--- a/text/ncat.tex Sat May 29 23:08:36 2010 -0700
+++ b/text/ncat.tex Sat May 29 23:13:37 2010 -0700
@@ -86,6 +86,7 @@
Morphisms are modeled on balls, so their boundaries are modeled on spheres:
\begin{axiom}[Boundaries (spheres)]
+\label{axiom:spheres}
For each $0 \le k \le n-1$, we have a functor $\cC_k$ from
the category of $k$-spheres and
homeomorphisms to the category of sets and bijections.
@@ -735,7 +736,7 @@
(actions of homeomorphisms);
define $k$-cat $\cC(\cdot\times W)$}
-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.
+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.
\begin{lem}
For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$