author | Scott Morrison <scott@tqft.net> |
Thu, 31 Mar 2011 14:13:58 -0700 | |
changeset 739 | da7ac7d30f30 |
parent 735 | bd0d5b2155a7 |
child 740 | 1708a3f23612 |
permissions | -rw-r--r-- |
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%% PNAStmpl.tex |
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%% Template file to use for PNAS articles prepared in LaTeX |
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%% Version: Apr 14, 2008 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%% BASIC CLASS FILE |
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%% PNAStwo for two column articles is called by default. |
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%% Uncomment PNASone for single column articles. One column class |
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%% and style files are available upon request from pnas@nas.edu. |
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%% (uncomment means get rid of the '%' in front of the command) |
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%\documentclass{pnasone} |
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\documentclass{pnastwo} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%% Changing position of text on physical page: |
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%% Since not all printers position |
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%% the printed page in the same place on the physical page, |
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%% you can change the position yourself here, if you need to: |
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% \advance\voffset -.5in % Minus dimension will raise the printed page on the |
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% physical page; positive dimension will lower it. |
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%% You may set the dimension to the size that you need. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%% OPTIONAL GRAPHICS STYLE FILE |
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%% Requires graphics style file (graphicx.sty), used for inserting |
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%% .eps files into LaTeX articles. |
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%% Note that inclusion of .eps files is for your reference only; |
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%% when submitting to PNAS please submit figures separately. |
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%% Type into the square brackets the name of the driver program |
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%% that you are using. If you don't know, try dvips, which is the |
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%% most common PC driver, or textures for the Mac. These are the options: |
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% [dvips], [xdvi], [dvipdf], [dvipdfm], [dvipdfmx], [pdftex], [dvipsone], |
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% [dviwindo], [emtex], [dviwin], [pctexps], [pctexwin], [pctexhp], [pctex32], |
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% [truetex], [tcidvi], [vtex], [oztex], [textures], [xetex] |
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%\usepackage[dvips]{graphicx} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%% OPTIONAL POSTSCRIPT FONT FILES |
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%% PostScript font files: You may need to edit the PNASoneF.sty |
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%% or PNAStwoF.sty file to make the font names match those on your system. |
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%% Alternatively, you can leave the font style file commands commented out |
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%% and typeset your article using the default Computer Modern |
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%% fonts (recommended). If accepted, your article will be typeset |
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%% at PNAS using PostScript fonts. |
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% Choose PNASoneF for one column; PNAStwoF for two column: |
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%\usepackage{PNASoneF} |
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%\usepackage{PNAStwoF} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%% ADDITIONAL OPTIONAL STYLE FILES |
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%% The AMS math files are commonly used to gain access to useful features |
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%% like extended math fonts and math commands. |
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\usepackage{amssymb,amsfonts,amsmath,amsthm} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%% OPTIONAL MACRO FILES |
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%% Insert self-defined macros here. |
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%% \newcommand definitions are recommended; \def definitions are supported |
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%\newcommand{\mfrac}[2]{\frac{\displaystyle #1}{\displaystyle #2}} |
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%\def\s{\sigma} |
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\def\bc{{\mathcal B}} |
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\def\btc{{\mathcal{BT}}} |
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\newcommand{\HC}{\operatorname{Hoch}} |
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\newcommand{\HH}{\operatorname{HH}} |
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\newcommand{\CM}[2]{C_*(\Maps(#1 \to #2))} |
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\newcommand{\CD}[1]{C_*(\Diff(#1))} |
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\newcommand{\CH}[1]{CH_*(#1)} |
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\newcommand{\cl}[1]{\underrightarrow{#1}} |
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\newcommand{\clh}[1]{\underrightarrow{#1}_{{}_{{}_{{}_h}}}} |
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\newcommand{\Set}{\text{\textbf{Set}}} |
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\newcommand{\Vect}{\text{\textbf{Vect}}} |
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\newcommand{\Kom}{\text{\textbf{Kom}}} |
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\newcommand{\Cat}{\mathcal{C}} |
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\newcommand{\cell}{\mathfrak{D}} |
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\newcommand{\into}{\hookrightarrow} |
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\newcommand{\onto}{\twoheadrightarrow} |
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\newcommand{\iso}{\cong} |
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\newcommand{\quism}{\underset{\text{q.i.}}{\simeq}} |
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\newcommand{\htpy}{\simeq} |
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\newcommand{\actsOn}{\circlearrowright} |
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\newcommand{\xto}[1]{\xrightarrow{#1}} |
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\newcommand{\isoto}{\xto{\iso}} |
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\newcommand{\quismto}{\xrightarrow[\text{q.i.}]{\iso}} |
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\newcommand{\diffeoto}{\xrightarrow[\text{diffeo}]{\iso}} |
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\newcommand{\htpyto}{\xrightarrow[\text{htpy}]{\htpy}} |
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\newcommand{\directSum}{\oplus} |
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\newcommand{\DirectSum}{\bigoplus} |
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\newcommand{\tensor}{\otimes} |
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\newcommand{\Tensor}{\bigotimes} |
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\newcommand{\Bord}{\operatorname{Bord}} |
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\newcommand{\selfarrow}{\ensuremath{\smash{\tikz[baseline]{\clip (0,0.36) rectangle (0.39,-0.16); \draw[->] (0,0.2) .. controls (0.5,0.6) and (0.5,-0.4) .. (0,0);}}}} |
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\newcommand{\bdy}{\partial} |
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\newcommand{\compose}{\circ} |
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\newcommand{\eset}{\emptyset} |
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\newcommand{\id}{\boldsymbol{1}} |
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% kw_macros |
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%!TEX root = ../blob1.tex |
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%%%%% excerpts from KW's include file of standard macros |
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%%% (with various new ones added) |
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\def\z{\mathbb{Z}} |
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\def\r{\mathbb{R}} |
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\def\c{\mathbb{C}} |
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\def\t{\mathbb{T}} |
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\def\ebb{\mathbb{E}} |
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\def\k{{\bf k}} |
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\def\du{\sqcup} |
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\def\bd{\partial} |
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\def\sub{\subset} |
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\def\subeq{\subseteq} |
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\def\sup{\supset} |
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%\def\setmin{\smallsetminus} |
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\def\setmin{\setminus} |
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\def\ep{\epsilon} |
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\def\sgl{_\mathrm{gl}} |
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\def\op{^\mathrm{op}} |
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\def\deq{\stackrel{\mathrm{def}}{=}} |
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\def\pd#1#2{\frac{\partial #1}{\partial #2}} |
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%\def\lf{\overline{\cC}} |
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\def\lf{\cC} |
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\def\ot{\otimes} |
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\def\vphi{\varphi} |
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\def\inv{^{-1}} |
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\def\ol{\overline} |
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\def\BD{BD} |
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\def\spl{_\pitchfork} |
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% equations |
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\newcommand{\eq}[1]{\begin{displaymath}#1\end{displaymath}} |
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\newcommand{\eqar}[1]{\begin{eqnarray*}#1\end{eqnarray*}} |
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\newcommand{\eqspl}[1]{\begin{displaymath}\begin{split}#1\end{split}\end{displaymath}} |
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\DeclareMathOperator*{\colim}{colim} |
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\DeclareMathOperator*{\hocolim}{hocolim} |
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\DeclareMathOperator{\most}{most} |
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\DeclareMathOperator{\rest}{rest} |
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\DeclareMathOperator{\kone}{cone} |
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% new theorems |
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\newtheorem{prop}{Proposition} |
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\newtheorem{property}[prop]{Property} |
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\newtheorem{thm}[prop]{Theorem} |
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\newtheorem{lem}[prop]{Lemma} |
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\newtheorem{defn}[prop]{Definition} |
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\newtheorem{axiom}[prop]{Axiom} |
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\newenvironment{rem}{\noindent\textsl{Remark.}}{} |
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% \mathrlap -- a horizontal \smash-------------------------------- |
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% For comparison, the existing overlap macros: |
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% \def\llap#1{\hbox to 0pt{\hss#1}} |
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% \def\rlap#1{\hbox to 0pt{#1\hss}} |
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\def\clap#1{\hbox to 0pt{\hss#1\hss}} |
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\def\mathllap{\mathpalette\mathllapinternal} |
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\def\mathrlap{\mathpalette\mathrlapinternal} |
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\def\mathclap{\mathpalette\mathclapinternal} |
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\def\mathllapinternal#1#2{% |
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\llap{$\mathsurround=0pt#1{#2}$}} |
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\def\mathrlapinternal#1#2{% |
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\rlap{$\mathsurround=0pt#1{#2}$}} |
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\def\mathclapinternal#1#2{% |
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\clap{$\mathsurround=0pt#1{#2}$}} |
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% tricky way to iterate macros over a list |
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\def\semicolon{;} |
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\def\applytolist#1{ |
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\expandafter\def\csname multi#1\endcsname##1{ |
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\def\multiack{##1}\ifx\multiack\semicolon |
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\def\next{\relax} |
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\else |
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\csname #1\endcsname{##1} |
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\def\next{\csname multi#1\endcsname} |
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\fi |
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\next} |
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\csname multi#1\endcsname} |
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% \def\cA{{\cal A}} for A..Z |
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\def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}} |
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\applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM; |
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% \def\bbA{{\mathbb A}} for A..Z |
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\def\bbc#1{\expandafter\def\csname bb#1\endcsname{{\mathbb #1}}} |
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\applytolist{bbc}QWERTYUIOPLKJHGFDSAZXCVBNM; |
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% \def\bA{{\mathbf A}} for A..Z |
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\def\bfc#1{\expandafter\def\csname bf#1\endcsname{{\mathbf #1}}} |
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\applytolist{bfc}QWERTYUIOPLKJHGFDSAZXCVBNM; |
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% \DeclareMathOperator{\pr}{pr} etc. |
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\def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} |
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\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}; |
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% \todo, \nn and \noop |
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\newcommand{\todo}[1]{\textbf{\color[rgb]{.8,.2,.5}\small TODO: #1}} |
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\def\nn#1{{{\color[rgb]{.2,.5,.6} \small [[#1]]}}} |
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\long\def\noop#1{} |
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% references |
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\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\tt arXiv:\nolinkurl{#1}}} |
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\newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{{\tt DOI:#1}}} |
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\newcommand{\euclid}[1]{\href{http://projecteuclid.org/euclid.cmp/#1}{{\tt at Project Euclid: #1}}} |
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\newcommand{\mathscinet}[1]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\tt #1}} |
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\newcommand{\googlebooks}[1]{(preview at \href{http://books.google.com/books?id=#1}{google books})} |
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% figures |
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\newcommand{\mathfig}[2]{\ensuremath{\hspace{-3pt}\begin{array}{c}% |
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\raisebox{-2.5pt}{\includegraphics[width=#1\textwidth]{#2}}% |
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\end{array}\hspace{-3pt}}} |
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% packages |
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\usepackage{tikz} |
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%\usetikzlibrary{shapes} |
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%\usetikzlibrary{backgrounds} |
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%\usetikzlibrary{decorations,decorations.pathreplacing} |
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%\usetikzlibrary{fit,calc,through} |
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\usepackage[all,color]{xy} |
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\SelectTips{cm}{} |
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\usepackage[pdftex,plainpages=false,hypertexnames=false,pdfpagelabels]{hyperref} |
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\usepackage{xcolor} |
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\definecolor{dark-red}{rgb}{0.7,0.25,0.25} |
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\definecolor{dark-blue}{rgb}{0.15,0.15,0.55} |
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\definecolor{medium-blue}{rgb}{0,0,0.65} |
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\hypersetup{ |
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colorlinks, linkcolor={dark-red}, |
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citecolor={dark-blue}, urlcolor={medium-blue} |
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} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%% Don't type in anything in the following section: |
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%%%%%%%%%%%% |
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%% For PNAS Only: |
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\contributor{Submitted to Proceedings |
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of the National Academy of Sciences of the United States of America} |
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%\url{www.pnas.org/cgi/doi/10.1073/pnas.0709640104} |
