122 the category of $k{-}1$-spheres and |
122 the category of $k{-}1$-spheres and |
123 homeomorphisms to the category of sets and bijections. |
123 homeomorphisms to the category of sets and bijections. |
124 \end{lem} |
124 \end{lem} |
125 |
125 |
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 some $k$ 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 at lower levels. |
128 along with the data described in the other axioms for smaller values of $k$. |
129 |
129 |
130 %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. |
130 Of course, Lemma \ref{lem:spheres}, as stated, is satisfied by the trivial functor. |
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131 What we really mean is that there is exists a functor which interacts with other data of $\cC$ as specified |
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132 in the other axioms below. |
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133 |
131 |
134 |
132 \begin{axiom}[Boundaries]\label{nca-boundary} |
135 \begin{axiom}[Boundaries]\label{nca-boundary} |
133 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 $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
134 These maps, for various $X$, comprise a natural transformation of functors. |
137 These maps, for various $X$, comprise a natural transformation of functors. |
135 \end{axiom} |
138 \end{axiom} |