189 

   190 A {\emph ball decomposition} of $W$ is a 

   191 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls

   192 $\du_a X_a$.

   193 

   194 If $X_a$ is some component of $M_0$, note that its image in $W$ need not be a ball; parts of $\bd X_a$ may have been glued together.

   195 Define a {\it permissible decomposition} of $W$ to be a map

   196 \[

   197 	\coprod_a X_a \to W,

   198 \]

   199 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$.

   200 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls

   201 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them.

   202 

   203 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement

   204 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$

   205 with $\du_b Y_b = M_i$ for some $i$.

   206 

   207 \begin{defn}

   208 The poset $\cell(W)$ has objects the permissible decompositions of $W$, 

   209 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.

   210 See Figure \ref{partofJfig} for an example.

   211 \end{defn}

   212 

   213 

   214 An $n$-category $\cC$ determines 

   215 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets 

   216 (possibly with additional structure if $k=n$).

   217 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls,

   218 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries

   219 are splittable along this decomposition.

   220 

   221 \begin{defn}

   222 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.

   223 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset

   224 \begin{equation}

   225 \label{eq:psi-C}

   226 	\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl

   227 \end{equation}

   228 where the restrictions to the various pieces of shared boundaries amongst the cells

   229 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells).

   230 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.

   231 \end{defn}

   232 

   233 

   521 

   522 \begin{figure}

   523 \begin{equation*}

   524 \mathfig{.23}{ncat/zz2}

   525 \end{equation*}

   526 \caption{A small part of $\cell(W)$}

   527 \label{partofJfig}

   528 \end{figure}

   529 

   530