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%%%%%%   General LaTeX commands for WMath
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%   Fonts
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\newcommand{\EuScript}[1]{\mathcal{#1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%   Brackets and Containers
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%
\newcommand{\AnglBr}[1]{\left\langle#1\right\rangle}	% angle bracket of #1
\newcommand{\SqBr}[1]{\left[#1\right]} % square brackets
\newcommand{\hy}{\text{-}}			% hypen in math environment
\newcommand{\imes}{\hskip -.2mm\times\hskip -.2mm} 	% narrower \times
% arrows
\newcommand{\llongrightarrow}{-\hskip -2mm \longrightarrow}		% gives ---->
\newcommand{\lllongrightarrow}{-\hskip -1mm-\hskip -2mm \longrightarrow}	% gives ------->
\newcommand{\XRA}[1]{\xrightarrow{\ #1\ }}
\newcommand{\XLA}[1]{\xleftarrow{\ #1\ }}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Symbols
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\DefEq}{:=}   		% := by definition equal
\Annotate[text]{\DefEq}{left hand side is defined to be equal to right hand side}
\Annotate[speech]{\DefEq}{is defined to be equal to}
%
\newcommand{\EqDef}{=:}   		% =: equal by definition
\Annotate[text]{\EqDef}{right hand side is defined to be equal to left hand side}
\Annotate[speech]{\EqDef}{is defined to be equal to}
%
\newcommand{\st}{:\hskip -.8pt\mid\hskip -.75pt :}	% "such that" symbol
\Annotate[text]{\st}{such that}
\Annotate[speech]{\st}{such that}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%		General Categorical
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\from}{\colon}			% colon in description of a function
\Annotate[text]{\from}{from}
\Annotate[speech]{\from}{from}
%
\newcommand{\Comp}{\circ}
\Annotate[text]{\Comp}{following}
\Annotate[speech]{\Comp}{following}
%
\newcommand{\prdct}{\hskip -.1em\times\hskip -.1em}
\Annotate[text]{\prdct}{product}
\Annotate[speech]{\prdct}{product}
%
\newcommand{\Prdct}[2]{#1\hskip -.1em\times\hskip -.1em #2}
\Annotate[text]{\Prdct}{\(#1\) product \(#2\)}
\Annotate[speech]{\Prdct}{\(#1\) product \(#2\)}
\newcommand{\PPrdct}[3]{#1\hskip -.1em\times\hskip -.1em #2\hskip -.1em\times\hskip -.1em #3}
\Annotate[text]{\PPrdct}{product of \(#1,#2,#3\)}
%
\newcommand{\coprdct}{\hskip -.1em\amalg\hskip -.1em}
\Annotate[text]{\coprdct}{coproduct}
\Annotate[speech]{\coprdct}{coproduct}
%
\newcommand{\CoPrdct}[2]{#1\hskip -.1em\amalg\hskip -.1em #2} % #1 product #2
\Annotate[text]{\CoPrdct}{\(#1\) coproduct \(#2\)}
\Annotate[speech]{\CoPrdct}{\(#1\) coproduct \(#2\)}
%
\newcommand{\dom}{\mathit{dom}}							% Domain of a morphism
\Annotate[text]{\dom}{domain}
\newcommand{\Domain}{\mathit{dom}}					% Domain function
\Annotate[text]{\Domain}{domain}
\newcommand{\DomainOf}[1]{\mathit{dom}(#1)}			% Domain of #1
\Annotate[text]{\DomainOf}{domain of \(#1\)}
\newcommand{\codom}{\mathit{codom}}					% Codomain of a morphism
\Annotate[text]{\codom}{codomain}
\newcommand{\CoDomain}{\mathit{codom}}			% Codomain function
\Annotate[text]{\CoDomain}{codomain}
\newcommand{\CoDomainOf}[1]{\mathit{codom}(#1)}	% Codomain of morphism #1
\Annotate[text]{\CoDomainOf}{codomain of \(#1\)}
%
%
%
\newcommand{\Mono}{\xrightarrow{\text{mono}}}
\Annotate[text]{\Mono}{monomorphism}
%
\newcommand{\Epi}{\xrightarrow{\text{epi}}}
\Annotate[text]{\Epi}{epimorphism}
%
\newcommand{\MonoL}{\xleftarrow{\text{mono}}}
\Annotate[text]{\MonoL}{monomorphism}
%
\newcommand{\EpiL}{\xleftarrow{\text{epi}}}
\Annotate[text]{\EpiL}{epimorphism}
%
\newcommand{\lepi}{\longrightarrow\hskip -3mm \rightarrow}	% epimorphism; gives --->>
\Annotate[text]{\lepi}{epimorphism}
%
\newcommand{\Iso}{\xrightarrow{\ \cong\ }}
\Annotate[text]{\Iso}{isomorphism}
%
\newcommand{\iso}{\rightarrowtail\hskip -9pt \rightarrow}		% monomorphism; gives >-->>
\Annotate[text]{\iso}{isomorphism}
%
%
\newcommand{\restrict}{\hskip -1mm \mid\hskip -1mm} 	% restriction (deprecated)
\newcommand{\RstrctnOf}[1]{#1\mid}
\Annotate[text]{\RstrctnOf}{restriction of \(#1\)}
\newcommand{\RstrctnOfTo}[2]{#1\mid_{#2}}
\Annotate[text]{\RstrctnOfTo}{restriction of \(#1\) to \(#2\)}
%
\newcommand{\IdMap}{\mathit{Id}}							% Identity map
\Annotate[text]{\IdMap}{identity map}
\newcommand{\IdMapOn}[1]{\mathit{Id}_{#1}}		% Identity on #1
\Annotate[text]{\IdMapOn}{identity map on \(#1\)}
\newcommand{\idmap}{\mathit{Id}}						% Identity (deprecated)
\Annotate[text]{\idmap}{identity map}				% Identity (deprecated)
\newcommand{\IsoMrphc}{\cong}								% isomorphic
\Annotate[text]{\IsoMrphc}{is isomorphic to}
%
\newcommand{\Prjctn}{\mathit{pr}}
\Annotate[text]{\Prjctn}{projection of a product}
\newcommand{\PrjctnOnto}[1]{\mathit{pr}_{#1}}
\Annotate[text]{\PrjctnOnto}{projection of a product onto \(#1\)-(factor)}
\newcommand{\PrjctnOntoOf}[2]{\mathit{pr}_{#1}(#2)}
\Annotate[text]{\PrjctnOntoOf}{projection of a product onto \(#1\)-(factor) evaluated at \(#2\)}
\newcommand{\PrjctnOntoOfCoord}[2]{\mathit{pr}_{#1}#2}
\Annotate[text]{\PrjctnOntoOfCoord}{projection of a product onto \(#1\)-(factor) evaluated at \(#2\)}
%%%
\newcommand{\Inclsn}{\mathit{inc}}
\Annotate[text]{\Inclsn}{inclusion mapping}
\newcommand{\InclsnOf}[1]{\mathit{inc}_{#1}}
\Annotate[text]{\InclsnOf}{inclusion of \(#1\)}
\newcommand{\InclsnOfAt}[2]{\mathit{inc}_{#1}\left(#2\right)}	% 
\Annotate[text]{\InclsnOfAt}{inclusion of \(#1\) evaluated at \(#2\)}
\newcommand{\CoordInclsn}[1]{\mathit{inc}_{#1}}
\Annotate[text]{\CoordInclsn}{\(#1\)-coordinate inclusion into product ...}
\newcommand{\CoordInclsnOf}[2]{\mathit{inc}_{#1}(#2)}		% #1-inclusion into a product of #2
\Annotate[text]{\CoordInclsnOf}{\(#1\)-coordinate inclusion into product ..., evaluated at \(#2\)}
\newcommand{\Eval}{\varepsilon}
\Annotate[text]{\Eval}{evaluation mapping}
%%%
\newcommand{\Dgnl}{\bigtriangleup}
\Annotate[text]{\Dgnl}{diagonal map}
\newcommand{\DgnlOf}[1]{\bigtriangleup_{#1}}
\Annotate[text]{\DgnlOf}{diagonal map of \(#1\)}
\newcommand{\DgnlOn}[1]{\bigtriangleup_{#1}}
\Annotate[text]{\DgnlOn}{diagonal map on \(#1\)}
\newcommand{\CoDgnl}{\nabla}
\Annotate[text]{\CoDgnl}{fold map}
\newcommand{\CoDgnlOn}[1]{\nabla_{#1}}
\Annotate[text]{\CoDgnlOn}{fold map of \(#1\)}
\newcommand{\CoDgnlOf}[1]{\nabla_{#1}}
\Annotate[text]{\CoDgnlOf}{fold map of \(#1\)}
%
\newcommand{\dgnl}{\bigtriangleup}					% Diagonal map (deprecated)
%%%
\newcommand{\CoKer}[1]{\mathit{coker}\left(#1\right)}
\Annotate[text]{\CoKer}{cokernel of \(#1\)}
\newcommand{\CoKerFnctr}{\mathit{coker}}
\Annotate[text]{\CoKerFnctr}{cokernel functor}
\newcommand{\CoKerOf}[1]{\mathit{coker}\left(#1\right)}	% deprecated
\Annotate[text]{\CoKerOf}{cokernel of \(#1\)}
\newcommand{\coker}{\mathit{coker}}					% (deprecated)
%
\newcommand{\Ker}[1]{\mathit{ker}\left( #1 \right)}
\Annotate[text]{\Ker}{kernel of \(#1\)}
\newcommand{\KerFnctr}{\mathit{ker}}
\Annotate[text]{\KerFnctr}{kernel functor}
\newcommand{\KerOf}[1]{\mathit{ker}\left( #1 \right)}		% deprecated
\Annotate[text]{\KerOf}{kernel of \(#1\)}
\newcommand{\ker}{\mathit{ker}}       			% Kernel (deprecated)
%
\newcommand{\AutGrp}[1]{\mathit{Aut}(#1)}		% Automorphism group of #1
\Annotate[text]{\AutGrp}{automorphism group of \(#1\)}
\newcommand{\IsoGrp}[1]{\mathit{Iso}(#1)}		% Isomorphism group of #1
\Annotate[text]{\IsoGrp}{isomorphism group of \(#1\)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%		Sets
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\Set}[1]{\left\{#1\right\}}		% set of elements #1
\Annotate[text]{\Set}{set of}
\newcommand{\SetSlct}[2]{\left\{ #1 \st #2 \right\}}		% set of #1 st #2
\Annotate[text]{\SetSlct}{set of all \( #1 \) such that \( #2 \)}
\Annotate[speech]{\SetSlct}{set of all \( #1 \) such that \( #2 \)}
%
\newcommand{\union}{\cup}
\Annotate[text]{\union}{union}
\newcommand{\Union}[2]{#1\cup #2}
\Annotate[text]{\Union}{\(#1\) union \(#2\)}
\newcommand{\dunion}{\bigcup}				% union of sets in display mode
\Annotate[text]{\dunion}{union}
\newcommand{\FamUnion}[2]{\bigcup_{#1}#2}	
\Annotate[text]{\FamUnion}{union of family \(#2\) over \(#1\)}
%
\newcommand{\intrsctn}{\cap}							% intersection of sets
\Annotate[text]{\intrsctn}{intersection}
\newcommand{\Intrsctn}[2]{#1\cap #2}			% #1 intersection with #2
\Annotate[text]{\Intrsctn}{\(#1\) intersection \(#2\)}
\newcommand{\SetIntrsctn}[2]{#1\cap #2}			
\Annotate[text]{\SetIntrsctn}{\(#1\) intersection \(#2\)}
\newcommand{\FamIntrsctn}[2]{\bigcap_{#1}#2}	% intersection of sets in display mode
\Annotate[text]{\FamIntrsctn}{intersection of \(#2\) over \(#1\)}
%
\newcommand{\stdffrnc}{-}
\Annotate[text]{\stdffrnc}{difference of sets}
\newcommand{\StDffrnc}[2]{#1 - #2}
\Annotate[text]{\StDffrnc}{set \(#1\) minus \(#2\)}
\newcommand{\SetDffrnc}[2]{#1 - #2}				% #1 minus set #2
\Annotate[text]{\SetDffrnc}{set \(#1\) minus \(#2\)}
%
\newcommand{\stprdct}{\hskip -.1em\times\hskip -.1em}  % product of sets
\Annotate[text]{\stprdct}{product}
\newcommand{\StPrdct}[2]{#1\hskip -.2em\times\hskip -.2em #2} % #1 set product #2
\Annotate[text]{\StPrdct}{set \(#1\) product \(#2\)}
%
\newcommand{\PwrSt}{\EuScript{P}}								% Powerset on #1
\Annotate[text]{\PwrSt}{power set}
\newcommand{\PwrStOf}[1]{\EuScript{P}(#1)}			% Powerset on #1
\Annotate[text]{\PwrStOf}{power set on set \(#1\)}
%
\newcommand{\Crdnlty}{\mathit{card}}					% Cardinality
\Annotate[text]{\Crdnlty}{cardinality}
\newcommand{\CrdnltyOf}[1]{\mathit{card}(#1)}  % Cardinality of #1
\Annotate[text]{\CrdnltyOf}{cardinality of set \(#1\)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%		Functions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\Fnctn}[1]{#1}										% function
\Annotate[text]{\Fnctn}{function \(#1\)}
\newcommand{\FnctnOf}[2]{#1\left( #2\right)}	% function #1 of #2
\Annotate[text]{\FnctnOf}{function \(#1\) of \( #2\)}
\newcommand{\FnctnOfTpl}[2]{#1 #2}						% function #1 of #2
\Annotate[text]{\FnctnOfTpl}{function \(#1\) of \( #2\)}
\newcommand{\FnctnDAT}[3]{#1\from #2\to #3}		% function #1:#2 -> #3
\Annotate[text]{\FnctnDAT}{function \(#1\) from \(#2\) to \(#3\)}
\newcommand{\FnctnDAAT}[3]{#1\from #2\longrightarrow #3}
\Annotate[text]{\FnctnDAAT}{function \(#1\) from \(#2\) to \(#3\)}
\newcommand{\FnctnSet}[2]{\mathit{Fun}\left(#1,#2\right)}
\Annotate[text]{\FnctnSet}{set of functions from \(#1\) to \(#2\)}
%
\newcommand{\Img}[1]{\mathit{im}\left(#1\right)}
\Annotate[text]{\Img}{image of \( #1 \)}
\newcommand{\ImgOf}[1]{\mathit{im}\left(#1\right)}		% deprecated
\Annotate[text]{\ImgOf}{image of the function \(#1\)}
\newcommand{\ImgFnctr}{\mathit{im}}
\Annotate[text]{\ImgFnctr}{image}
\newcommand{\TarFnctr}{\mathit{tar}}
\Annotate[text]{\TarFnctr}{target}
\newcommand{\Tar}[1]{\mathit{tar}(#1)}
\Annotate[text]{\Tar}{target of the function \(#1\)}
\newcommand{\TarOf}[1]{\mathit{tar}(#1)}	% deprecated
\Annotate[text]{\TarOf}{target of the function \(#1\)}
%
%	Number System
%
\newcommand{\NNr}{\mathbb{N}}							% natural numbers
