author | Kevin Walker <kevin@canyon23.net> |
Fri, 06 May 2011 14:11:43 -0700 | |
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%!TEX root = ../blob1.tex |
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\section{Introduction} |
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We construct a chain complex $\bc_*(M; \cC)$ --- the ``blob complex'' --- |
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associated to an $n$-manifold $M$ and a linear $n$-category $\cC$ with strong duality. |
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This blob complex provides a simultaneous generalization of several well known constructions: |
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\begin{itemize} |
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\item The 0-th homology $H_0(\bc_*(M; \cC))$ is isomorphic to the usual |
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topological quantum field theory invariant of $M$ associated to $\cC$. |
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(See Proposition \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
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\item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
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the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
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(See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) |
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%\item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category, we have |
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%that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
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%on the configuration space of unlabeled points in $M$. (See \S \ref{sec:comm_alg}.) |
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\item When $\cC$ is $\pi^\infty_{\leq n}(T)$, the $A_\infty$ version of the fundamental $n$-groupoid of |
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the space $T$ (Example \ref{ex:chains-of-maps-to-a-space}), |
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$\bc_*(M; \cC)$ is homotopy equivalent to $C_*(\Maps(M\to T))$, |
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the singular chains on the space of maps from $M$ to $T$. |
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(See Theorem \ref{thm:map-recon}.) |
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\end{itemize} |
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The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space |
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(replacing the quotient of fields by local relations with some sort of resolution), |
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and for a generalization of Hochschild homology to higher $n$-categories. |
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One can think of it as the push-out of these two familiar constructions. |
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More detailed motivations are described in \S \ref{sec:motivations}. |
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The blob complex has good formal properties, summarized in \S \ref{sec:properties}. |
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These include an action of $\CH{M}$, |
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extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (Theorem \ref{thm:evaluation}) and a gluing |
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formula allowing calculations by cutting manifolds into smaller parts (Theorem \ref{thm:gluing}). |
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We expect applications of the blob complex to contact topology and Khovanov homology |
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but do not address these in this paper. |
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%See \S \ref{sec:future} for slightly more detail. |
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Throughout, we have resisted the temptation to work in the greatest possible generality. |
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(Don't worry, it wasn't that hard.) |
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In most of the places where we say ``set" or ``vector space", any symmetric monoidal category |
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with sufficient limits and colimits would do. |
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We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories). |
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\subsection{Structure of the paper} |
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The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
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summarize the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), and outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}). |
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%and outline anticipated future directions (see \S \ref{sec:future}). |
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%\nn{recheck this list after done editing intro} |
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The first part of the paper (sections \S \ref{sec:fields}--\S \ref{sec:evaluation}) gives the definition of the blob complex, |
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and establishes some of its properties. |
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There are many alternative definitions of $n$-categories, and part of the challenge of defining the blob complex is |
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simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
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At first we entirely avoid this problem by introducing the notion of a ``system of fields", and define the blob complex |
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associated to an $n$-manifold and an $n$-dimensional system of fields. |
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We sketch the construction of a system of fields from a *-$1$-category and from a pivotal $2$-category. |
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Nevertheless, when we attempt to establish all of the observed properties of the blob complex, |
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we find this situation unsatisfactory. |
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Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another |
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definition of an $n$-category, or rather a definition of an $n$-category with strong duality. |
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(Removing the duality conditions from our definition would make it more complicated rather than less.) |
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We call these ``disk-like $n$-categories'', to differentiate them from previous versions. |
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Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. |
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(See \S \ref{n-cat-names} below for a discussion of $n$-category terminology.) |
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The basic idea is that each potential definition of an $n$-category makes a choice about the ``shape" of morphisms. |
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We try to be as lax as possible: a disk-like $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. |
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These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. |
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For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of |
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homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
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The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
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disk-like $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
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In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category |
