Next we define a map from $\phi: \bc_*^C(Y\times F) \to \bc_*^\cF(Y)$.

$\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with 

respect to some open cover

Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding

decomposition of $Y\times F$ into the pieces $X_i\times F$.

%We will define $\phi$ inductively, starting at blob degree 0.

%Given a 0-blob diagram $x$ on $Y\times F$, we can choose a decomposition $K$ of $Y$

%such that $x$ is splittable with respect to $K\times F$.

%This defines a filtration degree 0 element of $\bc_*^\cF(Y)$

We will define $\phi$ using a variant of the method of acyclic models.

Let $a\in S_m$ be a blob diagram on $Y\times F$.

For $m$ sufficiently small there exist decompositions of $K$ of $Y$ into $k$-balls such that the

codimension 1 cells of $K\times F$ miss the blobs of $a$, and more generally such that $a$ is splittable along $K\times F$.

Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \bar{K})$

such that each $K_i$ has the aforementioned splittable property

(see Subsection \ref{ss:ncat_fields}).

(By $(a, \bar{K})$ we really mean $(a', \bar{K})$, where $a^\sharp$ is 

$a$ split according to $K_0\times F$.

To simplify notation we will just write plain $a$ instead of $a^\sharp$.)

Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give

$a$, filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, 

filtration degree 2 stuff which kills the homology created by the 

filtration degree 1 stuff, and so on.

More formally,

\begin{lemma}

$D(a)$ is acyclic.

\end{lemma}

\begin{proof}

We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least}

leave the general case to the reader.

Let $K$ and $K'$ be two decompositions of $Y$ compatible with $a$.

We want to show that $(a, K)$ and $(a, K')$ are homologous

\nn{oops -- can't really ignore $\bd a$ like this}

\end{proof}