--- a/text/intro.tex	Wed Jun 22 11:13:51 2011 -0700
+++ b/text/intro.tex	Wed Jun 22 16:02:37 2011 -0700
@@ -96,6 +96,9 @@
 
 The relationship between all these ideas is sketched in Figure \ref{fig:outline}.
 
+% NB: the following tikz requires a *more recent* version of PGF than is distributed with MacTex 2010.
+% grab the latest build from http://www.texample.net/tikz/builds/
+% unzip it in your personal tex tree, and run "mktexlsr ." there
 \tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt]
 
 \begin{figure}[t]

--- a/text/ncat.tex	Wed Jun 22 11:13:51 2011 -0700
+++ b/text/ncat.tex	Wed Jun 22 16:02:37 2011 -0700
@@ -489,7 +489,7 @@
 \end{scope}
 \end{tikzpicture}
 $$
-\caption{Five examples of unions of pinched products}\label{pinched_prod_unions}
+\caption{Six examples of unions of pinched products}\label{pinched_prod_unions}
 \end{figure}
 
 Note that $\bd X$ has a (possibly trivial) subdivision according to 
@@ -1408,7 +1408,7 @@
 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$;
 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and
 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, 
-such that $b_i$ and $b_{i+1}$both map to (glue up to) $a_i$.
+such that $b_i$ and $b_{i+1}$ both map to (glue up to) $a_i$.
 \end{itemize}
 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$.
 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same