41 Also by isotopy invariance, composition is strictly associative. |
41 Also by isotopy invariance, composition is strictly associative. |
42 |
42 |
43 Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$. |
43 Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$. |
44 By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism. |
44 By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism. |
45 |
45 |
46 |
46 We have now defined the basic ingredients for the 1-category $c(\cX)$. |
47 If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors. |
47 As we explain below, $c(\cX)$ might have additional structure corresponding to the |
48 The base case is for oriented manifolds, where we obtain no extra algebraic data. |
48 unoriented, oriented, Spin, $\text{Pin}_+$ or $\text{Pin}_-$ structure on the 1-balls used to define $\cX$. |
49 |
49 |
50 For 1-categories based on unoriented manifolds, |
50 For 1-categories based on unoriented balls, |
51 there is a map $\dagger:c(\cX)^1\to c(\cX)^1$ |
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. |
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54 (Of course our $B^1$ is unoriented, i.e.\ not equipped with an orientation. |
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55 We mean the homeomorphism which would reverse the orientation if there were one; |
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56 $B^1$ is not oriented, but it is orientable.) |
54 Topological properties of this homeomorphism imply that |
57 Topological properties of this homeomorphism imply that |
55 $a^{\dagger\dagger} = a$ ($\dagger$ is order 2), $\dagger$ reverses domain and range, and $(ab)^\dagger = b^\dagger a^\dagger$ |
58 $a^{\dagger\dagger} = a$ ($\dagger$ is order 2), $\dagger$ reverses domain and range, and $(ab)^\dagger = b^\dagger a^\dagger$ |
56 ($\dagger$ is an anti-automorphism). |
59 ($\dagger$ is an anti-automorphism). |
57 |
60 Recall that in this context 0-balls should be thought of as equipped with a germ of a 1-dimensional neighborhood. |
58 For 1-categories based on Spin manifolds, |
61 There is a unique such 0-ball, up to homeomorphism, but it has a non-identity automorphism corresponding to reversing the |
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62 orientation of the germ. |
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63 Consequently, the objects of $c(\cX)$ are equipped with an involution, also denoted $\dagger$. |
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64 If $a:x\to y$ is a morphism of $c(\cX)$ then $a^\dagger: y^\dagger\to x^\dagger$. |
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65 |
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66 For 1-categories based on oriented balls, there are no non-trivial homeomorphisms of 0- or 1-balls, and thus no |
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67 additional structure on $c(\cX)$. |
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68 |
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69 For 1-categories based on Spin balls, |
59 the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity |
70 the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity |
60 gives an order 2 automorphism of $c(\cX)^1$. |
71 gives an order 2 automorphism of $c(\cX)^1$. |
61 |
72 |
62 For 1-categories based on $\text{Pin}_-$ manifolds, |
73 For 1-categories based on $\text{Pin}_-$ balls, |
63 we have an order 4 antiautomorphism of $c(\cX)^1$. |
74 we have an order 4 antiautomorphism of $c(\cX)^1$. |
64 For 1-categories based on $\text{Pin}_+$ manifolds, |
75 For 1-categories based on $\text{Pin}_+$ balls, |
65 we have an order 2 antiautomorphism and also an order 2 automorphism of $c(\cX)^1$, |
76 we have an order 2 antiautomorphism and also an order 2 automorphism of $c(\cX)^1$, |
66 and these two maps commute with each other. |
77 and these two maps commute with each other. |
67 %\nn{need to also consider automorphisms of $B^0$ / objects} |
78 |
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79 |
68 |
80 |
69 \noop{ |
81 \noop{ |
70 \medskip |
82 \medskip |
71 |
83 |
72 In the other direction, given a $1$-category $C$ |
84 In the other direction, given a $1$-category $C$ |