# HG changeset patch # User scott@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1215393914 0 # Node ID f5e553fbd69318d9fcc681baafb4a8581fca5c99 # Parent 0adb47730c7a342d1496ffe9d4498f0ebd86c23b ... diff -r 0adb47730c7a -r f5e553fbd693 bibliography/bibliography.bib --- a/bibliography/bibliography.bib Sun Jul 06 04:33:51 2008 +0000 +++ b/bibliography/bibliography.bib Mon Jul 07 01:25:14 2008 +0000 @@ -20,6 +20,24 @@ note = {draft available at \url{http://canyon23.net/math/tc.pdf}}, } +@article {MR1854636, + AUTHOR = {Keller, Bernhard}, + TITLE = {Introduction to {$A$}-infinity algebras and modules}, + JOURNAL = {Homology Homotopy Appl.}, + FJOURNAL = {Homology, Homotopy and Applications}, + VOLUME = {3}, + YEAR = {2001}, + NUMBER = {1}, + PAGES = {1--35 (electronic)}, + ISSN = {1512-0139}, + MRCLASS = {18E30 (18D50 18G40 55P43 55U35)}, + MRNUMBER = {MR1854636 (2004a:18008a)}, +MRREVIEWER = {Ulrike Tillmann}, + note = {\mathscinet{MR1854636} \arxiv{math.RA/9910179}}, +} + + + @article {MR1917056, AUTHOR = {Bar-Natan, Dror}, TITLE = {On {K}hovanov's categorification of the {J}ones polynomial}, diff -r 0adb47730c7a -r f5e553fbd693 blob1.tex --- a/blob1.tex Sun Jul 06 04:33:51 2008 +0000 +++ b/blob1.tex Mon Jul 07 01:25:14 2008 +0000 @@ -1180,9 +1180,9 @@ We now define $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$, first giving an opaque formula, then explaining the combinatorics behind it. \begin{align} \notag \bdy(\tm_k(a_1 & \tensor \cdots \tensor a_k)) = \\ -\label{eq:bdy-tm-k-1} & \phantom{+} \sum_{\ell'=0}^{k-1} (-1)^{\sum_{j=1}^{\ell'} \deg(a_j)} \tm_k(a_1 \tensor \cdots \tensor \bdy a_{\ell'+1} \tensor \cdots \tensor a_k) + \\ +\label{eq:bdy-tm-k-1} & \phantom{+} \sum_{\ell'=0}^{k-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_k(a_1 \tensor \cdots \tensor \bdy a_{\ell'+1} \tensor \cdots \tensor a_k) + \\ \label{eq:bdy-tm-k-2} & + \sum_{\ell=1}^{k-1} \tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k) + \\ -\label{eq:bdy-tm-k-3} & + \sum_{\ell=1}^{k-1} \sum_{\ell'=0}^{l-1} \tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell + 1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell + 1}) \tensor \cdots \tensor a_k) +\label{eq:bdy-tm-k-3} & + \sum_{\ell=1}^{k-1} \sum_{\ell'=0}^{l-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell + 1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell + 1}) \tensor \cdots \tensor a_k) \end{align} The first set of terms in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ just have $\bdy$ acting on each argument $a_i$. The terms appearing in \eqref{eq:bdy-tm-k-2} and \eqref{eq:bdy-tm-k-3} are indexed by trees with $2$ vertices on $k+1$ leaves. @@ -1198,12 +1198,11 @@ where again $\ell + 1$ is the number of branches entering the rightmost vertex, $k-\ell+1$ is the number of branches entering the other vertex, and $\ell'$ is the number of edges meeting the rightmost vertex which start to the left of the other vertex. For example, we have \begin{align*} -\bdy(\tm_2(a \tensor b)) & = \left(\tm_2(\bdy a \tensor b) + \tm_2(a \tensor \bdy b)\right) + \\ - & \qquad + a \tensor b + \\ - & \qquad + m_2(a \tensor b) \\ -\bdy(\tm_3(a \tensor b \tensor c)) & = \left(\tm_3(\bdy a \tensor b \tensor c) + \tm_3(a \tensor \bdy b \tensor