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1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{Definitions} |
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4 \label{sec:definitions} |
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5 |
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6 \subsection{Systems of fields} |
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7 \label{sec:fields} |
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8 |
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9 Let $\cM_k$ denote the category (groupoid, in fact) with objects |
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10 oriented PL manifolds of dimension |
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11 $k$ and morphisms homeomorphisms. |
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12 (We could equally well work with a different category of manifolds --- |
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13 unoriented, topological, smooth, spin, etc. --- but for definiteness we |
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14 will stick with oriented PL.) |
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15 |
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16 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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17 |
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18 A $n$-dimensional {\it system of fields} in $\cS$ |
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19 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
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20 together with some additional data and satisfying some additional conditions, all specified below. |
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21 |
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22 \nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris} |
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23 |
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24 Before finishing the definition of fields, we give two motivating examples |
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25 (actually, families of examples) of systems of fields. |
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26 |
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27 The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps |
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28 from X to $B$. |
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29 |
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30 The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be |
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31 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
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32 $j$-morphisms of $C$. |
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33 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
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34 This is described in more detail below. |
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35 |
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36 Now for the rest of the definition of system of fields. |
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37 \begin{enumerate} |
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38 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
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39 and these maps are a natural |
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40 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
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41 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
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42 $\cC(X)$ which restricts to $c$. |
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43 In this context, we will call $c$ a boundary condition. |
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44 \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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45 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps |
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46 again comprise a natural transformation of functors. |
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47 In addition, the orientation reversal maps are compatible with the boundary restriction maps. |
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48 \item $\cC_k$ is compatible with the symmetric monoidal |
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49 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
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50 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
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51 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
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52 restriction maps. |
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53 \item Gluing without corners. |
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54 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
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55 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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56 Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
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57 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
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58 copies of $Y$ in $\bd X$. |
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59 Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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60 Then (here's the axiom/definition part) there is an injective ``gluing" map |
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61 \[ |
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62 \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
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63 \] |
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64 and this gluing map is compatible with all of the above structure (actions |
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65 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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66 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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67 the gluing map is surjective. |
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68 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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69 gluing surface, we say that fields in the image of the gluing map |
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70 are transverse to $Y$ or cuttable along $Y$. |
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71 \item Gluing with corners. |
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72 Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
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73 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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74 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
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75 (without corners) along two copies of $\bd Y$. |
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76 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a cuttable field on $W\sgl$ and let |
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77 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
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78 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
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79 (This restriction map uses the gluing without corners map above.) |
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80 Using the boundary restriction, gluing without corners, and (in one case) orientation reversal |
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81 maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
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82 copies of $Y$ in $\bd X$. |
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83 Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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84 Then (here's the axiom/definition part) there is an injective ``gluing" map |
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85 \[ |
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86 \Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
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87 \] |
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88 and this gluing map is compatible with all of the above structure (actions |
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89 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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90 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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91 the gluing map is surjective. |
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92 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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93 gluing surface, we say that fields in the image of the gluing map |
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94 are transverse to $Y$ or cuttable along $Y$. |
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95 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
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96 $c \mapsto c\times I$. |
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97 These maps comprise a natural transformation of functors, and commute appropriately |
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98 with all the structure maps above (disjoint union, boundary restriction, etc.). |
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99 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
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100 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
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101 \end{enumerate} |
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102 |
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103 \nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$} |
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104 |
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105 \bigskip |
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106 Using the functoriality and $\bullet\times I$ properties above, together |
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107 with boundary collar homeomorphisms of manifolds, we can define the notion of |
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108 {\it extended isotopy}. |
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109 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
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110 of $\bd M$. |
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111 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$. |
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112 Let $c$ be $x$ restricted to $Y$. |
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113 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
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114 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
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115 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
