91 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
91 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
92 |
92 |
93 We will try to describe the construction in such a way the the generalization to $n>2$ is clear, |
93 We will try to describe the construction in such a way the the generalization to $n>2$ is clear, |
94 though this will make the $n=2$ case a little more complicated than necessary. |
94 though this will make the $n=2$ case a little more complicated than necessary. |
95 |
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 |
96 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard |
100 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard |
97 $k$-ball, which we also think of as the standard bihedron. |
101 $k$-ball, which we also think of as the standard bihedron. |
98 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
102 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
99 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
103 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
100 Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$ |
104 Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$ |
103 boundary and restriction maps of $\cC$. |
107 boundary and restriction maps of $\cC$. |
104 |
108 |
105 Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$. |
109 Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$. |
106 This is not associative, but we will see later that it is weakly associative. |
110 This is not associative, but we will see later that it is weakly associative. |
107 |
111 |
108 Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map on $C^2$. |
112 Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map |
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113 on $C^2$ (Figure \ref{fzo1}). |
109 Isotopy invariance implies that this is associative. |
114 Isotopy invariance implies that this is associative. |
110 We will define a ``horizontal" composition later. |
115 We will define a ``horizontal" composition later. |
111 |
116 |
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117 Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary). |
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118 Extended isotopy invariance for $\cC$ shows that this morphism is an identity for |
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119 vertical composition. |
112 |
120 |
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121 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
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122 We will show that this 1-morphism is a weak identity. |
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123 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
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124 Define let $a: y\to x$ be a 1-morphism. |
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125 Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
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126 as shown in Figure \ref{fzo2}. |
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127 In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon, |
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128 while the remained is a half-pinched version of $a\times I$. |
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129 We must show that the two compositions of these two maps give the identity 2-morphisms |
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130 on $a$ and $a\bullet \id_x$, as defined above. |
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131 Figure \ref{fzo3} shows one case. |
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132 In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}. |
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133 \nn{also need to talk about (somewhere above) |
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134 how this sort of insertion is allowed by extended isotopy invariance and gluing} |
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135 Figure \ref{fzo4} shows the other case. |
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136 \nn{At the moment, I don't see how the case follows from our candidate axioms for products. |
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137 Probably the axioms need to be strengthened a little bit.} |
113 |
138 |
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139 \nn{postponing horizontal composition of 2-morphisms until we make up our minds about product axioms.} |
114 |
140 |
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141 \nn{need to find a list of axioms for pivotal 2-cats to check} |
115 |
142 |
116 \nn{...} |
143 \nn{...} |
117 |
144 |
118 \medskip |
145 \medskip |
119 \hrule |
146 \hrule |