--- a/text/ncat.tex	Tue Aug 31 21:09:31 2010 -0700
+++ b/text/ncat.tex	Wed Sep 01 13:34:21 2010 -0700
@@ -1639,7 +1639,7 @@
 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by 
 gluing subintervals together and/or omitting some of the rightmost subintervals.
 (See Figure \ref{fig:lmar}.)
-\begin{figure}[t]$$
+\begin{figure}[t] \centering
 \definecolor{arcolor}{rgb}{.75,.4,.1}
 \begin{tikzpicture}[line width=1pt]
 \fill (0,0) circle (.1);
@@ -1679,7 +1679,6 @@
 }
 
 \end{tikzpicture}
-$$
 \caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure}
 
 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$.
@@ -1735,7 +1734,7 @@
 These are required to commute with gluing;
 for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes:
 \[ \xymatrix{
-	\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_0}\ot \id} 
+	\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_1}\ot \id} 
 							\ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) 
 								\ar[d]^{\gl} \\
 	\cX(K) \ar[r]^{h_{K}} & \cY(K)
@@ -1875,8 +1874,7 @@
 (see Figure \ref{feb21b}).
 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$.
 
-\begin{figure}[t]
-$$
+\begin{figure}[t] \centering
 \begin{tikzpicture}[blue,line width=2pt]
 \draw (0,1) -- (0,-1) node[below] {$X$};
 
@@ -1891,7 +1889,6 @@
 \path (bottom) node[below]{$X \times J$};
 
 \end{tikzpicture}
-$$
 \caption{The pinched product $X\times J$}
 \label{feb21b}
 \end{figure}
@@ -1904,8 +1901,7 @@
 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category.
 This amounts to a definition of taking tensor products of $0$-sphere modules over $n$-categories.
 
-\begin{figure}[t]
-$$
+\begin{figure}[t] \centering
 \begin{tikzpicture}[baseline,line width = 2pt]
 \draw[blue] (0,0) -- (6,0);
 \foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} {
@@ -1927,7 +1923,6 @@
 	\path (\q:2.4) node {\color{green!50!brown}$\cM_{\n}$};
 }
 \end{tikzpicture}
-$$
 \caption{Marked and labeled 1-manifolds}
 \label{feb21c}
 \end{figure}
@@ -1956,8 +1951,7 @@
 (See Figure \nn{need figure}.)
 We now proceed as in the above module definitions.
 
-\begin{figure}[!ht]
-$$
+\begin{figure}[t] \centering
 \begin{tikzpicture}[baseline,line width = 2pt]
 \draw[blue][fill=blue!15!white] (0,0) circle (2);
 \fill[red] (0,0) circle (0.1);
@@ -1968,7 +1962,6 @@
 	\draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3);
 }
 \end{tikzpicture}
-$$
 \caption{Cone on a marked circle}
 \label{feb21d}
 \end{figure}