--- a/text/ncat.tex	Tue Jul 28 18:52:39 2009 +0000
+++ b/text/ncat.tex	Wed Jul 29 18:23:18 2009 +0000
@@ -8,6 +8,8 @@
 \nn{experimental section.  maybe this should be rolled into other sections.
 maybe it should be split off into a separate paper.}
 
+\subsection{Definition of $n$-categories}
+
 Before proceeding, we need more appropriate definitions of $n$-categories, 
 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules.
 (As is the case throughout this paper, by ``$n$-category" we mean
@@ -24,7 +26,8 @@
 Still other definitions \nn{need refs for all these; maybe the Leinster book}
 model the $k$-morphisms on more complicated combinatorial polyhedra.
 
-We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to a $k$-ball.
+We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to 
+the standard $k$-ball.
 In other words,
 
 \xxpar{Morphisms (preliminary version):}
@@ -351,7 +354,66 @@
 
 \end{itemize}
 
-\medskip
+
+
+
+
+
+\subsection{From $n$-categories to systems of fields}
+
+We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows.
+
+Let $W$ be a $k$-manifold, $1\le k \le n$.
+We will define a set $\cC(W)$.
+(If $k = n$ and our $k$-categories are enriched, then
+$\cC(W)$ will have additional structure; see below.)
+$\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$,
+which we define next.
+
+Define a permissible decomposition of $W$ to be a decomposition
+\[
+	W = \bigcup_a X_a ,
+\]
+where each $X_a$ is a $k$-ball.
+Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
+of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
+This defines a partial ordering $\cJ(W)$, which we will think of as a category.
+(The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique
+morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.)
+\nn{need figures}
+
+$\cC$ determines 
+a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets 
+(possibly with additional structure if $k=n$).
+For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset
+\[
+	\psi_\cC(x) \sub \prod_a \cC(X_a)
+\]
+such that the restrictions to the various pieces of shared boundaries amongst the
+$X_a$ all agree.
+(Think fibered product.)
+If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$
+via the composition maps of $\cC$.
+
+Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$.
+In other words, for each decomposition $x$ there is a map
+$\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps
+above, and $\cC(W)$ is universal with respect to these properties.
+
+$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.
+
+It is easy to see that
+there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps
+comprise a natural transformation of functors.
+
+\nn{need to finish explaining why we have a system of fields;
+need to say more about ``homological" fields? 
+(actions of homeomorphisms);
+define $k$-cat $\cC(\cdot\times W)$}
+
+
+
+\subsection{Modules}
 
 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations,
 a.k.a.\ actions).
@@ -559,13 +621,63 @@
 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$),
 there is no left/right module distinction.
 
-\medskip
+
+\subsection{Modules as boundary labels}
+
+Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$),
+and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary 
+component $\bd_i W$ of $W$.
+
+We will define a set $\cC(W, \cN)$ using a colimit construction similar to above.
+\nn{give ref}
+(If $k = n$ and our $k$-categories are enriched, then
+$\cC(W, \cN)$ will have additional structure; see below.)
+
+Define a permissible decomposition of $W$ to be a decomposition
+\[
+	W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) ,
+\]
+where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and
+each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$,
+with $M_{ib}\cap\bd_i W$ being the marking.
+Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
+of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
+This defines a partial ordering $\cJ(W)$, which we will think of as a category.
+(The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique
+morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.)
+\nn{need figures}
+
+$\cN$ determines 
+a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets 
+(possibly with additional structure if $k=n$).
+For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset
+\[
+	\psi_\cN(x) \sub (\prod_a \cC(X_a)) \prod (\prod_{ib} \cN_i(M_{ib}))
+\]
+such that the restrictions to the various pieces of shared boundaries amongst the
+$X_a$ and $M_{ib}$ all agree.
+(Think fibered product.)
+If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$
+via the gluing (composition or action) maps from $\cC$ and the $\cN_i$.
+
+Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$.
+In other words, for each decomposition $x$ there is a map
+$\psi(x)\to \cC(W, \cN)$, these maps are compatible with the refinement maps
+above, and $\cC(W, \cN)$ is universal with respect to these properties.
+
+\nn{boundary restrictions, $k$-cat $\cC(\cdot\times W; N)$ etc.}
+
+\subsection{Tensor products}
 
 Next we consider tensor products (or, more generally, self tensor products
 or coends).
 
+\nn{maybe ``tensor product" is not the best name?}
+
 \nn{start with (less general) tensor products; maybe change this later}
 
+** \nn{stuff below needs to be rewritten (shortened), because of new subsections above}
+
 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$.
 (If $k=1$ and manifolds are oriented, then one should be 
 a left module and the other a right module.)