--- a/pnas/pnas.tex	Wed Nov 17 11:46:39 2010 -0800
+++ b/pnas/pnas.tex	Wed Nov 17 11:58:35 2010 -0800
@@ -246,21 +246,27 @@
 \section{Definitions}
 \subsection{$n$-categories} \mbox{}
 
-\nn{rough draft of n-cat stuff...}
+In this section we give a definition of $n$-categories designed to work well with TQFTs.
+The main idea is to base the definition on actual balls, rather combinatorial models of them.
+This has the advantages of avoiding a proliferation of coherency axioms and building in a strong
+version of duality from the start.
 
-\nn{maybe say something about goals: well-suited to TQFTs; avoid proliferation of coherency axioms;
-non-recursive (n-cats not defined n terms of (n-1)-cats; easy to show that the motivating
-examples satisfy the axioms; strong duality; both plain and infty case;
-(?) easy to see that axioms are correct, in the sense of nothing missing (need
-to say this better if we keep it)}
 
-\nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties
-which are weak enough to include the basic examples and strong enough to support the proofs
-of the main theorems; and (2) specify a minimal set of generators and/or axioms.
-We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second.
-More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.}
+%\nn{maybe say something about goals: well-suited to TQFTs; avoid proliferation of coherency axioms;
+%non-recursive (n-cats not defined n terms of (n-1)-cats; easy to show that the motivating
+%examples satisfy the axioms; strong duality; both plain and infty case;
+%(?) easy to see that axioms are correct, in the sense of nothing missing (need
+%to say this better if we keep it)}
+%
+%\nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties
+%which are weak enough to include the basic examples and strong enough to support the proofs
+%of the main theorems; and (2) specify a minimal set of generators and/or axioms.
+%We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second.
+%More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.}
 
-\nn{say something about defining plain and infty cases simultaneously}
+We will define plain and $A_\infty$ $n$-categories simultaneously, as all but one of the axioms are identical
+in the two cases.
+
 
 There are five basic ingredients 
 \cite{life-of-brian} of an $n$-category definition: