3(1), m_2(m_2(a,b),c)) - ev(\phi_3(0), m_2 (m_2(a,b),c)) - \\ & \qquad - ev(\phi_3, m_2(m_2(da, b), c) + (-1)^{\deg a} m_2(m_2(a, db), c) + \\ & \qquad \quad + (-1)^{\deg a+\deg b} m_2(m_2(a, b), dc) \\ |
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914 & = m_2(a , m_2(b,c)) - m_2(m_2(a,b),c) - \\ & \qquad - m_3(da,b,c) + (-1)^{\deg a + 1} m_3(a,db,c) + \\ & \qquad \quad + (-1)^{\deg a + \deg b + 1} m_3(a,b,dc), \\ |
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915 \intertext{and thus that} |
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916 m_1 \circ m_3 & = m_2 \circ (\id \tensor m_2) - m_2 \circ (m_2 \tensor \id) - \\ & \qquad - m_3 \circ (m_1 \tensor \id \tensor \id) - m_3 \circ (\id \tensor m_1 \tensor \id) - m_3 \circ (\id \tensor \id \tensor m_1) |
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917 \end{align*} |
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918 as required (c.f. \cite[p. 6]{MR1854636}). |
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919 \todo{then the general case.} |
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920 We won't describe a reverse construction (producing a topological $A_\infty$ category from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts. |
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921 |
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922 \section{Morphisms and duals of topological $A_\infty$ modules} |
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923 \label{sec:A-infty-hom-and-duals}% |
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924 |
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925 \begin{defn} |
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926 If $\cM$ and $\cN$ are topological $A_\infty$ left modules over a topological $A_\infty$ category $\cC$, then a morphism $f: \cM \to \cN$ consists of a chain map $f:\cM(J,p;b) \to \cN(J,p;b)$ for each right marked interval $(J,p)$ with a boundary condition $b$, such that for each interval $J'$ the diagram |
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927 \begin{equation*} |
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928 \xymatrix{ |
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929 \cC(J';a,b) \tensor \cM(J,p;b) \ar[r]^{\text{gl}} \ar[d]^{\id \tensor f} & \cM(J' cup J,a) \ar[d]^f \\ |
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930 \cC(J';a,b) \tensor \cN(J,p;b) \ar[r]^{\text{gl}} & \cN(J' cup J,a) |
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931 } |
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932 \end{equation*} |
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933 commutes on the nose, and the diagram |
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934 \begin{equation*} |
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935 \xymatrix{ |
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936 \CD{(J,p) \to (J',p')} \tensor \cM(J,p;a) \ar[r]^{\text{ev}} \ar[d]^{\id \tensor f} & \cM(J',p';a) \ar[d]^f \\ |
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937 \CD{(J,p) \to (J',p')} \tensor \cN(J,p;a) \ar[r]^{\text{ev}} & \cN(J',p';a) \\ |
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938 } |
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939 \end{equation*} |
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940 commutes up to a weakly unique homotopy. |
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941 \end{defn} |
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942 |
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943 The variations required for right modules and bimodules should be obvious. |
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944 |
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945 \todo{duals} |
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946 \todo{ the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic} |
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947 |
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948 |
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949 \input{text/obsolete.tex} |
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950 |
90 |
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