--- a/text/appendixes/comparing_defs.tex	Fri Jul 15 15:03:22 2011 -0700
+++ b/text/appendixes/comparing_defs.tex	Sat Jul 16 11:39:58 2011 -0600
@@ -126,7 +126,7 @@
 We will construct from $\cC$ 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,
+We will try to describe the construction in such a way that the generalization to $n>2$ is clear,
 though this will make the $n=2$ case a little more complicated than necessary.
 
 Before proceeding, we must decide whether the 2-morphisms of our
@@ -586,7 +586,7 @@
 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$).
 For simplicity we will now assume there is only one object and suppress it from the notation.
 
-A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\times A\to A$.
+A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\otimes A\to A$.
 We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic.
 Choose a specific 1-parameter family of homeomorphisms connecting them; this induces
 a degree 1 chain homotopy $m_3:A\ot A\ot A\to A$.