# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1294701953 28800
# Node ID 0cbef0258d724f0fd05ad3aab6dbaca7faca8bc2
# Parent  b8f7de7a42066714bec39b808a024f1bebee2759
minor changes to kevin's new paragraphs

diff -r b8f7de7a4206 -r 0cbef0258d72 text/intro.tex
--- a/text/intro.tex	Mon Jan 10 14:18:52 2011 -0800
+++ b/text/intro.tex	Mon Jan 10 15:25:53 2011 -0800
@@ -509,8 +509,8 @@
 that we are enriching in.
 We have decided to call them ``$A_\infty$ $n$-categories", since they are a natural generalization 
 of the familiar $A_\infty$ 1-categories.
-Other possible names include ``homotopy $n$-categories" and ``infinity $n$-categories".
-When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$
+We also considered the names ``homotopy $n$-categories" and ``infinity $n$-categories".
+When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$ in this sense
 we will say ``ordinary $n$-category".
 % small problem: our n-cats are of course strictly associative, since we have more morphisms.
 % when we say ``associative only up to homotopy" above we are thinking about
@@ -520,8 +520,8 @@
 more traditional and combinatorial definitions.
 We will call instances of our definition ``disk-like $n$-categories", since $n$-dimensional disks
 play a prominent role in the definition.
-(In general we prefer to ``$k$-ball" to ``$k$-disk", but ``ball-like" doesn't roll off 
-the tongue as well as "disk-like".)
+(In general we prefer ``$k$-ball" to ``$k$-disk", but ``ball-like" doesn't roll off 
+the tongue as well as ``disk-like''.)
 
 Another thing we need a name for is the ability to rotate morphisms around in various ways.
 For 2-categories, ``pivotal" is a standard term for what we mean.
@@ -530,9 +530,9 @@
 rotating $k$-morphisms correspond to all the ways of rotating $k$-balls.
 We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied
 by ``disk-like".
-(But beware: disks can come in various flavors, and some of them (such as framed disks)
+(But beware: disks can come in various flavors, and some of them, such as framed disks,
 don't actually imply much duality.)
-Another possibility here is ``pivotal $n$-category".
+Another possibility considered here was ``pivotal $n$-category", but we prefer to preserve pivotal for its usual sense. It will thus be a theorem that our disk-like 2-categories are equivalent to pivotal 2-categories, c.f. \S \ref{ssec:2-cats}.
 
 Finally, we need a general name for isomorphisms between balls, where the balls could be
 piecewise linear or smooth or topological or Spin or framed or etc., or some combination thereof.
diff -r b8f7de7a4206 -r 0cbef0258d72 text/ncat.tex
--- a/text/ncat.tex	Mon Jan 10 14:18:52 2011 -0800
+++ b/text/ncat.tex	Mon Jan 10 15:25:53 2011 -0800
@@ -678,12 +678,12 @@
 
 An $n$-category consists of the following data:
 \begin{itemize}
-\item Functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms})
-\item Boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary})
-\item Composition/gluing maps $\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B_1\cup_Y B_2)_E$ (Axiom \ref{axiom:composition})
-\item Product/identity maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product})
-\item If enriching in an auxiliary category, additional structure on $\cC_n(X; c)$
-\item In the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families})
+\item Functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}).
+\item Boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}).
+\item Composition/gluing maps $\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B_1\cup_Y B_2)_E$ (Axiom \ref{axiom:composition}).
+\item Product/identity maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}).
+\item If enriching in an auxiliary category, additional structure on $\cC_n(X; c)$.
+\item In the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}).
 \end{itemize}
 The above data must satisfy the following conditions:
 \begin{itemize}