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of sphere modules. |
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When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwinors. |
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In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a disk-like $n$-category |
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(using a colimit along certain decompositions of a manifold into balls). |
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With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ |
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with the system of fields constructed from the $n$-category $\cC$. |
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%\nn{KW: I don't think we use this notational convention any more, right?} |
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In \S \ref{sec:ainfblob} we give an alternative definition |
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of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
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Using these definitions, we show how to use the blob complex to ``resolve" any ordinary $n$-category as an |
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$A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
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We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
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in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. |
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The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
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\tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt] |
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\begin{figure}[!ht] |
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{\center |
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\beginpgfgraphicnamed{gadgets-external}% |
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\begin{tikzpicture}[align=center,line width = 1.5pt] |
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\newcommand{\xxa}{2} |
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\newcommand{\xxb}{8} |
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\newcommand{\yya}{14} |
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\newcommand{\yyb}{10} |
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\newcommand{\yyc}{6} |
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\node[box] at (-4,\yyb) (tC) {$C$ \\ a `traditional' \\ weak $n$-category}; |
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\node[box] at (\xxa,\yya) (C) {$\cC$ \\ a disk-like \\ $n$-category}; |
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\node[box] at (\xxb,\yya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space}; |
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\node[box] at (\xxa,\yyb) (FU) {$(\cF, U)$ \\ fields and\\ local relations}; |
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\node[box] at (\xxb,\yyb) (BC) {$\bc_*(M; \cF)$ \\ the blob complex}; |
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\node[box] at (\xxa,\yyc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
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\node[box] at (\xxb,\yyc) (BCs) {$\underrightarrow{\cC_*}(M)$}; |
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\draw[->] (C) -- node[above] {$\displaystyle \colim_{\cell(M)} \cC$} node[below] {\S\S \ref{sec:constructing-a-tqft} \& \ref{ss:ncat_fields}} (A); |
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\draw[->] (FU) -- node[below] {blob complex \\ for $M$} (BC); |
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\draw[->] (Cs) -- node[above] {$\displaystyle \hocolim_{\cell(M)} \cC_*$} node[below] {\S \ref{ss:ncat_fields}} (BCs); |
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\draw[->] (FU) -- node[right=10pt] {$\cF(M)/U$} (A); |
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\draw[->] (tC) -- node[above] {Example \ref{ex:traditional-n-categories(fields)}} (FU); |
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\draw[->] (C.-100) -- node[left] { |
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\S \ref{ss:ncat_fields} |
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%$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle U(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
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} (FU.100); |
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\draw[->] (C) -- node[above left=3pt] {restrict to \\ standard balls} (tC); |
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\draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); |
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\draw[->] (BC) -- node[right] {$H_0$ \\ c.f. Proposition \ref{thm:skein-modules}} (A); |
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\draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
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\draw[<->] (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
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\end{tikzpicture} |
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\endpgfgraphicnamed% |
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\mbox{} % <-- weird, doesn't compile unless I put something here after the \endpgfgraphicnamed...? -S |
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} |
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\caption{The main gadgets and constructions of the paper.} |
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\label{fig:outline} |
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\end{figure} |
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Later sections address other topics. |
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Section \S \ref{sec:deligne} gives |
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a higher dimensional generalization of the Deligne conjecture |
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(that the little discs operad acts on Hochschild cochains) in terms of the blob complex. |
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The appendices prove technical results about $\CH{M}$ and |
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make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
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as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
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%Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
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%thought of as a disk-like $n$-category, in terms of the topology of $M$. |
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|
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%%%% this is said later in the intro |
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%Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) |
|
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%even when we could work in greater generality (symmetric monoidal categories, model categories, etc.). |
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%\item ? one of the points we make (far) below is that there is not really much |
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%difference between (a) systems of fields and local relations and (b) $n$-cats; |
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%thus we tend to switch between talking in terms of one or the other |
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\subsection{Motivations} |
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\label{sec:motivations} |
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We will briefly sketch our original motivation for defining the blob complex. |
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As a starting point, consider TQFTs constructed via fields and local relations. |
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(See \S\ref{sec:tqftsviafields} or \cite{kw:tqft}.) |
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This gives a satisfactory treatment for semisimple TQFTs |
