...
--- a/blob1.tex Tue Mar 02 04:26:36 2010 +0000
+++ b/blob1.tex Tue Mar 02 20:07:31 2010 +0000
@@ -51,7 +51,10 @@
\input{text/intro}
-\input{text/definitions}
+%\input{text/definitions}
+\input{text/tqftreview}
+
+\input{text/blobdef}
\input{text/basic_properties}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/text/blobdef.tex Tue Mar 02 20:07:31 2010 +0000
@@ -0,0 +1,189 @@
+%!TEX root = ../blob1.tex
+
+\section{The blob complex}
+\label{sec:blob-definition}
+
+Let $X$ be an $n$-manifold.
+Assume a fixed system of fields and local relations.
+In this section we will usually suppress boundary conditions on $X$ from the notation
+(e.g. write $\lf(X)$ instead of $\lf(X; c)$).
+
+We want to replace the quotient
+\[
+ A(X) \deq \lf(X) / U(X)
+\]
+of the previous section with a resolution
+\[
+ \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) .
+\]
+
+We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.
+
+We of course define $\bc_0(X) = \lf(X)$.
+(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$.
+We'll omit this sort of detail in the rest of this section.)
+In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$.
+
+$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$.
+Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear
+combinations of 1-blob diagrams, where a 1-blob diagram to consists of
+\begin{itemize}
+\item An embedded closed ball (``blob") $B \sub X$.
+\item A field $r \in \cC(X \setmin B; c)$
+(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$).
+\item A local relation field $u \in U(B; c)$
+(same $c$ as previous bullet).
+\end{itemize}
+(See Figure \ref{blob1diagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{definition/single-blob}
+\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}
+In order to get the linear structure correct, we (officially) define
+\[
+ \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) .
+\]
+The first direct sum is indexed by all blobs $B\subset X$, and the second
+by all boundary conditions $c \in \cC(\bd B)$.
+Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$.
+
+Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by
+\[
+ (B, u, r) \mapsto u\bullet r,
+\]
+where $u\bullet r$ denotes the linear
+combination of fields on $X$ obtained by gluing $u$ to $r$.
+In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by
+just erasing the blob from the picture
+(but keeping the blob label $u$).
+
+Note that the skein space $A(X)$
+is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
+
+$\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the
+local relations encoded in $\bc_1(X)$.
+More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
+2-blob diagrams, of which there are two types, disjoint and nested.
+
+A disjoint 2-blob diagram consists of
+\begin{itemize}
+\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors.
+\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$
+(where $c_i \in \cC(\bd B_i)$).
+\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \nn{We're inconsistent with the indexes -- are they 0,1 or 1,2? I'd prefer 1,2.}
+\end{itemize}
+(See Figure \ref{blob2ddiagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{definition/disjoint-blobs}
+\end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure}
+We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$;
+reversing the order of the blobs changes the sign.
+Define $\bd(B_0, B_1, u_0, u_1, r) =
+(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$.
+In other words, the boundary of a disjoint 2-blob diagram
+is the sum (with alternating signs)
+of the two ways of erasing one of the blobs.
+It's easy to check that $\bd^2 = 0$.
+
+A nested 2-blob diagram consists of
+\begin{itemize}
+\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$.
+\item A field $r \in \cC(X \setmin B_0; c_0)$
+(for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$.
+\item A local relation field $u_0 \in U(B_0; c_0)$.
+\end{itemize}
+(See Figure \ref{blob2ndiagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{definition/nested-blobs}
+\end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure}
+Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
+(for some $c_1 \in \cC(B_1)$) and
+$r' \in \cC(X \setmin B_1; c_1)$.
+Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$.
+Note that the requirement that
+local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$.
+As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating
+sum of the two ways of erasing one of the blobs.
+If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$.
+It is again easy to check that $\bd^2 = 0$.
+
+As with the 1-blob diagrams, in order to get the linear structure correct it is better to define
+(officially)
+\begin{eqnarray*}
+ \bc_2(X) & \deq &
+ \left(
+ \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1}
+ U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1)
+ \right) \\
+ && \bigoplus \left(
+ \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0}
+ U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0)
+ \right) .
+\end{eqnarray*}
+The final $\lf(X\setmin B_0; c_0)$ above really means fields splittable along $\bd B_1$,
+but we didn't feel like introducing a notation for that.
+For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign
+(rather than a new, linearly independent 2-blob diagram).
+
+Now for the general case.
+A $k$-blob diagram consists of
+\begin{itemize}
+\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$.
+For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or
+$B_i \sub B_j$ or $B_j \sub B_i$.
+(The case $B_i = B_j$ is allowed.
+If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.)
+If a blob has no other blobs strictly contained in it, we call it a twig blob.
+\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
+(These are implied by the data in the next bullets, so we usually
+suppress them from the notation.)
+$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$
+if the latter space is not empty.
+\item A field $r \in \cC(X \setmin B^t; c^t)$,
+where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$
+is determined by the $c_i$'s.
+$r$ is required to be splittable along the boundaries of all blobs, twigs or not.
+\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$,
+where $c_j$ is the restriction of $c^t$ to $\bd B_j$.
+If $B_i = B_j$ then $u_i = u_j$.
+\end{itemize}
+(See Figure \ref{blobkdiagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{definition/k-blobs}
+\end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure}
+
+If two blob diagrams $D_1$ and $D_2$
+differ only by a reordering of the blobs, then we identify
+$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$.
+
+$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams.
+As before, the official definition is in terms of direct sums
+of tensor products:
+\[
+ \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
+ \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
+\]
+Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
+$\overline{c}$ runs over all boundary conditions, again as described above.
+$j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$.
+
+The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows.
+Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram.
+Let $E_j(b)$ denote the result of erasing the $j$-th blob.
+If $B_j$ is not a twig blob, this involves only decrementing
+the indices of blobs $B_{j+1},\ldots,B_{k-1}$.
+If $B_j$ is a twig blob, we have to assign new local relation labels
+if removing $B_j$ creates new twig blobs.
+If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$,
+where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.
+Finally, define
+\eq{
+ \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b).
+}
+The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel.
+Thus we have a chain complex.
+
+\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)}
+
+\nn{?? remark about dendroidal sets?; probably not}
+
+
--- a/text/definitions.tex Tue Mar 02 04:26:36 2010 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,541 +0,0 @@
-%!TEX root = ../blob1.tex
-
-\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
-\[
- A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) .
-\]
-
-
-\section{The blob complex}
-\label{sec:blob-definition}
-
-Let $X$ be an $n$-manifold.
-Assume a fixed system of fields and local relations.
-In this section we will usually suppress boundary conditions on $X$ from the notation
-(e.g. write $\lf(X)$ instead of $\lf(X; c)$).
-
-We want to replace the quotient
-\[
- A(X) \deq \lf(X) / U(X)
-\]
-of the previous section with a resolution
-\[
- \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) .
-\]
-
-We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.
-
-We of course define $\bc_0(X) = \lf(X)$.
-(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$.
-We'll omit this sort of detail in the rest of this section.)
-In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$.
-
-$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$.
-Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear
-combinations of 1-blob diagrams, where a 1-blob diagram to consists of
-\begin{itemize}
-\item An embedded closed ball (``blob") $B \sub X$.
-\item A field $r \in \cC(X \setmin B; c)$
-(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$).
-\item A local relation field $u \in U(B; c)$
-(same $c$ as previous bullet).
-\end{itemize}
-(See Figure \ref{blob1diagram}.)
-\begin{figure}[!ht]\begin{equation*}
-\mathfig{.9}{definition/single-blob}
-\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}
-In order to get the linear structure correct, we (officially) define
-\[
- \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) .
-\]
-The first direct sum is indexed by all blobs $B\subset X$, and the second
-by all boundary conditions $c \in \cC(\bd B)$.
-Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$.
-
-Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by
-\[
- (B, u, r) \mapsto u\bullet r,
-\]
-where $u\bullet r$ denotes the linear
-combination of fields on $X$ obtained by gluing $u$ to $r$.
-In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by
-just erasing the blob from the picture
-(but keeping the blob label $u$).
-
-Note that the skein space $A(X)$
-is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
-
-$\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the
-local relations encoded in $\bc_1(X)$.
-More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
-2-blob diagrams, of which there are two types, disjoint and nested.
-
-A disjoint 2-blob diagram consists of
-\begin{itemize}
-\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors.
-\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$
-(where $c_i \in \cC(\bd B_i)$).
-\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \nn{We're inconsistent with the indexes -- are they 0,1 or 1,2? I'd prefer 1,2.}
-\end{itemize}
-(See Figure \ref{blob2ddiagram}.)
-\begin{figure}[!ht]\begin{equation*}
-\mathfig{.9}{definition/disjoint-blobs}
-\end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure}
-We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$;
-reversing the order of the blobs changes the sign.
-Define $\bd(B_0, B_1, u_0, u_1, r) =
-(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$.
-In other words, the boundary of a disjoint 2-blob diagram
-is the sum (with alternating signs)
-of the two ways of erasing one of the blobs.
-It's easy to check that $\bd^2 = 0$.
-
-A nested 2-blob diagram consists of
-\begin{itemize}
-\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$.
-\item A field $r \in \cC(X \setmin B_0; c_0)$
-(for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$.
-\item A local relation field $u_0 \in U(B_0; c_0)$.
-\end{itemize}
-(See Figure \ref{blob2ndiagram}.)
-\begin{figure}[!ht]\begin{equation*}
-\mathfig{.9}{definition/nested-blobs}
-\end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure}
-Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
-(for some $c_1 \in \cC(B_1)$) and
-$r' \in \cC(X \setmin B_1; c_1)$.
-Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$.
-Note that the requirement that
-local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$.
-As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating
-sum of the two ways of erasing one of the blobs.
-If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$.
-It is again easy to check that $\bd^2 = 0$.
-
-As with the 1-blob diagrams, in order to get the linear structure correct it is better to define
-(officially)
-\begin{eqnarray*}
- \bc_2(X) & \deq &
- \left(
- \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1}
- U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1)
- \right) \\
- && \bigoplus \left(
- \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0}
- U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0)
- \right) .
-\end{eqnarray*}
-The final $\lf(X\setmin B_0; c_0)$ above really means fields splittable along $\bd B_1$,
-but we didn't feel like introducing a notation for that.
-For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign
-(rather than a new, linearly independent 2-blob diagram).
-
-Now for the general case.
-A $k$-blob diagram consists of
-\begin{itemize}
-\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$.
-For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or
-$B_i \sub B_j$ or $B_j \sub B_i$.
-(The case $B_i = B_j$ is allowed.
-If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.)
-If a blob has no other blobs strictly contained in it, we call it a twig blob.
-\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
-(These are implied by the data in the next bullets, so we usually
-suppress them from the notation.)
-$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$
-if the latter space is not empty.
-\item A field $r \in \cC(X \setmin B^t; c^t)$,
-where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$
-is determined by the $c_i$'s.
-$r$ is required to be splittable along the boundaries of all blobs, twigs or not.
-\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$,
-where $c_j$ is the restriction of $c^t$ to $\bd B_j$.
-If $B_i = B_j$ then $u_i = u_j$.
-\end{itemize}
-(See Figure \ref{blobkdiagram}.)
-\begin{figure}[!ht]\begin{equation*}
-\mathfig{.9}{definition/k-blobs}
-\end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure}
-
-If two blob diagrams $D_1$ and $D_2$
-differ only by a reordering of the blobs, then we identify
-$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$.
-
-$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams.
-As before, the official definition is in terms of direct sums
-of tensor products:
-\[
- \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
- \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
-\]
-Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
-$\overline{c}$ runs over all boundary conditions, again as described above.
-$j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$.
-
-The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows.
-Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram.
-Let $E_j(b)$ denote the result of erasing the $j$-th blob.
-If $B_j$ is not a twig blob, this involves only decrementing
-the indices of blobs $B_{j+1},\ldots,B_{k-1}$.
-If $B_j$ is a twig blob, we have to assign new local relation labels
-if removing $B_j$ creates new twig blobs.
-If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$,
-where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.
-Finally, define
-\eq{
- \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b).
-}
-The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel.
-Thus we have a chain complex.
-
-\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)}
-
-\nn{?? remark about dendroidal sets}
-
-
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/text/tqftreview.tex Tue Mar 02 20:07:31 2010 +0000
@@ -0,0 +1,353 @@
+%!TEX root = ../blob1.tex
+
+\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
+\[
+ A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) .
+\]
+