Binary file pnas/PNAS-final.pdf has changed

--- a/text/intro.tex	Thu Jun 23 09:19:42 2011 -0700
+++ b/text/intro.tex	Thu Jun 23 09:19:51 2011 -0700
@@ -80,7 +80,7 @@
 
 In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category 
 of sphere modules.
-When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwinors.
+When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwiners.
 
 In \S \ref{ss:ncat_fields}  we explain how to construct a system of fields from a disk-like $n$-category 
 (using a colimit along certain decompositions of a manifold into balls). 
@@ -96,6 +96,9 @@
 
 The relationship between all these ideas is sketched in Figure \ref{fig:outline}.
 
+% NB: the following tikz requires a *more recent* version of PGF than is distributed with MacTex 2010.
+% grab the latest build from http://www.texample.net/tikz/builds/
+% unzip it in your personal tex tree, and run "mktexlsr ." there
 \tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt]
 
 \begin{figure}[t]

--- a/text/ncat.tex	Thu Jun 23 09:19:42 2011 -0700
+++ b/text/ncat.tex	Thu Jun 23 09:19:51 2011 -0700
@@ -489,7 +489,7 @@
 \end{scope}
 \end{tikzpicture}
 $$
-\caption{Five examples of unions of pinched products}\label{pinched_prod_unions}
+\caption{Six examples of unions of pinched products}\label{pinched_prod_unions}
 \end{figure}
 
 Note that $\bd X$ has a (possibly trivial) subdivision according to 
@@ -1507,7 +1507,7 @@
 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$;
 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and
 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, 
-such that $b_i$ and $b_{i+1}$both map to (glue up to) $a_i$.
+such that $b_i$ and $b_{i+1}$ both map to (glue up to) $a_i$.
 \end{itemize}
 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$.
 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same
@@ -2079,13 +2079,13 @@
 
 In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules".
 The objects are $n$-categories, the $k$-morphisms are $k{-}1$-sphere modules for $1\le k \le n$,
-and the $n{+}1$-morphisms are intertwinors.
+and the $n{+}1$-morphisms are intertwiners.
 With future applications in mind, we treat simultaneously the big category
 of all $n$-categories and all sphere modules and also subcategories thereof.
 When $n=1$ this is closely related to familiar $2$-categories consisting of 
 algebras, bimodules and intertwiners (or a subcategory of that).
 The sphere module $n{+}1$-category is a natural generalization of the 
-algebra-bimodule-intertwinor 2-category to higher dimensions.
+algebra-bimodule-intertwiner 2-category to higher dimensions.
 
 Another possible name for this $n{+}1$-category is $n{+}1$-category of defects.
 The $n$-categories are thought of as representing field theories, and the 
@@ -2693,7 +2693,7 @@
 
 We end this subsection with some remarks about Morita equivalence of disklike $n$-categories.
 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.
+objects in the 2-category of (linear) 1-categories, bimodules, and intertwiners.
 Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the
 $n{+}1$-category of sphere modules.
 
@@ -2717,20 +2717,52 @@
 cell complexes (cups and caps) in $B^2$ shown in Figure \ref{morita-fig-1}.
 \begin{figure}[t]
 $$\mathfig{.65}{tempkw/morita1}$$
+
+
+$$
+\begin{tikzpicture}
+\node(L) at (0,0) {\tikz{
+	\draw[orange] (0,0) -- node[below] {$\cC$} (1,0);
+	\draw[blue] (1,0) -- node[below] {$\cD$} (2,0);
+	\draw[orange] (2,0) -- node[below] {$\cC$} (3,0);
+	\node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (1,0) {};
+	\node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (2,0) {};
+}};
+
+\node(R) at (6,0) {\tikz{
+	\draw[orange] (0,0) -- node[below] {$\cC$} (3,0);
+	\node[label={\phantom{$\cM$}}] at (1.5,0) {};
+}};
+
+\node at (-1,-1.5) { $\leftidx{_\cC}{(\cM \tensor_\cD \cM)}{_\cC}$ };
+\node at (7,-1.5) { $\leftidx{_\cC}{\cC}{_\cC}$ };
+
+\draw[->] (L) to[out=35, in=145] node[below] {$w$} node[above] { \tikz{
+	\draw (0,0) circle (16pt);
+}}(R);
+
+\draw[->] (R) to[out=-145, in=-35] node[above] {$x$} node[below] { \tikz{
+	\draw (0,0) circle (16pt);
+}}(L);
+
+
+\end{tikzpicture}
+$$
+
 \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
+Recall that the 3-morphisms of $\cS$ are intertwiners between representations of 1-categories associated
 to decorated circles.
 Figure \ref{morita-fig-2} 
 \begin{figure}[t]
 $$\mathfig{.55}{tempkw/morita2}$$
-\caption{Intertwinors for a Morita equivalence}\label{morita-fig-2}
+\caption{intertwiners for a Morita equivalence}\label{morita-fig-2}
 \end{figure}
-shows the intertwinors we need.
+shows the intertwiners 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.
@@ -2743,15 +2775,15 @@
 These are illustrated in Figure \ref{morita-fig-3}.
 \begin{figure}[t]
 $$\mathfig{.65}{tempkw/morita3}$$
-\caption{Identities for intertwinors}\label{morita-fig-3}
+\caption{Identities for intertwiners}\label{morita-fig-3}
 \end{figure}
-Each line shows a composition of two intertwinors which we require to be equal to the identity intertwinor.
+Each line shows a composition of two intertwiners which we require to be equal to the identity intertwiner.
 
 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 $\cC$, $\cD$ 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 
+The $n{+}1$-dimensional part of the equivalence is given by certain intertwiners, and these intertwiners must 
 be invertible and satisfy
 identities corresponding to Morse cancellations in $n$-manifolds.