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(i.e.\ TQFTs for which the cylinder 1-category associated to an |
|
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$n{-}1$-manifold $Y$ is semisimple for all $Y$). |
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|
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For non-semi-simple TQFTs, this approach is less satisfactory. |
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Our main motivating example (though we will not develop it in this paper) |
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is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
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It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
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with a link $L \subset \bd W$. |
|
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The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
|
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%\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S} |
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|
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How would we go about computing $A_{Kh}(W^4, L)$? |
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For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence) |
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relating resolutions of a crossing. |
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Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
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to compute $A_{Kh}(S^1\times B^3, L)$. |
|
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According to the gluing theorem for TQFTs, gluing along $B^3 \subset \bd B^4$ |
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corresponds to taking a coend (self tensor product) over the cylinder category |
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associated to $B^3$ (with appropriate boundary conditions). |
|
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The coend is not an exact functor, so the exactness of the triangle breaks. |
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The obvious solution to this problem is to replace the coend with its derived counterpart, |
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Hochschild homology. |
|
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This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology |
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of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
|
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If we build our manifold up via a handle decomposition, the computation |
|
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would be a sequence of derived coends. |
|
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A different handle decomposition of the same manifold would yield a different |
|
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sequence of derived coends. |
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To show that our definition in terms of derived coends is well-defined, we |
|
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would need to show that the above two sequences of derived coends yield |
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isomorphic answers, and that the isomorphism does not depend on any |
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choices we made along the way. |
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This is probably not easy to do. |
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Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
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which is manifestly invariant. |
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(That is, a definition that does not |
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involve choosing a decomposition of $W$. |
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After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
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is that it has just this sort of manifest invariance.) |
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The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
|
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\[ |
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\text{linear combinations of fields} \;\big/\; \text{local relations} , |
|
217 |
\] |
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with an appropriately free resolution (the blob complex) |
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\[ |
220 |
\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
|
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\] |
|
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Here $\bc_0$ is linear combinations of fields on $W$, |
|
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$\bc_1$ is linear combinations of local relations on $W$, |
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$\bc_2$ is linear combinations of relations amongst relations on $W$, |
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and so on. We now have a short exact sequence of chain complexes relating resolutions of the link $L$ (c.f. Lemma \ref{lem:hochschild-exact} which shows exactness with respect to boundary conditions in the context of Hochschild homology). |
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\subsection{Formal properties} |
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\label{sec:properties} |
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The blob complex enjoys the following list of formal properties. |
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\begin{property}[Functoriality] |
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\label{property:functoriality}% |
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The blob complex is functorial with respect to homeomorphisms. |
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That is, |
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for a fixed $n$-dimensional system of fields $\cF$, the association |
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\begin{equation*} |
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X \mapsto \bc_*(X; \cF) |
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\end{equation*} |
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is a functor from $n$-manifolds and homeomorphisms between them to chain |
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complexes and isomorphisms between them. |
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\end{property} |
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As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$; |
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this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below. |
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The blob complex is also functorial with respect to $\cF$, |
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although we will not address this in detail here. |
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\begin{property}[Disjoint union] |
250 |
\label{property:disjoint-union} |
|
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The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
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\begin{equation*} |
253 |
\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
|
254 |
\end{equation*} |
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\end{property} |
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||
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If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
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write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
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Note that this includes the case of gluing two disjoint manifolds together. |
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\begin{property}[Gluing map] |
98 | 261 |
\label{property:gluing-map}% |
136 | 262 |
%If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
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%\begin{equation*} |