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\copyrightyear{2008} |
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\issuedate{Issue Date} |
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\volume{Volume} |
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\issuenumber{Issue Number} |
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%%%%%%%%%%%% |
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\begin{document} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%% For titles, only capitalize the first letter |
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%% \title{Almost sharp fronts for the surface quasi-geostrophic equation} |
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\title{Higher categories, colimits, and the blob complex} |
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%% Enter authors via the \author command. |
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%% Use \affil to define affiliations. |
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%% (Leave no spaces between author name and \affil command) |
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%% Note that the \thanks{} command has been disabled in favor of |
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%% a generic, reserved space for PNAS publication footnotes. |
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%% \author{<author name> |
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%% \affil{<number>}{<Institution>}} One number for each institution. |
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%% The same number should be used for authors that |
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%% are affiliated with the same institution, after the first time |
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%% only the number is needed, ie, \affil{number}{text}, \affil{number}{} |
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%% Then, before last author ... |
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%% \and |
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%% \author{<author name> |
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%% \affil{<number>}{}} |
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%% For example, assuming Garcia and Sonnery are both affiliated with |
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%% Universidad de Murcia: |
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%% \author{Roberta Graff\affil{1}{University of Cambridge, Cambridge, |
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%% United Kingdom}, |
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%% Javier de Ruiz Garcia\affil{2}{Universidad de Murcia, Bioquimica y Biologia |
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%% Molecular, Murcia, Spain}, \and Franklin Sonnery\affil{2}{}} |
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\author{Scott Morrison\affil{1}{Miller Institute for Basic Research, UC Berkeley, CA 94704, USA} |
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\and Kevin Walker\affil{2}{Microsoft Station Q, 2243 CNSI Building, UC Santa Barbara, CA 93106, USA}} |
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\contributor{Submitted to Proceedings of the National Academy of Sciences |
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of the United States of America} |
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\begin{article} |
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\begin{abstract} |
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We summarize our axioms for higher categories, and describe the ``blob complex". |
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Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. |
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The $0$-th homology of this chain complex recovers the usual topological quantum field theory invariants of $W$. |
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The higher homology groups should be viewed as generalizations of Hochschild homology (indeed, when $W=S^1$ they coincide). |
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The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold $W$. |
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We outline the important properties of the blob complex, and sketch the proof of a generalization of |
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Deligne's conjecture on Hochschild cohomology and the little discs operad to higher dimensions. |
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\end{abstract} |
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\keywords{n-categories | topological quantum field theory | hochschild homology} |
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% \abbreviations{TQFT, topological quantum field theory} |
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\dropcap{T}he aim of this paper is to describe a derived category analogue of topological quantum field theories. |
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For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of |
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invariants of manifolds of dimensions 0 through $n{+}1$. In particular, |
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the TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category. |
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If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford |
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a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$. |
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(See \cite{1009.5025} and \cite{kw:tqft}; |
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for a more homotopy-theoretic point of view see \cite{0905.0465}.) |
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We now comment on some particular values of $k$ above. |
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A linear 0-category is a vector space, and a representation |
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of a vector space is an element of the dual space. |
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Thus a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$, |
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and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
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For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional |
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TQFTs, which are slightly weaker structures in that they assign |
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invariants to mapping cylinders of homeomorphisms between $n$-manifolds, but not to general $(n{+}1)$-manifolds. |
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When $k=n{-}1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
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and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
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The TQFT gluing rule in dimension $n$ states that |
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$A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
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where $Y_1$ and $Y_2$ are $n$-manifolds with common boundary $S$. |
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When $k=0$ we have an $n$-category $A(pt)$. |
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This can be thought of as the local part of the TQFT, and the full TQFT can be reconstructed from $A(pt)$ |
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via colimits (see below). |
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We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$ |
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and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$. |
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Examples of semisimple TQFTs include Witten-Reshetikhin-Turaev theories, |
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Turaev-Viro theories, and Dijkgraaf-Witten theories. |
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These can all be given satisfactory accounts in the framework outlined above. |
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(The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories with only a weak |
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dependence on interiors in order to be |
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extended all the way down to dimension 0.) |
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For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate. |
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For example, the gluing rule for 3-manifolds in Ozsv\'ath-Szab\'o/Seiberg-Witten theory |
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involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}. |
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Long exact sequences are important computational tools in these theories, |
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and also in Khovanov homology, but the colimit construction breaks exactness. |
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For these reasons and others, it is desirable to |
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extend the above framework to incorporate ideas from derived categories. |
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One approach to such a generalization might be to simply define a |
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TQFT via its gluing formulas, replacing tensor products with |
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derived tensor products (c.f. \cite{1011.1958}). |
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However, it is probably difficult to prove |
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the invariance of such a definition, as the object associated to a manifold |
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will a priori depend on the explicit presentation used to apply the gluing formulas. |
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We instead give a manifestly invariant construction, and |
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deduce from it the gluing formulas based on $A_\infty$ tensor products. |
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This paper is organized as follows. |
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We first give an account of our version of $n$-categories. |
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According to our definition, $n$-categories are, among other things, |
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functorial invariants of $k$-balls, $0\le k \le n$, which behave well with respect to gluing. |
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We then show how to extend an $n$-category from balls to arbitrary $k$-manifolds, |
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using colimits and homotopy colimits. |
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This extension, which we call the blob complex, has as $0$-th homology the usual TQFT invariant. |
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(The name comes from the ``blobs" which feature prominently |
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in a concrete version of the homotopy colimit.) |
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We then review some basic properties of the blob complex, and finish by showing how it |
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yields a higher categorical and higher dimensional generalization of Deligne's |
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conjecture on Hochschild cochains and the little 2-disks operad. |
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At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. |
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In this paper we attempt to give a clear view of the big picture without getting |
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bogged down in technical details. |
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\section{Definitions} |
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\subsection{$n$-categories} \mbox{} |
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In this section we give a definition of $n$-categories designed to work well with TQFTs. |
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The main idea is to base the definition on actual balls, rather than combinatorial models of them. |
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This has the advantages of avoiding a proliferation of coherency axioms and building in a strong |
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version of duality from the start. |
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%\nn{maybe say something about goals: well-suited to TQFTs; avoid proliferation of coherency axioms; |
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%non-recursive (n-cats not defined n terms of (n-1)-cats; easy to show that the motivating |
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%examples satisfy the axioms; strong duality; both plain and infty case; |
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%(?) easy to see that axioms are correct, in the sense of nothing missing (need |
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%to say this better if we keep it)} |
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% |
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%\nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties |
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%which are weak enough to include the basic examples and strong enough to support the proofs |
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%of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
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%We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
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%More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
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Of course, there are currently many interesting alternative notions of $n$-category. |
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We note that our $n$-categories are both more and less general |
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than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
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They are more general in that we make no duality assumptions in the top dimension $n{+}1$. |
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They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
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Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional {\it unoriented} or {\it oriented} TQFTs, while |
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Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs. |
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We will define two variations simultaneously, as all but one of the axioms are identical in the two cases. |
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These variations are ``ordinary $n$-categories", where homeomorphisms fixing the boundary |
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act trivially on the sets associated to $n$-balls |
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(and these sets are usually vector spaces or more generally modules over a commutative ring) |
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and ``$A_\infty$ $n$-categories", where there is a homotopy action of |
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$k$-parameter families of homeomorphisms on these sets |
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(which are usually chain complexes or topological spaces). |
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There are five basic ingredients |
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\cite{life-of-brian} of an $n$-category definition: |
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$k$-morphisms (for $0\le k \le n$), domain and range, composition, |
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identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
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in some auxiliary category, or strict associativity instead of weak associativity). |
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We will treat each of these in turn. |
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To motivate our morphism axiom, consider the venerable notion of the Moore loop space |
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\cite[\S 2.2]{MR505692}. |
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In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$, |
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so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation |
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of higher associativity relations. |
|
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While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory |
|
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of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
|
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In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
|
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{\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
|
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Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
|
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We wish to imitate this strategy in higher categories. |
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Because we are mainly interested in the case of pivotal $n$-categories, we replace the intervals $[0,r]$ not with |
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a product of $k$ intervals (c.f.\ \cite{0909.2212}) but rather with any $k$-ball, that is, |
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% \cite{ulrike-tillmann-2008,0909.2212} |
651 | 503 |
any $k$-manifold which is homeomorphic |
582 | 504 |
to the standard $k$-ball $B^k$. |
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|
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By default our balls are unoriented, |
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but it is useful at times to vary this, |
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for example by considering oriented or Spin balls. |
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We can also consider more exotic structures, such as balls with a map to some target space, |
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or equipped with $m$ independent vector fields. |
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(The latter structure would model $n$-categories with less duality than we usually assume.) |
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\begin{axiom}[Morphisms] |
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\label{axiom:morphisms} |
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For each $0 \le k \le n$, we have a functor $\cC_k$ from |
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the category of $k$-balls and |
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homeomorphisms to the category of sets and bijections. |
