
diff -r 1e50c1a5e8c0 -r 76f423a9c787 text/comparing_defs.tex
--- a/text/comparing_defs.tex	Tue Aug 18 19:27:44 2009 +0000
+++ b/text/comparing_defs.tex	Fri Aug 21 23:17:10 2009 +0000
@@ -78,10 +78,29 @@
 \nn{need better notation here}
 As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence.
 
+\medskip
+
+Similar arguments show that modules for topological 1-categories are essentially
+the same thing as traditional modules for traditional 1-categories.
 
 \subsection{Plain 2-categories}
 
-blah
+Let $\cC$ be a topological 2-category.
+We will construct a traditional pivotal 2-category.
+(The ``pivotal" corresponds to our assumption of strong duality for $\cC$.)
+
+We will try to describe the construction in such a way the the generalization to $n>2$ is clear,
+though this will make the $n=2$ case a little more complicated that necessary.
+
+Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard
+$k$-ball, which we also think of as the standard bihedron.
+Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$
+into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$.
+Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$
+whose boundary is splittable along $E$.
+This allows us to define the domain and range of morphisms of $C$ using
+boundary and restriction maps of $\cC$.
+
 \nn{...}
 
 \medskip