\Annotate[text]{\NNr}{natural numbers}
\newcommand{\ZNr}{\mathbb{Z}}							% integers
\Annotate[text]{\ZNr}{integers}
\newcommand{\ZMod}[1]{\mathbb{Z}/#1}
\Annotate[text]{\ZMod}{\(\ZNr\) modulo \(#1\cdot \ZNr\)}
\newcommand{\ZNrSpc}[1]{\mathbb{Z}^{#1}}
\Annotate[text]{\ZNrSpc}{\( #1\)-fold product of the integers}
\newcommand{\QNr}{\mathbb{Q}}
\Annotate[text]{\QNr}{rational numbers}
\newcommand{\QNrSpc}[1]{\mathbb{Q}^{#1}}			% integer lattice of dimension #1
\Annotate[text]{\QNrSpc}{\( #1\)-fold product of the rational numbers}
\newcommand{\RNr}{\mathbb R}							% real numbers
\Annotate[text]{\RNr}{real numbers}
\newcommand{\RNrSpc}[1]{\mathbb{R}^{#1}}	% Vector space R^{#1}
\Annotate[text]{\RNrSpc}{\( \mathbb{R}\) vector space of dimension \(#1\)}
\newcommand{\CNr}{\mathbb{C}}							% complex numbers, with one optional argument for dimension
\Annotate[text]{\CNr}{complex numbers}
\newcommand{\CNrSpc}[1]{\mathbb{C}^{#1}}	% complex numbers, with one optional argument for dimension
\Annotate[text]{\CNrSpc}{complex vector space of dimension \( #1\)}
\newcommand{\QuatNr}{\mathbb{H}}					% quaternion numbers
\Annotate[text]{\QuatNr}{quaternion numbers}
%
\newcommand{\CCIntrvl}[2]{\left[ #1 , #2 \right]} % Interval [#1,#2]
\Annotate[text]{\CCIntrvl}{closed interval from \(#1\) to \(#2\)}
\newcommand{\COIntrvl}[2]{\left[ #1 , #2 \right[} % Interval [#1,#2[
\Annotate[text]{\COIntrvl}{left closed, right open interval from \(#1\) to \(#2\)}
\newcommand{\OCIntrvl}[2]{\left] #1 , #2 \right]} % Interval ]#1,#2]
\Annotate[text]{\OCIntrvl}{left open, right closed interval from \(#1\) to \(#2\)}
\newcommand{\OOIntrvl}[2]{\left] #1 , #2 \right[} % Interval ]#1,#2[
\Annotate[text]{\OOIntrvl}{open interval from \(#1\) to \(#2\)}
%
\newcommand{\NrSgn}[1]{\mathit{sign}(#1)}							% Sign of number #1
\Annotate[text]{\NrSgn}{sign of number \(#1\)}
\newcommand{\Min}[1]{\mathit{min}\left\{ #1 \right\}} % Minimum of #1
\Annotate[text]{\Min}{minimum of \(#1\)}
\newcommand{\Max}[1]{\mathit{max}\left\{ #1 \right\}} % Maximum of #1
\Annotate[text]{\Max}{maximum of \(#1\)}
\newcommand{\Norm}[1]{\left\Vert#1\right\Vert}
\Annotate[text]{\Norm}{norm of \(#1\)}
\newcommand{\NormVect}[1]{\left\Vert \mathbf{#1}\right\Vert}
\Annotate[text]{\NormVect}{norm of \(\mathbf{#1}\)}
\newcommand{\Abs}[1]{\left\vert#1\right\vert}
\Annotate[text]{\Abs}{absolute value of \(#1\)}
%
\newcommand{\LstCmmnMltpl}[1]{\mathit{lcm}(#1)}			% least common multiple of #1
\Annotate[text]{\LstCmmnMltpl}{least common multiple of \(#1\)}
\newcommand{\GrtstCmmnDvsr}[1]{\mathit{gcd}(#1)}		% greatest common divisor of #1
\Annotate[text]{\GrtstCmmnDvsr}{greatest common divisor of \(#1\)}
%
\newcommand{\ZPoly}[1]{\ZNr\SqBr{#1}}			% PolynomialRing Z[#1]
