
diff -r d5cbbc87c340 -r 2a8aecc675c7 text/ncat.tex
--- a/text/ncat.tex	Sun Feb 21 06:40:00 2010 +0000
+++ b/text/ncat.tex	Sun Feb 21 22:49:18 2010 +0000
@@ -1202,7 +1202,7 @@
 
 It should now be clear how to define $n$-category $m$-sphere modules for $0\le m \le n-1$.
 For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere,
-and an $m$-sphere module is a representation of such an $n{-}2$-category.
+and a 2-sphere module is a representation of such an $n{-}2$-category.
 
 \medskip
 
@@ -1216,6 +1216,31 @@
 $L_0$ could contain infinitely many $n$-categories or just one.
 For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or 
 it could contain several.
+The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category
+constructed out of labels taken from $L_j$ for $j<k$.
+
+We now define $\cS(X)$, for $X$ of dimension at most $n$, to be the set of all 
+cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled
+by elements of $L_j$.
+As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module
+for the $n{-}k{+}1$-category associated to its decorated boundary.
+Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought 
+of as $n$-category $k{-}1$-sphere modules 
+(generalizations of bimodules).
+On the other hand, we can equally think of the $k$-morphisms as decorations on $k$-balls, 
+and from this (official) point of view it is clear that they satisfy all of the axioms of an
+$n{+}1$-category.
+(All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.)
+
+\medskip
+
+Next we define the $n{+}1$-morphisms of $\cS$.
+
+
+
+
+
+
 
 \nn{...}
 
@@ -1227,6 +1252,10 @@
 \medskip
 
 
+
+
+
+
 Stuff that remains to be done (either below or in an appendix or in a separate section or in
 a separate paper):
 \begin{itemize}




