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+%!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}
+
+