
diff -r 29beaf2e4577 -r cd2ebc293e6b text/comparing_defs.tex
--- a/text/comparing_defs.tex	Fri Oct 16 22:44:25 2009 +0000
+++ b/text/comparing_defs.tex	Fri Oct 16 23:28:51 2009 +0000
@@ -114,6 +114,14 @@
 Isotopy invariance implies that this is associative.
 We will define a ``horizontal" composition later.
 
+\begin{figure}[t]
+\begin{equation*}
+\mathfig{.73}{tempkw/zo1}
+\end{equation*}
+\caption{Vertical composition of 2-morphisms}
+\label{fzo1}
+\end{figure}
+
 Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary).
 Extended isotopy invariance for $\cC$ shows that this morphism is an identity for 
 vertical composition.
@@ -124,17 +132,41 @@
 Define 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]
+\begin{equation*}
+\mathfig{.73}{tempkw/zo2}
+\end{equation*}
+\caption{blah blah}
+\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 $a\times I$.
+\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
 on $a$ and $a\bullet \id_x$, as defined above.
 Figure \ref{fzo3} shows one case.
+\begin{figure}[t]
+\begin{equation*}
+\mathfig{.83}{tempkw/zo3}
+\end{equation*}
+\caption{blah blah}
+\label{fzo3}
+\end{figure}
 In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}.
 \nn{also need to talk about (somewhere above) 
-how this sort of insertion is allowed by extended isotopy invariance and gluing}
+how this sort of insertion is allowed by extended isotopy invariance and gluing.
+Also: maybe half-pinched and unpinched products can be derived from fully pinched
+products after all (?)}
 Figure \ref{fzo4} shows the other case.
-\nn{At the moment, I don't see how the case follows from our candidate axioms for products.
-Probably the axioms need to be strengthened a little bit.}
+\begin{figure}[t]
+\begin{equation*}
+\mathfig{.83}{tempkw/zo4}
+\end{equation*}
+\caption{blah blah}
+\label{fzo4}
+\end{figure}
+We first collapse the red region, then remove a product morphism from the boundary,
 
 \nn{postponing horizontal composition of 2-morphisms until we make up our minds about product axioms.}
 




