--- a/pnas/pnas.tex	Sun Oct 31 15:14:36 2010 -0700
+++ b/pnas/pnas.tex	Sun Oct 31 22:56:33 2010 -0700
@@ -210,10 +210,9 @@
 Thus we can have the simplicity of strict associativity in exchange for more morphisms.
 We wish to imitate this strategy in higher categories.
 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with
-a product of $n$ intervals \nn{cf xxxx} but rather with any $n$-ball, that is, any $n$-manifold which is homeomorphic
-to the standard $n$-ball $B^n$.
-
-\nn{...}
+a product of $k$ intervals \nn{cf xxxx} but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic
+to the standard $k$-ball $B^k$.
+\nn{maybe add that in addition we want funtoriality}
 
 \begin{axiom}[Morphisms]
 \label{axiom:morphisms}
@@ -221,16 +220,21 @@
 the category of $k$-balls and 
 homeomorphisms to the category of sets and bijections.
 \end{axiom}
+
+
+
 \begin{lem}
 \label{lem:spheres}
 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from 
 the category of $k{-}1$-spheres and 
 homeomorphisms to the category of sets and bijections.
 \end{lem}
+
 \begin{axiom}[Boundaries]\label{nca-boundary}
 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.
 These maps, for various $X$, comprise a natural transformation of functors.
 \end{axiom}
+
 \begin{lem}[Boundary from domain and range]
 \label{lem:domain-and-range}
 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$,
@@ -245,6 +249,7 @@
 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product
 becomes a normal product.)
 \end{lem}
+
 \begin{axiom}[Composition]
 \label{axiom:composition}
 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)
@@ -264,6 +269,7 @@
 we require that $\gl_Y$ is injective.
 (For $k=n$ in the plain (non-$A_\infty$) case, see below.)
 \end{axiom}
+
 \begin{axiom}[Strict associativity] \label{nca-assoc}
 The composition (gluing) maps above are strictly associative.
 Given any splitting of a ball $B$ into smaller balls