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116 Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
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117 More generally, we define extended isotopy to be the equivalence relation on fields |
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118 on $M$ generated by isotopy plus all instance of the above construction |
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119 (for all appropriate $Y$ and $x$). |
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120 |
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121 \nn{should also say something about pseudo-isotopy} |
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122 |
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123 %\bigskip |
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124 %\hrule |
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125 %\bigskip |
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126 % |
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127 %\input{text/fields.tex} |
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128 % |
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129 % |
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130 %\bigskip |
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131 %\hrule |
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132 %\bigskip |
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133 |
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134 \nn{note: probably will suppress from notation the distinction |
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135 between fields and their (orientation-reversal) duals} |
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136 |
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137 \nn{remark that if top dimensional fields are not already linear |
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138 then we will soon linearize them(?)} |
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139 |
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140 We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
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141 by $n$-category morphisms. |
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142 |
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143 Given an $n$-category $C$ with the right sort of duality |
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144 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
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145 we can construct a system of fields as follows. |
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146 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
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147 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
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148 We'll spell this out for $n=1,2$ and then describe the general case. |
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149 |
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150 If $X$ has boundary, we require that the cell decompositions are in general |
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151 position with respect to the boundary --- the boundary intersects each cell |
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152 transversely, so cells meeting the boundary are mere half-cells. |
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153 |
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154 Put another way, the cell decompositions we consider are dual to standard cell |
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155 decompositions of $X$. |
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156 |
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157 We will always assume that our $n$-categories have linear $n$-morphisms. |
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158 |
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159 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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160 an object (0-morphism) of the 1-category $C$. |
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161 A field on a 1-manifold $S$ consists of |
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162 \begin{itemize} |
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163 \item A cell decomposition of $S$ (equivalently, a finite collection |
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164 of points in the interior of $S$); |
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165 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
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166 by an object (0-morphism) of $C$; |
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167 \item a transverse orientation of each 0-cell, thought of as a choice of |
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168 ``domain" and ``range" for the two adjacent 1-cells; and |
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169 \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
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170 domain and range determined by the transverse orientation and the labelings of the 1-cells. |
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171 \end{itemize} |
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172 |
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173 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
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174 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
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175 interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
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176 of the algebra. |
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177 |
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178 \medskip |
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179 |
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180 For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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181 that are common in the literature. |
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182 We describe these carefully here. |
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183 |
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184 A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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185 an object of the 2-category $C$. |
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186 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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187 A field on a 2-manifold $Y$ consists of |
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188 \begin{itemize} |
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189 \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
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190 that each component of the complement is homeomorphic to a disk); |
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191 \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
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192 by a 0-morphism of $C$; |
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193 \item a transverse orientation of each 1-cell, thought of as a choice of |
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194 ``domain" and ``range" for the two adjacent 2-cells; |
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195 \item a labeling of each 1-cell by a 1-morphism of $C$, with |
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196 domain and range determined by the transverse orientation of the 1-cell |
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197 and the labelings of the 2-cells; |
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198 \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
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199 of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
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200 to $\pm 1 \in S^1$; and |
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201 \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
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202 determined by the labelings of the 1-cells and the parameterizations of the previous |
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203 bullet. |
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204 \end{itemize} |
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205 \nn{need to say this better; don't try to fit everything into the bulleted list} |
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206 |
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207 For general $n$, a field on a $k$-manifold $X^k$ consists of |
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208 \begin{itemize} |
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209 \item A cell decomposition of $X$; |
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210 \item an explicit general position homeomorphism from the link of each $j$-cell |
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211 to the boundary of the standard $(k-j)$-dimensional bihedron; and |
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212 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
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213 domain and range determined by the labelings of the link of $j$-cell. |
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214 \end{itemize} |
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215 |
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216 %\nn{next definition might need some work; I think linearity relations should |
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217 %be treated differently (segregated) from other local relations, but I'm not sure |
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218 %the next definition is the best way to do it} |
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219 |
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220 \medskip |
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221 |
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222 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
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223 in the linearized space of fields. |
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224 By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
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225 the vector space of finite |
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226 linear combinations of fields on $X$. |
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227 If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
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228 Thus the restriction (to boundary) maps are well defined because we never |
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229 take linear combinations of fields with differing boundary conditions. |
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230 |
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231 In some cases we don't linearize the default way; instead we take the |
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232 spaces $\lf(X; a)$ to be part of the data for the system of fields. |
