# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1317076434 21600
# Node ID 3f38383f26d3b97de21f004e1367d401319df5c5
# Parent  3bc9a91009520751de8e0b56433360ac41b51c84
straighten out 1-cat vs system of fields in Section 4, per referee

diff -r 3bc9a9100952 -r 3f38383f26d3 text/hochschild.tex
--- a/text/hochschild.tex	Sun Sep 25 22:35:24 2011 -0600
+++ b/text/hochschild.tex	Mon Sep 26 16:33:54 2011 -0600
@@ -8,18 +8,23 @@
 So far we have provided no evidence that blob homology is interesting in degrees 
 greater than zero.
 In this section we analyze the blob complex in dimension $n=1$.
-We find that $\bc_*(S^1, \cC)$ is homotopy equivalent to the 
-Hochschild complex of the 1-category $\cC$.
-(Recall from \S \ref{sec:example:traditional-n-categories(fields)} that a 
-$1$-category gives rise to a $1$-dimensional system of fields; as usual, 
-talking about the blob complex with coefficients in an $n$-category means 
-first passing to the corresponding $n$ dimensional system of fields.)
+
+Recall (\S \ref{sec:example:traditional-n-categories(fields)}) 
+that from a *-1-category $C$ we can construct a system of fields $\cC$.
+In this section we prove that $\bc_*(S^1, \cC)$ is homotopy equivalent to the 
+Hochschild complex of $C$.
+%(Recall from \S \ref{sec:example:traditional-n-categories(fields)} that a 
+%$1$-category gives rise to a $1$-dimensional system of fields; as usual, 
+%talking about the blob complex with coefficients in an $n$-category means 
+%first passing to the corresponding $n$ dimensional system of fields.)
 Thus the blob complex is a natural generalization of something already
 known to be interesting in higher homological degrees.
 
 It is also worth noting that the original idea for the blob complex came from trying
 to find a more ``local" description of the Hochschild complex.
 
+\medskip
+
 Let $C$ be a *-1-category.
 Then specializing the definition of the associated system of fields from \S \ref{sec:example:traditional-n-categories(fields)} above to the case $n=1$ we have:
 \begin{itemize}
@@ -53,7 +58,7 @@
 The fields have elements of $M_i$ labeling 
 the fixed points $p_i$ and elements of $C$ labeling other (variable) points.
 As before, the regions between the marked points are labeled by
-objects of $\cC$.
+objects of $C$.
 The blob twig labels lie in kernels of evaluation maps.
 (The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s,
 corresponding to the $p_i$'s that lie within the twig blob.)