--- a/text/appendixes/comparing_defs.tex	Wed Mar 23 15:29:31 2011 -0700
+++ b/text/appendixes/comparing_defs.tex	Wed Mar 23 15:30:38 2011 -0700
@@ -118,12 +118,12 @@
 Each approach has advantages and disadvantages.
 For better or worse, we choose bigons here.
 
-Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard
+Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k) \trans E$, where $B^k$ denotes the standard
 $k$-ball, which we also think of as the standard bihedron (a.k.a.\ globe).
 (For $k=1$ this is an interval, and for $k=2$ it is a bigon.)
 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$
 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$.
-Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$
+Recall that the subscript in $\cC(B^k) \trans E$ means that we consider the subset of $\cC(B^k)$
 whose boundary is splittable along $E$.
 This allows us to define the domain and range of morphisms of $C$ using
 boundary and restriction maps of $\cC$.

--- a/text/ncat.tex	Wed Mar 23 15:29:31 2011 -0700
+++ b/text/ncat.tex	Wed Mar 23 15:30:38 2011 -0700
@@ -207,16 +207,16 @@
 
 We do not insist on surjectivity of the gluing map, since this is not satisfied by all of the examples
 we are trying to axiomatize.
-If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$, then a $k$-morphism is
+If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$ (c.f. Example \ref{ex:traditional-n-categories} below), then a $k$-morphism is
 in the image of the gluing map precisely which the cell complex is in general position
-with respect to $E$.
+with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective
 
 If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union
 of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified
 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$.
 
 Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$.
-We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". 
+We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$".  When the gluing map is surjective every such element is splittable.
 
 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$
 as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$.