
diff -r 14e3124a48e8 -r 98b8559b0b7a text/tqftreview.tex
--- a/text/tqftreview.tex	Wed Jun 30 08:55:46 2010 -0700
+++ b/text/tqftreview.tex	Sat Jul 03 13:19:15 2010 -0600
@@ -4,16 +4,31 @@
 \label{sec:fields}
 \label{sec:tqftsviafields}
 
-In this section we review the notion of a ``system of fields and local relations".
+In this section we review the construction of TQFTs from 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}.
+For our purposes, a TQFT is {\it defined} to be something which arises
+from this construction.
+This is an alternative to the more common definition of a TQFT
+as a functor on cobordism categories satisfying various conditions.
+A fully local (``down to points") version of the cobordism-functor TQFT definition
+should be equivalent to the fields-and-local-relations definition.
+
 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.
+In one direction, Example \ref{ex:traditional-n-categories(fields)}
+shows how to construct a system of fields from a (traditional) $n$-category.
+We do this in detail for $n=1,2$ (Subsection \ref{sec:example:traditional-n-categories(fields)}) 
+and more informally for general $n$.
+In the other direction, 
+our preferred definition of an $n$-category in Section \ref{sec:ncats} is essentially
+just a system of fields restricted to balls of dimensions 0 through $n$;
+one could call this the ``local" part of a system of fields.
+
+Since this section is intended primarily to motivate
+the blob complex construction of Section \ref{sec:blob-definition}, 
+we suppress some technical details.
+In Section \ref{sec:ncats} the analogous details are treated more carefully.
+
+\medskip
 
 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
@@ -26,11 +41,12 @@
 unoriented PL manifolds of dimension
 $k$ and morphisms homeomorphisms.
 (We could equally well work with a different category of manifolds ---
-oriented, topological, smooth, spin, etc. --- but for definiteness we
+oriented, topological, smooth, spin, etc. --- but for simplicity 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 $\cS = \Vect$.
+Fields on $n$-manifolds will be enriched over $\cS$.
+Good examples to keep in mind are $\cS = \Set$ 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 $\cS$
@@ -64,10 +80,12 @@
 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 $\Kom$), 
+\item The subset $\cC_n(X;c)$ of top-dimensional 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 $\cS$.
+Any maps mentioned below between fields on $n$-manifolds 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.
@@ -86,22 +104,24 @@
 \]
 and this gluing map is compatible with all of the above structure (actions
 of homeomorphisms, boundary restrictions, disjoint union).
-Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
+Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity 
+and collaring maps,
 the gluing map is surjective.
 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 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 $Y$.
+Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$
+(Figure xxxx).
 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
 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$.
 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$.
 (This restriction map uses the gluing without corners map above.)
-Using the boundary restriction, gluing without corners, and (in one case) orientation reversal
-maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two
+Using the boundary restriction and gluing without corners maps, 
+we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two
 copies of $Y$ in $\bd X$.
 Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps.
 Then (here's the axiom/definition part) there is an injective ``gluing" map
@@ -109,12 +129,14 @@
 	\Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) ,
 \]
 and this gluing map is compatible with all of the above structure (actions
-of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).
-Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
+of homeomorphisms, boundary restrictions, disjoint union).
+Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity
+and collaring maps,
 the gluing map is surjective.
 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
+\item Product fields.
+There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted
 $c \mapsto c\times I$.
 These maps comprise a natural transformation of functors, and commute appropriately
 with all the structure maps above (disjoint union, boundary restriction, etc.).
@@ -136,9 +158,9 @@
 
 \medskip
 
-Using the functoriality and $\cdot\times I$ properties above, together
-with boundary collar homeomorphisms of manifolds, we can define the notion of 
-{\it extended isotopy}.
+Using the functoriality and product field properties above, together
+with boundary collar homeomorphisms of manifolds, we can define 
+{\it collar maps} $\cC(M)\to \cC(M)$.
 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold
 of $\bd M$.
 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$.
@@ -146,10 +168,16 @@
 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$.
 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$.
 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism.
-Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$.
-More generally, we define extended isotopy to be the equivalence relation on fields
-on $M$ generated by isotopy plus all instance of the above construction
-(for all appropriate $Y$ and $x$).
+Then we call the map $x \mapsto f(x \bullet (c\times I))$ a {\it collar map}.
+We call the equivalence relation generated by collar maps and
+homeomorphisms isotopic to the identity {\it extended isotopy}, since the collar maps
+can be thought of (informally) as the limit of homeomorphisms
+which expand an infinitesimally thin collar neighborhood of $Y$ to a thicker
+collar neighborhood.
+
+
+% all this linearizing stuff is unnecessary, I think
+\noop{
 
 \nn{the following discussion of linearizing fields is kind of lame.
 maybe just assume things are already linearized.}
@@ -195,6 +223,8 @@
 We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the
 above tensor products.
 
+} % end \noop
+
 
 \subsection{Systems of fields from $n$-categories}
 \label{sec:example:traditional-n-categories(fields)}