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233 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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234 Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
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235 obvious relations on 0-cell labels. |
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236 More specifically, let $L$ be a cell decomposition of $X$ |
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237 and let $p$ be a 0-cell of $L$. |
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238 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
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239 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
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240 Then the subspace $K$ is generated by things of the form |
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241 $\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
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242 to infer the meaning of $\alpha_{\lambda c + d}$. |
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243 Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
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244 |
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245 \nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
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246 will do something similar below; in general, whenever a label lives in a linear |
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247 space we do something like this; ? say something about tensor |
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248 product of all the linear label spaces? Yes:} |
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249 |
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250 For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
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251 Define an ``almost-field" to be a field without labels on the 0-cells. |
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252 (Recall that 0-cells are labeled by $n$-morphisms.) |
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253 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
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254 space determined by the labeling of the link of the 0-cell. |
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255 (If the 0-cell were labeled, the label would live in this space.) |
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256 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
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257 We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
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258 above tensor products. |
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259 |
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260 |
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261 |
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262 \subsection{Local relations} |
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263 \label{sec:local-relations} |
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264 |
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265 |
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266 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
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267 for all $n$-manifolds $B$ which are |
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268 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
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269 satisfying the following properties. |
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270 \begin{enumerate} |
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271 \item functoriality: |
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272 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
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273 \item local relations imply extended isotopy: |
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274 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
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275 to $y$, then $x-y \in U(B; c)$. |
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276 \item ideal with respect to gluing: |
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277 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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278 \end{enumerate} |
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279 See \cite{kw:tqft} for details. |
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280 |
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281 |
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282 For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, |
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283 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
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284 |
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285 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
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286 $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
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287 domain and range. |
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288 |
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289 \nn{maybe examples of local relations before general def?} |
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290 |
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291 Given a system of fields and local relations, we define the skein space |
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292 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
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293 the $n$-manifold $Y$ modulo local relations. |
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294 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
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295 is defined to be the dual of $A(Y; c)$. |
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296 (See \cite{kw:tqft} or xxxx for details.) |
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297 |
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298 \nn{should expand above paragraph} |
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299 |
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300 The blob complex is in some sense the derived version of $A(Y; c)$. |
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301 |
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302 |
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303 |
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304 \subsection{The blob complex} |
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305 \label{sec:blob-definition} |
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306 |
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307 Let $X$ be an $n$-manifold. |
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308 Assume a fixed system of fields and local relations. |
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309 In this section we will usually suppress boundary conditions on $X$ from the notation |
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310 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
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311 |
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312 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
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313 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
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314 $\overline{X \setmin Y}$. |
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315 |
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316 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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317 |
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318 Define $\bc_0(X) = \lf(X)$. |
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319 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
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320 We'll omit this sort of detail in the rest of this section.) |
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321 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
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322 |
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323 $\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
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324 Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
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325 combinations of 1-blob diagrams, where a 1-blob diagram to consists of |
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326 \begin{itemize} |
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327 \item An embedded closed ball (``blob") $B \sub X$. |
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328 \item A field $r \in \cC(X \setmin B; c)$ |
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329 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
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330 \item A local relation field $u \in U(B; c)$ |
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331 (same $c$ as previous bullet). |
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332 \end{itemize} |
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333 In order to get the linear structure correct, we (officially) define |
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334 \[ |
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335 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
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336 \] |
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337 The first direct sum is indexed by all blobs $B\subset X$, and the second |
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338 by all boundary conditions $c \in \cC(\bd B)$. |
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339 Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
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340 |
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341 Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
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342 \[ |
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343 (B, u, r) \mapsto u\bullet r, |
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344 \] |
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345 where $u\bullet r$ denotes the linear |
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346 combination of fields on $X$ obtained by gluing $u$ to $r$. |
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347 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
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348 just erasing the blob from the picture |
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349 (but keeping the blob label $u$). |
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350 |
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351 Note that the skein space $A(X)$ |
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352 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
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353 |
