
diff -r d2409e357801 -r dd4b4ac15023 text/ncat.tex
--- a/text/ncat.tex	Tue Jul 28 00:33:08 2009 +0000
+++ b/text/ncat.tex	Tue Jul 28 15:33:33 2009 +0000
@@ -564,7 +564,57 @@
 Next we consider tensor products (or, more generally, self tensor products
 or coends).
 
+\nn{start with (less general) tensor products; maybe change this later}
 
+Define a {\it doubly marked $k$-ball} to be a triple $(B, N, N')$, where $B$ is a $k$-ball
+and $N$ and $N'$ are disjoint $k{-}1$-balls in $\bd B$.
+
+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.)
+Let $D = (B, N, N')$ be a doubly marked $k$-ball, $1\le k \le n$.
+We will define a set $\cM\ot_\cC\cM'(D)$.
+(If $k = n$ and our $k$-categories are enriched, then
+$\cM\ot_\cC\cM'(D)$ will have additional structure; see below.)
+$\cM\ot_\cC\cM'(D)$ will be the colimit of a functor defined on a category $\cJ(D)$,
+which we define next.
+
+Define a permissible decomposition of $D$ to be a decomposition
+\[
+	D = (\cup_a X_a) \cup (\cup_b M_b) \cup (\cup_c M'_c) ,
+\]
+Where each $X_a$ is a plain $k$-ball (disjoint from the markings $N$ and $N'$ of $D$),
+each $M_b$ is a marked $k$-ball intersecting $N$, and
+each $M'_b$ is a marked $k$-ball intersecting $N'$.
+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 $\cJ(D)$, which we will think of as a category.
+(The objects of $\cJ(D)$ are permissible decompositions of $D$, and there is a unique
+morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.)
+\nn{need figures}
+
+$\cC$, $\cM$ and $\cM'$ determine 
+a functor $\psi$ from $\cJ(D)$ to the category of sets 
+(possibly with additional structure if $k=n$).
+For a decomposition $x = (X_a, M_b, M'_c)$ in $\cJ(D)$, define $\psi(x)$ to subset
+\[
+	\psi(x) \sub (\prod_a \cC(X_a)) \prod (\prod_b \cM(M_b)) \prod (\prod_c \cM'(M'_c))
+\]
+such that the restrictions to the various pieces of shared boundaries amongst the
+$X_a$, $M_b$ and $M'_c$ all agree.
+(Think fibered product.)
+If $x$ is a refinement of $y$, define a map $\psi(x)\to\psi(y)$
+via the gluing (composition or action) maps from $\cC$, $\cM$ and $\cM'$.
+
+Finally, define $\cM\ot_\cC\cM'(D)$ to be the colimit of $\psi$.
+In other words, for each decomposition $x$ there is a map
+$\psi(x)\to \cM\ot_\cC\cM'(D)$, these maps are compatible with the refinement maps
+above, and $\cM\ot_\cC\cM'(D)$ is universal with respect to these properties.
+
+Note that if $k=n$ and we fix boundary conditions $c$ on the unmarked boundary of $D$,
+then $\cM\ot_\cC\cM'(D; c)$ will be an object in the enriching category
+(e.g.\ vector space or chain complex).
+\nn{say this more precisely?}
 
 \medskip
 \hrule




