--- a/pnas/pnas.tex	Thu Nov 04 17:02:06 2010 +0900
+++ b/pnas/pnas.tex	Tue Nov 09 14:03:58 2010 +0900
@@ -239,7 +239,7 @@
 These maps, for various $X$, comprise a natural transformation of functors.
 \end{axiom}
 
-For $c\in \cl{\cC}_{k-1}(\bd X)$ we let $\cC_k(X; c)$ denote the preimage $\bd^{-1}(c)$.
+For $c\in \cl{\cC}_{k-1}(\bd X)$ we define $\cC_k(X; c) = \bd^{-1}(c)$.
 
 Many of the examples we are interested in are enriched in some auxiliary category $\cS$
 (e.g. $\cS$ is vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces).
@@ -756,16 +756,44 @@
 
 
 \begin{figure}
+\centering
+\begin{tikzpicture}[%every label/.style={green}
+]
+\node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {};
+\node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {};
+\draw (S) arc  (-90:90:1);
+\draw (N) arc  (90:270:1);
+\node[left] at (-1,1) {$B_1$};
+\node[right] at (1,1) {$B_2$};
+\end{tikzpicture}
+\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure}
+
+\begin{figure}
+\centering
+\begin{tikzpicture}[%every label/.style={green},
+				x=1.5cm,y=1.5cm]
+\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {};
+\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {};
+\draw (S) arc  (-90:90:1);
+\draw (N) arc  (90:270:1);
+\draw (N) -- (S);
+\node[left] at (-1/4,1) {$B_1$};
+\node[right] at (1/4,1) {$B_2$};
+\node at (1/6,3/2)  {$Y$};
+\end{tikzpicture}
+\caption{From two balls to one ball.}\label{blah5}\end{figure}
+
+\begin{figure}
 \begin{equation*}
 \mathfig{.23}{ncat/zz2}
 \end{equation*}
-\caption{A small part of $\cell(W)$}
+\caption{A small part of $\cell(W)$.}
 \label{partofJfig}
 \end{figure}
 
 \begin{figure}
 $$\mathfig{.4}{deligne/manifolds}$$
-\caption{An $n$-dimensional surgery cylinder}\label{delfig2}
+\caption{An $n$-dimensional surgery cylinder.}\label{delfig2}
 \end{figure}