
diff -r 7a4fc5a873ac -r 6088d0b8611b pnas/pnas.tex
--- a/pnas/pnas.tex	Tue Nov 30 11:24:05 2010 -0800
+++ b/pnas/pnas.tex	Tue Nov 30 11:30:33 2010 -0800
@@ -270,7 +270,7 @@
 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs.
 
 We will define two variations simultaneously,  as all but one of the axioms are identical in the two cases.
-These variations are ``plain $n$-categories", where homeomorphisms fixing the boundary
+These variations are ``ordinary $n$-categories", where homeomorphisms fixing the boundary
 act trivially on the sets associated to $n$-balls
 (and these sets are usually vector spaces or more generally modules over a commutative ring)
 and ``$A_\infty$ $n$-categories",  where there is a homotopy action of
@@ -375,7 +375,7 @@
 If $k < n$,
 or if $k=n$ and we are in the $A_\infty$ case, 
 we require that $\gl_Y$ is injective.
-(For $k=n$ in the plain $n$-category case, see Axiom \ref{axiom:extended-isotopies}.)
+(For $k=n$ in the ordinary $n$-category case, see Axiom \ref{axiom:extended-isotopies}.)
 \end{axiom}
 
 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity}
@@ -461,7 +461,7 @@
 to the identity on the boundary.
 
 
-\begin{axiom}[\textup{\textbf{[for plain  $n$-categories]}} Extended isotopy invariance in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[for ordinary  $n$-categories]}} Extended isotopy invariance in dimension $n$.]
 \label{axiom:extended-isotopies}
 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
 to the identity on $\bd X$ and isotopic (rel boundary) to the identity.
@@ -585,7 +585,7 @@
 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$.
 
 The natural construction achieving this is a colimit along the poset of permissible decompositions.
-Given a plain $n$-category $\cC$, 
+Given an ordinary $n$-category $\cC$, 
 we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, 
 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. 
 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} 
@@ -622,7 +622,7 @@
 %the flexibility available in the construction of a homotopy colimit allows
 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$.
 %\todo{either need to explain why this is the same, or significantly rewrite this section}
-When $\cC$ is the plain $n$-category based on string diagrams for a traditional
+When $\cC$ is the ordinary $n$-category based on string diagrams for a traditional
 $n$-category $C$,
 one can show \cite{1009.5025} that the above two constructions of the homotopy colimit
 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$.
@@ -746,7 +746,7 @@
 
 \begin{thm}[Skein modules]
 \label{thm:skein-modules}
-Suppose $\cC$ is a plain $n$-category.
+Suppose $\cC$ is an ordinary $n$-category.
 The $0$-th blob homology of $X$ is the usual 
 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
 by $\cC$.
@@ -890,7 +890,7 @@
 \begin{thm}[Product formula]
 \label{thm:product}
 Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold.
-Let $\cC$ be a plain $n$-category.
+Let $\cC$ be an ordinary $n$-category.
 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above.
 Then
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