98 |
98 |
99 An ambitious reader may want to keep in mind two other classes of balls. |
99 An ambitious reader may want to keep in mind two other classes of balls. |
100 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
100 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
101 This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with |
101 This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with |
102 base space $Y$. |
102 base space $Y$. |
103 The second is balls equipped with a section of the tangent bundle, or the frame |
103 The second is balls equipped (partially defined) sections of the tangent bundle, or the frame |
104 bundle (i.e.\ framed balls), or more generally some partial flag bundle associated to the tangent bundle. |
104 bundle (i.e.\ framed balls), or more generally some partial flag bundle associated to the tangent bundle. |
105 These can be used to define categories with less than the ``strong" duality we assume here, |
105 These can be used to define categories with less than the ``strong" duality we assume here, |
106 though we will not develop that idea fully in this paper. |
106 though we will not develop that idea in this paper. |
107 |
107 |
108 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
108 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
109 of morphisms). |
109 of morphisms). |
110 The 0-sphere is unusual among spheres in that it is disconnected. |
110 The 0-sphere is unusual among spheres in that it is disconnected. |
111 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
111 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |