mathcommon.sty

%%%
%
% mathcommon.sty
%
% My personal list of commonly used LaTeX math packages and macros.
%
% To use the macros defined in this file, add the following line to
% the preamble of a LaTeX document:
%
%   \uspackage{mathcommon}
%
% Authors: Sandro Stucki <sandro.stucki@gmail.com>
%          Ricardo Honorato-Zimmer <rikardo.horo@gmail.com>
%
%
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{mathcommon}%
  [2013/05/11 LaTeX package defining common math macros]


%%% Packages %%%

% AMS Math typesetting
\RequirePackage{amsmath}    % AMS math basics
\RequirePackage{amsfonts}   % Math fonts
\RequirePackage{amssymb}    % Math symbols
\RequirePackage{mathtools}  % Extensible arrows etc.

% Some styles may come with their own definitions proof/theorem
% environments that are not compatible with the amsthm package.
\@ifundefined{proof}{  %
  \RequirePackage{amsthm}  % Package for theorem environments
}

% Double stroke font (e.g. \mathds{1} for the indicator function)
\RequirePackage{dsfont}

% More math symbols (e.g. double-brackets: \llbracket, \rrbracket,
% triangle arrows: \leftarrowtriangle, \rightarrowtriangle, etc.)
\RequirePackage{stmaryrd}

% Using TikZ for diagrams
\RequirePackage{tikz}
\usetikzlibrary{arrows,fit,matrix,positioning,calc}


%%% Theorem environments %%%


% Only attempt to define new theorem environments if the amsthm
% package was loaded.  The use of the \@ifundefined macro below guards
% the theorem environment definitions to avoid overriding previous
% definitions (that are compatible with amsthm), e.g. when using
% special styles or packages such as beamer.
\ifcsname newtheoremstyle\endcsname%
%
% Theorem environments with slanted text.
% \newtheoremstyle{thmslanttext}%
%   {\topsep}% space above the theorem (default)
%   {\topsep}% space below the theorem (default)
%   {\slshape}% body font (slanted)
%   {}% indentation (none)
%   {\bfseries\sffamily}% head font (bold, sans serif)
%   {}% punctuation between head and body (none)
%   {1em}% space after theorem head (1em)
%   {}% head specification (default format)
% \theoremstyle{thmslanttext}%
\theoremstyle{plain}% (use default AMS "plain" style)
\newcounter{null}
\@ifundefined{theorem}{\newtheorem{theorem}{Theorem}[null]}{}
\@ifundefined{proposition}{\newtheorem{proposition}{Proposition}[null]}{}
\@ifundefined{definition}{\newtheorem{definition}{Definition}[null]}{}
\@ifundefined{remark}{\newtheorem{remark}{Remark}[null]}{}
% Uncomment to have section numbers in theorems.
% \@ifundefined{theorem}{\newtheorem{theorem}{Theorem}[section]}{}
% \@ifundefined{proposition}{\newtheorem{proposition}{Proposition}[section]}{}
% \@ifundefined{definition}{\newtheorem{definition}{Definition}[section]}{}
% \@ifundefined{remark}{\newtheorem{remark}{Remark}[section]}{}
%
% Theorem environments that share a counter with 'theorem'
\@ifundefined{corollary}{\newtheorem{corollary}[theorem]{Corollary}}{}
\@ifundefined{lemma}{\newtheorem{lemma}[theorem]{Lemma}}{}
\@ifundefined{conjecture}{\newtheorem{conjecture}[theorem]{Conjecture}}{}
% Theorem environment that doesn't use a counter
\@ifundefined{note}{\newtheorem*{note}{Note}}{}
% Simple hack to have simple numbering of theorems
% Comment this if using section numbers in theorems
\renewcommand{\thetheorem}{\arabic{theorem}}
\renewcommand{\theproposition}{\arabic{proposition}}
\renewcommand{\thedefinition}{\arabic{definition}}
\renewcommand{\theremark}{\arabic{remark}}
%
% Theorem environments with regular text.
% \newtheoremstyle{thmnormaltext}%
%   {\topsep}% space above the theorem (default)
%   {\topsep}% space below the theorem (default)
%   {}% body font (regular)
%   {}% indentation (none)
%   {\bfseries\sffamily}% head font (bold, sans serif)
%   {}% punctuation between head and body (none)
%   {1em}% space after theorem head (1em)
%   {}% head specification (default format)
% \theoremstyle{thmnormaltext}%
\theoremstyle{definition}% (use predefined AMS "definition" style)
\@ifundefined{fact}{\newtheorem{fact}{Fact}[section]}{}
\@ifundefined{hypothesis}{\newtheorem{hypothesis}[theorem]{Hypothesis}}{}
\@ifundefined{example}{\newtheorem{example}{Example}[section]}{}
\@ifundefined{exercise}{\newtheorem{exercise}{Exercise}[section]}{}
\@ifundefined{problem}{\newtheorem{problem}{Problem}[section]}{}
\@ifundefined{notation}{\newtheorem*{notation}{Notation}}{}
%
\fi% End of \ifcsnamenewtheorem\endcsname


%%% Probability theory %%%

% Simple bold versions
\DeclareMathOperator*{\pr}{\mathbf{P}}     % Probability
\renewcommand*{\exp}{\operatorname*{\mathbf{E}}}  % Expectation
\DeclareMathOperator*{\var}{\mathbf{V}}    % Variance
\DeclareMathOperator*{\ind}{\mathbf{1}}    % Indicator function

% "Blackboard bold"/double stroke versions
\DeclareMathOperator*{\PP}{\mathbb{P}}     % Probability
\DeclareMathOperator*{\EE}{\mathbb{E}}     % Expectation
\DeclareMathOperator*{\VV}{\mathbb{V}}     % Variance
\DeclareMathOperator*{\IInd}{\mathds{1}}   % Indicator function

% "Verbose" bold versions
\renewcommand*{\Pr}{\operatorname*{\mathbf{Pr}}}  % Probability
\DeclareMathOperator*{\Var}{\mathbf{Var}}  % Variance
\DeclareMathOperator*{\Cov}{\mathbf{Cov}}  % Covariance

% Average
\newcommand{\avg}[1]{\langle #1 \rangle}


%%% Sets %%%

% Common sets
\renewcommand{\emptyset}{\varnothing}  % Empty set
\newcommand{\NN}{\mathbb{N}}           % Natural numbers
\newcommand{\ZZ}{\mathbb{Z}}           % Integers
\newcommand{\QQ}{\mathbb{Q}}           % Rational numbers
\newcommand{\RR}{\mathbb{R}}           % Real numbers

% Absolute value/cardinality
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}

% Orientations in 3D euclidean space
\DeclareMathOperator{\SO}{SO(3)}

% Computability theory sets
\newcommand{\KK}{\mathbb{K}}     % Undecidable problems
\newcommand{\zero}{\varnothing}  % lowest Turing degree (0)

% Complexity classes
\newcommand{\Poly}{\mathbf{P}}         % P
\newcommand{\NP}{\mathbf{NP}}          % NP
\newcommand{\PSPACE}{\mathbf{PSPACE}}  % PSPACE
\newcommand{\AP}{\mathbf{AP}}          % AP
\newcommand{\PPoly}{\mathbf{P/Poly}}   % P/Poly

% Standalone "such that" colon
\newcommand{\st}{\; | \;}%{:}

% Simple set notation.
\newcommand{\set}[1]{\left\{ #1 \right\}}

% Command for generating "set of <x> s.t. <p(x)>" definitions.  The
% parameters are
%
% #1: the variable name(s) (to appear before the "s.t." colon), and
%
% #2: the predicate deciding set membership for particular instances
% of the variables (to appear after the colon).
%
% The output is of the form "{ #1 : #2 }" with the curly braces
% adjusted to the expression.
\newcommand{\setof}[2]{\set{#1 \st #2}}

% Indexed set
\newcommand{\iset}[1]{\set{#1}_{i \in I}}

% Tuples and sequences
\newcommand{\tuple}[1]{\langle #1 \rangle}
\newcommand{\seq}[1]{#1}

% Set operators
\newcommand{\duni}{+}              % Disjoint union.
\newcommand{\dsum}{\sum}           % Disjoint union ("sum").
\DeclareMathOperator{\inl}{inl}    % Left injection of disjoint union.
\DeclareMathOperator{\inr}{inr}    % Right injection of disjoint union.
% \renewcommand{\setminus}{-}      % Relative complement/difference.
\DeclareMathOperator*{\union}{\cup}        % Union
\DeclareMathOperator*{\bigunion}{\bigcup}  % Union
\DeclareMathOperator*{\intersection}{\cap} % Intersection

% Powerset
\newcommand{\pset}[1]{\mathcal{P}\left(#1\right)}

% Cardinality of a set
\DeclareMathOperator*{\card}{Card}


%%% Functions %%%

% Symbols for domain, co-domain and image of functions
\DeclareMathOperator*{\dom}{dom}    % Domain of a (partial) function
\DeclareMathOperator*{\cod}{cod}    % Co-domain of a function
\DeclareMathOperator*{\img}{img}    % Image/range of a function

