--- a/text/tqftreview.tex Tue Jun 01 21:44:09 2010 -0700
+++ b/text/tqftreview.tex Tue Jun 01 23:07:42 2010 -0700
@@ -4,8 +4,8 @@
\label{sec:fields}
\label{sec:tqftsviafields}
-In this section we review the construction of TQFTs from ``topological fields".
-For more details see \cite{kw:tqft}.
+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.
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
@@ -21,18 +21,17 @@
oriented, topological, smooth, spin, etc. --- but for definiteness we
will stick with unoriented PL.)
-%Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$.
+Fix a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$.
A $n$-dimensional {\it system of fields} in $\cS$
is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$
together with some additional data and satisfying some additional conditions, all specified below.
-Before finishing the definition of fields, we give two motivating examples
-(actually, families of examples) of systems of fields.
+Before finishing the definition of fields, we give two motivating examples of systems of fields.
\begin{example}
\label{ex:maps-to-a-space(fields)}
-Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps
+Fix a target space $T$, and let $\cC(X)$ be the set of continuous maps
from X to $B$.
\end{example}
@@ -42,7 +41,7 @@
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.
+This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}.
\end{example}
Now for the rest of the definition of system of fields.
@@ -144,6 +143,47 @@
\nn{remark that if top dimensional fields are not already linear
then we will soon linearize them(?)}
+For top dimensional ($n$-dimensional) manifolds, we're actually interested
+in the linearized space of fields.
+By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is
+the vector space of finite
+linear combinations of fields on $X$.
+If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$.
+Thus the restriction (to boundary) maps are well defined because we never
+take linear combinations of fields with differing boundary conditions.
+
+In some cases we don't linearize the default way; instead we take the
+spaces $\lf(X; a)$ to be part of the data for the system of fields.
+In particular, for fields based on linear $n$-category pictures we linearize as follows.
+Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
+obvious relations on 0-cell labels.
+More specifically, let $L$ be a cell decomposition of $X$
+and let $p$ be a 0-cell of $L$.
+Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
+$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
+Then the subspace $K$ is generated by things of the form
+$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
+to infer the meaning of $\alpha_{\lambda c + d}$.
+Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
+
+\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
+will do something similar below; in general, whenever a label lives in a linear
+space we do something like this; ? say something about tensor
+product of all the linear label spaces? Yes:}
+
+For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
+Define an ``almost-field" to be a field without labels on the 0-cells.
+(Recall that 0-cells are labeled by $n$-morphisms.)
+To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
+space determined by the labeling of the link of the 0-cell.
+(If the 0-cell were labeled, the label would live in this space.)
+We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
+We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the
+above tensor products.
+
+
+\subsection{Systems of fields from $n$-categories}
+\label{sec:example:traditional-n-categories(fields)}
We now describe in more detail systems of fields coming from sub-cell-complexes labeled
by $n$-category morphisms.
@@ -226,43 +266,6 @@
\medskip
-For top dimensional ($n$-dimensional) manifolds, we're actually interested
-in the linearized space of fields.
-By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is
-the vector space of finite
-linear combinations of fields on $X$.
-If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$.
-Thus the restriction (to boundary) maps are well defined because we never
-take linear combinations of fields with differing boundary conditions.
-
-In some cases we don't linearize the default way; instead we take the
-spaces $\lf(X; a)$ to be part of the data for the system of fields.
-In particular, for fields based on linear $n$-category pictures we linearize as follows.
-Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
-obvious relations on 0-cell labels.
-More specifically, let $L$ be a cell decomposition of $X$
-and let $p$ be a 0-cell of $L$.
-Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
-$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
-Then the subspace $K$ is generated by things of the form
-$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
-to infer the meaning of $\alpha_{\lambda c + d}$.
-Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
-
-\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
-will do something similar below; in general, whenever a label lives in a linear
-space we do something like this; ? say something about tensor
-product of all the linear label spaces? Yes:}
-
-For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
-Define an ``almost-field" to be a field without labels on the 0-cells.
-(Recall that 0-cells are labeled by $n$-morphisms.)
-To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
-space determined by the labeling of the link of the 0-cell.
-(If the 0-cell were labeled, the label would live in this space.)
-We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
-We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the
-above tensor products.