
diff -r 9698f584e732 -r f7da004e1f14 text/tqftreview.tex
--- a/text/tqftreview.tex	Fri Jun 04 08:15:08 2010 -0700
+++ b/text/tqftreview.tex	Fri Jun 04 11:42:07 2010 -0700
@@ -5,7 +5,15 @@
 \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 +29,9 @@
 oriented, topological, smooth, spin, etc. --- but for definiteness we
 will stick with unoriented PL.)
 
-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.
+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 $0 \leq k \leq n$
@@ -54,7 +64,10 @@
 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of 
 $\cC(X)$ which restricts to $c$.
 In this context, we will call $c$ a boundary condition.
-\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$.
+\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$.
+(This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), 
+then this extra structure is considered part of the definition of $\cC_n$.
+Any maps mentioned below between top level fields must be morphisms in $\cS$.
 \item $\cC_k$ is compatible with the symmetric monoidal
 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
 compatibly with homeomorphisms and restriction to boundary.
@@ -185,11 +198,12 @@
 
 \subsection{Systems of fields from $n$-categories}
 \label{sec:example:traditional-n-categories(fields)}
-We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, systems of fields coming from sub-cell-complexes labeled
+We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, 
+systems of fields coming from sub-cell-complexes labeled
 by $n$-category morphisms.
 
 Given an $n$-category $C$ with the right sort of duality
-(e.g. a pivotal 2-category, 1-category with duals, star 1-category),
+(e.g. a pivotal 2-category, *-1-category),
 we can construct a system of fields as follows.
 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$
 with codimension $i$ cells labeled by $i$-morphisms of $C$.
@@ -197,7 +211,8 @@
 
 If $X$ has boundary, we require that the cell decompositions are in general
 position with respect to the boundary --- the boundary intersects each cell
-transversely, so cells meeting the boundary are mere half-cells. Put another way, the cell decompositions we consider are dual to standard cell
+transversely, so cells meeting the boundary are mere half-cells.
+Put another way, the cell decompositions we consider are dual to standard cell
 decompositions of $X$.
 
 We will always assume that our $n$-categories have linear $n$-morphisms.
@@ -270,7 +285,8 @@
 
 \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.
+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.]
@@ -353,7 +369,8 @@
 Let $Y$ be an $n{-}1$-manifold.
 Define a (linear) 1-category $A(Y)$ as follows.
 The objects of $A(Y)$ are $\cC(Y)$.
-The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$.
+The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, 
+where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$.
 Composition is given by gluing of cylinders.
 
 Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces