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354 $\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the |
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355 local relations encoded in $\bc_1(X)$. |
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356 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
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357 2-blob diagrams, of which there are two types, disjoint and nested. |
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358 |
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359 A disjoint 2-blob diagram consists of |
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360 \begin{itemize} |
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361 \item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
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362 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
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363 (where $c_i \in \cC(\bd B_i)$). |
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364 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
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365 \end{itemize} |
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366 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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367 reversing the order of the blobs changes the sign. |
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368 Define $\bd(B_0, B_1, u_0, u_1, r) = |
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369 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
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370 In other words, the boundary of a disjoint 2-blob diagram |
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371 is the sum (with alternating signs) |
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372 of the two ways of erasing one of the blobs. |
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373 It's easy to check that $\bd^2 = 0$. |
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374 |
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375 A nested 2-blob diagram consists of |
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376 \begin{itemize} |
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377 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
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378 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
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379 (for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$. |
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380 \item A local relation field $u_0 \in U(B_0; c_0)$. |
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381 \end{itemize} |
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382 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
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383 (for some $c_1 \in \cC(B_1)$) and |
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384 $r' \in \cC(X \setmin B_1; c_1)$. |
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385 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
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386 Note that the requirement that |
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387 local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
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388 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
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389 sum of the two ways of erasing one of the blobs. |
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390 If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
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391 It is again easy to check that $\bd^2 = 0$. |
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392 |
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393 \nn{should draw figures for 1, 2 and $k$-blob diagrams} |
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394 |
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395 As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
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396 (officially) |
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397 \begin{eqnarray*} |
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398 \bc_2(X) & \deq & |
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399 \left( |
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400 \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} |
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401 U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) |
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402 \right) \\ |
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403 && \bigoplus \left( |
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404 \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
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405 U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
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406 \right) . |
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407 \end{eqnarray*} |
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408 The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$, |
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409 but we didn't feel like introducing a notation for that. |
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410 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
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411 (rather than a new, linearly independent 2-blob diagram). |
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412 |
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413 Now for the general case. |
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414 A $k$-blob diagram consists of |
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415 \begin{itemize} |
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416 \item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
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417 For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or |
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418 $B_i \sub B_j$ or $B_j \sub B_i$. |
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419 (The case $B_i = B_j$ is allowed. |
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420 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
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421 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
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422 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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423 (These are implied by the data in the next bullets, so we usually |
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424 suppress them from the notation.) |
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425 $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
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426 if the latter space is not empty. |
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427 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
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428 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
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429 is determined by the $c_i$'s. |
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430 $r$ is required to be cuttable along the boundaries of all blobs, twigs or not. |
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431 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
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432 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
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433 If $B_i = B_j$ then $u_i = u_j$. |
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434 \end{itemize} |
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435 |
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436 If two blob diagrams $D_1$ and $D_2$ |
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437 differ only by a reordering of the blobs, then we identify |
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438 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
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439 |
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440 $\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. |
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441 As before, the official definition is in terms of direct sums |
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442 of tensor products: |
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443 \[ |
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444 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
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445 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
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446 \] |
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447 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
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448 $\overline{c}$ runs over all boundary conditions, again as described above. |
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449 $j$ 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 $\overline{B}$. |
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450 |
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451 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
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452 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
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453 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
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454 If $B_j$ is not a twig blob, this involves only decrementing |
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455 the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
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456 If $B_j$ is a twig blob, we have to assign new local relation labels |
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457 if removing $B_j$ creates new twig blobs. |
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458 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
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459 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
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460 Finally, define |
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461 \eq{ |
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462 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
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463 } |
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464 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
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465 Thus we have a chain complex. |
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466 |
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467 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
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468 |
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469 \nn{?? remark about dendroidal sets} |
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470 |
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471 |