16 Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$. The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$. |
16 Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$. The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$. |
17 |
17 |
18 Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$. |
18 Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$. |
19 Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$ (defined only when range and domain agree). |
19 Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$ (defined only when range and domain agree). |
20 By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule. |
20 By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule. |
21 Also by isotopy invariance, composition is associative on the nose. |
21 Also by isotopy invariance, composition is strictly associative. |
22 |
22 |
23 Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$. |
23 Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$. |
24 By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism. |
24 By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism. |
25 |
25 |
26 |
26 |
124 vertical composition. |
124 vertical composition. |
125 |
125 |
126 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
126 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
127 We will show that this 1-morphism is a weak identity. |
127 We will show that this 1-morphism is a weak identity. |
128 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
128 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
129 Define let $a: y\to x$ be a 1-morphism. |
129 Let $a: y\to x$ be a 1-morphism. |
130 Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
130 Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
131 as shown in Figure \ref{fzo2}. |
131 as shown in Figure \ref{fzo2}. |
132 \begin{figure}[t] |
132 \begin{figure}[t] |
133 \begin{equation*} |
133 \begin{equation*} |
134 \mathfig{.73}{tempkw/zo2} |
134 \mathfig{.73}{tempkw/zo2} |
135 \end{equation*} |
135 \end{equation*} |
136 \caption{blah blah} |
136 \caption{blah blah} |
137 \label{fzo2} |
137 \label{fzo2} |
138 \end{figure} |
138 \end{figure} |
139 In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon, |
139 In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon, |
140 while the remained is a half-pinched version of $a\times I$. |
140 while the remainder is a half-pinched version of $a\times I$. |
141 \nn{the red region is unnecessary; remove it? or does it help? |
141 \nn{the red region is unnecessary; remove it? or does it help? |
142 (because it's what you get if you bigonify the natural rectangular picture)} |
142 (because it's what you get if you bigonify the natural rectangular picture)} |
143 We must show that the two compositions of these two maps give the identity 2-morphisms |
143 We must show that the two compositions of these two maps give the identity 2-morphisms |
144 on $a$ and $a\bullet \id_x$, as defined above. |
144 on $a$ and $a\bullet \id_x$, as defined above. |
145 Figure \ref{fzo3} shows one case. |
145 Figure \ref{fzo3} shows one case. |