...
--- a/text/appendixes/comparing_defs.tex Wed Jan 27 19:34:48 2010 +0000
+++ b/text/appendixes/comparing_defs.tex Mon Feb 01 06:11:18 2010 +0000
@@ -18,7 +18,7 @@
Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$.
Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$ (defined only when range and domain agree).
By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule.
-Also by isotopy invariance, composition is associative on the nose.
+Also by isotopy invariance, composition is strictly associative.
Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$.
By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism.
@@ -126,7 +126,7 @@
Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$.
We will show that this 1-morphism is a weak identity.
This would be easier if our 2-morphisms were shaped like rectangles rather than bigons.
-Define let $a: y\to x$ be a 1-morphism.
+Let $a: y\to x$ be a 1-morphism.
Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$
as shown in Figure \ref{fzo2}.
\begin{figure}[t]
@@ -137,7 +137,7 @@
\label{fzo2}
\end{figure}
In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon,
-while the remained is a half-pinched version of $a\times I$.
+while the remainder is a half-pinched version of $a\times I$.
\nn{the red region is unnecessary; remove it? or does it help?
(because it's what you get if you bigonify the natural rectangular picture)}
We must show that the two compositions of these two maps give the identity 2-morphisms