--- a/pnas/pnas.tex Mon Nov 29 10:01:34 2010 -0700
+++ b/pnas/pnas.tex Tue Nov 30 11:07:24 2010 -0800
@@ -136,12 +136,10 @@
\begin{article}
\begin{abstract}
-\nn{needs revision}
-We explain the need for new axioms for topological quantum field theories that include ideas from derived
-categories and homotopy theory. We summarize our axioms for higher categories, and describe the ``blob complex".
+We summarize our axioms for higher categories, and describe the ``blob complex".
Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$.
-The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$.
-The higher homology groups should be viewed as generalizations of Hochschild homology.
+The $0$-th homology of this chain complex recovers the usual topological quantum field theory invariants of $W$.
+The higher homology groups should be viewed as generalizations of Hochschild homology (indeed, when $W=S^1$ they coincide).
The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold $W$.
We outline the important properties of the blob complex, and sketch the proof of a generalization of
Deligne's conjecture on Hochschild cohomology and the little discs operad to higher dimensions.
@@ -654,7 +652,7 @@
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 fields $s_i$ on the initial pieces $X_i$ of the decomposition
+ $s$ is the result of gluing together 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 fields (linear combinations with only one term).