%!TEX root = ../blob1.tex

\section{TQFTs via fields}

\label{sec:fields}

\label{sec:tqftsviafields}

In this section we review the construction of TQFTs from fields and local relations.

For more details see \cite{kw: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 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 $n=1,2$ (\S\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 \S\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 \S\ref{sec:blob-definition}, 

we suppress some technical details.

In \S\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

$\overline{X \setmin Y}$.

\subsection{Systems of fields}

\label{ss:syst-o-fields}

Let $\cM_k$ denote the category with objects 

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 simplicity we

will stick with unoriented PL.)

Fix a symmetric monoidal category $\cS$.

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 $\cS$

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 of systems of fields.

\begin{example}

\label{ex:maps-to-a-space(fields)}

Fix a target space $T$, and let $\cC(X)$ be the set of continuous maps

from $X$ to $T$.

\end{example}

\begin{example}

\label{ex:traditional-n-categories(fields)}

Fix an $n$-category $C$, and let $\cC(X)$ be 

the set of embedded cell complexes in $X$ with codimension-$j$ cells labeled by

$j$-morphisms of $C$.

One can think of such embedded cell complexes as dual to pasting diagrams for $C$.

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.

(Readers desiring a more precise definition should refer to \S\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 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 $c$.

In this context, we will call $c$ a boundary condition.

\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 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.

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 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.

Then (here's the axiom/definition part) there is an injective ``gluing" map

\[

	\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, 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 on $X\sgl$ in the image of the gluing map

are transverse to $Y$ or splittable along $Y$.

\item Gluing with corners.

Let $\bd X = (Y \du Y) \cup W$, where the two copies of $Y$ 

are disjoint from each other and $\bd(Y\du Y) = \bd W$.

Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$

(Figure \ref{fig:gluing-with-corners}).

\begin{figure}[t]

\begin{center}

\begin{tikzpicture}

\node(A) at (-4,0) {

\begin{tikzpicture}[scale=.8, fill=blue!15!white]

\filldraw[line width=1.5pt] (-.4,1) .. controls +(-1,-.1) and +(-1,0) .. (0,-1)

		.. controls +(1,0) and +(1,-.1) .. (.4,1) -- (.4,3)

		.. controls +(3,-.4) and +(3,0) .. (0,-3)

		.. controls +(-3,0) and +(-3,-.1) .. (-.4,3) -- cycle;

\node at (0,-2) {$X$};

\node (W) at (-2.7,-2) {$W$};

\node (Y1) at (-1.2,3.5) {$Y$};

\node (Y2) at (1.4,3.5) {$Y$};

\node[outer sep=2.3] (y1e) at (-.4,2) {};

\node[outer sep=2.3] (y2e) at (.4,2) {};

\node (we1) at (-2.2,-1.1) {};

\node (we2) at (-.6,-.7) {};

\draw[->] (Y1) -- (y1e);

\draw[->] (Y2) -- (y2e);

\draw[->] (W) .. controls +(0,.5) and +(-.5,-.2) .. (we1);

\draw[->] (W) .. controls +(.5,0) and +(-.2,-.5) .. (we2);

\end{tikzpicture}

};

\node(B) at (4,0) {

\begin{tikzpicture}[scale=.8, fill=blue!15!white]

\fill (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1)

		.. controls +(1,0) and +(1,0) .. (0,1) -- (0,3)

		.. controls +(3,0) and +(3,0) .. (0,-3)

		.. controls +(-3,0) and +(-3,0) .. (0,3) -- cycle;

\draw[line width=1.5pt] (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1)

		.. controls +(1,0) and +(1,0) .. (0,1);

\draw[line width=1.5pt] (0,3) .. controls +(3,0) and +(3,0) .. (0,-3)

		.. controls +(-3,0) and +(-3,0) .. (0,3);

\draw[line width=.5pt, black!65!white] (0,1) -- (0,3);

\node at (0,-2) {$X\sgl$};

\node (W) at (2.7,-2) {$W\sgl$};

\node (we1) at (2.2,-1.1) {};

\node (we2) at (.6,-.7) {};

\draw[->] (W) .. controls +(0,.5) and +(.5,-.2) .. (we1);

\draw[->] (W) .. controls +(-.5,0) and +(.2,-.5) .. (we2);

\end{tikzpicture}

};

\draw[->, red!50!green, line width=2pt] (A) -- node[above, black] {glue} (B);

\end{tikzpicture}

\end{center}

\caption{Gluing with corners}

\label{fig:gluing-with-corners}

\end{figure}

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 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

\[

	\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, 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 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.).

Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism

covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$.

\end{enumerate}

There are two notations we commonly use for gluing.

One is 

\[

	x\sgl \deq \gl(x) \in \cC(X\sgl) , 

\]

for $x\in\cC(X)$.

The other is

\[

	x_1\bullet x_2 \deq \gl(x_1\otimes x_2) \in \cC(X\sgl) , 

\]

in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$.

\medskip

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 $\bd Y$.

Let $c$ be $x$ restricted to $Y$.