--- a/pnas/pnas.tex Sun Nov 14 15:01:53 2010 -0800
+++ b/pnas/pnas.tex Sun Nov 14 15:54:11 2010 -0800
@@ -239,7 +239,11 @@
\nn{say something about defining plain and infty cases simultaneously}
-There are five basic ingredients of an $n$-category definition:
+There are five basic ingredients
+(not two, or four, or seven, but {\bf five} basic ingredients,
+which he shall wield all wretched sinners and that includes on you, sir, there in the front row!
+(cf.\ Monty Python, Life of Brian, http://www.youtube.com/watch?v=fIRb8TigJ28))
+of an $n$-category definition:
$k$-morphisms (for $0\le k \le n$), domain and range, composition,
identity morphisms, and special behavior in dimension $n$ (e.g. enrichment
in some auxiliary category, or strict associativity instead of weak associativity).
@@ -540,7 +544,7 @@
%\todo{either need to explain why this is the same, or significantly rewrite this section}
When $\cC$ is the topological $n$-category based on string diagrams for a traditional
$n$-category $C$,
-one can show \nn{cite us} that the above two constructions of the homotopy colimit
+one can show \cite{1009.5025} that the above two constructions of the homotopy colimit
are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$.
Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with
a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested.
@@ -718,31 +722,17 @@
Let $\cC$ be a topological $n$-category.
Let $Y$ be an $n{-}k$-manifold.
There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$,
-to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set
+to be the set $$\bc_*(Y;\cC)(D) = \cl{\cC}(Y \times D)$$ and on $k$-balls $D$ to be the set
$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$
(When $m=k$ the subsets with fixed boundary conditions form a chain complex.)
These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in
Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}.
\end{thm}
\begin{rem}
-When $Y$ is a point this gives $A_\infty$ $n$-category from a topological $n$-category, which can be thought of as a free resolution.
+When $Y$ is a point this produces an $A_\infty$ $n$-category from a topological $n$-category,
+which can be thought of as a free resolution.
\end{rem}
-This result is described in more detail as Example 6.2.8 of \cite{1009.5025}
-
-We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above.
-
-\begin{thm}[Product formula]
-\label{thm:product}
-Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.
-Let $\cC$ be an $n$-category.
-Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above.
-Then
-\[
- \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W).
-\]
-\end{thm}
-The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps
-(see \cite[\S7.1]{1009.5025}).
+This result is described in more detail as Example 6.2.8 of \cite{1009.5025}.
Fix a topological $n$-category $\cC$, which we'll now omit from notation.
Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category.
@@ -762,6 +752,22 @@
\end{itemize}
\end{thm}
+
+We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above.
+
+\begin{thm}[Product formula]
+\label{thm:product}
+Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.
+Let $\cC$ be an $n$-category.
+Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above.
+Then
+\[
+ \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W).
+\]
+\end{thm}
+The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps
+(see \cite[\S7.1]{1009.5025}).
+
\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.}
\section{Applications}
@@ -788,15 +794,23 @@
this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball.
\end{thm}
-An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another.
+An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}),
+modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism.
+Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another.
By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module.
\begin{proof}
-We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, and the action of surgeries is just composition of maps of $A_\infty$-modules. We only need to check that the relations of the $n$-SC operad are satisfied. This follows immediately from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity.
+We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation},
+and the action of surgeries is just composition of maps of $A_\infty$-modules.
+We only need to check that the relations of the $n$-SC operad are satisfied.
+This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity.
\end{proof}
-The little disks operad $LD$ is homotopy equivalent to 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 usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map
+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 usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map
\[
C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}
\to Hoch^*(C, C),
@@ -821,7 +835,15 @@
%% \appendix[Appendix Title]
\begin{acknowledgments}
-\nn{say something here}
+It is a pleasure to acknowledge helpful conversations with
+Kevin Costello,
+Mike Freedman,
+Justin Roberts,
+and
+Peter Teichner.
+\nn{not full list from big paper, but only most significant names}
+We also thank the Aspen Center for Physics for providing a pleasant and productive
+environment during the last stages of this project.
\end{acknowledgments}
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