--- a/pnas/pnas.tex Tue Nov 02 06:38:40 2010 -0700
+++ b/pnas/pnas.tex Thu Nov 04 17:02:06 2010 +0900
@@ -508,14 +508,13 @@
\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
@@ -524,7 +523,6 @@
$x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$.
\end{proof}
-
\subsection{Specializations}
\label{sec:specializations}
@@ -532,13 +530,15 @@
\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}
--- a/text/intro.tex Tue Nov 02 06:38:40 2010 -0700
+++ b/text/intro.tex Thu Nov 04 17:02:06 2010 +0900
@@ -277,7 +277,7 @@
\end{equation*}
\end{property}
-Properties \ref{property:functoriality} will be immediate from the definition given in
+Property \ref{property:functoriality} will be immediate from the definition given in
\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there.
Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and
\ref{property:contractibility} are established in \S \ref{sec:basic-properties}.