c) + \tm_3(a \tensor b \tensor \bdy c)\right) + \\ - & \qquad + \left(\tm_2(a \tensor b) \tensor c + a \tensor \tm_2(b \tensor c)\right) + \\ - & \qquad + \left(\tm_2(m_2(a \tensor b) \tensor c) + \tm_2(a, m_2(b \tensor c)) + m_3(a \tensor b \tensor c)\right) +\bdy(\tm_2(a \tensor b)) & = \left(\tm_2(\bdy a \tensor b) + (-1)^{\abs{a}} \tm_2(a \tensor \bdy b)\right) + \\ + & \qquad - a \tensor b + m_2(a \tensor b) \\ +\bdy(\tm_3(a \tensor b \tensor c)) & = \left(- \tm_3(\bdy a \tensor b \tensor c) + (-1)^{\abs{a} + 1} \tm_3(a \tensor \bdy b \tensor c) + (-1)^{\abs{a} + \abs{b} + 1} \tm_3(a \tensor b \tensor \bdy c)\right) + \\ + & \qquad + \left(- \tm_2(a \tensor b) \tensor c + a \tensor \tm_2(b \tensor c)\right) + \\ + & \qquad + \left(- \tm_2(m_2(a \tensor b) \tensor c) + \tm_2(a, m_2(b \tensor c)) + m_3(a \tensor b \tensor c)\right) \end{align*} \begin{align*} \bdy(& \tm_4(a \tensor b \tensor c \tensor d)) = \left(\tm_4(\bdy a \tensor b \tensor c \tensor d) + \cdots + \tm_4(a \tensor b \tensor c \tensor \bdy d)\right) + \\ @@ -1237,8 +1236,8 @@ \bdy \tm(\T) & = \ssum{2} \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} + \ssum{3} \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\ \intertext{and we calculate} \notag -\bdy^2 \tm(\T) & = \ssum{2} (\bdy \tm(\T)) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} \\ -\notag & \qquad + \ssum{2} \tm(\T) \tensor (\bdy \tm(\T)) \times \sigma_{0;l_1,l_2} \\ +\bdy^2 \tm(\T) & = \ssum{2} \bdy \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} \\ +\notag & \qquad + \ssum{2} \tm(\T) \tensor \bdy \tm(\T) \times \sigma_{0;l_1,l_2} \\ \notag & \qquad + \ssum{3} \bdy \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\ \label{eq:d21} & = \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2,l_3} \sigma_{0;l_1,l_2} \\ \label{eq:d22} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2+l_3,l_4} \tau_{0;l_1,l_2,l_3} \\ diff -r 0adb47730c7a -r f5e553fbd693 code/signs.nb --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/code/signs.nb Mon Jul 07 01:25:14 2008 +0000 @@ -0,0 +1,1668 @@ +(* Content-type: application/mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* CreatedBy='Mathematica 6.0' *) + +(*CacheID: 234*) +(* Internal cache information: +NotebookFileLineBreakTest +NotebookFileLineBreakTest +NotebookDataPosition[ 145, 7] +NotebookDataLength[ 51807, 1659] +NotebookOptionsPosition[ 49044, 1566] +NotebookOutlinePosition[ 49407, 1582] +CellTagsIndexPosition[ 49364, 1579] +WindowFrame->Normal +ContainsDynamic->False*) + +(* Beginning of Notebook Content *) +Notebook[{ +Cell[BoxData[ 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-\usepackage{ifpdf} - -%\ifpdf -%\usepackage[pdftex,all,color]{xy} -%\else -\usepackage[all,color]{xy} -%\fi - -\SelectTips{cm}{} -% This may speed up compilation of complex documents with many xymatrices. -%\CompileMatrices - -% ---------------------------------------------------------------- -\vfuzz2pt % Don't report over-full v-boxes if over-edge is small -\hfuzz2pt % Don't report over-full h-boxes if over-edge is small -% ---------------------------------------------------------------- - -\def\RCS$#1: #2 ${\expandafter\def\csname RCS#1\endcsname{#2}} -\RCS$Revision$ -\RCS$Date$ -\newcommand{\versioninfo}{Version \RCSRevision; Last commit \RCSDate} - -% diagrams ------------------------------------------------------- -% figures --------------------------------------------------------- -%%% borrowed from Dror's cobordisms paper, use this to include eps or pdf graphics. -\ifpdf -\newcommand{\pathtodiagrams}{\pathtotrunk diagrams/pdf/} -\else -\newcommand{\pathtodiagrams}{\pathtotrunk diagrams/eps/} -\fi - -\newcommand{\mathfig}[2]{{\hspace{-3pt}\begin{array}{c}% - \raisebox{-2.5pt}{\includegraphics[width=#1\textwidth]{\pathtodiagrams #2}}% -\end{array}\hspace{-3pt}}} -\newcommand{\reflectmathfig}[2]{{\hspace{-3pt}\begin{array}{c}% - \raisebox{-2.5pt}{\reflectbox{\includegraphics[width=#1\textwidth]{\pathtodiagrams #2}}}% -\end{array}\hspace{-3pt}}} -\newcommand{\rotatemathfig}[3]{{\hspace{-3pt}\begin{array}{c}% - \raisebox{-2.5pt}{\rotatebox{#2}{\includegraphics[height=#1\textwidth]{\pathtodiagrams #3}}}% -\end{array}\hspace{-3pt}}} -\newcommand{\placefig}[2]{\includegraphics[width=#1\linewidth]{\pathtodiagrams #2}} - -\ifpdf -\usepackage[pdftex,plainpages=false,hypertexnames=false,pdfpagelabels]{hyperref} -\else -\usepackage[dvips,plainpages=false,hypertexnames=false]{hyperref} -\fi -\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\tt arXiv:\nolinkurl{#1}}} -\newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{{\tt DOI:#1}}} -\newcommand{\mathscinet}[1]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\tt #1}} - - -% THEOREMS ------------------------------------------------------- -\theoremstyle{plain} -\newtheorem*{fact}{Fact} -\newtheorem{prop}{Proposition}[section] -\newtheorem{conj}[prop]{Conjecture} -\newtheorem{thm}[prop]{Theorem} -\newtheorem{lem}[prop]{Lemma} -\newtheorem{lemma}[prop]{Lemma} -\newtheorem{cor}[prop]{Corollary} -\newtheorem*{cor*}{Corollary} -\newtheorem*{exc}{Exercise} -%\theoremstyle{definition} -\newtheorem{defn}[prop]{Definition} % numbered definition -\newtheorem*{defn*}{Definition} % unnumbered definition -\newtheorem{question}{Question} -\newtheorem{property}{Property} -\newenvironment{rem}{\noindent\textsl{Remark.}}{} % perhaps looks better than rem above? -\numberwithin{equation}{section} -%\numberwithin{figure}{section} - -% Marginal notes in draft mode ----------------------------------- -\newcommand{\scott}[1]{\stepcounter{comment}{{\color{blue} $\star^{(\arabic{comment})}$}}\marginpar{\color{blue} $\star^{(\arabic{comment})}$ \usefont{T1}{scott}{m}{n} #1 --S}} % draft mode +%auto-ignore +%this ensures the arxiv doesn't try to start TeXing here. + +\usepackage{amsmath,amssymb,amsfonts} +\usepackage{ifpdf} + +%\ifpdf +%\usepackage[pdftex,all,color]{xy} +%\else +\usepackage[all,color]{xy} +%\fi + +\SelectTips{cm}{} +% This may speed up compilation of complex documents with many xymatrices. +%\CompileMatrices + +% ---------------------------------------------------------------- +\vfuzz2pt % Don't report over-full v-boxes if over-edge is small +\hfuzz2pt % Don't report over-full h-boxes if over-edge is small +% ---------------------------------------------------------------- + +\def\RCS$#1: #2 ${\expandafter\def\csname RCS#1\endcsname{#2}} +\RCS$Revision$ +\RCS$Date$ +\newcommand{\versioninfo}{Version \RCSRevision; Last commit \RCSDate} + +% diagrams ------------------------------------------------------- +% figures --------------------------------------------------------- +%%% borrowed from Dror's cobordisms paper, use this to include eps or pdf graphics. +\ifpdf +\newcommand{\pathtodiagrams}{\pathtotrunk diagrams/pdf/} +\else +\newcommand{\pathtodiagrams}{\pathtotrunk diagrams/eps/} +\fi + +\newcommand{\mathfig}[2]{{\hspace{-3pt}\begin{array}{c}% + \raisebox{-2.5pt}{\includegraphics[width=#1\textwidth]{\pathtodiagrams #2}}% +\end{array}\hspace{-3pt}}} +\newcommand{\reflectmathfig}[2]{{\hspace{-3pt}\begin{array}{c}% + \raisebox{-2.5pt}{\reflectbox{\includegraphics[width=#1\textwidth]{\pathtodiagrams #2}}}% +\end{array}\hspace{-3pt}}} +\newcommand{\rotatemathfig}[3]{{\hspace{-3pt}\begin{array}{c}% + \raisebox{-2.5pt}{\rotatebox{#2}{\includegraphics[height=#1\textwidth]{\pathtodiagrams #3}}}% +\end{array}\hspace{-3pt}}} +\newcommand{\placefig}[2]{\includegraphics[width=#1\linewidth]{\pathtodiagrams #2}} + +\ifpdf +\usepackage[pdftex,plainpages=false,hypertexnames=false,pdfpagelabels]{hyperref} +\else +\usepackage[dvips,plainpages=false,hypertexnames=false]{hyperref} +\fi +\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\tt arXiv:\nolinkurl{#1}}} +\newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{{\tt DOI:#1}}} +\newcommand{\mathscinet}[1]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\tt #1}} + + +% THEOREMS ------------------------------------------------------- +\theoremstyle{plain} +\newtheorem*{fact}{Fact} +\newtheorem{prop}{Proposition}[section] +\newtheorem{conj}[prop]{Conjecture} +\newtheorem{thm}[prop]{Theorem} +\newtheorem{lem}[prop]{Lemma} +\newtheorem{lemma}[prop]{Lemma} +\newtheorem{cor}[prop]{Corollary} +\newtheorem*{cor*}{Corollary} +\newtheorem*{exc}{Exercise} +%\theoremstyle{definition} +\newtheorem{defn}[prop]{Definition} % numbered definition +\newtheorem*{defn*}{Definition} % unnumbered definition +\newtheorem{question}{Question} +\newtheorem{property}{Property} +\newenvironment{rem}{\noindent\textsl{Remark.}}{} % perhaps looks better than rem above? +\numberwithin{equation}{section} +%\numberwithin{figure}{section} + +% Marginal notes in draft mode ----------------------------------- +\newcommand{\scott}[1]{\stepcounter{comment}{{\color{blue} $\star^{(\arabic{comment})}$}}\marginpar{\color{blue} $\star^{(\arabic{comment})}$ \usefont{T1}{scott}{m}{n} #1 --S}} % draft mode \newcommand{\kevin}[1]{\stepcounter{comment}{\color{green} $\star^{(\arabic{comment})}$}\marginpar{\color{green} $\star^{(\arabic{comment})}$ #1 --K}} % draft mode -\newcommand{\comment}[1]{\stepcounter{comment}$\star^{(\arabic{comment})}$\marginpar{\tiny $\star^{(\arabic{comment})}$ #1}} % draft mode -\newcounter{comment} -\newcommand{\noop}[1]{} -\newcommand{\todo}[1]{\textbf{TODO: #1}} - -% \mathrlap -- a horizontal \smash-------------------------------- -% For comparison, the existing overlap macros: -% \def\llap#1{\hbox to 0pt{\hss#1}} -% \def\rlap#1{\hbox to 0pt{#1\hss}} -\def\clap#1{\hbox to 0pt{\hss#1\hss}} -\def\mathllap{\mathpalette\mathllapinternal} -\def\mathrlap{\mathpalette\mathrlapinternal} -\def\mathclap{\mathpalette\mathclapinternal} -\def\mathllapinternal#1#2{% -\llap{$\mathsurround=0pt#1{#2}$}} -\def\mathrlapinternal#1#2{% -\rlap{$\mathsurround=0pt#1{#2}$}} -\def\mathclapinternal#1#2{% -\clap{$\mathsurround=0pt#1{#2}$}} - -% MATH ----------------------------------------------------------- -\newcommand{\Natural}{\mathbb N} -\newcommand{\Integer}{\mathbb Z} -\newcommand{\Rational}{\mathbb Q} -\newcommand{\Real}{\mathbb R} -\newcommand{\Complex}{\mathbb C} -\newcommand{\Field}{\mathbb F} - -\newcommand{\Id}{\boldsymbol{1}} -\renewcommand{\imath}{\mathfrak{i}} -\renewcommand{\jmath}{\mathfrak{j}} - -\newcommand{\qRing}{\Integer[q,q^{-1}]} -\newcommand{\qMod}{\qRing-\operatorname{Mod}} -\newcommand{\ZMod}{\Integer-\operatorname{Mod}} - -\newcommand{\into}{\hookrightarrow} -\newcommand{\onto}{\twoheadrightarrow} -\newcommand{\iso}{\cong} -\newcommand{\actsOn}{\circlearrowright} -\newcommand{\isoto}{\xrightarrow{\iso}} -\newcommand{\quismto}{\xrightarrow[\text{q.i.}]{\iso}} - -\newcommand{\htpy}{\simeq} - -\newcommand{\restrict}[2]{#1{}_{\mid #2}{}} -\newcommand{\set}[1]{\left\{#1\right\}} -\newcommand{\setc}[2]{\left\{#1 \;\left| \; #2 \right. \right\}} -\newcommand{\relations}[2]{\left<#1 \;\left| \; #2 \right. \right>} -\newcommand{\cone}[3]{C\left(#1 \xrightarrow{#2} #3\right)} -\newcommand{\pairing}[2]{\left\langle#1 ,#2 \right\rangle} - -\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} -\newcommand{\norm}[1]{\left|\left|#1\right|\right|} - -\newcommand{\card}[1]{\sharp{#1}} - -\newcommand{\bdy}{\partial} -\newcommand{\compose}{\circ} -\newcommand{\eset}{\emptyset} - -\newcommand{\Cat}{\mathcal{C}} - -\newcommand{\psmallmatrix}[1]{\left(\begin{smallmatrix} #1 \end{smallmatrix}\right)} - -\newcommand{\qiq}[2]{[#1]_{#2}} -\newcommand{\qi}[1]{\qiq{#1}{q}} -\newcommand{\qdim}{\operatorname{dim_q}} - -\newcommand{\directSum}{\oplus} -\newcommand{\DirectSum}{\bigoplus} -\newcommand{\tensor}{\otimes} -\newcommand{\Tensor}{\bigotimes} - -\newcommand{\db}[1]{\left(\left(#1\right)\right)} - -\newcommand{\Mat}[1]{\mathbf{Mat}\left(#1\right)} -\newcommand{\Inv}[1]{\operatorname{Inv}\left(#1\right)} -\newcommand{\Hom}[3]{\operatorname{Hom}_{#1}\left(#2,#3\right)} -\newcommand{\End}[1]{\operatorname{End}\left(#1\right)} - -\newcommand{\CD}[1]{C_*(\Diff(#1))} - -\newcommand{\directSumStack}[2]{{\begin{matrix}#1 \\ \DirectSum \\#2\end{matrix}}} -\newcommand{\directSumStackThree}[3]{{\begin{matrix}#1 \\ \DirectSum \\#2 \\ \DirectSum \\#3\end{matrix}}} - -\newcommand{\grading}[1]{{\color{blue}\{#1\}}} -\newcommand{\shift}[1]{\left[#1\right]} - -\newenvironment{narrow}[2]{% -\vspace{-0.4cm}% horrible hack, by scott % this only seems to be appropriate in beamer mode... -\begin{list}{}{% -\setlength{\topsep}{0pt}% -\setlength{\leftmargin}{#1}% -\setlength{\rightmargin}{#2}% -\setlength{\listparindent}{\parindent}% -\setlength{\itemindent}{\parindent}% -\setlength{\parsep}{\parskip}}% -\item[]}{\end{list}} -% ---------------------------------------------------------------- +\newcommand{\comment}[1]{\stepcounter{comment}$\star^{(\arabic{comment})}$\marginpar{\tiny $\star^{(\arabic{comment})}$ #1}} % draft mode +\newcounter{comment} +\newcommand{\noop}[1]{} +\newcommand{\todo}[1]{\textbf{TODO: #1}} + +% \mathrlap -- a horizontal \smash-------------------------------- +% For comparison, the existing overlap macros: +% \def\llap#1{\hbox