70 %\nn{need to check whether this makes much difference} |
70 %\nn{need to check whether this makes much difference} |
71 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
71 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
72 to be fussier about corners.) |
72 to be fussier about corners.) |
73 For each flavor of manifold there is a corresponding flavor of $n$-category. |
73 For each flavor of manifold there is a corresponding flavor of $n$-category. |
74 We will concentrate on the case of PL unoriented manifolds. |
74 We will concentrate on the case of PL unoriented manifolds. |
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75 |
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76 (The ambitious reader may want to keep in mind two other classes of balls. |
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77 The first is balls equipped with a map to some other space $Y$. |
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78 This will be used below to describe the blob complex of a fiber bundle with |
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79 base space $Y$. |
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80 The second is balls equipped with a section of the the tangent bundle, or the frame |
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81 bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle. |
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82 These can be used to define categories with less than the ``strong" duality we assume here, |
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83 though we will not develop that idea fully in this paper.) |
75 |
84 |
76 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
85 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
77 of morphisms). |
86 of morphisms). |
78 The 0-sphere is unusual among spheres in that it is disconnected. |
87 The 0-sphere is unusual among spheres in that it is disconnected. |
79 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
88 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |