%!TEX root = ../blob1.tex

\section{Definitions}

\label{sec:definitions}

\subsection{Systems of fields}

\label{sec:fields}

Let $\cM_k$ denote the category (groupoid, in fact) with objects 

oriented PL manifolds of dimension

$k$ and morphisms homeomorphisms.

(We could equally well work with a different category of manifolds ---

unoriented, topological, smooth, spin, etc. --- but for definiteness we

will stick with oriented 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 There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps

again comprise a natural transformation of functors.

In addition, the orientation reversal maps are compatible with the boundary restriction maps.

\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 cuttable 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 cuttable 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 cuttable 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}

\nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$}

\bigskip

Using the functoriality and $\bullet\times I$ 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 cuttable 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}

%\bigskip

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%\input{text/fields.tex}

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\nn{note: probably will suppress from notation the distinction

between fields and their (orientation-reversal) duals}

\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?}

Given a system of fields and local relations, we define the skein space

$A(Y^n; c)$ to be the space of all finite linear combinations of fields on

the $n$-manifold $Y$ modulo local relations.

The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations

is defined to be the dual of $A(Y; c)$.

(See \cite{kw:tqft} or xxxx for details.)

\nn{should expand above paragraph}

The blob complex is in some sense the derived version of $A(Y; c)$.

\subsection{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 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

$\overline{X \setmin Y}$.

We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.

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}

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 $u_i \in U(B_i; c_i)$, $i=1,2$.

\end{itemize}

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 cuttable along $\bd B_1$.

\item A local relation field $u_0 \in U(B_0; c_0)$.

\end{itemize}

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 $\bd^2 = 0$.

\nn{should draw figures for 1, 2 and $k$-blob diagrams}

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 cuttable 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 cuttable 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}

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 cuttable 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}