--- a/text/ncat.tex	Tue Mar 15 17:22:44 2011 -0700
+++ b/text/ncat.tex	Sun Mar 20 06:26:04 2011 -0700
@@ -825,9 +825,9 @@
 \label{ex:bord-cat}
 \rm
 \label{ex:bordism-category}
-For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional
-submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse
-to $\bd X$.
+For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional PL
+submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W$ is 
+contained in $\bd X \times \Real^\infty$.
 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds;
 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism
 $W \to W'$ which restricts to the identity on the boundary.
@@ -896,8 +896,8 @@
 \label{ex:bordism-category-ainf}
 As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,\infty}(X)$
 to be the set of all $k$-dimensional
-submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse
-to $\bd X$.
+submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W$ is 
+contained in $\bd X \times \Real^\infty$.
 For an $n$-ball $X$ with boundary condition $c$ 
 define $\Bord^{n,\infty}(X; c)$ to be the space of all $k$-dimensional
 submanifolds $W$ of $X\times \Real^\infty$ such that