
diff -r 5bb1cbe49c40 -r ef8fac44a8aa text/tqftreview.tex
--- a/text/tqftreview.tex	Mon May 31 17:27:17 2010 -0700
+++ b/text/tqftreview.tex	Mon May 31 23:42:37 2010 -0700
@@ -30,14 +30,20 @@
 Before finishing the definition of fields, we give two motivating examples
 (actually, families of examples) of systems of fields.
 
-The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps
+\begin{example}
+\label{ex:maps-to-a-space(fields)}
+Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps
 from X to $B$.
+\end{example}
 
-The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be 
+\begin{example}
+\label{ex:traditional-n-categories(fields)}
+Fix an $n$-category $C$, and let $\cC(X)$ be 
 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by
 $j$-morphisms of $C$.
 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$.
 This is described in more detail below.
+\end{example}
 
 Now for the rest of the definition of system of fields.
 \begin{enumerate}
@@ -262,8 +268,23 @@
 
 \subsection{Local relations}
 \label{sec:local-relations}
+Local relations are certain subspaces of the fields on balls, which form an ideal under gluing. Again, we give the examples first.
 
+\addtocounter{prop}{-2}
+\begin{example}[contd.]
+For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$,
+where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
+\end{example}
 
+\begin{example}[contd.]
+For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map
+$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
+domain and range.
+\end{example}
+
+These motivate the following definition.
+
+\begin{defn}
 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$,
 for all $n$-manifolds $B$ which are
 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, 
@@ -277,17 +298,9 @@
 \item ideal with respect to gluing:
 if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$
 \end{enumerate}
-See \cite{kw:tqft} for details.
-
-
-For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$,
-where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
+\end{defn}
+See \cite{kw:tqft} for further details.
 
-For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map
-$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
-domain and range.
-
-\nn{maybe examples of local relations before general def?}
 
 \subsection{Constructing a TQFT}
 \label{sec:constructing-a-tqft}




