# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1289788383 28800
# Node ID 53aed9fdfcd9cc32808e0e45a6b6e378dc5108a6
# Parent  dda6d3a00b0960f4d7c59a1727d042a83f283884
proof of product thm

diff -r dda6d3a00b09 -r 53aed9fdfcd9 pnas/pnas.tex
--- a/pnas/pnas.tex	Sun Nov 14 17:28:04 2010 -0800
+++ b/pnas/pnas.tex	Sun Nov 14 18:33:03 2010 -0800
@@ -788,12 +788,24 @@
 (see \cite[\S7.1]{1009.5025}).
 
 \begin{proof} (Sketch.)
+The proof is similar to that of the second part of Theorem \ref{thm:gluing}.
+There is a natural map from the 0-simplices of $\clh{\bc_*(Y;\cC)}(W)$ to $\bc_*(Y\times W; \cC)$,
+given by reinterpreting a decomposition of $W$ labeled by $(n{-}k)$-morphisms of $\bc_*(Y; \cC)$ as a blob 
+diagram on $W\times Y$.
+This map can be extended to all of $\clh{\bc_*(Y;\cC)}(W)$ by sending higher simplices to zero.
 
+To construct the homotopy inverse of the above map one first shows that
+$\bc_*(Y\times W; \cC)$ is homotopy equivalent to the subcomplex generated by blob diagrams which
+are small with respect any fixed open cover of $Y\times W$.
+For a sufficiently fine open cover the generators of this ``small" blob complex are in the image of the map
+of the previous paragraph, and furthermore the preimage in $\clh{\bc_*(Y;\cC)}(W)$ of such small diagrams
+lie in contractible subcomplexes.
+A standard acyclic models argument now constructs the homotopy inverse.
 \end{proof}
 
 %\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.}
 
-\section{Higher Deligne conjecture}
+\section{Deligne conjecture for $n$-categories}
 \label{sec:applications}
 
 \begin{thm}[Higher dimensional Deligne conjecture]
@@ -818,7 +830,7 @@
 
 The little disks operad $LD$ is homotopy equivalent to 
 \nn{suboperad of}
-the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. 
+the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cochains $Hoch^*(C, C)$. 
 The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map
 \[
 	C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}}