--- a/pnas/pnas.tex	Fri Nov 12 15:07:00 2010 -0800
+++ b/pnas/pnas.tex	Sat Nov 13 12:14:55 2010 -0800
@@ -399,6 +399,20 @@
 \end{enumerate}
 } %%% end \noop %%%
 \end{axiom}
+
+To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms.
+Let $X$ be a $k$-ball and $Y\sub\bd X$ be a $(k{-}1)$-ball.
+Let $J$ be a 1-ball.
+Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$.
+A collar map is an instance of the composition
+\[
+	\cC(X) \to \cC(X\cup_Y (Y\times_p J)) \to \cC(X) ,
+\]
+where the first arrow is gluing with a product morphism on $Y\times_p J$ and
+the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts
+to the identity on the boundary.
+
+
 \begin{axiom}[\textup{\textbf{[plain  version]}} Extended isotopy invariance in dimension $n$.]
 \label{axiom:extended-isotopies}
 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
@@ -407,8 +421,6 @@
 In addition, collar maps act trivially on $\cC(X)$.
 \end{axiom}
 
-\nn{need to define collar maps}
-
 \smallskip
 
 For $A_\infty$ $n$-categories, we replace