
diff -r c21da249a015 -r 7a4fc5a873ac pnas/pnas.tex
--- a/pnas/pnas.tex	Tue Nov 30 11:07:24 2010 -0800
+++ b/pnas/pnas.tex	Tue Nov 30 11:24:05 2010 -0800
@@ -524,6 +524,7 @@
 
 \subsection{The blob complex}
 \subsubsection{Decompositions of manifolds}
+Our description of an $n$-category associates data to each $k$-ball for $k\leq n$. In order to define invariants of $n$-manifolds, we will need a class of decompositions of manifolds into balls. We present one choice here, but alternatives of varying degrees of generality exist, for example handle decompositions or piecewise-linear CW-complexes \cite{1009.4227}.
 
 A \emph{ball decomposition} of a $k$-manifold $W$ is a 
 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls
@@ -535,7 +536,7 @@
 \]
 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$.
 A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls
-are glued up to yield $W$, and just require that there is some non-pathological way to do this.
+are glued up to yield $W$, and just require that there is some non-pathological way to do this. 
 
 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement
 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$




