--- a/preamble.tex	Wed Sep 15 13:30:15 2010 -0500
+++ b/preamble.tex	Wed Sep 15 13:33:14 2010 -0500
@@ -62,8 +62,8 @@
 \newtheorem*{defn*}{Definition}             % unnumbered definition
 \newtheorem{question}{Question}
 \newtheorem{property}{Property}
-\newtheorem{axiom}{Axiom}
-\newtheorem{module-axiom}{Module Axiom}
+\newtheorem{axiom}{Axiom}[section]
+\newtheorem{module-axiom}{Module Axiom}[section]
 \newenvironment{rem}{\noindent\textsl{Remark.}}{}  % perhaps looks better than rem above?
 \newtheorem{rem*}[prop]{Remark}
 \newtheorem{remark}[prop]{Remark}

--- a/text/ncat.tex	Wed Sep 15 13:30:15 2010 -0500
+++ b/text/ncat.tex	Wed Sep 15 13:33:14 2010 -0500
@@ -1093,8 +1093,7 @@
 \end{itemize}
 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$.
 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same
-disjoint union of balls, and then invoke the associativity axiom \ref{nca-assoc}.
-\nn{hmmm... it would be nicer if this were ``7.xx" instead of ``4"}
+disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity.
 
 Let $z$ be a decomposition of $W$ which is in general position with respect to all of the 
 $x_i$'s and $v_i$'s.