--- a/text/ncat.tex	Mon Jul 20 22:50:59 2009 +0000
+++ b/text/ncat.tex	Tue Jul 21 05:56:45 2009 +0000
@@ -160,7 +160,7 @@
 Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.
 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram
 \[ \xymatrix{
-	X\times D \ar[r]^{\tilde{f}} \ar[d]^{\pi} & X'\times D' \ar[d]^{\pi} \\
+	X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\
 	X \ar[r]^{f} & X'
 } \]
 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.}
@@ -176,8 +176,43 @@
 \xxpar{Isotopy invariance in dimension $n$ (preliminary version):}
 {Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts
 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.
-Then $f(a) = a$ for all $a\in \cC(X)$.}
+Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.}
+
+We will strengthen the above axiom in two ways.
+(Amusingly, these two ways are related to each of the two senses of the term
+``pseudo-isotopy".)
+
+First, we require that $f$ act trivially on $\cC(X)$ if it is pseudo-isotopic to the identity
+in the sense of homeomorphisms of mapping cylinders.
+This is motivated by TQFT considerations:
+If the mapping cylinder of $f$ is homeomorphic to the mapping cylinder of the identity,
+then these two $n{+}1$-manifolds should induce the same map from $\cC(X)$ to itself.
+\nn{is there a non-TQFT reason to require this?}
 
+Second, we require that product (a.k.a.\ identity) $n$-morphisms act as the identity.
+Let $X$ be an $n$-ball and $Y\sub\bd X$ be at $n{-}1$-ball.
+Let $J$ be a 1-ball (interval).
+We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$.
+We define a map
+\begin{eqnarray*}
+	\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
+	a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) .
+\end{eqnarray*}
+\nn{need to say something somewhere about pinched boundary convention for products}
+We will call $\psi_{Y,J}$ an extended isotopy.
+It can be thought of as the action of the inverse of
+a map which projects a collar neighborhood of $Y$ onto $Y$.
+(This sort of collapse map is the other sense of ``pseudo-isotopy".)
+\nn{need to check this}
+
+The revised axiom is
+
+\xxpar{Pseudo and extended isotopy invariance in dimension $n$:}
+{Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts
+to the identity on $\bd X$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity.
+Then $f$ acts trivially on $\cC(X)$.}
+
+\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.}