1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{TQFTs via fields} |
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4 \label{sec:fields} |
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5 \label{sec:tqftsviafields} |
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6 |
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7 In this section we review the construction of TQFTs from ``topological fields". |
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8 For more details see \cite{kw:tqft}. |
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9 |
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10 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
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11 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
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12 $\overline{X \setmin Y}$. |
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13 |
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14 |
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15 \subsection{Systems of fields} |
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16 |
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17 Let $\cM_k$ denote the category with objects |
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18 unoriented PL manifolds of dimension |
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19 $k$ and morphisms homeomorphisms. |
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20 (We could equally well work with a different category of manifolds --- |
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21 oriented, topological, smooth, spin, etc. --- but for definiteness we |
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22 will stick with unoriented PL.) |
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23 |
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24 %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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25 |
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26 A $n$-dimensional {\it system of fields} in $\cS$ |
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27 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
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28 together with some additional data and satisfying some additional conditions, all specified below. |
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29 |
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30 Before finishing the definition of fields, we give two motivating examples |
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31 (actually, families of examples) of systems of fields. |
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32 |
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33 The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps |
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34 from X to $B$. |
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35 |
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36 The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be |
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37 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
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38 $j$-morphisms of $C$. |
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39 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
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40 This is described in more detail below. |
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41 |
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42 Now for the rest of the definition of system of fields. |
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43 \begin{enumerate} |
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44 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
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45 and these maps are a natural |
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46 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
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47 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
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48 $\cC(X)$ which restricts to $c$. |
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49 In this context, we will call $c$ a boundary condition. |
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50 \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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51 \item $\cC_k$ is compatible with the symmetric monoidal |
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52 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
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53 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
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54 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
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55 restriction maps. |
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56 \item Gluing without corners. |
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57 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
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58 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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59 Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
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60 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
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61 copies of $Y$ in $\bd X$. |
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62 Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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63 Then (here's the axiom/definition part) there is an injective ``gluing" map |
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64 \[ |
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65 \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
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66 \] |
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67 and this gluing map is compatible with all of the above structure (actions |
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68 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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69 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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70 the gluing map is surjective. |
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71 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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72 gluing surface, we say that fields in the image of the gluing map |
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73 are transverse to $Y$ or splittable along $Y$. |
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74 \item Gluing with corners. |
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75 Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
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76 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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77 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
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78 (without corners) along two copies of $\bd Y$. |
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79 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let |
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80 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
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81 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
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82 (This restriction map uses the gluing without corners map above.) |
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83 Using the boundary restriction, gluing without corners, and (in one case) orientation reversal |
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84 maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
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85 copies of $Y$ in $\bd X$. |
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86 Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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87 Then (here's the axiom/definition part) there is an injective ``gluing" map |
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88 \[ |
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89 \Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
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90 \] |
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91 and this gluing map is compatible with all of the above structure (actions |
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92 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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93 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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94 the gluing map is surjective. |
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95 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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96 gluing surface, we say that fields in the image of the gluing map |
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97 are transverse to $Y$ or splittable along $Y$. |
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98 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
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99 $c \mapsto c\times I$. |
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100 These maps comprise a natural transformation of functors, and commute appropriately |
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101 with all the structure maps above (disjoint union, boundary restriction, etc.). |
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102 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
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103 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
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104 \end{enumerate} |
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105 |
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106 There are two notations we commonly use for gluing. |
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107 One is |
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108 \[ |
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109 x\sgl \deq \gl(x) \in \cC(X\sgl) , |
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110 \] |
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111 for $x\in\cC(X)$. |
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112 The other is |
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113 \[ |
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114 x_1\bullet x_2 \deq \gl(x_1\otimes x_2) \in \cC(X\sgl) , |
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115 \] |
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116 in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
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117 |
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118 \medskip |
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119 |
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120 Using the functoriality and $\cdot\times I$ properties above, together |
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121 with boundary collar homeomorphisms of manifolds, we can define the notion of |
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122 {\it extended isotopy}. |
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123 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
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124 of $\bd M$. |
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125 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |
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126 Let $c$ be $x$ restricted to $Y$. |
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127 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
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128 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
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129 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
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130 Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
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131 More generally, we define extended isotopy to be the equivalence relation on fields |
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132 on $M$ generated by isotopy plus all instance of the above construction |
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133 (for all appropriate $Y$ and $x$). |
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134 |
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135 \nn{should also say something about pseudo-isotopy} |
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136 |
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137 |
