# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1323280978 28800
# Node ID cd26c49d673cc58e948eed31fef965b1df4fa272
# Parent  bfae3636133ee59dbd81b8d3b77e64cb3175069a
C -> D in C.2

diff -r bfae3636133e -r cd26c49d673c text/appendixes/comparing_defs.tex
--- a/text/appendixes/comparing_defs.tex	Fri Dec 02 22:09:48 2011 -0800
+++ b/text/appendixes/comparing_defs.tex	Wed Dec 07 10:02:58 2011 -0800
@@ -123,7 +123,7 @@
 \subsection{Pivotal 2-categories}
 \label{ssec:2-cats}
 Let $\cC$ be a disk-like 2-category.
-We will construct from $\cC$ a traditional pivotal 2-category.
+We will construct from $\cC$ a traditional pivotal 2-category $D$.
 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.)
 
 We will try to describe the construction in such a way that the generalization to $n>2$ is clear,
@@ -134,21 +134,21 @@
 Each approach has advantages and disadvantages.
 For better or worse, we choose bigons here.
 
-Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k) \trans E$, where $B^k$ denotes the standard
+Define the $k$-morphisms $D^k$ of $D$ to be $\cC(B^k) \trans E$, where $B^k$ denotes the standard
 $k$-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 $\cC(B^k) \trans E$ means that we consider the subset of $\cC(B^k)$
 whose boundary is splittable along $E$.
-This allows us to define the domain and range of morphisms of $C$ using
+This allows us to define the domain and range of morphisms of $D$ using
 boundary and restriction maps of $\cC$.
 
-Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$.
+Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $D^1$.
 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 $C^2$ (Figure \ref{fzo1}).
+on $D^2$ (Figure \ref{fzo1}).
 Isotopy invariance implies that this is associative.
 We will define a ``horizontal" composition later.
 
@@ -203,7 +203,7 @@
 \label{fzo1}
 \end{figure}
 
-Given $a\in C^1$, define $\id_a = a\times I \in C^2$ (pinched boundary).
+Given $a\in D^1$, define $\id_a = a\times I \in D^2$ (pinched boundary).
 Extended isotopy invariance for $\cC$ shows that this morphism is an identity for 
 vertical composition.