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\end{axiom} |
582 | 519 |
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Note that the functoriality in the above axiom allows us to operate via |
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homeomorphisms which are not the identity on the boundary of the $k$-ball. |
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The action of these homeomorphisms gives the pivotal structure. |
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For this reason we don't subdivide the boundary of a morphism |
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into domain and range in the next axiom --- the duality operations can convert between domain and range. |
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|
651 | 526 |
Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ |
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defined on arbitrary manifolds. |
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We need these functors for $k$-spheres, for $k<n$, for the next axiom. |
582 | 529 |
|
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\begin{axiom}[Boundaries]\label{nca-boundary} |
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531 |
For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
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These maps, for various $X$, comprise a natural transformation of functors. |
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\end{axiom} |
582 | 534 |
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For $c\in \cl{\cC}_{k-1}(\bd X)$ we define $\cC_k(X; c) = \bd^{-1}(c)$. |
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|
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Many of the examples we are interested in are enriched in some auxiliary category $\cS$ |
597 | 538 |
(e.g. vector spaces or rings, or, in the $A_\infty$ case, chain complexes or topological spaces). |
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This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
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of an object of $\cS$, and all of the structure maps of the category (above and below) are |
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compatible with the $\cS$ structure on $\cC_n(X; c)$. |
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|
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|
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Given two hemispheres (a ``domain" and ``range") that agree on the equator, we need to be able to |
651 | 545 |
assemble them into a boundary value of the entire sphere. |
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|
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\begin{lem} |
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548 |
\label{lem:domain-and-range} |
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549 |
Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
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$B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
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551 |
Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
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552 |
two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
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553 |
Then we have an injective map |
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554 |
\[ |
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555 |
\gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
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556 |
\] |
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557 |
which is natural with respect to the actions of homeomorphisms. |
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558 |
%(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
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559 |
%becomes a normal product.) |
575
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560 |
\end{lem} |
582 | 561 |
|
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|
562 |
If $\bdy B = S$, we denote $\bdy^{-1}(\im(\gl_E))$ by $\cC(B)_E$. |
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563 |
|
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564 |
\begin{axiom}[Gluing] |
575
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565 |
\label{axiom:composition} |
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566 |
Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
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567 |
and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
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568 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
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569 |
%Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
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570 |
We have restriction maps $\cC(B_i)_E \to \cC(Y)$. |
575
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571 |
Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
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572 |
We have a map |
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573 |
\[ |
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574 |
\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
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575 |
\] |
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576 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
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577 |
to the intersection of the boundaries of $B$ and $B_i$. |
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578 |
If $k < n$, |
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579 |
or if $k=n$ and we are in the $A_\infty$ case, |
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580 |
we require that $\gl_Y$ is injective. |
671 | 581 |
(For $k=n$ in the ordinary $n$-category case, see Axiom \ref{axiom:extended-isotopies}.) |
575
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582 |
\end{axiom} |
582 | 583 |
|
646 | 584 |
\begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
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585 |
The gluing maps above are strictly associative. |
584 | 586 |
Given any decomposition of a ball $B$ into smaller balls |
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587 |
$$\bigsqcup B_i \to B,$$ |
584 | 588 |
any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
575
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589 |
\end{axiom} |
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590 |
%This axiom is only reasonable because the definition assigns a set to every ball; |
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591 |
%any identifications would limit the extent to which we can demand associativity. |
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592 |
%%%% KW: It took me quite a while figure out what you [or I??] meant by the above, so I'm attempting a rewrite. |
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593 |
Note that even though our $n$-categories are ``weak" in the traditional sense, we can require |
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594 |
strict associativity because we have more morphisms (cf.\ discussion of Moore loops above). |
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|
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596 |
For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
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597 |
\begin{axiom}[Product (identity) morphisms] |
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598 |
\label{axiom:product} |
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599 |
For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
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600 |
there is a map $\pi^*:\cC(X)\to \cC(E)$. |
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601 |
These maps must be |
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602 |
\begin{enumerate} |
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603 |
\item natural with respect to maps of pinched products, |
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604 |
\item functorial with respect to composition of pinched products, |
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605 |
\item compatible with gluing and restriction of pinched products. |
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606 |
\end{enumerate} |
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607 |
|
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608 |
%%% begin noop %%% |
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609 |
% this was the original list of conditions, which I've replaced with the much terser list above -S |
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610 |
\noop{ |
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611 |
These maps must satisfy the following conditions. |
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612 |
\begin{enumerate} |
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613 |
\item |
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614 |
If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
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615 |
if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
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616 |
\[ \xymatrix{ |
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617 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
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618 |
X \ar[r]^{f} & X' |
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619 |
} \] |
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620 |
commutes, then we have |
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621 |
\[ |
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622 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
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623 |
\] |
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624 |
\item |
585
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625 |
Product morphisms are compatible with gluing. |
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626 |
Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
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627 |
be pinched products with $E = E_1\cup E_2$. |
611 | 628 |
Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\subset X$. |
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629 |
Then |
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630 |
\[ |
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631 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
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632 |
\] |
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633 |
\item |
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634 |
Product morphisms are associative. |
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635 |
If $\pi:E\to X$ and $\rho:D\to E$ are pinched products then |
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636 |
\[ |
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637 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
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638 |
\] |
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639 |
\item |
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640 |
Product morphisms are compatible with restriction. |
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641 |
If we have a commutative diagram |
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642 |
\[ \xymatrix{ |
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643 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
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644 |
Y \ar@{^(->}[r] & X |
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645 |
} \] |
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646 |
such that $\rho$ and $\pi$ are pinched products, then |
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647 |
\[ |
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648 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
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649 |
\] |
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650 |
\end{enumerate} |
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651 |
} %%% end \noop %%% |
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652 |
\end{axiom} |
604
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653 |
|
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654 |
To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms. |
611 | 655 |
Let $X$ be a $k$-ball and $Y\subset\bd X$ be a $(k{-}1)$-ball. |
604
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656 |
Let $J$ be a 1-ball. |
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657 |
Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$. |
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658 |
A collar map is an instance of the composition |
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659 |
\[ |
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660 |
\cC(X) \to \cC(X\cup_Y (Y\times_p J)) \to \cC(X) , |
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661 |
\] |
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662 |
where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
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663 |
the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
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664 |
to the identity on the boundary. |
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665 |
|
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666 |
|
671 | 667 |
\begin{axiom}[\textup{\textbf{[for ordinary $n$-categories]}} Extended isotopy invariance in dimension $n$.] |
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668 |
\label{axiom:extended-isotopies} |
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669 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
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670 |
to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
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671 |
Then $f$ acts trivially on $\cC(X)$. |
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672 |
In addition, collar maps act trivially on $\cC(X)$. |
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673 |
\end{axiom} |
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674 |
|
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675 |
\smallskip |
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676 |
|
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677 |
For $A_\infty$ $n$-categories, we replace |
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678 |
isotopy invariance with the requirement that families of homeomorphisms act. |
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679 |
For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
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680 |
Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
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681 |
$C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
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682 |
|
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683 |
|
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684 |
\begin{axiom}[\textup{\textbf{[for $A_\infty$ $n$-categories]}} Families of homeomorphisms act in dimension $n$.] |
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685 |
\label{axiom:families} |
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686 |
For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
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687 |
\[ |
611 | 688 |
C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) . |
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689 |
\] |
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690 |
These action maps are required to restrict to the usual action of homeomorphisms on $C_0$, be associative up to homotopy, |
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691 |
and also be compatible with composition (gluing) in the sense that |
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692 |
a diagram like the one in Theorem \ref{thm:CH} commutes. |
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693 |
\end{axiom} |
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694 |
|
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695 |
\subsection{Example (the fundamental $n$-groupoid)} \mbox{} |
601
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|
696 |
We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$. |
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diff
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|
697 |
When $X$ is a $k$-ball with $k<n$, define $\pi_{\le n}(T)(X)$ |
600
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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parents:
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|
698 |
to be the set of continuous maps from $X$ to $T$. |
601
6bfa35fb758a
minor changes to cone-product polyhedra discussion
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600
diff
changeset
|
699 |
When $X$ is an $n$-ball, define $\pi_{\le n}(T)(X)$ to be homotopy classes (rel boundary) of such maps. |
600
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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parents:
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|
700 |
Define boundary restrictions and gluing in the obvious way. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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parents:
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diff
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|
701 |
If $\rho:E\to X$ is a pinched product and $f:X\to T$ is a $k$-morphism, |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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parents:
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diff
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|
702 |
define the product morphism $\rho^*(f)$ to be $f\circ\rho$. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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parents:
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diff
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|
703 |
|
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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parents:
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|
704 |
We can also define an $A_\infty$ version $\pi_{\le n}^\infty(T)$ of the fundamental $n$-groupoid. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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diff
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|
705 |
For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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diff
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|
706 |
(if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
707 |
|
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
708 |
|
658
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changes from proof-read, 1st installment
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|
709 |
\subsection{Example (string diagrams)} \mbox{} |
672
3f0f4f4ad048
preferring pivotal over 'strong duality'. may want to search for 'duality' and think about phrasing...
