65 |
65 |
66 For 1-categories based on oriented balls, there are no non-trivial homeomorphisms of 0- or 1-balls, and thus no |
66 For 1-categories based on oriented balls, there are no non-trivial homeomorphisms of 0- or 1-balls, and thus no |
67 additional structure on $c(\cX)$. |
67 additional structure on $c(\cX)$. |
68 |
68 |
69 For 1-categories based on Spin balls, |
69 For 1-categories based on Spin balls, |
70 the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity |
70 the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity |
71 gives an order 2 automorphism of $c(\cX)^1$. |
71 gives an order 2 automorphism of $c(\cX)^1$. |
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72 There is a similar involution on the objects $c(\cX)^0$. |
72 |
73 |
73 For 1-categories based on $\text{Pin}_-$ balls, |
74 For 1-categories based on $\text{Pin}_-$ balls, |
74 we have an order 4 antiautomorphism of $c(\cX)^1$. |
75 we have an order 4 antiautomorphism of $c(\cX)^1$. |
75 For 1-categories based on $\text{Pin}_+$ balls, |
76 For 1-categories based on $\text{Pin}_+$ balls, |
76 we have an order 2 antiautomorphism and also an order 2 automorphism of $c(\cX)^1$, |
77 we have an order 2 antiautomorphism and also an order 2 automorphism of $c(\cX)^1$, |
77 and these two maps commute with each other. |
78 and these two maps commute with each other. |
78 |
79 In both cases there is a similar map on objects. |
79 |
80 |
80 |
81 |
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83 \medskip |
83 |
84 |