%!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