Scott Morrison <scott@tqft.net>
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diff
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|
710 |
Fix a ``traditional" pivotal $n$-category $C$ (e.g.\ a pivotal 2-category). |
600
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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parents:
599
diff
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|
711 |
Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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parents:
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diff
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|
712 |
that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
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diff
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|
713 |
If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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parents:
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diff
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|
714 |
Boundary restrictions and gluing are again straightforward to define. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
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parents:
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diff
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|
715 |
Define product morphisms via product cell decompositions. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
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diff
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|
716 |
|
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changes from proof-read, 1st installment
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|
717 |
\subsection{Example (bordism)} \mbox{} |
612 | 718 |
When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional |
719 |
submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely |
|
720 |
to $\bd X$. |
|
721 |
For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds. |
|
600
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Examples and misc.; quality of writing perhaps not so great.
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|
722 |
|
651 | 723 |
There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take |
724 |
all such submanifolds, rather than homeomorphism classes. |
|
652
821d79885bfe
minor changes from proofreading
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diff
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|
725 |
For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we |
651 | 726 |
topologize the set of submanifolds by ambient isotopy rel boundary. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
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diff
changeset
|
727 |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
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571
diff
changeset
|
728 |
\subsection{The blob complex} |
e0f5ec582725
incorporating statements of results in PNAS article
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diff
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|
729 |
\subsubsection{Decompositions of manifolds} |
670
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adding a sentence about types of decompositions of manifolds, including a cite to kirillov. Feel free to revert.
Scott Morrison <scott@tqft.net>
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669
diff
changeset
|
730 |
Our description of an $n$-category associates data to each $k$-ball for $k\leq n$. In order to define invariants of $n$-manifolds, we will need a class of decompositions of manifolds into balls. We present one choice here, but alternatives of varying degrees of generality exist, for example handle decompositions or piecewise-linear CW-complexes \cite{1009.4227}. |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
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diff
changeset
|
731 |
|
646 | 732 |
A \emph{ball decomposition} of a $k$-manifold $W$ is a |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
733 |
sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
574 | 734 |
$\du_a X_a$ and each $M_i$ is a manifold. |
735 |
If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
|
736 |
A {\it permissible decomposition} of $W$ is a map |
|
573
8378e03d3c7f
starting on cell decompositions
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parents:
572
diff
changeset
|
737 |
\[ |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
738 |
\coprod_a X_a \to W, |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
739 |
\] |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
740 |
which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
574 | 741 |
A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls |
670
7a4fc5a873ac
adding a sentence about types of decompositions of manifolds, including a cite to kirillov. Feel free to revert.
Scott Morrison <scott@tqft.net>
parents:
669
diff
changeset
|
742 |
are glued up to yield $W$, and just require that there is some non-pathological way to do this. |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
743 |
|
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
744 |
Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
745 |
of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
735
bd0d5b2155a7
correctly definition of ball decomposition in PNAS article
Scott Morrison <scott@tqft.net>
parents:
714
diff
changeset
|
746 |
with $\du_b Y_b = M_i$ for some $i$, and each $M_j$ with $j<i$ is also a disjoint union of balls. |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
747 |
|
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
748 |
\begin{defn} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
749 |
The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
750 |
and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
751 |
See Figure \ref{partofJfig} for an example. |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
752 |
\end{defn} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
753 |
|
651 | 754 |
This poset in fact has more structure, since we can glue together permissible decompositions of |
755 |
$W_1$ and $W_2$ to obtain a permissible decomposition of $W_1 \sqcup W_2$. |
|
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
756 |
|
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
757 |
An $n$-category $\cC$ determines |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
758 |
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
759 |
(possibly with additional structure if $k=n$). |
653 | 760 |
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-manifolds, |
611 | 761 |
and there is a subset $\cC(X)\spl \subset \cC(X)$ of morphisms whose boundaries |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
762 |
are splittable along this decomposition. |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
763 |
|
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
764 |
\begin{defn} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
765 |
Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
766 |
For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
574 | 767 |
\begin{equation*} |
768 |
%\label{eq:psi-C} |
|
611 | 769 |
\psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
574 | 770 |
\end{equation*} |
653 | 771 |
where the restrictions to the various pieces of shared boundaries amongst the balls |
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
772 |
$X_a$ all agree (similar to a fibered product). |
657
9fbd8e63ab2e
fixing single quotes and long lines
Kevin Walker <kevin@canyon23.net>
parents:
656
diff
changeset
|
773 |
When $k=n$, the ``subset" and ``product" in the above formula should be |
651 | 774 |
interpreted in the appropriate enriching category. |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
775 |
If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
776 |
\end{defn} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
777 |
|
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
778 |
%We will use the term ``field on $W$" to refer to a point of this functor, |
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
779 |
%that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
780 |
|
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
781 |
|
632 | 782 |
\subsubsection{Colimits} |
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
783 |
Recall that our definition of an $n$-category is essentially a collection of functors |
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
784 |
defined on the categories of homeomorphisms of $k$-balls |
651 | 785 |
for $k \leq n$ satisfying certain axioms. |
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
786 |
It is natural to hope to extend such functors to the |
651 | 787 |
larger categories of all $k$-manifolds (again, with homeomorphisms). |
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
788 |
In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$. |
638
6a7f2a6295d1
very paltry start on colimits, out of time for now
Scott Morrison <scott@tqft.net>
parents:
637
diff
changeset
|
789 |
|
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
790 |
The natural construction achieving this is a colimit along the poset of permissible decompositions. |
671 | 791 |
Given an ordinary $n$-category $\cC$, |
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
792 |
we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
656
28592849a474
some more fixes in the colimit section
Scott Morrison <scott@tqft.net>
parents:
655
diff
changeset
|
793 |
this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. |
651 | 794 |
Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
795 |
imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
|
660 | 796 |
Suppose that $\cC$ is enriched in vector spaces: this means that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
797 |
the set $\cC(X;c)$ is a vector space. |
|
798 |
In this case, for $W$ an arbitrary $n$-manifold and $c \in \cl{\cC}(\bdy W)$, |
|
799 |
the set $\cl{\cC}(W;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. |
|
651 | 800 |
These are the usual TQFT skein module invariants on $n$-manifolds. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
801 |
|
598
20de3d710f77
writing inconclusively about homotopy colimits, but have to run
Scott Morrison <scott@tqft.net>
parents:
597
diff
changeset
|
802 |
We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
803 |
with coefficients in the $n$-category $\cC$ as the {\it homotopy} colimit along $\cell(W)$ |
598
20de3d710f77
writing inconclusively about homotopy colimits, but have to run
Scott Morrison <scott@tqft.net>
parents:
597
diff
changeset
|
804 |
of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
20de3d710f77
writing inconclusively about homotopy colimits, but have to run
Scott Morrison <scott@tqft.net>
parents:
597
diff
changeset
|
805 |
|
651 | 806 |
An explicit realization of the homotopy colimit is provided by the simplices of the |
807 |
functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ |
|
808 |
where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. |
|
809 |
The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as |
|
599 | 810 |
$$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
811 |
where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
|
598
20de3d710f77
writing inconclusively about homotopy colimits, but have to run
Scott Morrison <scott@tqft.net>
parents:
597
diff
changeset
|
812 |
|
651 | 813 |
Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the |
814 |
homotopy colimit via the cone-product polyhedra in $\cell(W)$. |
|
815 |
A cone-product polyhedra is obtained from a point by successively taking the cone or taking the |
|
816 |
product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, |
