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1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{Comparing $n$-category definitions} |
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4 \label{sec:comparing-defs} |
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5 |
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6 In this appendix we relate the ``topological" category definitions of Section \ref{sec:ncats} |
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7 to more traditional definitions, for $n=1$ and 2. |
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8 |
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9 \subsection{Plain 1-categories} |
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10 |
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11 Given a topological 1-category $\cC$, we construct a traditional 1-category $C$. |
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12 (This is quite straightforward, but we include the details for the sake of completeness and |
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13 to shed some light on the $n=2$ case.) |
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14 |
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15 Let the objects of $C$ be $C^0 \deq \cC(B^0)$ and the morphisms of $C$ be $C^1 \deq \cC(B^1)$, |
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16 where $B^k$ denotes the standard $k$-ball. |
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17 The boundary and restriction maps of $\cC$ give domain and range maps from $C^1$ to $C^0$. |
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18 |
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19 Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$. |
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20 Define composition in $C$ to be the induced map $C^1\times C^1 \to C^1$ (defined only when range and domain agree). |
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21 By isotopy invariance in $\cC$, any other choice of homeomorphism gives the same composition rule. |
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22 Also by isotopy invariance, composition is associative. |
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23 |
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24 Given $a\in C^0$, define $\id_a \deq a\times B^1$. |
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25 By extended isotopy invariance in $\cC$, this has the expected properties of an identity morphism. |
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26 |
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27 \nn{(slash)id seems to rendering a a boldface 1 --- is this what we want?} |
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28 |
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29 \medskip |
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30 |
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31 For 1-categories based on oriented manifolds, there is no additional structure. |
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32 |
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33 For 1-categories based on unoriented manifolds, there is a map $*:C^1\to C^1$ |
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34 coming from $\cC$ applied to an orientation-reversing homeomorphism (unique up to isotopy) |
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35 from $B^1$ to itself. |
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36 Topological properties of this homeomorphism imply that |
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37 $a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$ |
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38 (* is an anti-automorphism). |
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39 |
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40 For 1-categories based on Spin manifolds, |
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41 the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity |
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42 gives an order 2 automorphism of $C^1$. |
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43 |
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44 For 1-categories based on $\text{Pin}_-$ manifolds, |
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45 we have an order 4 antiautomorphism of $C^1$. |
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46 |
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47 For 1-categories based on $\text{Pin}_+$ manifolds, |
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48 we have an order 2 antiautomorphism and also an order 2 automorphism of $C^1$, |
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49 and these two maps commute with each other. |
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50 |
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51 \nn{need to also consider automorphisms of $B^0$ / objects} |
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52 |
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53 \medskip |
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54 |
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55 In the other direction, given a traditional 1-category $C$ |
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56 (with objects $C^0$ and morphisms $C^1$) we will construct a topological |
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57 1-category $\cC$. |
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58 |
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59 If $X$ is a 0-ball (point), let $\cC(X) \deq C^0$. |
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60 If $S$ is a 0-sphere, let $\cC(S) \deq C^0\times C^0$. |
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61 If $X$ is a 1-ball, let $\cC(X) \deq C^1$. |
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62 Homeomorphisms isotopic to the identity act trivially. |
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63 If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure |
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64 to define the action of homeomorphisms not isotopic to the identity |
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65 (and get, e.g., an unoriented topological 1-category). |
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66 |
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67 The domain and range maps of $C$ determine the boundary and restriction maps of $\cC$. |
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68 |
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69 Gluing maps for $\cC$ are determined my composition of morphisms in $C$. |
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70 |
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71 For $X$ a 0-ball, $D$ a 1-ball and $a\in \cC(X)$, define the product morphism |
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72 $a\times D \deq \id_a$. |
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73 It is not hard to verify that this has the desired properties. |
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74 |
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75 \medskip |
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76 |
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77 The compositions of the above two ``arrows" ($\cC\to C\to \cC$ and $C\to \cC\to C$) give back |
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78 more or less exactly the same thing we started with. |
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79 \nn{need better notation here} |
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80 As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence. |
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81 |
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82 \medskip |
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83 |
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84 Similar arguments show that modules for topological 1-categories are essentially |
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85 the same thing as traditional modules for traditional 1-categories. |
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86 |
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87 \subsection{Plain 2-categories} |
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88 |
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89 Let $\cC$ be a topological 2-category. |
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90 We will construct a traditional pivotal 2-category. |
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91 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
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92 |
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93 We will try to describe the construction in such a way the the generalization to $n>2$ is clear, |
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94 though this will make the $n=2$ case a little more complicated than necessary. |