|
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%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
|
265 |
%\end{equation*} |
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Given a gluing $X \to X_\mathrm{gl}$, there is |
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a natural map |
268 |
\[ |
|
437 | 269 |
\bc_*(X) \to \bc_*(X_\mathrm{gl}) |
131 | 270 |
\] |
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(natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
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\end{property} |
273 |
||
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\begin{property}[Contractibility] |
|
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\label{property:contractibility}% |
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With field coefficients, the blob complex on an $n$-ball is contractible in the sense |
277 |
that it is homotopic to its $0$-th homology. |
|
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Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces |
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associated by the system of fields $\cF$ to balls. |
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\begin{equation*} |
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\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)} |
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\end{equation*} |
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\end{property} |
284 |
||
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Property \ref{property:functoriality} will be immediate from the definition given in |
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\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
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Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and |
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\ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
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\subsection{Specializations} |
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\label{sec:specializations} |
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The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
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\newtheorem*{thm:skein-modules}{Proposition \ref{thm:skein-modules}} |
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|
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\begin{thm:skein-modules}[Skein modules] |
|
98 | 298 |
The $0$-th blob homology of $X$ is the usual |
299 |
(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
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by $\cF$. |
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(See \S \ref{sec:local-relations}.) |
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\begin{equation*} |
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H_0(\bc_*(X;\cF)) \iso A_{\cF}(X) |
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\end{equation*} |
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\end{thm:skein-modules} |
98 | 306 |
|
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\newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}} |
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\begin{thm:hochschild}[Hochschild homology when $X=S^1$] |
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The blob complex for a $1$-category $\cC$ on the circle is |
311 |
quasi-isomorphic to the Hochschild complex. |
|
312 |
\begin{equation*} |
|
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\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
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\end{equation*} |
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\end{thm:hochschild} |
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Proposition \ref{thm:skein-modules} is immediate from the definition, and |
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Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
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%We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of |
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%certain commutative algebras thought of as $n$-categories. |
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\subsection{Structure of the blob complex} |
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\label{sec:structure} |
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In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
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|
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\newtheorem*{thm:CH}{Theorem \ref{thm:CH}} |
|
329 |
||
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\begin{thm:CH}[$C_*(\Homeo(-))$ action] |
437 | 331 |
There is a chain map |
98 | 332 |
\begin{equation*} |
482 | 333 |
e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
98 | 334 |
\end{equation*} |
437 | 335 |
such that |
336 |
\begin{enumerate} |
|
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\item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
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313 | 339 |
\item For |
147 | 340 |
any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
313 | 341 |
(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
147 | 342 |
\begin{equation*} |
343 |
\xymatrix@C+2cm{ |
|
166 | 344 |
\CH{X} \otimes \bc_*(X) |
482 | 345 |
\ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
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\bc_*(X) \ar[d]_{\gl_Y} \\ |
482 | 347 |
\CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
147 | 348 |
} |
349 |
\end{equation*} |
|
437 | 350 |
\end{enumerate} |
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%Moreover any such chain map is unique, up to an iterated homotopy. |
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%(That is, any pair of homotopies have a homotopy between them, and so on.) |
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%\nn{revisit this after proof below has stabilized} |
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\end{thm:CH} |
355 |
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\newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}} |
|
357 |
||
358 |
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359 |
Further, |
|
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\begin{thm:CH-associativity} |
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The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy). |
313 | 362 |
\begin{equation*} |
363 |
\xymatrix{ |
|
482 | 364 |
\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\ |
365 |
\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
|
313 | 366 |
} |
367 |
\end{equation*} |
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\end{thm:CH-associativity} |
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313 | 370 |
Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
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$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
313 | 372 |
for any homeomorphic pair $X$ and $Y$, |
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satisfying corresponding conditions. |
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In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, from which we can construct systems of fields. |
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Below, when we talk about the blob complex for a disk-like $n$-category, we are implicitly passing first to this associated system of fields. |
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Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. In that section we describe how to use the blob complex to construct $A_\infty$ $n$-categories from ordinary $n$-categories: |
484 | 378 |
|
536 | 379 |
\newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}} |
151 | 380 |
|
536 | 381 |
\begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
382 |
%\label{thm:blobs-ainfty} |
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Let $\cC$ be an ordinary $n$-category. |
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Let $Y$ be an $n{-}k$-manifold. |