|
817 |
cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, |
|
818 |
and taking product identifies the roots of several trees. |
|
657
9fbd8e63ab2e
fixing single quotes and long lines
Kevin Walker <kevin@canyon23.net>
parents:
656
diff
changeset
|
819 |
The ``local homotopy colimit" is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. |
661 | 820 |
We further require that all (compositions of) morphisms in a directed tree are not expressible as a product. |
651 | 821 |
The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
657
9fbd8e63ab2e
fixing single quotes and long lines
Kevin Walker <kevin@canyon23.net>
parents:
656
diff
changeset
|
822 |
A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
823 |
|
605
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
824 |
%When $\cC$ is a topological $n$-category, |
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
825 |
%the flexibility available in the construction of a homotopy colimit allows |
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
826 |
%us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
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|
827 |
%\todo{either need to explain why this is the same, or significantly rewrite this section} |
671 | 828 |
When $\cC$ is the ordinary $n$-category based on string diagrams for a traditional |
605
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|
829 |
$n$-category $C$, |
610 | 830 |
one can show \cite{1009.5025} that the above two constructions of the homotopy colimit |
606 | 831 |
are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
832 |
Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
|
605
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|
833 |
a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
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|
834 |
The restriction of the string diagram to innermost blobs is required to be ``null" in the sense that |
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|
835 |
it evaluates to a zero $n$-morphism of $C$. |
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|
836 |
The next few paragraphs describe this in more detail. |
575
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|
837 |
|
667
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|
838 |
We will call a string diagram on a manifold a ``field". |
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|
839 |
(See \cite{1009.5025} for a more general notion of field.) |
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|
840 |
|
575
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|
841 |
We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
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|
842 |
if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
651 | 843 |
each $B_i$ appears as a connected component of one of the $M_j$. |
659
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diff
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|
844 |
Note that this forces the balls to be pairwise either disjoint or nested. |
651 | 845 |
Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. |
663 | 846 |
These pieces need not be manifolds, |
847 |
but they can be further subdivided into pieces which are manifolds |
|
848 |
and which fit into a permissible decomposition of $W$. |
|
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|
849 |
|
662
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|
850 |
The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. |
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|
851 |
A $k$-blob diagram consists of |
575
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|
852 |
\begin{itemize} |
663 | 853 |
\item a permissible collection of $k$ embedded balls, and |
854 |
\item a linear combination $s$ of string diagrams on $W$, |
|
575
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|
855 |
\end{itemize} |
663 | 856 |
such that |
857 |
\begin{itemize} |
|
858 |
\item there is a permissible decomposition of $W$, compatible with the $k$ blobs, such that |
|
669
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|
859 |
$s$ is the result of gluing together linear combinations of fields $s_i$ on the initial pieces $X_i$ of the decomposition |
663 | 860 |
(for fixed restrictions to the boundaries of the pieces), |
861 |
\item the $s_i$'s corresponding to innermost blobs evaluate to zero in $\cC$, and |
|
667
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diff
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|
862 |
\item the $s_i$'s corresponding to the other pieces are single fields (linear combinations with only one term). |
663 | 863 |
\end{itemize} |
864 |
%that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
|
865 |
We call such linear combinations which evaluate to zero on a blob $B$ a ``null field on $B$". |
|
668 | 866 |
% could maybe say something here like "if blobs have nice complements then this is just...." |
575
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diff
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|
867 |
|
608 | 868 |
The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
575
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|
869 |
|
662
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diff
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|
870 |
We now spell this out for some small values of $k$. |
667
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diff
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|
871 |
For $k=0$, the $0$-blob group is simply linear combinations of fields (string diagrams) on $W$. |
651 | 872 |
For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. |
873 |
The differential simply forgets the ball. |
|
874 |
Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball. |
|
580
99611dfed1f3
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diff
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|
875 |
|
651 | 876 |
For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. |
877 |
In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. |
|
878 |
In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. |
|
879 |
Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. |
|
880 |
This ensures that the differential is well-defined. |
|
575
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|
881 |
|
572
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|
882 |
\section{Properties of the blob complex} |
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diff
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|
883 |
\subsection{Formal properties} |
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|
884 |
\label{sec:properties} |
602
109ecc26c50d
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|
885 |
The blob complex enjoys the following list of formal properties. The first three are immediate from the definitions. |
572
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|
886 |
|
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|
887 |
\begin{property}[Functoriality] |
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diff
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|
888 |
\label{property:functoriality}% |
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diff
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|
889 |
The blob complex is functorial with respect to homeomorphisms. |
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diff
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|
890 |
That is, |
574 | 891 |
for a fixed $n$-category $\cC$, the association |
572
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diff
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|
892 |
\begin{equation*} |
574 | 893 |
X \mapsto \bc_*(X; \cC) |
572
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|
894 |
\end{equation*} |
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diff
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|
895 |
is a functor from $n$-manifolds and homeomorphisms between them to chain |
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diff
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|
896 |
complexes and isomorphisms between them. |
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diff
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|
897 |
\end{property} |
574 | 898 |
As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cC)$; |
572
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|
899 |
this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below. |
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|
900 |
|
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|
901 |
\begin{property}[Disjoint union] |
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diff
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|
902 |
\label{property:disjoint-union} |
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diff
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|
903 |
The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
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diff
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|
904 |
\begin{equation*} |
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diff
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|
905 |
\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
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|
906 |
\end{equation*} |
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|
907 |
\end{property} |
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diff
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|
908 |
|
574 | 909 |
If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary, |
910 |
write $X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
|
572
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|
911 |
\begin{property}[Gluing map] |
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diff
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|
912 |
\label{property:gluing-map}% |
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diff
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|
913 |
%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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diff
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|
914 |
%\begin{equation*} |
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diff
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|
915 |
%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
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diff
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|
916 |
%\end{equation*} |
607 | 917 |
Given a gluing $X \to X \bigcup_{Y}\selfarrow$, there is |
574 | 918 |
a map |
572
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diff
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|
919 |
\[ |
574 | 920 |
\bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow), |
572
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diff
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|
921 |
\] |
574 | 922 |
natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
572
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|
923 |
\end{property} |
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|
924 |
|
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|
925 |
\begin{property}[Contractibility] |
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diff
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|
926 |
\label{property:contractibility}% |
589
14b7d867e423
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Scott Morrison <scott@tqft.net>
parents:
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diff
changeset
|
927 |
The blob complex on an $n$-ball is contractible in the sense |
14b7d867e423
a few changes, maybe bad ones...
