
diff -r 7fc1a7ff9667 -r 6de8871d5786 pnas/pnas.tex
--- a/pnas/pnas.tex	Tue Nov 02 08:41:11 2010 +0900
+++ b/pnas/pnas.tex	Thu Nov 04 17:01:56 2010 +0900
@@ -499,14 +499,13 @@
 
 \begin{property}[Contractibility]
 \label{property:contractibility}%
-With field coefficients, the blob complex on an $n$-ball is contractible in the sense 
-that it is homotopic to its $0$-th homology.
-Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces 
-associated by the system of fields $\cF$ to balls.
+The blob complex on an $n$-ball is contractible in the sense 
+that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category.
 \begin{equation*}
-\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)}
+\xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)}
 \end{equation*}
 \end{property}
+\nn{maybe should say something about the $A_\infty$ case}
 
 \begin{proof}(Sketch)
 For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram
@@ -515,7 +514,6 @@
 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$.
 \end{proof}
 
-
 \subsection{Specializations}
 \label{sec:specializations}
 
@@ -523,13 +521,15 @@
 
 \begin{thm}[Skein modules]
 \label{thm:skein-modules}
+\nn{Plain n-categories only?}
 The $0$-th blob homology of $X$ is the usual 
 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
-by $\cF$.
+by $\cC$.
 \begin{equation*}
-H_0(\bc_*(X;\cF)) \iso A_{\cF}(X)
+H_0(\bc_*(X;\cC)) \iso A_{\cC}(X)
 \end{equation*}
 \end{thm}
+This follows from the fact that the $0$-th homology of a homotopy colimit of \todo{something plain} is the usual colimit, or directly from the explicit description of the blob complex.
 
 \begin{thm}[Hochschild homology when $X=S^1$]
 \label{thm:hochschild}




