
diff -r bc22926d4fb0 -r eb03c4a92f98 text/tqftreview.tex
--- a/text/tqftreview.tex	Thu Jun 03 09:47:18 2010 -0700
+++ b/text/tqftreview.tex	Thu Jun 03 12:33:47 2010 -0700
@@ -5,7 +5,7 @@
 \label{sec:tqftsviafields}
 
 In this section we review the notion of a ``system of fields and local relations".
-For more details see \cite{kw:tqft}. From a system of fields and local relations we can readily construct TQFT invariants of manifolds. This is described in \S \ref{sec:constructing-a-TQFT}. A system of fields is very closely related to an $n$-category. In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section, we sketch the construction of a system of fields from an $n$-category. We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)}, and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats}, we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations.
+For more details see \cite{kw:tqft}. From a system of fields and local relations we can readily construct TQFT invariants of manifolds. This is described in \S \ref{sec:constructing-a-tqft}. A system of fields is very closely related to an $n$-category. In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section, we sketch the construction of a system of fields from an $n$-category. We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)}, and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats}, we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations.
 
 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
 submanifold of $X$, then $X \setmin Y$ implicitly means the closure
@@ -21,7 +21,7 @@
 oriented, topological, smooth, spin, etc. --- but for definiteness we
 will stick with unoriented PL.)
 
-Fix a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$.
+Fix a symmetric monoidal category $\cS$. While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$. The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired.
 
 A $n$-dimensional {\it system of fields} in $\cS$
 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$
@@ -326,13 +326,14 @@
 %To harmonize notation with the next section, 
 %let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so
 %$\bc_0(X) = \lf(X)$.
-Define $U(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$;
-$U(X)$ is generated by things of the form $u\bullet r$, where
-$u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.
-Define
-\[
-	A(X) \deq \lf(X) / U(X) .
-\]
+\begin{defn}
+\label{defn:TQFT-invariant}
+The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is 
+	$$A(X) \deq \lf(X) / U(X),$$
+where $\cU(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$;
+$\cU(X)$ is generated by things of the form $u\bullet r$, where
+$u\in \cU(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.
+\end{defn}
 (The blob complex, defined in the next section, 
 is in some sense the derived version of $A(X)$.)
 If $X$ has boundary we can similarly define $A(X; c)$ for each 