\Annotate[text]{\ZPoly}{\(\ZNr\)-polynomial ring in variable(s) \(#1\)}
\newcommand{\QPoly}[1]{\QNr\SqBr{#1}}			% PolynomialRing Q[#1]
\Annotate[text]{\QPoly}{\(\QNr\)-polynomial ring in variable(s) \(#1\)}
\newcommand{\RPoly}[1]{\RNr\SqBr{#1}}			% PolynomialRing R[#1]
\Annotate[text]{\RPoly}{\(\RNr\)-polynomial ring in variable(s) \(#1\)}
\newcommand{\CPoly}[1]{\CNr\hskip -.2em\left[#1\right]}	% PolynomialRing C[#1]
\Annotate[text]{\CPoly}{\(\CNr\)-polynomial ring in variable(s) \(#1\)}
\newcommand{\PolyDegree}[1]{\mathit{deg}\left(#1\right)}	% degree of polynomial #1
\Annotate[text]{\PolyDegree}{degree of polynomial \(#1\)}
%  Basic complex number constructs
%
\newcommand{\ImPrt}[1]{\mathit{Im}(#1)} 		% imaginary part of complex number #1
\Annotate[text]{\ImPrt}{imaginary part of complex number \(#1\)}
\newcommand{\RePrt}[1]{\mathit{Re}(#1)} 		% real part of complex number #1
\Annotate[text]{\RePrt}{real part of complex number \(#1\)}
\newcommand{\CCnjgt}[1]{\overline{#1}}							% complex conjugate of #1
\Annotate[text]{\CCnjgt}{conjugate of complex number \(#1\)}
\newcommand{\CAbs}[1]{\left\vert #1 \right\vert}		% absolute value of complex number #1
\Annotate[text]{\CAbs}{modulus of complex number \(#1\)}
%
%	Algebraic Constructs
%
\newcommand{\Dim}{\mathit{dim}}
\Annotate[text]{\Dim}{dimension}
\newcommand{\DimOf}[1]{\mathit{dim}(#1)}
\Annotate[text]{\DimOf}{dimension of \(#1\)}
\newcommand{\RDim}{\mathit{dim}_{\mathbb{R}}}
\Annotate[text]{\RDim}{dimension of \(\mathbb{R}\)-vector space}
\newcommand{\RDimOf}[1]{\mathit{dim}_{\mathbb{R}}\left(#1\right)}
\Annotate[text]{\RDimOf}{\(\mathbb{R}\)-dimension of \(#1\)}
\newcommand{\CDim}{\mathit{dim}_{\mathbb{C}}}
\Annotate[text]{\CDim}{dimension of \(\mathbb{C}\)-vector space}
\newcommand{\CDimOf}[1]{\mathit{dim}_{\mathbb{C}}\left(#1\right)}
\Annotate[text]{\CDimOf}{\(\mathbb{C}\)-dimension of \(#1\)}
% below: deprecated
\newcommand{\Dmnsn}{\mathit{dim}}
\Annotate[text]{\Dmnsn}{dimension}
\newcommand{\DmnsnOf}[1]{\mathit{dim}(#1)}
\Annotate[text]{\DmnsnOf}{dimension of \(#1\)}
\newcommand{\DmnsnOver}[1]{\mathit{dim}_{#1}}
\Annotate[text]{\DmnsnOver}{dimension over field \(#1\)}
\Annotate[speech]{\DmnsnOver}{dimension over field \(#1\)}
\newcommand{\DmnsnOverOf}[2]{\mathit{dim}_{#1}(#2)}
\Annotate[text]{\DmnsnOverOf}{dimension of \(#2\) over field \(#1\)}
\Annotate[speech]{\DmnsnOverOf}{dimension of vector space \(#2\) over field \(#1\)}
\newcommand{\Dimnsn}[1]{\mathit{dim}(#1)}		% dimension of #1
\Annotate[text]{\Dimnsn}{dimension of \(#1\)}
\newcommand{\DimnsnOver}[2]{\mathit{dim}_{#2}(#1)}	% dimension of #1 over #2
\Annotate[text]{\DimnsnOver}{dimension of \(#1\) over field \(#2\)}
\Annotate[speech]{\DimnsnOver}{dimension of vector space \(#1\) over field \(#2\)}
