112 Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map |
112 Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map |
113 on $C^2$ (Figure \ref{fzo1}). |
113 on $C^2$ (Figure \ref{fzo1}). |
114 Isotopy invariance implies that this is associative. |
114 Isotopy invariance implies that this is associative. |
115 We will define a ``horizontal" composition later. |
115 We will define a ``horizontal" composition later. |
116 |
116 |
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117 \begin{figure}[t] |
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118 \begin{equation*} |
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119 \mathfig{.73}{tempkw/zo1} |
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120 \end{equation*} |
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121 \caption{Vertical composition of 2-morphisms} |
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122 \label{fzo1} |
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123 \end{figure} |
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124 |
117 Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary). |
125 Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary). |
118 Extended isotopy invariance for $\cC$ shows that this morphism is an identity for |
126 Extended isotopy invariance for $\cC$ shows that this morphism is an identity for |
119 vertical composition. |
127 vertical composition. |
120 |
128 |
121 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
129 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
122 We will show that this 1-morphism is a weak identity. |
130 We will show that this 1-morphism is a weak identity. |
123 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
131 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
124 Define let $a: y\to x$ be a 1-morphism. |
132 Define let $a: y\to x$ be a 1-morphism. |
125 Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
133 Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
126 as shown in Figure \ref{fzo2}. |
134 as shown in Figure \ref{fzo2}. |
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135 \begin{figure}[t] |
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136 \begin{equation*} |
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137 \mathfig{.73}{tempkw/zo2} |
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138 \end{equation*} |
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139 \caption{blah blah} |
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140 \label{fzo2} |
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141 \end{figure} |
127 In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon, |
142 In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon, |
128 while the remained is a half-pinched version of $a\times I$. |
143 while the remained is a half-pinched version of $a\times I$. |
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144 \nn{the red region is unnecessary; remove it? or does it help? |
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145 (because it's what you get if you bigonify the natural rectangular picture)} |
129 We must show that the two compositions of these two maps give the identity 2-morphisms |
146 We must show that the two compositions of these two maps give the identity 2-morphisms |
130 on $a$ and $a\bullet \id_x$, as defined above. |
147 on $a$ and $a\bullet \id_x$, as defined above. |
131 Figure \ref{fzo3} shows one case. |
148 Figure \ref{fzo3} shows one case. |
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149 \begin{figure}[t] |
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150 \begin{equation*} |
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151 \mathfig{.83}{tempkw/zo3} |
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152 \end{equation*} |
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153 \caption{blah blah} |
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154 \label{fzo3} |
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155 \end{figure} |
132 In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}. |
156 In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}. |
133 \nn{also need to talk about (somewhere above) |
157 \nn{also need to talk about (somewhere above) |
134 how this sort of insertion is allowed by extended isotopy invariance and gluing} |
158 how this sort of insertion is allowed by extended isotopy invariance and gluing. |
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159 Also: maybe half-pinched and unpinched products can be derived from fully pinched |
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160 products after all (?)} |
135 Figure \ref{fzo4} shows the other case. |
161 Figure \ref{fzo4} shows the other case. |
136 \nn{At the moment, I don't see how the case follows from our candidate axioms for products. |
162 \begin{figure}[t] |
137 Probably the axioms need to be strengthened a little bit.} |
163 \begin{equation*} |
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164 \mathfig{.83}{tempkw/zo4} |
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165 \end{equation*} |
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166 \caption{blah blah} |
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167 \label{fzo4} |
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168 \end{figure} |
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169 We first collapse the red region, then remove a product morphism from the boundary, |
138 |
170 |
139 \nn{postponing horizontal composition of 2-morphisms until we make up our minds about product axioms.} |
171 \nn{postponing horizontal composition of 2-morphisms until we make up our minds about product axioms.} |
140 |
172 |
141 \nn{need to find a list of axioms for pivotal 2-cats to check} |
173 \nn{need to find a list of axioms for pivotal 2-cats to check} |
142 |
174 |