--- a/text/tqftreview.tex Thu Jun 03 23:08:47 2010 -0700
+++ b/text/tqftreview.tex Fri Jun 04 08:15:08 2010 -0700
@@ -45,9 +45,11 @@
\end{example}
Now for the rest of the definition of system of fields.
+(Readers desiring a more precise definition should refer to Subsection \ref{ss:n-cat-def}
+and replace $k$-balls with $k$-manifolds.)
\begin{enumerate}
\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$,
-and these maps are a natural
+and these maps comprise a natural
transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$.
For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of
$\cC(X)$ which restricts to $c$.
@@ -55,13 +57,13 @@
\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, restriction to boundary, and orientation reversal.
+compatibly with homeomorphisms and restriction to boundary.
We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$
restriction maps.
\item Gluing without corners.
-Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.
-Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
-Using the boundary restriction, disjoint union, and (in one case) orientation reversal
+Let $\bd X = Y \du Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.
+Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$.
+Using the boundary restriction and disjoint union
maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two
copies of $Y$ in $\bd X$.
Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps.
@@ -70,15 +72,15 @@
\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) ,
\]
and this gluing map is compatible with all of the above structure (actions
-of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).
+of homeomorphisms, boundary restrictions, disjoint union).
Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
the gluing map is surjective.
-From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the
-gluing surface, we say that fields in the image of the gluing map
+We say that fields on $X\sgl$ in the image of the gluing map
are transverse to $Y$ or splittable along $Y$.
\item Gluing with corners.
-Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries.
-Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
+Let $\bd X = Y \cup Y \cup W$, where the two copies of $Y$ and
+$W$ might intersect along their boundaries.
+Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$.
Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself
(without corners) along two copies of $\bd Y$.
Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let
@@ -97,8 +99,7 @@
of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).
Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
the gluing map is surjective.
-From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the
-gluing surface, we say that fields in the image of the gluing map
+We say that fields in the image of the gluing map
are transverse to $Y$ or splittable along $Y$.
\item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted
$c \mapsto c\times I$.
@@ -137,8 +138,8 @@
on $M$ generated by isotopy plus all instance of the above construction
(for all appropriate $Y$ and $x$).
-\nn{should also say something about pseudo-isotopy}
-
+\nn{the following discussion of linearizing fields is kind of lame.
+maybe just assume things are already linearized.}
\nn{remark that if top dimensional fields are not already linear
then we will soon linearize them(?)}
@@ -188,7 +189,7 @@
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, disklike $n$-category),
+(e.g. a pivotal 2-category, 1-category with duals, star 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 $C$.