preamble.tex

% Of the highest importance:
\renewcommand{\phi}{\varphi}

% Saner abbreviations for font faces (take an argument rather than
% applying to the current context).
\renewcommand{\rm}[1]{\mathrm{#1}}
\newcommand{\bb}[1]{\mathbb{#1}}
\renewcommand{\bf}[1]{\mathbf{#1}}

% Define \cA..\cZ
\definealphabet{c}{\mathcal}{A}{Z}
% Define \bA..\bZ
\definealphabet{b}{\mathbb}{A}{Z}

% Notation for lambda abstractions and function application. $~$
% is the spacing that should be used between arguments.
\newcommand{\lam}[2]{\lambda #1.\ #2}

% Notation for paths, the interval, and its operations.
\newcommand{\is}{\equiv}
\newcommand{\PathP}[3]{\rm{PathP}~ {#1}~ {#2}~ {#3}}
\newcommand{\Path}[3]{\rm{Path}~ {#1}~ {#2}~ {#3}}
\newcommand{\Square}[4]{\rm{Square}~ {#1}~ {#2}~ {#3}~ {#4}}
\newcommand{\ap}[2]{\rm{ap}~ {#1}~ {#2}}

\newcommand{\subst}{\operatorname{subst}}
\newcommand{\ua}{\operatorname{ua}}

\newcommand{\iZ}{\rm{i0}}
\newcommand{\iO}{\rm{i1}}

\newcommand{\ineg}{\lnot}
\newcommand{\imin}{\land}
\newcommand{\imax}{\lor}

\newcommand{\transport}[2]{\rm{transport}~ #1~ #2}
\newcommand{\transp}[3]{\rm{transp}~ #1~ #2~ #3}
\newcommand{\Partial}[2]{\rm{Partial}~ #1~ #2}
\newcommand{\dcomp}{\mathrel{\cdot\cdot}}
\newcommand{\Extn}[2]{{#1}[#2]}

\DeclareMathOperator{\inS}{inS}
\DeclareMathOperator{\outS}{outS}

\newcommand{\thecat}[1]{\mathbf{#1}}     % Names of "concrete" categories
\newcommand{\thebicat}[1]{\mathfrak{#1}} % Names of "concrete" bicategories
\newcommand{\bicat}[1]{\mathbf{#1}}  % Names of variable bicategories
\newcommand{\ca}[1]{\mathcal{#1}}    % Names of variable categories
\newcommand{\ica}[1]{\mathbb{#1}}    % Names of internal categories

\newcommand{\Arr}[1]{%% The arrow category of a category.
  \rm{Arr}(#1)
}
\newcommand{\FAlg}[1]{%% The category of F-algebras
  \rm{Alg}(#1)
}

% \liesover arrow is only allowed in KaTeX:
\newcommand{\liesover}\katex{\mathrel{\htmlClass{liesover}{\hspace{1.366em}}}}

% \yo for よ is only allowed in KaTeX:
\newcommand{\yo}\katex{よ}
\newcommand{\zap}\katex{↯}

\newcommand{\knowncat}[1]{\newcommand{\csname #1\endcsname}{\thecat{#1}}}
\newcommand{\knownbicat}[1]{\newcommand{\csname #1\endcsname}{\thebicat{#1}}}

\newcommand{\ty}{\rm{Type}}
\newcommand{\set}{\rm{Set}}
\newcommand{\prop}{\rm{Prop}}
\newcommand{\psh}{\rm{PSh}}

\DeclareMathOperator{\Sh}{Sh}

\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Id}{Id}

\knowncat{Sets}
\knowncat{Ab}
\knowncat{Graphs}
\knowncat{MarkedGraphs}
\knowncat{Props}
\knowncat{FinSets}
\knowncat{Rings}
\knowncat{Grp}
\knowncat{Rel}
\knowncat{Par}
\knowncat{Pos}
\knownbicat{Cat}
\knownbicat{Topos}
\knownbicat{Grpd}