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There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
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to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
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$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
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(When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
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These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
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Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
536 | 391 |
\end{ex:blob-complexes-of-balls} |
151 | 392 |
\begin{rem} |
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Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from an ordinary $n$-category. |
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We think of this $A_\infty$ $n$-category as a free resolution. |
151 | 395 |
\end{rem} |
484 | 396 |
|
136 | 397 |
There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
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398 |
instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}. |
536 | 399 |
The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The next theorem describes the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example. |
484 | 400 |
%The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
136 | 401 |
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\newtheorem*{thm:product}{Theorem \ref{thm:product}} |
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403 |
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\begin{thm:product}[Product formula] |
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Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
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Let $\cC$ be an $n$-category. |
536 | 407 |
Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}). |
136 | 408 |
Then |
409 |
\[ |
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\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
136 | 411 |
\] |
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\end{thm:product} |
484 | 413 |
The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
526 | 414 |
(see \S \ref{ss:product-formula}). |
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415 |
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Fix a disk-like $n$-category $\cC$, which we'll omit from the notation. |
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Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
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(See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
98 | 419 |
|
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\newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
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421 |
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\begin{thm:gluing}[Gluing formula] |
98 | 423 |
\mbox{}% <-- gets the indenting right |
424 |
\begin{itemize} |
|
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\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
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$A_\infty$ module for $\bc_*(Y)$. |
98 | 427 |
|
437 | 428 |
\item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ is the $A_\infty$ self-tensor product of |
429 |
$\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
|
98 | 430 |
\begin{equation*} |
437 | 431 |
\bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
98 | 432 |
\end{equation*} |
433 |
\end{itemize} |
|
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434 |
\end{thm:gluing} |
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435 |
|
484 | 436 |
Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}. |
98 | 437 |
|
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438 |
\subsection{Applications} |
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439 |
\label{sec:applications} |
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440 |
Finally, we give two applications of the above machinery. |
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441 |
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442 |
\newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}} |
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443 |
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444 |
\begin{thm:map-recon}[Mapping spaces] |
187 | 445 |
Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
446 |
$B^n \to T$. |
|
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447 |
(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
136 | 448 |
Then |
484 | 449 |
$$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
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450 |
\end{thm:map-recon} |
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|
484 | 452 |
This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
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453 |
Note that there is no restriction on the connectivity of $T$. |
484 | 454 |
The proof appears in \S \ref{sec:map-recon}. |
187 | 455 |
|
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456 |
\newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}} |
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457 |
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458 |
\begin{thm:deligne}[Higher dimensional Deligne conjecture] |
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459 |
The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
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Since the little $n{+}1$-balls operad is a suboperad of the $n$-dimensional surgery cylinder operad, |
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461 |
this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
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462 |
\end{thm:deligne} |
484 | 463 |
See \S \ref{sec:deligne} for a full explanation of the statement, and the proof. |
98 | 464 |
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465 |
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466 |
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467 |
\noop{ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
148 | 468 |
\subsection{Future directions} |
151 | 469 |
\label{sec:future} |
484 | 470 |
\nn{KW: Perhaps we should delete this subsection and salvage only the first few sentences.} |
471 |
Throughout, we have resisted the temptation to work in the greatest generality possible. |
|
472 |
(Don't worry, it wasn't that hard.) |
|
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473 |
In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
484 | 474 |
We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories). |
475 |
%%%%%% |
|
476 |
And likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories. |
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477 |
More could be said about finite characteristic |
437 | 478 |
(there appears in be $2$-torsion in $\bc_1(S^2; \cC)$ for any spherical $2$-category $\cC$, for example). |
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479 |
Much more could be said about other types of manifolds, in particular oriented, |
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480 |
$\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. |
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481 |
(We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) |
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482 |
We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; |
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483 |
there may be some differences for topological manifolds and smooth manifolds. |
148 | 484 |
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485 |
The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be |
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486 |
interesting to investigate if there is a connection with the material here. |
314 | 487 |
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488 |
Many results in Hochschild homology can be understood ``topologically" via the blob complex. |
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489 |
For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
437 | 490 |
