blob: comparison pnas/pnas.tex

equal deleted inserted replaced

-:ddf9c4daf210
+:357f8673564f
 We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second.
 More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.}
 \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).
 We will treat each of these in turn.
 %the flexibility available in the construction of a homotopy colimit allows
 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$.
 %\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.
 The restriction of the string diagram to innermost blobs is required to be ``null" in the sense that
 it evaluates to a zero $n$-morphism of $C$.
 \begin{thm}
 \label{thm:blobs-ainfty}
 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}
+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.
+\begin{thm}[Gluing formula]
+\label{thm:gluing}
+\mbox{}% <-- gets the indenting right
+\begin{itemize}
+\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an
+$A_\infty$ module for $\bc_*(Y)$.
+\item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of
+$\bc_*(X)$ as an $\bc_*(Y)$-bimodule:
+\begin{equation*}
+\bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow
+\end{equation*}
+\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}
 	\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}).
-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.
-\begin{thm}[Gluing formula]
-\label{thm:gluing}
-\mbox{}% <-- gets the indenting right
-\begin{itemize}
-\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an
-$A_\infty$ module for $\bc_*(Y)$.
-\item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of
-$\bc_*(X)$ as an $\bc_*(Y)$-bimodule:
-\begin{equation*}
-\bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow
-\end{equation*}
-\end{itemize}
-\end{thm}
 \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}
 \label{sec:applications}
 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains.
 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad,
 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
 \[
 	C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}
 			\to  Hoch^*(C, C),
 \]
 which we now see to be a specialization of Theorem \ref{thm:deligne}.
 %% \appendix Appendix text...
 %% or, for appendix with title, use square brackets:
 %% \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}
 %% PNAS does not support submission of supporting .tex files such as BibTeX.
 %% Instead all references must be included in the article .tex document.
 %% If you currently use BibTeX, your bibliography is formed because the

changeset 610	357f8673564f
parent 609	ddf9c4daf210
child 613	5700634d8442