# HG changeset patch # User Kevin Walker # Date 1289778851 28800 # Node ID 357f8673564f052b9bd5e84ce00eba938fdd5916 # Parent ddf9c4daf210337a21814cef95868f1cecb9e875 misc diff -r ddf9c4daf210 -r 357f8673564f pnas/pnas.tex --- 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} %% PNAS does not support submission of supporting .tex files such as BibTeX.