(see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1]; A)$, |
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491 |
but haven't investigated the details. |
148 | 492 |
|
437 | 493 |
Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} \nn{stable categories, generalized cohomology theories} |
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494 |
} %%% end \noop %%%%%%%%%%%%%%%%%%%%% |
148 | 495 |
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\subsection{\texorpdfstring{$n$}{n}-category terminology} |
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497 |
\label{n-cat-names} |
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Section \S \ref{sec:ncats} adds to the zoo of $n$-category definitions, and the new creatures need names. |
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Unfortunately, we have found it difficult to come up with terminology which satisfies all |
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of the colleagues whom we have consulted, or even satisfies just ourselves. |
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One distinction we need to make is between $n$-categories which are associative in dimension $n$ and those |
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that are associative only up to higher homotopies. |
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The latter are closely related to $(\infty, n)$-categories (i.e.\ $\infty$-categories where all morphisms |
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of dimension greater than $n$ are invertible), but we don't want to use that name |
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since we think of the higher homotopies not as morphisms of the $n$-category but |
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rather as belonging to some auxiliary category (like chain complexes) |
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that we are enriching in. |
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We have decided to call them ``$A_\infty$ $n$-categories", since they are a natural generalization |
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of the familiar $A_\infty$ 1-categories. |
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We also considered the names ``homotopy $n$-categories" and ``infinity $n$-categories". |
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When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$ in this sense |
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we will say ``ordinary $n$-category". |
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% small problem: our n-cats are of course strictly associative, since we have more morphisms. |
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% when we say ``associative only up to homotopy" above we are thinking about |
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% what would happen we we tried to convert to a more traditional n-cat with fewer morphisms |
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Another distinction we need to make is between our style of definition of $n$-categories and |
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more traditional and combinatorial definitions. |
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We will call instances of our definition ``disk-like $n$-categories", since $n$-dimensional disks |
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play a prominent role in the definition. |
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(In general we prefer ``$k$-ball" to ``$k$-disk", but ``ball-like" doesn't roll off |
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the tongue as well as ``disk-like''.) |
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Another thing we need a name for is the ability to rotate morphisms around in various ways. |
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For 2-categories, ``pivotal" is a standard term for what we mean. |
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A more general term is ``duality", but duality comes in various flavors and degrees. |
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We are mainly interested in a very strong version of duality, where the available ways of |
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rotating $k$-morphisms correspond to all the ways of rotating $k$-balls. |
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We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied |
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by ``disk-like". |
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(But beware: disks can come in various flavors, and some of them, such as framed disks, |
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don't actually imply much duality.) |
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Another possibility considered here was ``pivotal $n$-category", but we prefer to preserve pivotal for its usual sense. It will thus be a theorem that our disk-like 2-categories are equivalent to pivotal 2-categories, c.f. \S \ref{ssec:2-cats}. |
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Finally, we need a general name for isomorphisms between balls, where the balls could be |
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piecewise linear or smooth or topological or Spin or framed or etc., or some combination thereof. |
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We have chosen to use ``homeomorphism" for the appropriate sort of isomorphism, so the reader should |
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keep in mind that ``homeomorphism" could mean PL homeomorphism or diffeomorphism (and so on) |
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depending on context. |
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\subsection{Thanks and acknowledgements} |
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% attempting to make this chronological rather than alphabetical |
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We'd like to thank |
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Justin Roberts, |
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Michael Freedman, |
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Peter Teichner, |
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David Ben-Zvi, |
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Vaughan Jones, |
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Chris Schommer-Pries, |
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Thomas Tradler, |
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Kevin Costello, |
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Chris Douglas, |
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Alexander Kirillov, |
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and |
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Michael Shulman |
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for many interesting and useful conversations. |
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Peter Teichner ran a reading course based on an earlier draft of this paper, and the detailed feedback |
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we got from the student lecturers lead to very many improvements in later drafts. |
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So big thanks to |
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Aaron Mazel-Gee, |
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Nate Watson, |
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Alan Wilder, |
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Dmitri Pavlov, |
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Ansgar Schneider, |
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and |
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Dan Berwick-Evans. |
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569 |
\nn{need to double-check this list once the reading course is over} |
522 | 570 |
During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. We'd like to thank the Aspen Center for Physics for the pleasant and productive |
571 |
environment provided there during the final preparation of this manuscript. |
|
155 | 572 |