46 |
46 |
47 If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors. |
47 If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors. |
48 The base case is for oriented manifolds, where we obtain no extra algebraic data. |
48 The base case is for oriented manifolds, where we obtain no extra algebraic data. |
49 |
49 |
50 For 1-categories based on unoriented manifolds, |
50 For 1-categories based on unoriented manifolds, |
51 there is a map $*:c(\cX)^1\to c(\cX)^1$ |
51 there is a map $\dagger:c(\cX)^1\to c(\cX)^1$ |
52 coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy) |
52 coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy) |
53 from $B^1$ to itself. |
53 from $B^1$ to itself. |
54 Topological properties of this homeomorphism imply that |
54 Topological properties of this homeomorphism imply that |
55 $a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$ |
55 $a^{\dagger\dagger} = a$ ($\dagger$ is order 2), $\dagger$ reverses domain and range, and $(ab)^\dagger = b^\dagger a^\dagger$ |
56 (* is an anti-automorphism). |
56 ($\dagger$ is an anti-automorphism). |
57 |
57 |
58 For 1-categories based on Spin manifolds, |
58 For 1-categories based on Spin manifolds, |
59 the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity |
59 the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity |
60 gives an order 2 automorphism of $c(\cX)^1$. |
60 gives an order 2 automorphism of $c(\cX)^1$. |
61 |
61 |