% Function composition
\DeclareMathOperator*{\comp}{\circ}

% Restriction of a function #1 to domain #2
\newcommand{\rest}[2]{{#1}|_{#2}}

% Restriction of a function #1 to domain #2 (for big #1's)
\newcommand{\bigrest}[2]{\left.#1\right|_{#2}}

% Symbols for argmax/argmin
\DeclareMathOperator*{\argmax}{argmax}
\DeclareMathOperator*{\argmin}{argmin}

% Derivatives
\newcommand{\ddt}{\frac{d}{dt}}


%%% Vectors %%%

% Typeset vectors using a bold, regular math font
\renewcommand{\vec}[1]{\boldsymbol{\mathbf{#1}}}

% Vector norm
\newcommand{\vnorm}[1]{\lvert #1 \rvert}

% Inner product
\newcommand{\innerprod}[2]{\langle #1, #2 \rangle}


%%% Logic %%%

% Boolean constants
\newcommand{\false}{\mathbf{false}}
\newcommand{\true}{\mathbf{true}}

% Implication
\DeclareMathOperator*{\then}{\Rightarrow}


%%% Categories %%%

% Typeset categories using a bold, regular math font
\newcommand{\cat}[1]{\ensuremath{\boldsymbol{\mathbf{#1}}}}

% Classes in categories
\DeclareMathOperator{\ob}{ob}      % Objects of a category.
\def\hom{\operatorname{hom}}       % Morphisms of a category.
\newcommand{\Span}{\mathbf{Span}}
\newcommand{\Cospan}{\mathbf{Cospan}}
\newcommand{\Cocone}{\mathbf{Co\text{-}cone}}
\newcommand{\Prod}{\mathbf{Prod}}


%% Objects

\newcommand{\init}{\mathbf{0}} % Initial object
\newcommand{\term}{\mathbf{1}} % Terminal object


%% Arrows

% Simple arrows with one argument (the name of the morphism).
\newcommand{\toby}[1]{\stackrel{#1}{\longrightarrow}}
\newcommand{\getsby}[1]{\stackrel{#1}{\longleftarrow}}

% Automatically extended arrows for long names.
\newcommand{\xtoby}[1]{\xrightarrow{#1}}
\newcommand{\xgetsby}[1]{\xleftarrow{#1}}

% Inclusion arrows
\newcommand{\incds}{\hookleftarrow}
\newcommand{\incin}{\hookrightarrow}

% Inclusion arrows with one argument (the name of the morphism).
\newcommand{\incdsby}[1]{\xhookleftarrow{#1}}
\newcommand{\incinby}[1]{\xhookrightarrow{#1}}

% Epimorphism
\newcommand{\toepi}{\twoheadrightarrow}


%%% TikZ %%%

%% Arrows

% Custom, bigger "to" arrowheads for commutative diagrams.
\pgfarrowsdeclare{bigto}{bigto}
{
  \arrowsize=-1.54pt%
  \advance\arrowsize by -1.23\pgflinewidth%
  \pgfarrowsleftextend{+\arrowsize}
  \arrowsize=0.28pt%
  \advance\arrowsize by .55\pgflinewidth%
  \pgfarrowsrightextend{\arrowsize}
}
{
  \arrowsize=0.56pt%
  \advance\arrowsize by .5\pgflinewidth%
  \pgfsetlinewidth{0.8\pgflinewidth}
  \pgfsetdash{}{+0pt}
  \pgfsetroundcap
  \pgfsetroundjoin
  \pgfpathmoveto{\pgfqpoint{-3\arrowsize}{4\arrowsize}}
  \pgfpathcurveto
  {\pgfqpoint{-2.75\arrowsize}{2.5\arrowsize}}
  {\pgfqpoint{-1\arrowsize}{0.5\arrowsize}}
  {\pgfqpoint{0.5\arrowsize}{0pt}}
  \pgfpathcurveto
  {\pgfqpoint{-1\arrowsize}{-0.5\arrowsize}}
  {\pgfqpoint{-2.75\arrowsize}{-2.5\arrowsize}}
  {\pgfqpoint{-3\arrowsize}{-4\arrowsize}}
  \pgfusepathqstroke
}

% Custom, bigger "right to" arrowheads for partial functions.
\pgfarrowsdeclare{right bigto}{right bigto}
{
  \arrowsize=-1.54pt%
  \advance\arrowsize by -1.23\pgflinewidth%
  \pgfarrowsleftextend{+\arrowsize}
  \arrowsize=0.28pt%
  \advance\arrowsize by .55\pgflinewidth%
  \pgfarrowsrightextend{\arrowsize}
}
{
  \arrowsize=0.56pt%
  \advance\arrowsize by .5\pgflinewidth%
  \pgfsetlinewidth{0.8\pgflinewidth}
  \pgfsetdash{}{0pt}
  \pgfsetroundjoin
  \pgfsetroundcap
  \pgfpathmoveto{\pgfqpoint{0.5\arrowsize}{0pt}}
  \pgfpathcurveto
  {\pgfqpoint{-1\arrowsize}{-0.5\arrowsize}}
  {\pgfqpoint{-2.75\arrowsize}{-2.5\arrowsize}}
  {\pgfqpoint{-3\arrowsize}{-4\arrowsize}}
  \pgfusepathqstroke
}

