# HG changeset patch
# User kevin@6e1638ff-ae45-0410-89bd-df963105f760
# Date 1267647472 0
# Node ID d13df7f3b2de1ce0bb97976cb0c61d6420f7baa5
# Parent  1b3ebb7793c9e821ba143ec1a60df0649e883979
...

diff -r 1b3ebb7793c9 -r d13df7f3b2de text/hochschild.tex
--- a/text/hochschild.tex	Tue Mar 02 21:52:01 2010 +0000
+++ b/text/hochschild.tex	Wed Mar 03 20:17:52 2010 +0000
@@ -5,19 +5,16 @@
 
 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 $n=1$
-and find that for $S^1$ the blob complex is homotopy equivalent to the 
-Hochschild complex of the category (algebroid) that we started with.
+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 $\cC$.
+\nn{cat vs fields --- need to make sure this is clear}
 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.
 
-\nn{need to be consistent about quasi-isomorphic versus homotopy equivalent
-in this section.
-since the various complexes are free, q.i. implies h.e.}
-
 Let $C$ be a *-1-category.
 Then specializing the definitions from above to the case $n=1$ we have:
 \begin{itemize}
diff -r 1b3ebb7793c9 -r d13df7f3b2de text/intro.tex
--- a/text/intro.tex	Tue Mar 02 21:52:01 2010 +0000
+++ b/text/intro.tex	Wed Mar 03 20:17:52 2010 +0000
@@ -5,7 +5,7 @@
 We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. This blob complex provides a simultaneous generalisation of several well-understood constructions:
 \begin{itemize}
 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. (See \S \ref{sec:fields} \nn{more specific}.)
-\item When $n=1$, $\cC$ is just an associative algebroid, and $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See \S \ref{sec:hochschild}.)
+\item When $n=1$, $\cC$ is just a 1-category (e.g.\ an associative algebra), and $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See \S \ref{sec:hochschild}.)
 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have 
 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains
 on the configurations space of unlabeled points in $M$.