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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. |
75 |
75 |
76 (The ambitious reader may want to keep in mind two other classes of balls. |
76 (The ambitious reader may want to keep in mind two other classes of balls. |
77 The first is balls equipped with a map to some other space $Y$. |
77 The first is balls equipped with a map to some other space $Y$. \todo{cite something of Teichner's?} |
78 This will be used below to describe the blob complex of a fiber bundle with |
78 This will be used below to describe the blob complex of a fiber bundle with |
79 base space $Y$. |
79 base space $Y$. |
80 The second is balls equipped with a section of the the tangent bundle, or the frame |
80 The second is balls equipped with a section of the the tangent bundle, or the frame |
81 bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle. |
81 bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle. |
82 These can be used to define categories with less than the ``strong" duality we assume here, |
82 These can be used to define categories with less than the ``strong" duality we assume here, |
105 For each $1 \le k \le n$, we have a functor $\cC_{k-1}$ from |
105 For each $1 \le k \le n$, we have a functor $\cC_{k-1}$ from |
106 the category of $k{-}1$-spheres and |
106 the category of $k{-}1$-spheres and |
107 homeomorphisms to the category of sets and bijections. |
107 homeomorphisms to the category of sets and bijections. |
108 \end{prop} |
108 \end{prop} |
109 |
109 |
110 We postpone the proof of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in other Axioms at lower levels. |
110 We postpone the proof of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels. |
111 |
111 |
112 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
112 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
113 |
113 |
114 \begin{axiom}[Boundaries]\label{nca-boundary} |
114 \begin{axiom}[Boundaries]\label{nca-boundary} |
115 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cC_{k-1}(\bd X)$. |
115 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cC_{k-1}(\bd X)$. |