116 Before proceeding, we must decide whether the 2-morphisms of our |
116 Before proceeding, we must decide whether the 2-morphisms of our |
117 pivotal 2-category are shaped like rectangles or bigons. |
117 pivotal 2-category are shaped like rectangles or bigons. |
118 Each approach has advantages and disadvantages. |
118 Each approach has advantages and disadvantages. |
119 For better or worse, we choose bigons here. |
119 For better or worse, we choose bigons here. |
120 |
120 |
121 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard |
121 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k) \trans E$, where $B^k$ denotes the standard |
122 $k$-ball, which we also think of as the standard bihedron (a.k.a.\ globe). |
122 $k$-ball, which we also think of as the standard bihedron (a.k.a.\ globe). |
123 (For $k=1$ this is an interval, and for $k=2$ it is a bigon.) |
123 (For $k=1$ this is an interval, and for $k=2$ it is a bigon.) |
124 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
124 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
125 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
125 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
126 Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$ |
126 Recall that the subscript in $\cC(B^k) \trans E$ means that we consider the subset of $\cC(B^k)$ |
127 whose boundary is splittable along $E$. |
127 whose boundary is splittable along $E$. |
128 This allows us to define the domain and range of morphisms of $C$ using |
128 This allows us to define the domain and range of morphisms of $C$ using |
129 boundary and restriction maps of $\cC$. |
129 boundary and restriction maps of $\cC$. |
130 |
130 |
131 Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$. |
131 Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$. |