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+
+\section{TQFTs via fields}
+\label{sec:fields}
+\label{sec:tqftsviafields}
+
+In this section we review the construction of TQFTs from ``topological fields".
+For more details see \cite{kw:tqft}.
+
+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}
+
+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 definiteness we
+will stick with unoriented PL.)
+
+%Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$.
+
+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
+(actually, families of examples) of systems of fields.
+
+The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps
+from X to $B$.
+
+The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be 
+the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by
+$j$-morphisms of $C$.
+One can think of such sub-cell-complexes as dual to pasting diagrams for $C$.
+This is described in more detail below.
+
+Now for the rest of the definition of system of fields.
+\begin{enumerate}
+\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, 
+and these maps are 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 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.
+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
+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, 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
+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 $\pm Y$.
+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
+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, 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
+are transverse to $Y$ or splittable along $Y$.
+\item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted
+$c \mapsto c\times I$.
+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 $\cdot\times I$ properties above, together
+with boundary collar homeomorphisms of manifolds, we can define the notion of 
+{\it extended isotopy}.
+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 $Y$.
+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 $x$).
+
+\nn{should also say something about pseudo-isotopy}
+
+
+\nn{remark that if top dimensional fields are not already linear
+then we will soon linearize them(?)}
+
+We now describe in more detail systems of fields coming from sub-cell-complexes labeled
+by $n$-category morphisms.
+
+Given an $n$-category $C$ with the right sort of duality
+(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-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 $C$.
+We'll spell this out for $n=1,2$ and then describe the general case.
+
+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
+decompositions of $X$.
+
+We will always assume that our $n$-categories have linear $n$-morphisms.
+
+For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with
+an object (0-morphism) of the 1-category $C$.
+A field on a 1-manifold $S$ consists of
+\begin{itemize}
+    \item A cell decomposition of $S$ (equivalently, a finite collection
+of points in the interior of $S$);
+    \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$)
+by an object (0-morphism) of $C$;
+    \item a transverse orientation of each 0-cell, thought of as a choice of
+``domain" and ``range" for the two adjacent 1-cells; and
+    \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with
+domain and range determined by the transverse orientation and the labelings of the 1-cells.
+\end{itemize}
+
+If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
+of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the
+interior of $S$, each transversely oriented and each labeled by an element (1-morphism)
+of the algebra.
+
+\medskip
+
+For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories)
+that are common in the literature.
+We describe these carefully here.
+
+A field on a 0-manifold $P$ is a labeling of each point of $P$ with
+an object of the 2-category $C$.
+A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$.
+A field on a 2-manifold $Y$ consists of
+\begin{itemize}
+    \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such
+that each component of the complement is homeomorphic to a disk);
+    \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$)
+by a 0-morphism of $C$;
+    \item a transverse orientation of each 1-cell, thought of as a choice of
+``domain" and ``range" for the two adjacent 2-cells;
+    \item a labeling of each 1-cell by a 1-morphism of $C$, with
+domain and range determined by the transverse orientation of the 1-cell
+and the labelings of the 2-cells;
+    \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood
+of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped
+to $\pm 1 \in S^1$; and
+    \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range
+determined by the labelings of the 1-cells and the parameterizations of the previous
+bullet.
+\end{itemize}
+\nn{need to say this better; don't try to fit everything into the bulleted list}
+
+For general $n$, a field on a $k$-manifold $X^k$ consists of
+\begin{itemize}
+    \item A cell decomposition of $X$;
+    \item an explicit general position homeomorphism from the link of each $j$-cell
+to the boundary of the standard $(k-j)$-dimensional bihedron; and
+    \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with
+domain and range determined by the labelings of the link of $j$-cell.
+\end{itemize}
+
+%\nn{next definition might need some work; I think linearity relations should
+%be treated differently (segregated) from other local relations, but I'm not sure
+%the next definition is the best way to do it}
+
+\medskip
+
+For top dimensional ($n$-dimensional) manifolds, we're actually interested
+in the linearized space of fields.
+By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is
+the vector space of finite
+linear combinations of fields on $X$.
+If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$.
+Thus the restriction (to boundary) maps are well defined because we never
+take linear combinations of fields with differing boundary conditions.
+
+In some cases we don't linearize the default way; instead we take the
+spaces $\lf(X; a)$ to be part of the data for the system of fields.
+In particular, for fields based on linear $n$-category pictures we linearize as follows.
+Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
+obvious relations on 0-cell labels.
+More specifically, let $L$ be a cell decomposition of $X$
+and let $p$ be a 0-cell of $L$.
+Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
+$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
+Then the subspace $K$ is generated by things of the form
+$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
+to infer the meaning of $\alpha_{\lambda c + d}$.
+Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
+
+\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
+will do something similar below; in general, whenever a label lives in a linear
+space we do something like this; ? say something about tensor
+product of all the linear label spaces?  Yes:}
+
+For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
+Define an ``almost-field" to be a field without labels on the 0-cells.
+(Recall that 0-cells are labeled by $n$-morphisms.)
+To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
+space determined by the labeling of the link of the 0-cell.
+(If the 0-cell were labeled, the label would live in this space.)
+We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
+We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the
+above tensor products.
+
+
+
+\subsection{Local relations}
+\label{sec:local-relations}
+
+
+A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$,
+for all $n$-manifolds $B$ which are
+homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, 
+satisfying the following properties.
+\begin{enumerate}
+\item functoriality: 
+$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$
+\item local relations imply extended isotopy: 
+if $x, y \in \cC(B; c)$ and $x$ is extended isotopic 
+to $y$, then $x-y \in U(B; c)$.
+\item ideal with respect to gluing:
+if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$
+\end{enumerate}
+See \cite{kw:tqft} for details.
+
+
+For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$,
+where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
+
+For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map
+$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
+domain and range.
+
+\nn{maybe examples of local relations before general def?}
+
+\subsection{Constructing a TQFT}
+
+In this subsection we briefly review the construction of a TQFT from a system of fields and local relations.
+(For more details, see \cite{kw:tqft}.)
+
+Let $W$ be an $n{+}1$-manifold.
+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 $\lf(\bd W)\to \c$.
+
+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.
+(In other words, $A(\bd W)$ is a sort of generalized skein module.)
+This is the motivation behind the definition of fields and local relations above.
+
+In more detail, let $X$ be an $n$-manifold.
+%To harmonize notation with the next section, 
+%let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so
+%$\bc_0(X) = \lf(X)$.
+Define $U(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$;
+$U(X)$ is generated by things of the form $u\bullet r$, where
+$u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.
+Define
+\[
+	A(X) \deq \lf(X) / U(X) .
+\]
+(The blob complex, defined in the next section, 
+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) 
+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.)
+
+Let $Y$ be an $n{-}1$-manifold.
+Define a (linear) 1-category $A(Y)$ as follows.
+The objects of $A(Y)$ are $\cC(Y)$.
+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 $\cC(\bd X)$.
+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 $A(X_2; \cdot)$.
+The gluing theorem for $n$-manifolds states that there is a natural isomorphism
+\[
+	A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) .
+\]
+