Scott Morrison <scott@tqft.net>
parents:
577
diff
changeset
|
928 |
that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category. |
572
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|
929 |
\begin{equation*} |
649 | 930 |
\xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\htpy} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)} |
572
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|
931 |
\end{equation*} |
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diff
changeset
|
932 |
\end{property} |
627
b0ed73b141d8
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diff
changeset
|
933 |
%\nn{maybe should say something about the $A_\infty$ case} |
572
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changeset
|
934 |
|
583 | 935 |
\begin{proof}(Sketch) |
936 |
For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram |
|
937 |
obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
|
938 |
For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
|
939 |
$x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
|
940 |
\end{proof} |
|
572
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|
941 |
|
649 | 942 |
If $\cC$ is an $A_\infty$ $n$-category then $\bc_*(B^n;\cC)$ is still homotopy equivalent to $\cC(B^n)$, |
627
b0ed73b141d8
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parents:
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diff
changeset
|
943 |
but this is no longer concentrated in degree zero. |
b0ed73b141d8
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diff
changeset
|
944 |
|
572
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|
945 |
\subsection{Specializations} |
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|
946 |
\label{sec:specializations} |
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|
947 |
|
615
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
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diff
changeset
|
948 |
The blob complex has several important special cases. |
572
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|
949 |
|
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|
950 |
\begin{thm}[Skein modules] |
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|
951 |
\label{thm:skein-modules} |
671 | 952 |
Suppose $\cC$ is an ordinary $n$-category. |
572
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|
953 |
The $0$-th blob homology of $X$ is the usual |
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|
954 |
(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
589
14b7d867e423
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|
955 |
by $\cC$. |
572
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|
956 |
\begin{equation*} |
649 | 957 |
H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
572
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|
958 |
\end{equation*} |
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|
959 |
\end{thm} |
651 | 960 |
This follows from the fact that the $0$-th homology of a homotopy colimit is the usual colimit, |
961 |
or directly from the explicit description of the blob complex. |
|
572
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|
962 |
|
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|
963 |
\begin{thm}[Hochschild homology when $X=S^1$] |
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964 |
\label{thm:hochschild} |
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|
965 |
The blob complex for a $1$-category $\cC$ on the circle is |
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|
966 |
quasi-isomorphic to the Hochschild complex. |
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|
967 |
\begin{equation*} |
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|
968 |
\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
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|
969 |
\end{equation*} |
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|
970 |
\end{thm} |
651 | 971 |
This theorem is established by extending the statement to bimodules as well as categories, |
972 |
then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$. |
|
572
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|
973 |
|
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|
974 |
\begin{thm}[Mapping spaces] |
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|
975 |
\label{thm:map-recon} |
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|
976 |
Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
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|
977 |
$B^n \to T$. |
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|
978 |
(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
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|
979 |
Then |
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|
980 |
$$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
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|
981 |
\end{thm} |
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|
982 |
|
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|
983 |
This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
651 | 984 |
Note that there is no restriction on the connectivity of $T$ as there is for |
985 |
the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. |
|
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|
986 |
The result is proved in \cite[\S 7.3]{1009.5025}. |
615
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|
987 |
|
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|
988 |
\subsection{Structure of the blob complex} |
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|
989 |
\label{sec:structure} |
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|
990 |
|
651 | 991 |
In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space |
992 |
of homeomorphisms of $X$, fixed on $\bdy X$. |
|
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|
993 |
|
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|
994 |
\begin{thm} |
572
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|
995 |
\label{thm:CH}\label{thm:evaluation} |
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|
996 |
There is a chain map |
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|
997 |
\begin{equation*} |
585
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|
998 |
e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X) |
572
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|
999 |
\end{equation*} |
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|
1000 |
such that |
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|
1001 |
\begin{enumerate} |
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|
1002 |
\item Restricted to $CH_0(X)$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
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|
1003 |
|
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|
1004 |
\item For |
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|
1005 |
any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
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|
1006 |
(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
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|
1007 |
\begin{equation*} |
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|
1008 |
\xymatrix@C+0.3cm{ |
611 | 1009 |
\CH{X} \tensor \bc_*(X) |
1010 |
\ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \tensor \gl_Y} & |
|
572
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1011 |
\bc_*(X) \ar[d]_{\gl_Y} \\ |
611 | 1012 |
\CH{X \bigcup_Y \selfarrow} \tensor \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
572
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|
1013 |
} |
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|
1014 |
\end{equation*} |
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|
1015 |
\end{enumerate} |
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|
1016 |
|
609 | 1017 |
Further, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
572
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|
1018 |
\begin{equation*} |
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|
1019 |
\xymatrix{ |
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|
1020 |
\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} \\ |
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|
1021 |
\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
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|
1022 |
} |
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|
1023 |
\end{equation*} |
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|
1024 |
\end{thm} |
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|
1025 |
|
609 | 1026 |
\begin{proof}(Sketch.) |
622
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|
1027 |
We introduce yet another homotopy equivalent version of |
609 | 1028 |
the blob complex, $\cB\cT_*(X)$. |
1029 |
Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
|
1030 |
In $\cB\cT_*(X)$ we take this topology into account, treating the blob diagrams as something |
|
1031 |
analogous to a simplicial space (but with cone-product polyhedra replacing simplices). |
|
651 | 1032 |
More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$. |
1033 |
An essential step in the proof of this equivalence is a result to the effect that a $k$-parameter |
|
652
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|
1034 |
family of homeomorphisms can be localized to at most $k$ small sets. |
609 | 1035 |
|
652
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|
1036 |
With this alternate version in hand, the theorem is straightforward. |
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|
1037 |
By functoriality (Property \ref{property:functoriality}) $\Homeo(X)$ acts on the set $BD_j(X)$ of $j$-blob diagrams, and this |
614 | 1038 |
induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$ |
1039 |
and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$. |
|
609 | 1040 |
It is easy to check that $e_X$ thus defined has the desired properties. |
1041 |
\end{proof} |
|
575
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|
1042 |
|
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|
1043 |
\begin{thm} |
572
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|
1044 |
\label{thm:blobs-ainfty} |
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|
1045 |
Let $\cC$ be a topological $n$-category. |
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|
1046 |
Let $Y$ be an $n{-}k$-manifold. |
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|
1047 |
There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
610 | 1048 |
to be the set $$\bc_*(Y;\cC)(D) = \cl{\cC}(Y \times D)$$ and on $k$-balls $D$ to be the set |
572
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|
1049 |
$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
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|
1050 |
(When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
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|
1051 |
These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
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|
1052 |
Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
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|
1053 |
\end{thm} |
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|
1054 |
\begin{rem} |
610 | 1055 |
When $Y$ is a point this produces an $A_\infty$ $n$-category from a topological $n$-category, |
1056 |
which can be thought of as a free resolution. |
|
572
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|
1057 |
\end{rem} |
610 | 1058 |
This result is described in more detail as Example 6.2.8 of \cite{1009.5025}. |
572
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|
1059 |
|
618 | 1060 |
Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
661 | 1061 |
From the above, associated to any $(n{-}1)$-manifold $Y$ is an $A_\infty$ category $\bc_*(Y)$. |
572
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|
1062 |
|
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|
1063 |
\begin{thm}[Gluing formula] |
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|
1064 |
\label{thm:gluing} |
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|
1065 |
\mbox{}\vspace{-0.2cm}% <-- gets the indenting right |
572
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|
1066 |
\begin{itemize} |
651 | 1067 |
\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, |
1068 |
the blob complex of $X$ is naturally an |
|
572
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|
1069 |
$A_\infty$ module for $\bc_*(Y)$. |
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|
1070 |
|
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|
1071 |
\item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of |
659
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|
1072 |
$\bc_*(X)$ as a $\bc_*(Y)$-bimodule: |
572
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|
1073 |
\begin{equation*} |
585
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|
1074 |
\bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
572
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|
1075 |
\end{equation*} |
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|
1076 |
\end{itemize} |
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|
1077 |
\end{thm} |
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|
1078 |
|
618 | 1079 |
\begin{proof} (Sketch.) |
620
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|
1080 |
The $A_\infty$ action of $\bc_*(Y)$ follows from the naturality of the blob complex with respect to gluing |
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|
1081 |
and the $C_*(\Homeo(-))$ action of Theorem \ref{thm:evaluation}. |
618 | 1082 |
|
620
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|
1083 |
Let $T_*$ denote the self tensor product of $\bc_*(X)$, which is a homotopy colimit. |
622
dda6d3a00b09
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|
1084 |
There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X\bigcup_Y \selfarrow)$, |
620
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|
1085 |