%
%	Basic Topological Concepts
%
\newcommand{\Dstnc}[2]{\mathit{dist}\left(#1,#2\right)}	% Distance in a metric space
\Annotate[text]{\Dstnc}{distance from \(#1\) to \(#2\)}
\newcommand{\uInt}{\mathbb{I}}						% unit interval
\Annotate[text]{\uInt}{unit interval \( [0,1]\)}
\newcommand{\uIntSpc}[1]{\mathbb{I}^{#1}}	% unit cube of dimension #1
\Annotate[text]{\uIntSpc}{unit cube in \(\mathbb{R}^{#1}\)}
\newcommand{\uBll}[1]{\mathbb{B}^{#1}}
\Annotate[text]{\uBll}{closed unit ball in \(\mathbb{R}^{#1}\)}
\newcommand{\uDskSpc}[1]{\mathbb{D}^{#1}}	% closed unit disk of in R^#1
\Annotate[text]{\uDskSpc}{closed unit disk in \(\mathbb{R}^{#1}\)}
\newcommand{\uSphr}[1]{\mathbb{S}^{#1}} 	% unit sphere in R^#1
\Annotate[text]{\uSphr}{unit sphere in \(\RNrSpc{#1+1}\)}
\newcommand{\AffSmplx}[1]{\bigtriangleup^{#1}}
\Annotate[text]{\AffSmplx}{affine simplex, dimension  \(#1\)}
\newcommand{\AffSmplxFace}[2]{\bigtriangleup^{#1}_{#2}} % #2-face of Aff{#1}
\Annotate[text]{\AffSmplxFace}{\(#2\)-face of affine \(#1\)-simplex}
\newcommand{\uCrcl}{C}
\Annotate[text]{\uCrcl}{unit circle in \(\CNr\)}
\newcommand{\OBll}[2]{B(#1;#2)}
\Annotate[text]{\OBll}{open ball centered at \(#1\), of radius \(#2\)}
\newcommand{\CBll}[2]{\bar{B}(#1;#2)}			% closed ball centered at #1, radius #2
\Annotate[text]{\CBll}{closed ball centered at \(#1\), radius \(#2\)}
\newcommand{\Sphr}[2]{S(#1;#2)}						% sphere centered at #1, radius #2
\Annotate[text]{\Sphr}{sphere centered at \(#1\), radius \(#2\)}
\newcommand{\Trs}[1]{\mathbb{T}^{#1}}		  % torus of dimension #1
\Annotate[text]{\Trs}{torus of dimension \(#1\)}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% deprecated
\newcommand{\abs}[1]{\left\vert#1\right\vert}			% absolute value of #1
\newcommand{\abr}[1]{\left\langle#1\right\rangle}	% angle bracket of #1
\newcommand{\Id}[1][]{\mathit{Id}_{#1}} 	% Identity on #1
\newcommand{\inc}{\mathit{inc}}						% Inclusion ...
\newcommand{\im}{\mathit{im}}									% image of a map
\newcommand{\lcm}{\operatorname{lcm}}	    % least common multiple
\newcommand{\Degree}[1]{\mathit{deg}\left(#1\right)}	%
\newcommand{\Hom}{\operatorname{Hom}}			% Homomorphisms
\newcommand{\sign}{\operatorname{sign}}		% sign of a number or a permutation
\newcommand{\Im}{\mathit{Im}} 		% imaginary part of complex number
\newcommand{\Re}{\mathit{Re}} 		% real part of complex number
\newcommand{\map}{\operatorname{map}}				    % Mapping space
\newcommand{\inclsn}{\mathit{inc}}					% Inclusion of #1
\newcommand{\prjctn}{\mathit{pr}}						% projection 
\newcommand{\dtrmnnt}{\mathit{det}}     		% determinant
\newcommand{\uDsk}{\mathbb{D}}						% unit disk in C
\newcommand{\uDSK}[1]{\mathbb{D}^{#1}}		% unit disk of dimension #1
% end deprecated
