--- a/text/ncat.tex	Fri Dec 09 17:01:53 2011 -0800
+++ b/text/ncat.tex	Fri Dec 09 18:43:11 2011 -0800
@@ -57,14 +57,16 @@
 Still other definitions (see, for example, \cite{MR2094071})
 model the $k$-morphisms on more complicated combinatorial polyhedra.
 
-For our definition, we will allow our $k$-morphisms to have any shape, so long as it is 
+For our definition, we will allow our $k$-morphisms to have {\it any} shape, so long as it is 
 homeomorphic to the standard $k$-ball.
 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic 
 to the standard $k$-ball.
-By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the 
+
+Below, we will use ``a $k$-ball" to mean any $k$-manifold which is homeomorphic to the 
 standard $k$-ball.
-We {\it do not} assume that it is equipped with a 
-preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below.
+We {\it do not} assume that such $k$-balls are equipped with a 
+preferred homeomorphism to the standard $k$-ball.
+The same applies to ``a $k$-sphere" below.
 
 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on 
 the boundary), we want a corresponding
@@ -240,8 +242,9 @@
 .$$
 These restriction maps can be thought of as 
 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$.
-These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$,
-and so to emphasize this we will sometimes write the restriction map as $\cC(X)\trans E \to \cC(B_i)\trans E$.
+%%%% the next sentence makes no sense to me, even though I'm probably the one who wrote it -- KW
+\noop{These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$,
+and so to emphasize this we will sometimes write the restriction map as $\cC(X)\trans E \to \cC(B_i)\trans E$.}
 
 
 Next we consider composition of morphisms.
@@ -409,7 +412,7 @@
 The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs}
 where we construct a traditional 2-category from a disk-like 2-category.
 For example, ``half-pinched" products of 1-balls are used to construct weak identities for 1-morphisms
-in 2-categories.
+in 2-categories (see \S\ref{ssec:2-cats}).
 We also need fully-pinched products to define collar maps below (see Figure \ref{glue-collar}).
 
 Define a {\it pinched product} to be a map