
diff -r 0c681fbb7b85 -r ff5483a2f789 text/ncat.tex
--- a/text/ncat.tex	Tue Jun 21 18:10:31 2011 -0700
+++ b/text/ncat.tex	Wed Jun 22 11:06:33 2011 -0700
@@ -2593,26 +2593,27 @@
 \medskip
 
 We end this subsection with some remarks about Morita equivalence of disklike $n$-categories.
-Recall that two 1-categories $C$ and $D$ are Morita equivalent if and only if they are equivalent
+Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent
 objects in the 2-category of (linear) 1-categories, bimodules, and intertwinors.
 Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the
 $n{+}1$-category of sphere modules.
 
 Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in 
-dimensions 1 and $n+1$ (the middle dimensions come along for free), and this data must satisfy
-identities corresponding to Morse cancellations in $n{+}1$-manifolds.
+dimensions 1 and $n+1$ (the middle dimensions come along for free).
+The $n{+}1$-dimensional part of the data must be invertible and satisfy
+identities corresponding to Morse cancellations in $n$-manifolds.
 We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar.
 
-Let $C$ and $D$ be (unoriented) disklike 2-categories.
+Let $\cC$ and $\cD$ be (unoriented) disklike 2-categories.
 Let $\cS$ denote the 3-category of 2-category sphere modules.
-The 1-dimensional part of the data for a Morita equivalence between $C$ and $D$ is a 0-sphere module $M = {}_CM_D$ 
-(categorified bimodule) connecting $C$ and $D$.
+The 1-dimensional part of the data for a Morita equivalence between $\cC$ and $\cD$ is a 0-sphere module $\cM = {}_\cC\cM_\cD$ 
+(categorified bimodule) connecting $\cC$ and $\cD$.
 Because of the full unoriented symmetry, this can also be thought of as a 
-0-sphere module ${}_DM_C$ connecting $D$ and $C$.
+0-sphere module ${}_\cD\cM_\cC$ connecting $\cD$ and $\cC$.
 
-We want $M$ to be an equivalence, so we need 2-morphisms in $\cS$ 
-between ${}_CM_D \otimes_D {}_DM_C$ and the identity 0-sphere module ${}_CC_C$, and similarly
-with the roles of $C$ and $D$ reversed.
+We want $\cM$ to be an equivalence, so we need 2-morphisms in $\cS$ 
+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}.
 
@@ -2624,34 +2625,46 @@
 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.
-\nn{?? note that, by symmetry, the x and y arrows of Fig xxxx are the same (up to rotation), as are the z and w arrows}
+(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.)
 
-In order for these 3-morphisms to be equivalences, they must satisfy identities corresponding to Morse cancellations
-on 3-manifolds.
+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}.
 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 $M$ which is the data for the 1-dimensional
+For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional
 part of the Morita equivalence.
 For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds
-labeled by $C$, $D$ and $M$; no additional data is needed for these parts.
-The $n{+}1$-dimensional part of the equivalence is given by certain intertwinors, and these intertwinors must satisfy
-identities corresponding to Morse cancellations in $n{+}1$-manifolds. 
+labeled by $\cC$, $\cD$ and $\cM$; no additional data is needed for these parts.
+The $n{+}1$-dimensional part of the equivalence is given by certain intertwinors, and these intertwinors must 
+be invertible and satisfy
+identities corresponding to Morse cancellations in $n$-manifolds. 
 
+\noop{
+One way of thinking of these conditions is as follows.
+Given a decorated $n{+}1$-manifold, with a codimension 1 submanifold labeled by $\cM$ and 
+codimension 0 submanifolds labeled by $\cC$ and $\cD$, we can make any local modification we like without 
+changing
+}
 
+If $\cC$ and $\cD$ are Morita equivalent $n$-categories, then it is easy to show that for any $n-j$-manifold
+$Y$ the $j$-categories $\cC(Y)$ and $\cD(Y)$ are Morita equivalent.
+When $j=0$ this means that the TQFT Hilbert spaces $\cC(Y)$ and $\cD(Y)$ are isomorphic 
+(if we are enriching over vector spaces).
 
 
 
 
 \noop{ % the following doesn't work; need 2^(k+1) different N's, not 2*(k+1)
-More specifically, the 1-dimensional part of the data is a 0-sphere module $M = {}_CM_D$ 
-(categorified bimodule) connecting $C$ and $D$.
-From $M$ we can construct various $k$-sphere modules $N^k_{j,E}$ for $0 \le k \le n$, $0\le j \le k$, and $E = C$ or $D$.
+More specifically, the 1-dimensional part of the data is a 0-sphere module $\cM = {}_\cCM_\cD$ 
+(categorified bimodule) connecting $\cC$ and $\cD$.
+From $\cM$ we can construct various $k$-sphere modules $N^k_{j,E}$ for $0 \le k \le n$, $0\le j \le k$, and $E = \cC$ or $\cD$.
 $N^k_{j,E}$ can be thought of as the graph of an index $j$ Morse function on the $k$-ball $B^k$
 (so the graph lives in $B^k\times I = B^{k+1}$).
 The positive side of the graph is labeled by $E$, the negative side by $E'$
-(where $C' = D$ and $D' = C$), and the codimension-1 
-submanifold separating the positive and negative regions is labeled by $M$.
+(where $\cC' = \cD$ and $\cD' = \cC$), and the codimension-1 
+submanifold separating the positive and negative regions is labeled by $\cM$.
 We think of $N^k_{j,E}$ as a $k{+}1$-morphism connecting 
 We plan on treating this in more detail in a future paper.
 \nn{should add a few more details}




