
diff -r 299b404b3bc0 -r 8bb0f0c51a6e text/ncat.tex
--- a/text/ncat.tex	Wed Nov 04 21:55:47 2009 +0000
+++ b/text/ncat.tex	Wed Nov 04 22:27:48 2009 +0000
@@ -280,18 +280,7 @@
 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.
 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.
+This axiom needs to be strengthened to force product morphisms to act as the identity.
 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball.
 Let $J$ be a 1-ball (interval).
 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$.
@@ -306,21 +295,21 @@
 \begin{figure}[!ht]\begin{equation*}
 \mathfig{.9}{tempkw/glue-collar}
 \end{equation*}\caption{Extended homeomorphism.}\label{glue-collar}\end{figure}
-We will call $\psi_{Y,J}$ an extended isotopy.
-\nn{or extended homeomorphism?  see below.}
+We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map.
+\nn{bad terminology; fix it later}
+\nn{also need to make clear that plain old isotopic to the identity implies
+extended isotopic}
 \nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) 
 extended isotopies are also plain isotopies, so
 no extension necessary}
 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 $n$:}
+\xxpar{Extended isotopy invariance in dimension $n$:}
 {Let $X$ be an $n$-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.
+to the identity on $\bd X$ and is 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.}




