76 The compositions of the above two ``arrows" ($\cC\to C\to \cC$ and $C\to \cC\to C$) give back |
76 The compositions of the above two ``arrows" ($\cC\to C\to \cC$ and $C\to \cC\to C$) give back |
77 more or less exactly the same thing we started with. |
77 more or less exactly the same thing we started with. |
78 \nn{need better notation here} |
78 \nn{need better notation here} |
79 As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence. |
79 As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence. |
80 |
80 |
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81 \medskip |
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82 |
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83 Similar arguments show that modules for topological 1-categories are essentially |
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84 the same thing as traditional modules for traditional 1-categories. |
81 |
85 |
82 \subsection{Plain 2-categories} |
86 \subsection{Plain 2-categories} |
83 |
87 |
84 blah |
88 Let $\cC$ be a topological 2-category. |
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89 We will construct a traditional pivotal 2-category. |
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90 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
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91 |
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92 We will try to describe the construction in such a way the the generalization to $n>2$ is clear, |
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93 though this will make the $n=2$ case a little more complicated that necessary. |
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94 |
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95 Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard |
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96 $k$-ball, which we also think of as the standard bihedron. |
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97 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
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98 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
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99 Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$ |
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100 whose boundary is splittable along $E$. |
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101 This allows us to define the domain and range of morphisms of $C$ using |
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102 boundary and restriction maps of $\cC$. |
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103 |
85 \nn{...} |
104 \nn{...} |
86 |
105 |
87 \medskip |
106 \medskip |
88 \hrule |
107 \hrule |
89 \medskip |
108 \medskip |