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111 |
111 |
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. |
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116 \nn{maybe no need to postpone?} |
116 |
117 |
117 \begin{figure}[t] |
118 \begin{figure}[t] |
118 \begin{equation*} |
119 \begin{equation*} |
119 \mathfig{.73}{tempkw/zo1} |
120 \mathfig{.73}{tempkw/zo1} |
120 \end{equation*} |
121 \end{equation*} |
166 \caption{blah blah} |
167 \caption{blah blah} |
167 \label{fzo4} |
168 \label{fzo4} |
168 \end{figure} |
169 \end{figure} |
169 We first collapse the red region, then remove a product morphism from the boundary, |
170 We first collapse the red region, then remove a product morphism from the boundary, |
170 |
171 |
171 \nn{postponing horizontal composition of 2-morphisms until we make up our minds about product axioms.} |
172 We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}. |
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173 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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174 \begin{figure}[t] |
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175 \begin{equation*} |
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176 \mathfig{.83}{tempkw/zo5} |
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177 \end{equation*} |
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178 \caption{Horizontal composition of 2-morphisms} |
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179 \label{fzo5} |
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180 \end{figure} |
172 |
181 |
173 \nn{need to find a list of axioms for pivotal 2-cats to check} |
182 \nn{need to find a list of axioms for pivotal 2-cats to check} |
174 |
183 |
175 \nn{...} |
184 \nn{...} |
176 |
185 |