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138 \nn{remark that if top dimensional fields are not already linear |
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139 then we will soon linearize them(?)} |
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140 |
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141 We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
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142 by $n$-category morphisms. |
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143 |
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144 Given an $n$-category $C$ with the right sort of duality |
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145 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
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146 we can construct a system of fields as follows. |
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147 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
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148 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
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149 We'll spell this out for $n=1,2$ and then describe the general case. |
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150 |
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151 If $X$ has boundary, we require that the cell decompositions are in general |
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152 position with respect to the boundary --- the boundary intersects each cell |
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153 transversely, so cells meeting the boundary are mere half-cells. |
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154 |
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155 Put another way, the cell decompositions we consider are dual to standard cell |
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156 decompositions of $X$. |
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157 |
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158 We will always assume that our $n$-categories have linear $n$-morphisms. |
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159 |
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160 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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161 an object (0-morphism) of the 1-category $C$. |
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162 A field on a 1-manifold $S$ consists of |
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163 \begin{itemize} |
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164 \item A cell decomposition of $S$ (equivalently, a finite collection |
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165 of points in the interior of $S$); |
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166 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
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167 by an object (0-morphism) of $C$; |
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168 \item a transverse orientation of each 0-cell, thought of as a choice of |
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169 ``domain" and ``range" for the two adjacent 1-cells; and |
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170 \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
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171 domain and range determined by the transverse orientation and the labelings of the 1-cells. |
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172 \end{itemize} |
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173 |
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174 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
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175 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
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176 interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
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177 of the algebra. |
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178 |
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179 \medskip |
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180 |
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181 For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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182 that are common in the literature. |
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183 We describe these carefully here. |
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184 |
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185 A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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186 an object of the 2-category $C$. |
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187 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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188 A field on a 2-manifold $Y$ consists of |
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189 \begin{itemize} |
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190 \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
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191 that each component of the complement is homeomorphic to a disk); |
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192 \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
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193 by a 0-morphism of $C$; |
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194 \item a transverse orientation of each 1-cell, thought of as a choice of |
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195 ``domain" and ``range" for the two adjacent 2-cells; |
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196 \item a labeling of each 1-cell by a 1-morphism of $C$, with |
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197 domain and range determined by the transverse orientation of the 1-cell |
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198 and the labelings of the 2-cells; |
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199 \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
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200 of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
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201 to $\pm 1 \in S^1$; and |
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202 \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
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203 determined by the labelings of the 1-cells and the parameterizations of the previous |
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204 bullet. |
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205 \end{itemize} |
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206 \nn{need to say this better; don't try to fit everything into the bulleted list} |
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207 |
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208 For general $n$, a field on a $k$-manifold $X^k$ consists of |
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209 \begin{itemize} |
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210 \item A cell decomposition of $X$; |
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211 \item an explicit general position homeomorphism from the link of each $j$-cell |
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212 to the boundary of the standard $(k-j)$-dimensional bihedron; and |
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213 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
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214 domain and range determined by the labelings of the link of $j$-cell. |
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215 \end{itemize} |
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216 |
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217 %\nn{next definition might need some work; I think linearity relations should |
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218 %be treated differently (segregated) from other local relations, but I'm not sure |
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219 %the next definition is the best way to do it} |
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220 |
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221 \medskip |
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222 |
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223 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
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224 in the linearized space of fields. |
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225 By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
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226 the vector space of finite |
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227 linear combinations of fields on $X$. |
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228 If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
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229 Thus the restriction (to boundary) maps are well defined because we never |
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230 take linear combinations of fields with differing boundary conditions. |
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231 |
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232 In some cases we don't linearize the default way; instead we take the |
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233 spaces $\lf(X; a)$ to be part of the data for the system of fields. |
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234 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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235 Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
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236 obvious relations on 0-cell labels. |
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237 More specifically, let $L$ be a cell decomposition of $X$ |
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238 and let $p$ be a 0-cell of $L$. |
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239 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
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240 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
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241 Then the subspace $K$ is generated by things of the form |
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242 $\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
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243 to infer the meaning of $\alpha_{\lambda c + d}$. |
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244 Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
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245 |
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246 \nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
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247 will do something similar below; in general, whenever a label lives in a linear |
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248 space we do something like this; ? say something about tensor |
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249 product of all the linear label spaces? Yes:} |
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250 |
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251 For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
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252 Define an ``almost-field" to be a field without labels on the 0-cells. |
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253 (Recall that 0-cells are labeled by $n$-morphisms.) |
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254 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
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255 space determined by the labeling of the link of the 0-cell. |
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256 (If the 0-cell were labeled, the label would live in this space.) |
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257 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
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258 We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
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259 above tensor products. |
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260 |
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261 |
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262 |
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263 \subsection{Local relations} |
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264 \label{sec:local-relations} |
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265 |
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266 |
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267 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
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268 for all $n$-manifolds $B$ which are |
