adding a sentence about types of decompositions of manifolds, including a cite to kirillov. Feel free to revert.
--- 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 $\du_a X_a = M_0\to\cdots\to M_m = W$