
diff -r c24e59300fca -r b76b4b79dbe1 text/ncat.tex
--- a/text/ncat.tex	Tue May 10 14:30:23 2011 -0700
+++ b/text/ncat.tex	Thu May 12 21:42:34 2011 -0700
@@ -1037,7 +1037,7 @@
 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement
 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$
 with $\du_b Y_b = M_i$ for some $i$,
-and with $M_0,\ldots, M_i$ each being a disjoint union of balls.
+and with $M_0, M_1, \ldots, M_i$ each being a disjoint union of balls.
 
 \begin{defn}
 The poset $\cell(W)$ has objects the permissible decompositions of $W$, 
@@ -1056,20 +1056,31 @@
 An $n$-category $\cC$ determines 
 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets 
 (possibly with additional structure if $k=n$).
-Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls,
+For pedagogical reasons, let us first the case where a decomposition $y$ of $W$ is a nice, non-pathological
+cell decomposition.
+Then each $k$-ball $X$ of $y$ has its boundary decomposed into $k{-}1$-balls,
 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries
 are splittable along this decomposition.
 
-\begin{defn}
-Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
+We can now
+define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset
 \begin{equation}
-\label{eq:psi-C}
+%\label{eq:psi-C}
 	\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl
 \end{equation}
 where the restrictions to the various pieces of shared boundaries amongst the cells
 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n{-}1$-cells).
 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.
+
+In general, $y$ might be more general than a cell decomposition
+(see Example \ref{sin1x-example}), so we must define $\psi_{\cC;W}$ in a more roundabout way.
+\nn{...}
+
+\begin{defn}
+Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
+\nn{...}
+If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.
 \end{defn}
 
 If $k=n$ in the above definition and we are enriching in some auxiliary category, 