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269 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
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270 satisfying the following properties. |
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271 \begin{enumerate} |
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272 \item functoriality: |
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273 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
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274 \item local relations imply extended isotopy: |
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275 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
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276 to $y$, then $x-y \in U(B; c)$. |
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277 \item ideal with respect to gluing: |
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278 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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279 \end{enumerate} |
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280 See \cite{kw:tqft} for details. |
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281 |
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282 |
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283 For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, |
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284 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
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285 |
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286 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
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287 $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
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288 domain and range. |
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289 |
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290 \nn{maybe examples of local relations before general def?} |
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291 |
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292 \subsection{Constructing a TQFT} |
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293 |
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294 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations. |
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295 (For more details, see \cite{kw:tqft}.) |
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296 |
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297 Let $W$ be an $n{+}1$-manifold. |
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298 We can think of the path integral $Z(W)$ as assigning to each |
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299 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$. |
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300 In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear |
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301 maps $\lf(\bd W)\to \c$. |
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302 |
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303 The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace |
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304 $Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections. |
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305 The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$, |
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306 can be thought of as finite linear combinations of fields modulo local relations. |
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307 (In other words, $A(\bd W)$ is a sort of generalized skein module.) |
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308 This is the motivation behind the definition of fields and local relations above. |
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309 |
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310 In more detail, let $X$ be an $n$-manifold. |
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311 %To harmonize notation with the next section, |
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312 %let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so |
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313 %$\bc_0(X) = \lf(X)$. |
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314 Define $U(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$; |
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315 $U(X)$ is generated by things of the form $u\bullet r$, where |
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316 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
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317 Define |
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318 \[ |
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319 A(X) \deq \lf(X) / U(X) . |
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320 \] |
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321 (The blob complex, defined in the next section, |
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322 is in some sense the derived version of $A(X)$.) |
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323 If $X$ has boundary we can similarly define $A(X; c)$ for each |
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324 boundary condition $c\in\cC(\bd X)$. |
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325 |
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326 The above construction can be extended to higher codimensions, assigning |
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327 a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$. |
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328 These invariants fit together via actions and gluing formulas. |
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329 We describe only the case $k=1$ below. |
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330 (The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
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331 requires that the starting data (fields and local relations) satisfy additional |
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332 conditions. |
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333 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT |
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334 that lacks its $n{+}1$-dimensional part.) |
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335 |
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336 Let $Y$ be an $n{-}1$-manifold. |
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337 Define a (linear) 1-category $A(Y)$ as follows. |
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338 The objects of $A(Y)$ are $\cC(Y)$. |
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339 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$. |
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340 Composition is given by gluing of cylinders. |
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341 |
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342 Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces |
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343 $A(X; \cdot) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$. |
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344 This collection of vector spaces affords a representation of the category $A(\bd X)$, where |
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345 the action is given by gluing a collar $\bd X\times I$ to $X$. |
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346 |
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347 Given a splitting $X = X_1 \cup_Y X_2$ of a closed $n$-manifold $X$ along an $n{-}1$-manifold $Y$, |
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348 we have left and right actions of $A(Y)$ on $A(X_1; \cdot)$ and $A(X_2; \cdot)$. |
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349 The gluing theorem for $n$-manifolds states that there is a natural isomorphism |
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350 \[ |
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351 A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) . |
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352 \] |
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353 |
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354 |
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355 \section{The blob complex} |
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356 \label{sec:blob-definition} |
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357 |
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358 Let $X$ be an $n$-manifold. |
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359 Assume a fixed system of fields and local relations. |
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360 In this section we will usually suppress boundary conditions on $X$ from the notation |
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361 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
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362 |
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363 We want to replace the quotient |
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364 \[ |
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365 A(X) \deq \lf(X) / U(X) |
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366 \] |
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367 of the previous section with a resolution |
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368 \[ |
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369 \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
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370 \] |
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371 |
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372 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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373 |
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374 We of course define $\bc_0(X) = \lf(X)$. |
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375 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
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376 We'll omit this sort of detail in the rest of this section.) |
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377 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
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378 |
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379 $\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
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380 Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
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381 combinations of 1-blob diagrams, where a 1-blob diagram to consists of |
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382 \begin{itemize} |
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383 \item An embedded closed ball (``blob") $B \sub X$. |
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384 \item A field $r \in \cC(X \setmin B; c)$ |
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385 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
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386 \item A local relation field $u \in U(B; c)$ |
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387 (same $c$ as previous bullet). |
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388 \end{itemize} |
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389 (See Figure \ref{blob1diagram}.) |
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390 \begin{figure}[!ht]\begin{equation*} |
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391 \mathfig{.9}{definition/single-blob} |
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392 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
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393 In order to get the linear structure correct, we (officially) define |
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394 \[ |
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395 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
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396 \] |
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397 The first direct sum is indexed by all blobs $B\subset X$, and the second |
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398 by all boundary conditions $c \in \cC(\bd B)$. |
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399 Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
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400 |
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401 Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
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402 \[ |
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403 (B, u, r) \mapsto u\bullet r, |
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404 \] |
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405 where $u\bullet r$ denotes the linear |
