# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1308766431 25200
# Node ID c2d1620c56dfd48cc7249aecb6d5cb964d5de6bc
# Parent  ff5483a2f789eea3a4d499280d26d0d5d254c372
morita figs

diff -r ff5483a2f789 -r c2d1620c56df text/ncat.tex
--- a/text/ncat.tex	Wed Jun 22 11:06:33 2011 -0700
+++ b/text/ncat.tex	Wed Jun 22 11:13:51 2011 -0700
@@ -2615,22 +2615,37 @@
 between ${}_\cC\cM_\cD \otimes_\cD {}_\cD\cM_\cC$ and the identity 0-sphere module ${}_\cC\cC_\cC$, and similarly
 with the roles of $\cC$ and $\cD$ reversed.
 These 2-morphisms come for free, in the sense of not requiring additional data, since we can take them to be the labeled 
-cell complexes (cups and caps) in $B^2$ shown in Figure \nn{need figure}.
+cell complexes (cups and caps) in $B^2$ shown in Figure \ref{morita-fig-1}.
+\begin{figure}[t]
+$$\mathfig{.65}{tempkw/morita1}$$
+\caption{Cups and caps for free}\label{morita-fig-1}
+\end{figure}
+
 
 We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms
 between various compositions of these 2-morphisms and various identity 2-morphisms.
 Recall that the 3-morphisms of $\cS$ are intertwinors between representations of 1-categories associated
 to decorated circles.
-Figure \nn{need Figure} shows the intertwinors we need.
+Figure \ref{morita-fig-2} 
+\begin{figure}[t]
+$$\mathfig{.55}{tempkw/morita2}$$
+\caption{Intertwinors for a Morita equivalence}\label{morita-fig-2}
+\end{figure}
+shows the intertwinors we need.
 Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle
 on the boundary.
 This is the 3-dimensional part of the data for the Morita equivalence.
-(Note that, by symmetry, the $c$ and $d$ arrows of Figure \ref{} are the same (up to rotation), as are the $h$ and $g$ arrows.)
+(Note that, by symmetry, the $c$ and $d$ arrows of Figure \ref{morita-fig-2} 
+are the same (up to rotation), as are the $h$ and $g$ arrows.)
 
 In order for these 3-morphisms to be equivalences, 
 they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition
 they must satisfy identities corresponding to Morse cancellations on 2-manifolds.
-These are illustrated in Figure \nn{need figure}.
+These are illustrated in Figure \ref{morita-fig-3}.
+\begin{figure}[t]
+$$\mathfig{.65}{tempkw/morita3}$$
+\caption{Identities for intertwinors}\label{morita-fig-3}
+\end{figure}
 Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor.
 
 For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional