Binary file diagrams/tempkw/zo2.pdf has changed
Binary file diagrams/tempkw/zo3.pdf has changed
Binary file diagrams/tempkw/zo4.pdf has changed
--- a/text/appendixes/comparing_defs.tex Mon Jul 19 12:27:19 2010 -0700
+++ b/text/appendixes/comparing_defs.tex Mon Jul 19 15:38:35 2010 -0600
@@ -108,20 +108,20 @@
\subsection{Plain 2-categories}
\label{ssec:2-cats}
Let $\cC$ be a topological 2-category.
-We will construct a traditional pivotal 2-category.
+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,
though this will make the $n=2$ case a little more complicated than necessary.
-\nn{Note: We have to decide whether our 2-morphsism are shaped like rectangles or bigons.
+Before proceeding, we must decide whether the 2-morphisms of our
+pivotal 2-category are shaped like rectangles or bigons.
Each approach has advantages and disadvantages.
-For better or worse, we choose bigons here.}
-
-\nn{maybe we should do both rectangles and bigons?}
+For better or worse, we choose bigons here.
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.
+$k$-ball, which we also think of as the standard bihedron (a.k.a.\ globe).
+(For $k=1$ this is an interval, and for $k=2$ it is a bigon.)
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 $\cC(B^k)$
@@ -136,7 +136,6 @@
on $C^2$ (Figure \ref{fzo1}).
Isotopy invariance implies that this is associative.
We will define a ``horizontal" composition later.
-\nn{maybe no need to postpone?}
\begin{figure}[t]
\begin{equation*}
@@ -146,15 +145,20 @@
\label{fzo1}
\end{figure}
-Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary).
+Given $a\in C^1$, define $\id_a = a\times I \in C^2$ (pinched boundary).
Extended isotopy invariance for $\cC$ shows that this morphism is an identity for
vertical composition.
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.
+
+In showing that identity 1-morphisms have the desired properties, we will
+rely heavily on the extended isotopy invariance of 2-morphisms in $\cC$.
+This means we are free to add or delete product regions from 2-morphisms.
+
Let $a: y\to x$ be a 1-morphism.
-Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$
+Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$
as shown in Figure \ref{fzo2}.
\begin{figure}[t]
\begin{equation*}
@@ -163,10 +167,8 @@
\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 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)}
+As suggested by the figure, these are two different reparameterizations
+of a half-pinched version of $a\times I$.
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.
@@ -177,11 +179,7 @@
\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.
-Also: maybe half-pinched and unpinched products can be derived from fully pinched
-products after all (?)}
+In the first step we have inserted a copy of $(x\times I)\times I$.
Figure \ref{fzo4} shows the other case.
\begin{figure}[t]
\begin{equation*}
@@ -190,7 +188,7 @@
\caption{blah blah}
\label{fzo4}
\end{figure}
-We first collapse the red region, then remove a product morphism from the boundary,
+We identify a product region and remove it.
We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}.
It is not hard to show that this is independent of the arbitrary (left/right)
@@ -203,16 +201,8 @@
\label{fzo5}
\end{figure}
-\nn{need to find a list of axioms for pivotal 2-cats to check}
-
-\nn{...}
+%\nn{need to find a list of axioms for pivotal 2-cats to check}
-\medskip
-\hrule
-\medskip
-
-\nn{to be continued...}
-\medskip
\subsection{$A_\infty$ $1$-categories}
\label{sec:comparing-A-infty}