# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1290617861 25200
# Node ID 0f45668726dd8d9e0259a546b5733b84d4cc3417
# Parent  6b6c565bd76eb0a87582e2259f6507f8b6d7db05
more string diagram / field nonsense

diff -r 6b6c565bd76e -r 0f45668726dd pnas/pnas.tex
--- a/pnas/pnas.tex	Wed Nov 24 09:51:28 2010 -0700
+++ b/pnas/pnas.tex	Wed Nov 24 09:57:41 2010 -0700
@@ -633,6 +633,9 @@
 it evaluates to a zero $n$-morphism of $C$.
 The next few paragraphs describe this in more detail.
 
+We will call a string diagram on a manifold a ``field".
+(See \cite{1009.5025} for a more general notion of field.)
+
 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible}
 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that
 each $B_i$ appears as a connected component of one of the $M_j$. 
@@ -651,10 +654,10 @@
 such that
 \begin{itemize}
 	\item there is a permissible decomposition of $W$, compatible with the $k$ blobs, such that
-	$s$ is the product of linear combinations of string diagrams $s_i$ on the initial pieces $X_i$ of the decomposition
+	$s$ is the product of linear combinations of fields $s_i$ on the initial pieces $X_i$ of the decomposition
 	(for fixed restrictions to the boundaries of the pieces),
 	\item the $s_i$'s corresponding to innermost blobs evaluate to zero in $\cC$, and
-	\item the $s_i$'s corresponding to the other pieces are single string diagrams (linear combinations with only one term).
+	\item the $s_i$'s corresponding to the other pieces are single fields (linear combinations with only one term).
 \end{itemize}
 %that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. 
 \nn{yech}
@@ -662,10 +665,8 @@
 
 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs.
 
-\nn{KW: I have not finished changng terminology from ``field" to ``string diagram"}
-
 We now spell this out for some small values of $k$. 
-For $k=0$, the $0$-blob group is simply linear combinations of string diagrams on $W$. 
+For $k=0$, the $0$-blob group is simply linear combinations of fields (string diagrams) 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.