\newcommand{\strcat}{\Cat_\rm{strict}}
\newcommand{\intcat}{\{0 \le 1\}}
\newcommand{\twocat}{\mathbf{2}}
\newcommand{\igpd}{\text{$\infty$-\textbf{Grpd}}}
\newcommand{\io}{(\infty,1)}

% "Postfix" operators that have a subscript and are annoying to write
% out:
\newcommand{\inv}{^{-1}}
\newcommand{\op}{^{\rm{op}}}

\newcommand{\ot}{\leftarrow}
\renewcommand{\epsilon}{\varepsilon}
\newcommand{\eps}{\varepsilon}

\newcommand{\To}{\Rightarrow}
\newcommand{\later}{\mathop{\blacktriangleright}}
\newcommand{\adj}{\rightleftarrows}
\newcommand{\epi}{\twoheadrightarrow}
\newcommand{\mono}{\hookrightarrow}
\newcommand{\iso}{\cong}
\newcommand{\tto}{\rightrightarrows}
\newcommand{\rel}{\mathrel{\looparrowright}}
\newcommand{\heyt}{\mathrel{\scriptsize{⇨}}} % Heyting implication

\newcommand{\xadj}[2]{\xrightleftarrows[#2]{#1}}
\newcommand{\xto}[1]{\xrightarrow{#1}}
\newcommand{\xot}[1]{\xleftarrow{#1}}
\newcommand{\xepi}[1]{\xtwoheadrightarrow{#1}}
\newcommand{\xmono}[1]{\xhookrightarrow{#1}}
\newcommand{\within}[1]{\overset{#1}{\approx}}

\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator*{\colim}{colim}
\DeclareMathOperator*{\im}{im}
\DeclareMathOperator*{\Sub}{Sub}
\DeclareMathOperator*{\baut}{\mathbf{B}}
\newcommand{\B}[1]{\mathbf{B} #1}
\newcommand{\point}[1]{\bullet_{#1}}
\newcommand{\List}[1]{\operatorname{List}(#1)}

\DeclareMathOperator{\Lan}{Lan}
\DeclareMathOperator{\Ran}{Ran}

\DeclareMathOperator{\src}{src}
\DeclareMathOperator{\tgt}{tgt}
\newcommand{\restrict}[1]{#1^{\downarrow}}

\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\II}{\mathbb{I}}

\renewcommand{\hom}{\mathbf{Hom}}
\newcommand{\ortho}{\mathrel{\bot}}

\newcommand{\Mod}[1][R]{{#1}\text{-}\thecat{Mod}}
\newcommand{\refl}{\mathrm{refl}}
\newcommand{\lsq}{\sqsubseteq}

\DeclareMathOperator{\isiso}{is-iso}
\DeclareMathOperator{\isequiv}{is-equiv}

\DeclareMathOperator{\Spans}{Spans}

\newcommand{\sem}[1]{\llbracket #1 \rrbracket}
\newcommand{\forces}{\Vdash}
\DeclareMathOperator{\Ne}{Ne}
\DeclareMathOperator{\Nf}{Nf}
\DeclareMathOperator{\reify}{reify}
\DeclareMathOperator{\reflect}{reflect}

\DeclareMathOperator{\Susp}{\Sigma}
\DeclareMathOperator{\merid}{merid}

\DeclareMathOperator{\Glue}{Glue}
\DeclareMathOperator{\glue}{glue}
\DeclareMathOperator{\unglue}{unglue}

\renewcommand{\day}[1]{\operatorname{day}(#1)}

\newcommand{\loc}[1]{[{#1}\inv]}

\newcommand{\indicatortwod}[2]{
  \begin{tikzpicture}
  \draw[->] (0,0) -- (1,0)  node[midway, above] {$#1$};
  \draw[->] (0,0) -- (0,-1) node[midway, left]  {$#2$};
  \end{tikzpicture}
}

\newcommand{\abs}[1]{\lvert x \rvert}
\DeclareMathOperator{\sgn}{sgn}
\newcommand{\ndiv}{\mathbin{/}}