16 where $B^k$ denotes the standard $k$-ball. |
16 where $B^k$ denotes the standard $k$-ball. |
17 The boundary and restriction maps of $\cC$ give domain and range maps from $C^1$ to $C^0$. |
17 The boundary and restriction maps of $\cC$ give domain and range maps from $C^1$ to $C^0$. |
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19 Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$. |
19 Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$. |
20 Define composition in $C$ to be the induced map $C^1\times C^1 \to C^1$ (defined only when range and domain agree). |
20 Define composition in $C$ to be the induced map $C^1\times C^1 \to C^1$ (defined only when range and domain agree). |
21 By isotopy invariance in $C$, any other choice of homeomorphism gives the same composition rule. |
21 By isotopy invariance in $\cC$, any other choice of homeomorphism gives the same composition rule. |
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22 Also by isotopy invariance, composition is associative. |
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23 Given $a\in C^0$, define $\id_a \deq a\times B^1$. |
24 Given $a\in C^0$, define $\id_a \deq a\times B^1$. |
24 By extended isotopy invariance in $\cC$, this has the expected properties of an identity morphism. |
25 By extended isotopy invariance in $\cC$, this has the expected properties of an identity morphism. |
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26 |
26 \nn{(slash)id seems to rendering a a boldface 1 --- is this what we want?} |
27 \nn{(slash)id seems to rendering a a boldface 1 --- is this what we want?} |
88 Let $\cC$ be a topological 2-category. |
89 Let $\cC$ be a topological 2-category. |
89 We will construct a traditional pivotal 2-category. |
90 We will construct a traditional pivotal 2-category. |
90 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
91 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
91 |
92 |
92 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, |
93 though this will make the $n=2$ case a little more complicated that necessary. |
94 though this will make the $n=2$ case a little more complicated than necessary. |
94 |
95 |
95 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard |
96 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard |
96 $k$-ball, which we also think of as the standard bihedron. |
97 $k$-ball, which we also think of as the standard bihedron. |
97 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
98 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
98 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
99 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
99 Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$ |
100 Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$ |
100 whose boundary is splittable along $E$. |
101 whose boundary is splittable along $E$. |
101 This allows us to define the domain and range of morphisms of $C$ using |
102 This allows us to define the domain and range of morphisms of $C$ using |
102 boundary and restriction maps of $\cC$. |
103 boundary and restriction maps of $\cC$. |
103 |
104 |
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105 Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$. |
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106 This is not associative, but we will see later that it is weakly associative. |
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107 |
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108 Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map on $C^2$. |
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109 Isotopy invariance implies that this is associative. |
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110 We will define a ``horizontal" composition later. |
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113 |
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114 |
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115 |
104 \nn{...} |
116 \nn{...} |
105 |
117 |
106 \medskip |
118 \medskip |
107 \hrule |
119 \hrule |
108 \medskip |
120 \medskip |