and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero. |
652
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|
1086 |
Constructing a homotopy inverse to this natural map involves making various choices, but one can show that the |
659
cc0c2dfe61f3
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|
1087 |
choices form contractible subcomplexes and apply the acyclic models theorem. |
618 | 1088 |
\end{proof} |
610 | 1089 |
|
652
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|
1090 |
We next describe the blob complex for product manifolds, in terms of the |
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|
1091 |
blob complexes for the $A_\infty$ $n$-categories constructed as above. |
610 | 1092 |
|
1093 |
\begin{thm}[Product formula] |
|
1094 |
\label{thm:product} |
|
652
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|
1095 |
Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold. |
671 | 1096 |
Let $\cC$ be an ordinary $n$-category. |
610 | 1097 |
Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
1098 |
Then |
|
1099 |
\[ |
|
1100 |
\bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
|
1101 |
\] |
|
659
cc0c2dfe61f3
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diff
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|
1102 |
That is, the blob complex of $Y\times W$ with coefficients in $\cC$ is homotopy equivalent |
cc0c2dfe61f3
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diff
changeset
|
1103 |
to the blob complex of $W$ with coefficients in $\bc_*(Y;\cC)$. |
610 | 1104 |
\end{thm} |
1105 |
The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
|
1106 |
(see \cite[\S7.1]{1009.5025}). |
|
1107 |
||
620
28b016b716b1
adding some proof sketches
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619
diff
changeset
|
1108 |
\begin{proof} (Sketch.) |
623 | 1109 |
The proof is similar to that of the second part of Theorem \ref{thm:gluing}. |
1110 |
There is a natural map from the 0-simplices of $\clh{\bc_*(Y;\cC)}(W)$ to $\bc_*(Y\times W; \cC)$, |
|
1111 |
given by reinterpreting a decomposition of $W$ labeled by $(n{-}k)$-morphisms of $\bc_*(Y; \cC)$ as a blob |
|
1112 |
diagram on $W\times Y$. |
|
1113 |
This map can be extended to all of $\clh{\bc_*(Y;\cC)}(W)$ by sending higher simplices to zero. |
|
620
28b016b716b1
adding some proof sketches
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619
diff
changeset
|
1114 |
|
623 | 1115 |
To construct the homotopy inverse of the above map one first shows that |
1116 |
$\bc_*(Y\times W; \cC)$ is homotopy equivalent to the subcomplex generated by blob diagrams which |
|
652
821d79885bfe
minor changes from proofreading
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651
diff
changeset
|
1117 |
are small with respect to any fixed open cover of $Y\times W$. |
623 | 1118 |
For a sufficiently fine open cover the generators of this ``small" blob complex are in the image of the map |
1119 |
of the previous paragraph, and furthermore the preimage in $\clh{\bc_*(Y;\cC)}(W)$ of such small diagrams |
|
1120 |
lie in contractible subcomplexes. |
|
1121 |
A standard acyclic models argument now constructs the homotopy inverse. |
|
620
28b016b716b1
adding some proof sketches
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diff
changeset
|
1122 |
\end{proof} |
28b016b716b1
adding some proof sketches
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619
diff
changeset
|
1123 |
|
28b016b716b1
adding some proof sketches
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619
diff
changeset
|
1124 |
%\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
572
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|
1125 |
|
662
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diff
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|
1126 |
\section{Extending Deligne's conjecture to $n$-categories} |
572
e0f5ec582725
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diff
changeset
|
1127 |
\label{sec:applications} |
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diff
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|
1128 |
|
625
c6d069b8f931
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diff
changeset
|
1129 |
Let $M$ and $N$ be $n$-manifolds with common boundary $E$. |
c6d069b8f931
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diff
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|
1130 |
Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$ |
c6d069b8f931
starting on Deligne section
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changeset
|
1131 |
acts on $\bc_*(M)$ and $\bc_*(N)$. |
c6d069b8f931
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diff
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|
1132 |
Let $\hom_A(\bc_*(M), \bc_*(N))$ denote the chain complex of $A_\infty$ module maps |
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
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diff
changeset
|
1133 |
from $\bc_*(M)$ to $\bc_*(N)$. |
652
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|
1134 |
Let $R$ be another $n$-manifold with boundary $E^\text{op}$. |
625
c6d069b8f931
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diff
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|
1135 |
There is a chain map |
661 | 1136 |
\begin{equation*} |
625
c6d069b8f931
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diff
changeset
|
1137 |
\hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) . |
661 | 1138 |
\end{equation*} |
625
c6d069b8f931
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diff
changeset
|
1139 |
We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
c6d069b8f931
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diff
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|
1140 |
replaces it with $N$, yielding $N\cup_E R$. |
714
e541b222dceb
fixing rob's typo (that->than)
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parents:
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diff
changeset
|
1141 |
(This is a more general notion of surgery than usual: $M$ and $N$ can be any manifolds |
626 | 1142 |
which share a common boundary.) |
627
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|
1143 |
In analogy to Hochschild cochains, we will call elements of $\hom_A(\bc_*(M), \bc_*(N))$ ``blob cochains". |
625
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|
1144 |
|
626 | 1145 |
Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
625
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|
1146 |
blob complex. |
626 | 1147 |
An $n$-dimensional surgery cylinder is |
1148 |
defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), |
|
1149 |
modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. |
|
659
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diff
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|
1150 |
One can associate to this data an $(n{+}1)$-manifold with a foliation by intervals, |
626 | 1151 |
and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds |
1152 |
which preserve the foliation. |
|
625
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|
1153 |
|
626 | 1154 |
Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
625
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|
1155 |
|
572
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|
1156 |
\begin{thm}[Higher dimensional Deligne conjecture] |
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diff
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|
1157 |
\label{thm:deligne} |
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|
1158 |
The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
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|
1159 |
\end{thm} |
577
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|
1160 |
|
627
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|
1161 |
More specifically, let $M_0, N_0, \ldots, M_k, N_k$ be $n$-manifolds and let $SC^n_{\overline{M}, \overline{N}}$ |
b0ed73b141d8
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diff
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|
1162 |
denote the component of the operad with outer boundary $M_0\cup N_0$ and inner boundaries |
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|
1163 |
$M_1\cup N_1,\ldots, M_k\cup N_k$. |
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|
1164 |
Then there is a collection of chain maps |
b0ed73b141d8
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diff
changeset
|
1165 |
\begin{multline*} |
b0ed73b141d8
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changeset
|
1166 |
C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots \\ |
b0ed73b141d8
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|
1167 |
\otimes \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
b0ed73b141d8
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changeset
|
1168 |
\end{multline*} |
b0ed73b141d8
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|
1169 |
which satisfy the operad compatibility conditions. |
577
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Scott Morrison <scott@tqft.net>
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diff
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|
1170 |
|
661 | 1171 |
\begin{proof} (Sketch.) |
610 | 1172 |
We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
1173 |
and the action of surgeries is just composition of maps of $A_\infty$-modules. |
|
662
57bd9fab3827
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|
1174 |
We only need to check that the relations of the surgery cylinder operad are satisfied. |
610 | 1175 |
This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
595
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
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594
diff
changeset
|
1176 |
\end{proof} |
577
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diff
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|
1177 |
|
652
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|
1178 |
Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are intervals. |
627
b0ed73b141d8
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|
1179 |
We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |
b0ed73b141d8
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|
1180 |
disks operad and $\hom(\bc_*(M_i), \bc_*(N_i))$ is homotopy equivalent to Hochschild cochains. |
b0ed73b141d8
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diff
changeset
|
1181 |
This special case is just the usual Deligne conjecture |
628
4cce595ae1d3
adding Gerstenhaber-Voronov, explicitly not proving the mapping spaces result, and slight tweaks
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diff
changeset
|
1182 |
(see \cite{hep-th/9403055, MR1328534, MR1805894, MR1805923, MR2064592}). |
627
b0ed73b141d8
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diff
changeset
|
1183 |
|
b0ed73b141d8
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|
1184 |
The general case when $n=1$ goes beyond the original Deligne conjecture, as the $M_i$'s and $N_i$'s |
652
821d79885bfe
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|
1185 |
could be disjoint unions of intervals and circles, and the surgery cylinders could be high genus surfaces. |
627
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diff
changeset
|
1186 |
|
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|
1187 |
If all of the $M_i$'s and $N_i$'s are $n$-balls, then $SC^n_{\overline{M}, \overline{N}}$ |
b0ed73b141d8
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|
1188 |
contains a copy of the little $(n{+}1)$-balls operad. |
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diff
changeset
|
1189 |
Thus the little $(n{+}1)$-balls operad acts on blob cochains of the $n$-ball. |
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|
1190 |
|
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|
1191 |
|
566 | 1192 |
|
1193 |
%% == end of paper: |
|
1194 |
||
1195 |
%% Optional Materials and Methods Section |
|
1196 |
%% The Materials and Methods section header will be added automatically. |
|
1197 |
||
1198 |
%% Enter any subheads and the Materials and Methods text below. |
|
1199 |
%\begin{materials} |
|
1200 |
% Materials text |
|
1201 |
%\end{materials} |
|
1202 |
||
1203 |
||
1204 |
%% Optional Appendix or Appendices |
|
1205 |
%% \appendix Appendix text... |
|
1206 |
%% or, for appendix with title, use square brackets: |
|
1207 |
%% \appendix[Appendix Title] |
|
1208 |
||
1209 |
\begin{acknowledgments} |
|
610 | 1210 |
It is a pleasure to acknowledge helpful conversations with |
1211 |
Kevin Costello, |
|
625
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|
1212 |
Michael Freedman, |
610 | 1213 |
Justin Roberts, |
1214 |
and |
|
1215 |
Peter Teichner. |
|
1216 |
We also thank the Aspen Center for Physics for providing a pleasant and productive |
|
1217 |
environment during the last stages of this project. |
|
566 | 1218 |
\end{acknowledgments} |
1219 |
||
1220 |
%% PNAS does not support submission of supporting .tex files such as BibTeX. |
|
1221 |
%% Instead all references must be included in the article .tex document. |
|
1222 |
%% If you currently use BibTeX, your bibliography is formed because the |
|
1223 |
%% command \verb+\bibliography{}+ brings the <filename>.bbl file into your |
|
1224 |
%% .tex document. To conform to PNAS requirements, copy the reference listings |
|
1225 |
%% from your .bbl file and add them to the article .tex file, using the |
|
1226 |
%% bibliography environment described above. |
|
1227 |
||
1228 |
%% Contact pnas@nas.edu if you need assistance with your |
|
1229 |
%% bibliography. |
|
1230 |
||
1231 |
% Sample bibliography item in PNAS format: |
|
1232 |
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|
1233 |
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|
1234 |
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|
1235 |
%% ie, |
|
1236 |
% \bibitem{Neuhaus} Neuhaus J-M, Sitcher L, Meins F, Jr, Boller T (1991) |
|
1237 |
% A short C-terminal sequence is necessary and sufficient for the |
|
1238 |
% targeting of chitinases to the plant vacuole. |
|
1239 |
