--- a/text/comparing_defs.tex	Thu Oct 15 23:29:45 2009 +0000
+++ b/text/comparing_defs.tex	Fri Oct 16 14:41:07 2009 +0000
@@ -18,7 +18,8 @@
 
 Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$.
 Define composition in $C$ to be the induced map $C^1\times C^1 \to C^1$ (defined only when range and domain agree).
-By isotopy invariance in $C$, any other choice of homeomorphism gives the same composition rule.
+By isotopy invariance in $\cC$, any other choice of homeomorphism gives the same composition rule.
+Also by isotopy invariance, composition is associative.
 
 Given $a\in C^0$, define $\id_a \deq a\times B^1$.
 By extended isotopy invariance in $\cC$, this has the expected properties of an identity morphism.
@@ -90,7 +91,7 @@
 (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.
+though this will make the $n=2$ case a little more complicated than 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.
@@ -101,6 +102,17 @@
 This allows us to define the domain and range of morphisms of $C$ using
 boundary and restriction maps of $\cC$.
 
+Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$.
+This is not associative, but we will see later that it is weakly associative.
+
+Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map on $C^2$.
+Isotopy invariance implies that this is associative.
+We will define a ``horizontal" composition later.
+
+
+
+
+
 \nn{...}
 
 \medskip