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95 |
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96 \nn{Note: We have to decide whether our 2-morphsism are shaped like rectangles or bigons. |
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97 Each approach has advantages and disadvantages. |
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98 For better or worse, we choose bigons here.} |
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99 |
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100 \nn{maybe we should do both rectangles and bigons?} |
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101 |
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102 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard |
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103 $k$-ball, which we also think of as the standard bihedron. |
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104 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
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105 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
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106 Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$ |
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107 whose boundary is splittable along $E$. |
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108 This allows us to define the domain and range of morphisms of $C$ using |
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109 boundary and restriction maps of $\cC$. |
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110 |
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111 Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$. |
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112 This is not associative, but we will see later that it is weakly associative. |
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113 |
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114 Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map |
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115 on $C^2$ (Figure \ref{fzo1}). |
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116 Isotopy invariance implies that this is associative. |
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117 We will define a ``horizontal" composition later. |
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118 \nn{maybe no need to postpone?} |
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119 |
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120 \begin{figure}[t] |
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121 \begin{equation*} |
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122 \mathfig{.73}{tempkw/zo1} |
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123 \end{equation*} |
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124 \caption{Vertical composition of 2-morphisms} |
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125 \label{fzo1} |
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126 \end{figure} |
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127 |
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128 Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary). |
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129 Extended isotopy invariance for $\cC$ shows that this morphism is an identity for |
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130 vertical composition. |
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131 |
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132 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
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133 We will show that this 1-morphism is a weak identity. |
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134 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
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135 Define let $a: y\to x$ be a 1-morphism. |
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136 Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
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137 as shown in Figure \ref{fzo2}. |
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138 \begin{figure}[t] |
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139 \begin{equation*} |
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140 \mathfig{.73}{tempkw/zo2} |
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141 \end{equation*} |
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142 \caption{blah blah} |
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143 \label{fzo2} |
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144 \end{figure} |
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145 In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon, |
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146 while the remained is a half-pinched version of $a\times I$. |
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147 \nn{the red region is unnecessary; remove it? or does it help? |
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148 (because it's what you get if you bigonify the natural rectangular picture)} |
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149 We must show that the two compositions of these two maps give the identity 2-morphisms |
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150 on $a$ and $a\bullet \id_x$, as defined above. |
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151 Figure \ref{fzo3} shows one case. |
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152 \begin{figure}[t] |
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153 \begin{equation*} |
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154 \mathfig{.83}{tempkw/zo3} |
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155 \end{equation*} |
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156 \caption{blah blah} |
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157 \label{fzo3} |
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158 \end{figure} |
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159 In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}. |
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160 \nn{also need to talk about (somewhere above) |
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161 how this sort of insertion is allowed by extended isotopy invariance and gluing. |
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162 Also: maybe half-pinched and unpinched products can be derived from fully pinched |
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163 products after all (?)} |
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164 Figure \ref{fzo4} shows the other case. |
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165 \begin{figure}[t] |
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166 \begin{equation*} |
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167 \mathfig{.83}{tempkw/zo4} |
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168 \end{equation*} |
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169 \caption{blah blah} |
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170 \label{fzo4} |
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171 \end{figure} |
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172 We first collapse the red region, then remove a product morphism from the boundary, |
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173 |
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174 We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}. |
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175 It is not hard to show that this is independent of the arbitrary (left/right) choice made in the definition, and that it is associative. |
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176 \begin{figure}[t] |
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177 \begin{equation*} |
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178 \mathfig{.83}{tempkw/zo5} |
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179 \end{equation*} |
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180 \caption{Horizontal composition of 2-morphisms} |
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181 \label{fzo5} |
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182 \end{figure} |
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183 |
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184 \nn{need to find a list of axioms for pivotal 2-cats to check} |
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185 |
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186 \nn{...} |
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187 |
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188 \medskip |
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189 \hrule |
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190 \medskip |
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191 |
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192 \nn{to be continued...} |
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193 \medskip |