
diff -r daa522adb488 -r 84bb5ab4c85c text/ncat.tex
--- a/text/ncat.tex	Fri Aug 05 12:27:11 2011 -0600
+++ b/text/ncat.tex	Tue Aug 09 19:28:39 2011 -0600
@@ -945,8 +945,13 @@
 
 Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. 
 In fact, the alternative construction $\btc_*(X)$ of the blob complex described in \S \ref{ss:alt-def} 
-gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; 
-since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.
+gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom. 
+%since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.
+For future reference we make the following definition.
+
+\begin{defn}
+A {\em strict $A_\infty$ $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative.
+\end{defn}
 
 \noop{
 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category
@@ -1220,7 +1225,7 @@
 Let $A$ be an $\cE\cB_n$-algebra.
 Note that this implies a $\Diff(B^n)$ action on $A$, 
 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$.
-We will define an $A_\infty$ $n$-category $\cC^A$.
+We will define a strict $A_\infty$ $n$-category $\cC^A$.
 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point.
 In other words, the $k$-morphisms are trivial for $k<n$.
 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction.
@@ -1237,10 +1242,13 @@
 also comes from the $\cE\cB_n$ action on $A$.
 %\nn{should we spell this out?}
 
-Conversely, one can show that a disk-like $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
+Conversely, one can show that a disk-like strict $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to 
 an $\cE\cB_n$-algebra.
 %\nn{The paper is already long; is it worth giving details here?}
+% According to the referee, yes it is...
+Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball.
+\nn{need to finish this}
 
 If we apply the homotopy colimit construction of the next subsection to this example, 
 we get an instance of Lurie's topological chiral homology construction.




