80 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
80 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
81 of morphisms). |
81 of morphisms). |
82 The 0-sphere is unusual among spheres in that it is disconnected. |
82 The 0-sphere is unusual among spheres in that it is disconnected. |
83 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
83 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
84 (Actually, this is only true in the oriented case, with 1-morphisms parameterized |
84 (Actually, this is only true in the oriented case, with 1-morphisms parameterized |
85 by oriented 1-balls.) |
85 by {\it oriented} 1-balls.) |
86 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. |
86 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. |
87 For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. |
87 For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. |
88 (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. |
88 (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. |
89 We prefer to not make the distinction in the first place. |
89 We prefer to not make the distinction in the first place. |
90 |
90 |