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 as a ``string diagram".
 It can be thought of as (geometrically) dual to a pasting diagram.
 One of the advantages of string diagrams over pasting diagrams is that one has more
 flexibility in slicing them up in various ways.
 In addition, string diagrams are traditional in quantum topology.
-The diagrams predate by many years the terms ``string diagram" and ``quantum topology".
+The diagrams predate by many years the terms ``string diagram" and ``quantum topology", e.g. \cite{
-\nn{?? cite penrose, kauffman, jones(?)}
+MR0281657,MR776784 % penrose
+}
 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
 As in the $n=1$ case, we allow formal linear combinations of fields on 2-manifolds,
 so long as their restrictions to the boundary coincide.
 In addition, we regard the labelings as being equivariant with respect to the * structure
 on 1-morphisms and pivotal structure on 2-morphisms.
-That is, we mod out my the relation which flips the transverse orientation of a 1-cell
+That is, we mod out by the relation which flips the transverse orientation of a 1-cell
 and replaces its label $a$ by $a^*$, as well as the relation which changes the parameterization of the link
 of a 0-cell and replaces its label by the appropriate pivotal conjugate.
 \medskip
 \label{sec:constructing-a-tqft}
 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations.
 As usual, see \cite{kw:tqft} for more details.
-Let $W$ be an $n{+}1$-manifold.
+We can think of a path integral $Z(W)$ of an $n+1$-manifold (which we're not defining in this context; this is just motivation) as assigning to each
-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 $\lf(\bd W)\to \c$.
-(We haven't defined a path integral in this context; this is just for motivation.)
 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.
 %$\bc_0(X) = \lf(X)$.
 \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 $\lf(X)$;
+where $\cU(X) \sub \lf(X)$ is the space of local relations in $\lf(X)$:
-$\cU(X)$ is generated by things of the form $u\bullet r$, where
+$\cU(X)$ is generated by fields of the form $u\bullet r$, where
 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.
 \end{defn}
-(The blob complex, defined in the next section,
+The blob complex, defined in the next section,
-is in some sense the derived version of $A(X)$.)
+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 $c\in\cC(\bd X)$.
 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)
+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.)
+that lacks its $n{+}1$-dimensional part. Such a ``decapitated'' TQFT is sometimes also called an $n+\epsilon$ or $n+\frac{1}{2}$ dimensional TQFT, referring to the fact that it assigns maps to mapping cylinders between $n$-manifolds, but nothing to arbitrary $n{+}1$-manifolds.
 Let $Y$ be an $n{-}1$-manifold.
-Define a (linear) 1-category $A(Y)$ as follows.
+Define a linear 1-category $A(Y)$ as follows.
-The objects of $A(Y)$ are $\cC(Y)$.
+The set of objects of $A(Y)$ is $\cC(Y)$.
 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 $\cC(\bd X)$.
+$A(X; -) \deq \{A(X; c)\}$ 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 $A(X_2; \cdot)$.
+we have left and right actions of $A(Y)$ on $A(X_1; -)$ and $A(X_2; -)$.
 The gluing theorem for $n$-manifolds states that there is a natural isomorphism
 \[
-	A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) .
+	A(X) \cong A(X_1; -) \otimes_{A(Y)} A(X_2; -) .
 \]
+A proof of this gluing formula appears in \cite{kw:tqft}, but it also becomes a special case of Theorem \ref{thm:gluing} by taking $0$-th homology.

changeset 437	93ce0ba3d2d7
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child 454	3377d4db80d9