
diff -r f3da9af1d8c7 -r 38532ba5bd0f pnas/pnas.tex
--- a/pnas/pnas.tex	Thu Nov 18 10:43:06 2010 -0800
+++ b/pnas/pnas.tex	Thu Nov 18 10:45:52 2010 -0800
@@ -257,8 +257,8 @@
 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second.
 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.}
 
-We will define plain and $A_\infty$ $n$-categories simultaneously, as all but one of the axioms are identical
-in the two cases.
+We will define two variations simultaneously,  as all but one of the axioms are identical
+in the two cases. These variations are `linear $n$-categories', where the sets associated to $n$-balls with specified boundary conditions are in fact vector spaces, and `$A_\infty$ $n$-categories', where these sets are chain complexes.
 
 
 There are five basic ingredients 
@@ -356,7 +356,7 @@
 If $k < n$,
 or if $k=n$ and we are in the $A_\infty$ case, 
 we require that $\gl_Y$ is injective.
-(For $k=n$ in the plain (non-$A_\infty$) case, see below.)
+(For $k=n$ in the linear case, see below.)
 \end{axiom}
 
 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity}
@@ -436,7 +436,7 @@
 to the identity on the boundary.
 
 
-\begin{axiom}[\textup{\textbf{[plain  version]}} Extended isotopy invariance in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[linear  version]}} Extended isotopy invariance in dimension $n$.]
 \label{axiom:extended-isotopies}
 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
 to the identity on $\bd X$ and isotopic (rel boundary) to the identity.
@@ -661,7 +661,7 @@
 
 \begin{thm}[Skein modules]
 \label{thm:skein-modules}
-\nn{Plain n-categories only?}
+\nn{linear n-categories only?}
 The $0$-th blob homology of $X$ is the usual 
 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
 by $\cC$.