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406 combination of fields on $X$ obtained by gluing $u$ to $r$. |
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407 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
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408 just erasing the blob from the picture |
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409 (but keeping the blob label $u$). |
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410 |
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411 Note that the skein space $A(X)$ |
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412 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
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413 |
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414 $\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the |
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415 local relations encoded in $\bc_1(X)$. |
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416 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
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417 2-blob diagrams, of which there are two types, disjoint and nested. |
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418 |
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419 A disjoint 2-blob diagram consists of |
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420 \begin{itemize} |
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421 \item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
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422 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
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423 (where $c_i \in \cC(\bd B_i)$). |
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424 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \nn{We're inconsistent with the indexes -- are they 0,1 or 1,2? I'd prefer 1,2.} |
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425 \end{itemize} |
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426 (See Figure \ref{blob2ddiagram}.) |
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427 \begin{figure}[!ht]\begin{equation*} |
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428 \mathfig{.9}{definition/disjoint-blobs} |
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429 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
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430 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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431 reversing the order of the blobs changes the sign. |
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432 Define $\bd(B_0, B_1, u_0, u_1, r) = |
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433 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
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434 In other words, the boundary of a disjoint 2-blob diagram |
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435 is the sum (with alternating signs) |
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436 of the two ways of erasing one of the blobs. |
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437 It's easy to check that $\bd^2 = 0$. |
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438 |
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439 A nested 2-blob diagram consists of |
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440 \begin{itemize} |
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441 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
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442 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
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443 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. |
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444 \item A local relation field $u_0 \in U(B_0; c_0)$. |
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445 \end{itemize} |
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446 (See Figure \ref{blob2ndiagram}.) |
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447 \begin{figure}[!ht]\begin{equation*} |
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448 \mathfig{.9}{definition/nested-blobs} |
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449 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
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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 $r' \in \cC(X \setmin B_1; c_1)$. |
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453 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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454 Note that the requirement that |
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455 local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
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456 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
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457 sum of the two ways of erasing one of the blobs. |
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458 If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
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459 It is again easy to check that $\bd^2 = 0$. |
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460 |
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461 As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
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462 (officially) |
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463 \begin{eqnarray*} |
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464 \bc_2(X) & \deq & |
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465 \left( |
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466 \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} |
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467 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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468 \right) \\ |
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469 && \bigoplus \left( |
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470 \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
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471 U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
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472 \right) . |
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473 \end{eqnarray*} |
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474 The final $\lf(X\setmin B_0; c_0)$ above really means fields splittable along $\bd B_1$, |
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475 but we didn't feel like introducing a notation for that. |
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476 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
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477 (rather than a new, linearly independent 2-blob diagram). |
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478 |
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479 Now for the general case. |
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480 A $k$-blob diagram consists of |
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481 \begin{itemize} |
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482 \item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
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483 For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or |
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484 $B_i \sub B_j$ or $B_j \sub B_i$. |
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485 (The case $B_i = B_j$ is allowed. |
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486 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
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487 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
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488 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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489 (These are implied by the data in the next bullets, so we usually |
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490 suppress them from the notation.) |
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491 $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
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492 if the latter space is not empty. |
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493 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
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494 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
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495 is determined by the $c_i$'s. |
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496 $r$ is required to be splittable along the boundaries of all blobs, twigs or not. |
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497 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
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498 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
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499 If $B_i = B_j$ then $u_i = u_j$. |
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500 \end{itemize} |
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501 (See Figure \ref{blobkdiagram}.) |
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502 \begin{figure}[!ht]\begin{equation*} |
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503 \mathfig{.9}{definition/k-blobs} |
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504 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
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505 |
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506 If two blob diagrams $D_1$ and $D_2$ |
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507 differ only by a reordering of the blobs, then we identify |
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508 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
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509 |
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510 $\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. |
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511 As before, the official definition is in terms of direct sums |
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512 of tensor products: |
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513 \[ |
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514 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
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515 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
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516 \] |
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517 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
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518 $\overline{c}$ runs over all boundary conditions, again as described above. |
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519 $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
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520 |
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521 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
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522 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
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523 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
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524 If $B_j$ is not a twig blob, this involves only decrementing |
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525 the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
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526 If $B_j$ is a twig blob, we have to assign new local relation labels |
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527 if removing $B_j$ creates new twig blobs. |
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528 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
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529 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
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530 Finally, define |
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531 \eq{ |
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532 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
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533 } |
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534 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
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535 Thus we have a chain complex. |
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536 |
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537 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
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538 |
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539 \nn{?? remark about dendroidal sets} |
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540 |
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541 |
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