
diff -r 62e8cc479953 -r 57291331fd82 text/definitions.tex
--- a/text/definitions.tex	Tue Oct 27 05:19:10 2009 +0000
+++ b/text/definitions.tex	Tue Oct 27 22:27:07 2009 +0000
@@ -277,20 +277,65 @@
 
 \subsection{Constructing a TQFT}
 
-\nn{need to expand this; use $\bc_0/\bc_1$ notation (maybe); also introduce
-cylinder categories and gluing formula}
+In this subsection we briefly review the construction of a TQFT from a system of fields and local relations.
+(For more details, see \cite{kw:tqft}.)
+
+Let $W$ be an $n{+}1$-manifold.
+We can think of the path integral $Z(W)$ as assigning to each
+boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$.
+In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear
+maps $\lf(\bd W)\to \c$.
+
+The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace
+$Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections.
+The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$,
+can be thought of as finite linear combinations of fields modulo local relations.
+(In other words, $A(\bd W)$ is a sort of generalized skein module.)
+This is the motivation behind the definition of fields and local relations above.
 
-Given a system of fields and local relations, we define the skein space
-$A(Y^n; c)$ to be the space of all finite linear combinations of fields on
-the $n$-manifold $Y$ modulo local relations.
-The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations
-is defined to be the dual of $A(Y; c)$.
-(See \cite{kw:tqft} or xxxx for details.)
+In more detail, let $X$ be an $n$-manifold.
+%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
+\[
+	A(X) \deq \lf(X) / U(X) .
+\]
+(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 
+boundary condition $c\in\cC(\bd X)$.
 
-\nn{should expand above paragraph}
+The above construction can be extended to higher codimensions, assigning
+a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$.
+These invariants fit together via actions and gluing formulas.
+We describe only the case $k=1$ below.
+(The construction of the $n{+}1$-dimensional part of the theory (the path integral) 
+requires that the starting data (fields and local relations) satisfy additional
+conditions.
+We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT
+that lacks its $n{+}1$-dimensional part.)
 
-The blob complex is in some sense the derived version of $A(Y; c)$.
+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 $Y\times I$.
+Composition is given by gluing of cylinders.
 
+Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces
+$A(X; \cdot) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$.
+This collection of vector spaces affords a representation of the category $A(\bd X)$, where
+the action is given by gluing a collar $\bd X\times I$ to $X$.
+
+Given a splitting $X = X_1 \cup_Y X_2$ of a closed $n$-manifold $X$ along an $n{-}1$-manifold $Y$,
+we have left and right actions of $A(Y)$ on $A(X_1; \cdot)$ and $A(X_2; \cdot)$.
+The gluing theorem for $n$-manifolds states that there is a natural isomorphism
+\[
+	A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) .
+\]
 
 
 \section{The blob complex}




