--- a/text/kw_macros.tex	Thu Dec 17 04:37:12 2009 +0000
+++ b/text/kw_macros.tex	Fri Dec 18 06:06:43 2009 +0000
@@ -24,6 +24,8 @@
 \def\ot{\otimes}
 \def\inv{^{-1}}
 
+\def\spl{_\pitchfork}
+
 %\def\nn#1{{{\it \small [#1]}}}
 \def\nn#1{{{\color[rgb]{.2,.5,.6} \small [#1]}}}
 \long\def\noop#1{}

--- a/text/ncat.tex	Thu Dec 17 04:37:12 2009 +0000
+++ b/text/ncat.tex	Fri Dec 18 06:06:43 2009 +0000
@@ -213,6 +213,8 @@
 $$\mathfig{.65}{tempkw/blah6}$$
 \caption{An example of strict associativity}\label{blah6}\end{figure}
 
+\nn{figure \ref{blah6} (blah6) needs a dotted line in the south split ball}
+
 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$.
 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ 
 a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$.
@@ -220,11 +222,13 @@
 For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a
 restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$.
 
-%More notation and terminology:
-%We will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ a {\it restriction}
-%map
+We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$.
+We will call $\cC(B)_Y$ morphisms which are splittable along $Y$ or transverse to $Y$.
+We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$.
 
-The above two axioms are equivalent to the following axiom,
+More generally, if $X$ is a sphere or ball subdivided \nn{...}
+
+The above two composition axioms are equivalent to the following one,
 which we state in slightly vague form.
 
 \xxpar{Multi-composition:}