
diff -r 6f0ad8c4f8e2 -r 455106e40a61 pnas/pnas.tex
--- a/pnas/pnas.tex	Sat Nov 13 20:58:23 2010 -0800
+++ b/pnas/pnas.tex	Sat Nov 13 20:58:40 2010 -0800
@@ -464,7 +464,7 @@
 Define product morphisms via product cell decompositions.
 
 
-\nn{also do bordism category?}
+\nn{also do bordism category}
 
 \subsection{The blob complex}
 \subsubsection{Decompositions of manifolds}
@@ -515,10 +515,11 @@
 We will use the term `field on $W$' to refer to a point of this functor,
 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.
 
-\todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?}
 
 \subsubsection{Homotopy colimits}
 \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?}
+\todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?}
+\nn{Explain codimension colimits here too}
 
 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$
 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$
@@ -551,13 +552,12 @@
 
 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of
 \begin{itemize}
-\item a permissible collection of $k$ embedded balls,
-\item an ordering of the balls, and \nn{what about reordering?}
+\item a permissible collection of $k$ embedded balls, and
 \item for each resulting piece of $W$, a field,
 \end{itemize}
 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. We call such a field a `null field on $B$'.
 
-The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with signs given by the ordering.
+The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs.
 
 We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. The differential simply forgets the ball. Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball.
 