to 0pt{\hss#1}} +% \def\rlap#1{\hbox to 0pt{#1\hss}} +\def\clap#1{\hbox to 0pt{\hss#1\hss}} +\def\mathllap{\mathpalette\mathllapinternal} +\def\mathrlap{\mathpalette\mathrlapinternal} +\def\mathclap{\mathpalette\mathclapinternal} +\def\mathllapinternal#1#2{% +\llap{$\mathsurround=0pt#1{#2}$}} +\def\mathrlapinternal#1#2{% +\rlap{$\mathsurround=0pt#1{#2}$}} +\def\mathclapinternal#1#2{% +\clap{$\mathsurround=0pt#1{#2}$}} + +% MATH ----------------------------------------------------------- +\newcommand{\Natural}{\mathbb N} +\newcommand{\Integer}{\mathbb Z} +\newcommand{\Rational}{\mathbb Q} +\newcommand{\Real}{\mathbb R} +\newcommand{\Complex}{\mathbb C} +\newcommand{\Field}{\mathbb F} + +\newcommand{\Id}{\boldsymbol{1}} +\renewcommand{\imath}{\mathfrak{i}} +\renewcommand{\jmath}{\mathfrak{j}} + +\newcommand{\qRing}{\Integer[q,q^{-1}]} +\newcommand{\qMod}{\qRing-\operatorname{Mod}} +\newcommand{\ZMod}{\Integer-\operatorname{Mod}} + +\newcommand{\into}{\hookrightarrow} +\newcommand{\onto}{\twoheadrightarrow} +\newcommand{\iso}{\cong} +\newcommand{\actsOn}{\circlearrowright} +\newcommand{\isoto}{\xrightarrow{\iso}} +\newcommand{\quismto}{\xrightarrow[\text{q.i.}]{\iso}} + +\newcommand{\htpy}{\simeq} + +\newcommand{\restrict}[2]{#1{}_{\mid #2}{}} +\newcommand{\set}[1]{\left\{#1\right\}} +\newcommand{\setc}[2]{\left\{#1 \;\left| \; #2 \right. \right\}} +\newcommand{\relations}[2]{\left<#1 \;\left| \; #2 \right. \right>} +\newcommand{\cone}[3]{C\left(#1 \xrightarrow{#2} #3\right)} +\newcommand{\pairing}[2]{\left\langle#1 ,#2 \right\rangle} + +\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} +\newcommand{\norm}[1]{\left|\left|#1\right|\right|} +\newcommand{\abs}[1]{\left|#1\right|} + +\newcommand{\card}[1]{\sharp{#1}} + +\newcommand{\bdy}{\partial} +\newcommand{\compose}{\circ} +\newcommand{\eset}{\emptyset} + +\newcommand{\Cat}{\mathcal{C}} + +\newcommand{\psmallmatrix}[1]{\left(\begin{smallmatrix} #1 \end{smallmatrix}\right)} + +\newcommand{\qiq}[2]{[#1]_{#2}} +\newcommand{\qi}[1]{\qiq{#1}{q}} +\newcommand{\qdim}{\operatorname{dim_q}} + +\newcommand{\directSum}{\oplus} +\newcommand{\DirectSum}{\bigoplus} +\newcommand{\tensor}{\otimes} +\newcommand{\Tensor}{\bigotimes} + +\newcommand{\db}[1]{\left(\left(#1\right)\right)} + +\newcommand{\Mat}[1]{\mathbf{Mat}\left(#1\right)} +\newcommand{\Inv}[1]{\operatorname{Inv}\left(#1\right)} +\newcommand{\Hom}[3]{\operatorname{Hom}_{#1}\left(#2,#3\right)} +\newcommand{\End}[1]{\operatorname{End}\left(#1\right)} + +\newcommand{\CD}[1]{C_*(\Diff(#1))} + +\newcommand{\directSumStack}[2]{{\begin{matrix}#1 \\ \DirectSum \\#2\end{matrix}}} +\newcommand{\directSumStackThree}[3]{{\begin{matrix}#1 \\ \DirectSum \\#2 \\ \DirectSum \\#3\end{matrix}}} + +\newcommand{\grading}[1]{{\color{blue}\{#1\}}} +\newcommand{\shift}[1]{\left[#1\right]} + +\newenvironment{narrow}[2]{% +\vspace{-0.4cm}% horrible hack, by scott % this only seems to be appropriate in beamer mode... +\begin{list}{}{% +\setlength{\topsep}{0pt}% +\setlength{\leftmargin}{#1}% +\setlength{\rightmargin}{#2}% +\setlength{\listparindent}{\parindent}% +\setlength{\itemindent}{\parindent}% +\setlength{\parsep}{\parskip}}% +\item[]}{\end{list}} +% ----------------------------------------------------------------