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text/ncat.tex
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    84 Instead, we combine the domain and range into a single entity which we call the 
    84 Instead, we combine the domain and range into a single entity which we call the 
    85 boundary of a morphism.
 
    85 boundary of a morphism.
 
    86 Morphisms are modeled on balls, so their boundaries are modeled on spheres:
 
    86 Morphisms are modeled on balls, so their boundaries are modeled on spheres:
 
    87 
    87 
    88 \begin{axiom}[Boundaries (spheres)]
 
    88 \begin{axiom}[Boundaries (spheres)]
 
       
    89 \label{axiom:spheres}
 
    89 For each $0 \le k \le n-1$, we have a functor $\cC_k$ from 
    90 For each $0 \le k \le n-1$, we have a functor $\cC_k$ from 
    90 the category of $k$-spheres and 
    91 the category of $k$-spheres and 
    91 homeomorphisms to the category of sets and bijections.
 
    92 homeomorphisms to the category of sets and bijections.
 
    92 \end{axiom}
 
    93 \end{axiom}
 
    93 
    94 
   733 \nn{need to finish explaining why we have a system of fields;
 
   734 \nn{need to finish explaining why we have a system of fields;
 
   734 need to say more about ``homological" fields? 
   735 need to say more about ``homological" fields? 
   735 (actions of homeomorphisms);
 
   736 (actions of homeomorphisms);
 
   736 define $k$-cat $\cC(\cdot\times W)$}
 
   737 define $k$-cat $\cC(\cdot\times W)$}
 
   737 
   738 
   738 Recall that Axiom \ref{} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction.
 
   739 Recall that Axiom \ref{axiom:spheres} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction.
 
   739 
   740 
   740 \begin{lem}
 
   741 \begin{lem}
 
   741 For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$
 
   742 For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$
 
   742 \end{lem}
 
   743 \end{lem}
 
   743 
   744
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