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126 We postpone the proof of this result until after we've actually given all the axioms. |
126 We postpone the proof of this result until after we've actually given all the axioms. |
127 Note that defining this functor for fixed $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, |
127 Note that defining this functor for fixed $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, |
128 along with the data described in the other axioms for smaller values of $k$. |
128 along with the data described in the other axioms for smaller values of $k$. |
129 |
129 |
130 Of course, Lemma \ref{lem:spheres}, as stated, is satisfied by the trivial functor. |
130 Of course, Lemma \ref{lem:spheres}, as stated, is satisfied by the trivial functor. |
131 What we really mean is that there is exists a functor which interacts with other data of $\cC$ as specified |
131 What we really mean is that there exists a functor which interacts with the other data of $\cC$ as specified |
132 in the other axioms below. |
132 in the axioms below. |
133 |
133 |
134 |
134 |
135 \begin{axiom}[Boundaries]\label{nca-boundary} |
135 \begin{axiom}[Boundaries]\label{nca-boundary} |
136 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
136 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
137 These maps, for various $X$, comprise a natural transformation of functors. |
137 These maps, for various $X$, comprise a natural transformation of functors. |