% Custom, bigger "left to" arrowheads for partial functions.
\pgfarrowsdeclare{left bigto}{left bigto}
{
  \arrowsize=-1.54pt%
  \advance\arrowsize by -1.23\pgflinewidth%
  \pgfarrowsleftextend{+\arrowsize}
  \arrowsize=0.28pt%
  \advance\arrowsize by .55\pgflinewidth%
  \pgfarrowsrightextend{\arrowsize}
}
{
  \arrowsize=0.56pt%
  \advance\arrowsize by .5\pgflinewidth%
  \pgfsetlinewidth{0.8\pgflinewidth}
  \pgfsetdash{}{0pt}
  \pgfsetroundjoin
  \pgfsetroundcap
  \pgfpathmoveto{\pgfqpoint{0.5\arrowsize}{0pt}}
  \pgfpathcurveto
  {\pgfqpoint{-1\arrowsize}{0.5\arrowsize}}
  {\pgfqpoint{-2.75\arrowsize}{2.5\arrowsize}}
  {\pgfqpoint{-3\arrowsize}{4\arrowsize}}
  \pgfusepathqstroke
}


%% Styles

% Set the default arrow type to 'stealth' for all drawings in the
% document.
\tikzset{>=stealth}


%% TikZ styles for commutative diagrams

% Normal arrows/homomorphisms (sets defaults for other arrows)
\tikzstyle{hom}=[>=bigto, ->, line width=0.5pt]
\tikzstyle{partialhom}=[hom, >=right bigto]

% Monomorphisms and epimorphisms
\tikzstyle{mono}=[hom, >->]
\tikzstyle{epi}=[hom, ->>]

% Identity arrows/equivalences
\tikzstyle{eq}=[-, double, double equal sign distance]

% Implications/natural transformations
\tikzstyle{nat}=[-implies, double, double equal sign distance]

% TikZ style for the matrices underlying commutation diagrams.
\tikzstyle{cd-mat}=[matrix of math nodes, nodes={anchor=center}]

% Style for edge labels on top of the edge.
\tikzstyle{ontop}=[fill=white, inner sep=3pt]


%% Commutative diagrams

% Commands for drawing pushout/pullback corners next to a node.
% The only parameter specifies the name of the node next to which
% the corner should be drawn.
\newcommand{\cornersize}{.6em}
\newcommand{\cornerNW}[1]{
  \draw[hom,-] #1 ++($(-2*\cornersize, \cornersize)$) --
    ++(0, \cornersize) -- ++($( .95*\cornersize,0)$);}
\newcommand{\cornerNE}[1]{
  \draw[hom,-] #1 ++($( 2*\cornersize, \cornersize)$) --
    ++(0, \cornersize) -- ++($(-.95*\cornersize,0)$);}
\newcommand{\cornerSW}[1]{
  \draw[hom,-] #1 ++($(-2*\cornersize,-\cornersize)$) --
    ++(0,-\cornersize) -- ++($( .95*\cornersize,0)$);}
\newcommand{\cornerSE}[1]{
  \draw[hom,-] #1 ++($( 2*\cornersize,-\cornersize)$) --
    ++(0,-\cornersize) -- ++($(-.95*\cornersize,0)$);}

\newcommand{\fpbcNW}[1]{
  \draw[hom,-] #1 ++($(-2*\cornersize, \cornersize)$) --
    ++($(0, .7*\cornersize)$) -- ++($( .8*\cornersize,0)$);}
\newcommand{\fpbcNE}[1]{
  \draw[hom,-] #1 ++($( 2*\cornersize, \cornersize)$) --
    ++($(0, .7*\cornersize)$) -- ++($(-.8*\cornersize,0)$);}
\newcommand{\fpbcSW}[1]{
  \draw[hom,-] #1 ++($(-2*\cornersize,-\cornersize)$) --
    ++($(0,-.7*\cornersize)$) -- ++($( .8*\cornersize,0)$);}
\newcommand{\fpbcSE}[1]{
  \draw[hom,-] #1 ++($( 2*\cornersize,-\cornersize)$) --
    ++($(0,-.7*\cornersize)$) -- ++($(-.8*\cornersize,0)$);}