% {\it Proc Natl Acad Sci USA} 88:10362-10366. |
|
1240 |
||
1241 |
||
1242 |
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1243 |
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1244 |
||
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1245 |
%%%% BIBTEX |
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1246 |
%\bibliographystyle{unsrt} |
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1247 |
%\bibliography{../bibliography/bibliography} |
566 | 1248 |
|
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1249 |
%%%% non-BIBTEX |
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1250 |
\newcommand{\noopsort}[1]{}\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$} |
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1251 |
\begin{thebibliography}{10} |
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1252 |
|
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1253 |
\bibitem{1009.5025} |
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|
1254 |
S.~{Morrison} and K.~{Walker}. |
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|
1255 |
\newblock {The blob complex}, September 2010. |
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|
1256 |
\newblock \arxiv{1009.5025}. |
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|
1257 |
|
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1258 |
\bibitem{kw:tqft} |
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|
1259 |
Kevin Walker. |
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|
1260 |
\newblock Topological quantum field theories, 2006. |
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1261 |
\newblock Available at \url{http://canyon23.net/math/}. |
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|
1262 |
|
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1263 |
\bibitem{0905.0465} |
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|
1264 |
Jacob Lurie. |
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|
1265 |
\newblock On the classification of topological field theories, 2009. |
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1266 |
\newblock \arxiv{0905.0465}. |
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|
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1268 |
\bibitem{1003.0598} |
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1269 |
R.~{Lipshitz}, P.~S. {Ozsvath}, and D.~P. {Thurston}. |
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1270 |
\newblock {Bimodules in bordered Heegaard Floer homology}, March 2010. |
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1271 |
\newblock \arxiv{1003.0598}. |
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1272 |
|
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1273 |
\bibitem{1005.1248} |
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1274 |
R.~{Lipshitz}, P.~S. {Ozsv{\'a}th}, and D.~P. {Thurston}. |
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1275 |
\newblock {Heegaard Floer homology as morphism spaces}, May 2010. |
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1276 |
\newblock \arxiv{1005.1248}. |
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1277 |
|
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1278 |
\bibitem{1011.1958} |
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1279 |
Lev Rozansky. |
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1280 |
\newblock A categorification of the stable su(2) |
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1281 |
{W}itten-{R}eshetikhin-{T}uraev invariant of links in $s^2 \times s^1$, Nov |
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1282 |
2010. |
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1283 |
\newblock \arxiv{1011.1958}. |
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1284 |
|
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1285 |
\bibitem{life-of-brian} |
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1286 |
Monty Python. |
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1287 |
\newblock {L}ife of {B}rian, 1979. |
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1288 |
\newblock \url{http://www.youtube.com/watch?v=fIRb8TigJ28}. |
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1289 |
|
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1290 |
\bibitem{MR505692} |
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1291 |
John~Frank Adams. |
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1292 |
\newblock {\em Infinite loop spaces}, volume~90 of {\em Annals of Mathematics |
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1293 |
Studies}. |
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1294 |
\newblock Princeton University Press, Princeton, N.J., 1978. |
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1295 |
\newblock \mathscinet{MR505692} \googlebooks{e2rYkg9lGnsC}. |
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|
678
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1297 |
%\bibitem{ulrike-tillmann-2008} |
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1298 |
%Ulrike Tillmann, 2008. |
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1299 |
%\newblock personal communication. |
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1300 |
|
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1301 |
\bibitem{0909.2212} |
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1302 |
Ronald {Brown}. |
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1303 |
\newblock {Moore hyperrectangles on a space form a strict cubical |
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1304 |
omega-category}, September 2009. |
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1305 |
\newblock \arxiv{0909.2212}. |
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1306 |
|
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1307 |
\bibitem{1009.4227} |
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1308 |
Alexander Kirillov, Jr. |
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1309 |
\newblock On piecewise linear cell decompositions, 2010. |
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1310 |
\newblock \arxiv{1009.4227}. |
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1311 |
|
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1312 |
\bibitem{0911.0018} |
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1313 |
Jacob Lurie. |
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1314 |
\newblock {D}erived {A}lgebraic {G}eometry {VI}: {$E_k$} {A}lgebras, 2009. |
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1315 |
\newblock \arxiv{0911.0018}. |
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1316 |
|
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1317 |
\bibitem{hep-th/9403055} |
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1318 |
E.~Getzler and J.~D.~S. Jones. |
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1319 |
\newblock {Operads, homotopy algebra, and iterated integrals for double loop |
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1320 |
spaces}, 1994. |
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1321 |
\newblock \arxiv{hep-th/9403055}. |
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1322 |
|
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1323 |
\bibitem{MR1328534} |
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1324 |
A.~A. Voronov and M.~Gerstenkhaber. |
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1325 |
\newblock Higher-order operations on the {H}ochschild complex. |
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1326 |
\newblock {\em Funktsional. Anal. i Prilozhen.}, 29(1):1--6, 96, 1995. |
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1327 |
\newblock \mathscinet{MR1328534} \doi{10.1007/BF01077036}. |
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1328 |
|
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1329 |
\bibitem{MR1805894} |
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1330 |
Maxim Kontsevich and Yan Soibelman. |
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1331 |
\newblock Deformations of algebras over operads and the {D}eligne conjecture. |
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1332 |
\newblock In {\em Conf\'erence {M}osh\'e {F}lato 1999, {V}ol. {I} ({D}ijon)}, |
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1333 |
volume~21 of {\em Math. Phys. Stud.}, pages 255--307. Kluwer Acad. Publ., |
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1334 |
Dordrecht, 2000. |
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1335 |
\newblock \mathscinet{MR1805894} \arxiv{math.QA/0001151}. |
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1336 |
|
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1337 |
\bibitem{MR1805923} |
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1338 |
Alexander~A. Voronov. |
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1339 |
\newblock Homotopy {G}erstenhaber algebras. |
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1340 |
\newblock In {\em Conf\'erence {M}osh\'e {F}lato 1999, {V}ol. {II} ({D}ijon)}, |
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1341 |
volume~22 of {\em Math. Phys. Stud.}, pages 307--331. Kluwer Acad. Publ., |
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1342 |
Dordrecht, 2000. |
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1343 |
\newblock \mathscinet{MR1805923} \arxiv{math.QA/9908040}. |
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1344 |
|
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1345 |
\bibitem{MR2064592} |
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1346 |
Dmitry~E. Tamarkin. |
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1347 |
\newblock Formality of chain operad of little discs. |
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1348 |
\newblock {\em Lett. Math. Phys.}, 66(1-2):65--72, 2003. |
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\newblock \mathscinet{MR2064592} \doi{10.1023/B:MATH.0000017651.12703.a1}. |
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|
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1351 |
\end{thebibliography} |
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566 | 1353 |
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|
566 | 1355 |
\end{article} |
1356 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1357 |
||
1358 |
%% Adding Figure and Table References |
|
1359 |
%% Be sure to add figures and tables after \end{article} |
|
1360 |
%% and before \end{document} |
|
1361 |
||
1362 |
%% For figures, put the caption below the illustration. |
|
1363 |
%% |
|
1364 |
%% \begin{figure} |
|
1365 |
%% \caption{Almost Sharp Front}\label{afoto} |
|
1366 |
%% \end{figure} |
|
1367 |
||
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|
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\begin{figure} |
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\begin{equation*} |
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\mathfig{0.18}{figure1} |
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1372 |
\end{equation*} |
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\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
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|
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\begin{figure} |
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\begin{equation*} |
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1377 |
\mathfig{0.14}{figure2} |
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1378 |
\end{equation*} |
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1379 |
\caption{From two balls to one ball.}\label{blah5}\end{figure} |
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\begin{figure} |
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\begin{equation*} |
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1383 |
\mathfig{.23}{zz2} |
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1384 |
\end{equation*} |
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1385 |
\caption{A small part of $\cell(W)$.} |
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1386 |
\label{partofJfig} |
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1387 |
\end{figure} |
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1389 |
\begin{figure} |
626 | 1390 |
%$$\mathfig{.4}{deligne/manifolds}$$ |
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$$\mathfig{.4}{mapping-cylinders}$$ |
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1392 |
\caption{An $n$-dimensional surgery cylinder.}\label{delfig2} |
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1393 |
\end{figure} |
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|
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|
566 | 1396 |
%% For Tables, put caption above table |
1397 |
%% |
|
1398 |
%% Table caption should start with a capital letter, continue with lower case |
|
1399 |
%% and not have a period at the end |
|
1400 |
%% Using @{\vrule height ?? depth ?? width0pt} in the tabular preamble will |
|
1401 |
%% keep that much space between every line in the table. |
|
1402 |
||
1403 |
%% \begin{table} |
|
1404 |
%% \caption{Repeat length of longer allele by age of onset class} |
|
1405 |
%% \begin{tabular}{@{\vrule height 10.5pt depth4pt width0pt}lrcccc} |
|
1406 |
%% table text |
|
1407 |
%% \end{tabular} |
|
1408 |
%% \end{table} |
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1409 |
||
1410 |
%% For two column figures and tables, use the following: |
|
1411 |
||
1412 |
%% \begin{figure*} |
|
1413 |
%% \caption{Almost Sharp Front}\label{afoto} |
|
1414 |
%% \end{figure*} |
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1415 |
||
1416 |
%% \begin{table*} |
|
1417 |
%% \caption{Repeat length of longer allele by age of onset class} |
|
1418 |
%% \begin{tabular}{ccc} |
|
1419 |
%% table text |
|
1420 |
%% \end{tabular} |
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1421 |
%% \end{table*} |
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1422 |
||
1423 |
\end{document} |
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1424 |