# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1288857726 -32400
# Node ID 294c6b2ab72323709b7370404edf1e0ed9609de0
# Parent  38ec3d05d0d88afe49e77442c5e9723489966c8b# Parent  6de8871d578654c2e984770439160ce751af41fe
Automated merge with https://tqft.net/hg/blob/

diff -r 38ec3d05d0d8 -r 294c6b2ab723 pnas/pnas.tex
--- 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}
diff -r 38ec3d05d0d8 -r 294c6b2ab723 text/intro.tex
--- 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}.