\newcommand{\subsubsubsection}[1]{\textbf{#1}}
\newcommand{\WIP}{\textbf{WIP}}

\newcommand{\coloneqq}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
                     \hbox{\scriptsize.}\hbox{\scriptsize.}}}%
  =}

\newcommand{\defeq}{\mathrel{\triangleq}}
%% Alternatively:
%% \newcommand{\defeq}{≔}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bC}{ℂ}
\newcommand{\bD}{𝔻}
\newcommand{\bX}{𝕏}
% \newcommand{\to}{\rightarrow}
%% \newcommand{\mto}{\mapsto}
\newcommand{\mto}{\rightarrow}
\newcommand{\UU}{\ensuremath{\mathcal{U}}\xspace}
\let\type\UU
\newcommand{\MCU}{\UU}
\newcommand{\todo}[1]{\textit{#1}}
\newcommand{\comp}{\circ}
\newcommand{\x}{\times}
\newcommand\inv[1]{#1\raisebox{1.15ex}{$\scriptscriptstyle-\!1$}}
\newcommand{\tp}{\mathrel{:}}
\newcommand{\Type}{\mathcal{U}}

\usepackage{graphicx}
\makeatletter
\newcommand{\shorteq}{%
  \settowidth{\@tempdima}{-}% Width of hyphen
  \resizebox{\@tempdima}{\height}{=}%
}
\makeatother
\newcommand{\var}[1]{\ensuremath{\mathit{#1}}}
\newcommand{\varindex}[1]{\ensuremath{\var{#1}}\index{$\var{#1}$}}
\newcommand{\nomen}[2]{\emph{#1}\index{#2}}
\newcommand{\nomenindex}[1]{\nomen{#1}{#1}}

\newcommand{\Hom}{\varindex{Hom}}
\newcommand{\fmap}{\varindex{fmap}}
\newcommand{\bind}{\varindex{bind}}
\newcommand{\join}{\varindex{join}}
\newcommand{\omap}{\varindex{omap}}
\newcommand{\pure}{\varindex{pure}}
\newcommand{\idFun}{\varindex{id}}
\newcommand{\Sets}{\varindex{Sets}}
\newcommand{\Set}{\varindex{Set}}
\newcommand{\hSet}{\varindex{hSet}}
\newcommand{\id}{\varindex{id}}
\newcommand{\isEquiv}{\varindex{isEquiv}}
\newcommand{\idToIso}{\varindex{idToIso}}
\newcommand{\idIso}{\varindex{idIso}}
\newcommand{\isSet}{\varindex{isSet}}
\newcommand{\isContr}{\varindex{isContr}}
\newcommand{\isGroupoid}{\varindex{isGroupoid}}
\newcommand{\pathJ}{\varindex{pathJ}}
\newcommand\Object{\varindex{Object}}
\newcommand\Functor{\varindex{Functor}}
\newcommand\isProp{\varindex{isProp}}
\newcommand\propPi{\varindex{propPi}}
\newcommand\propSig{\varindex{propSig}}
\newcommand\PreCategory{\varindex{PreCategory}}
\newcommand\IsPreCategory{\varindex{IsPreCategory}}
\newcommand\isIdentity{\varindex{isIdentity}}
\newcommand\propIsIdentity{\varindex{propIsIdentity}}
\newcommand\IsCategory{\varindex{IsCategory}}
\newcommand\Gl{\varindex{\lambda}}
\newcommand\lemPropF{\varindex{lemPropF}}
\newcommand\isPreCategory{\varindex{isPreCategory}}
\newcommand\congruence{\varindex{cong}}
\newcommand\identity{\varindex{identity}}
\newcommand\isequiv{\varindex{isequiv}}
\newcommand\qinv{\varindex{qinv}}
\newcommand\fiber{\varindex{fiber}}
\newcommand\shufflef{\varindex{shuffle}}
\newcommand\Univalent{\varindex{Univalent}}
\newcommand\refl{\varindex{refl}}
\newcommand\isoToId{\varindex{isoToId}}
\newcommand\Isomorphism{\varindex{Isomorphism}}
\newcommand\rrr{\ggg}
%% \newcommand\fish{\mathbin{↣}}
%% \newcommand\fish{\mathbin{⤅}}
\newcommand\fish{\mathbin{⤇}}
%% \newcommand\fish{\mathbin{⤜}}
%% \newcommand\fish{\mathrel{\wideoverbar{\rrr}}}
\newcommand\fst{\varindex{fst}}
\newcommand\snd{\varindex{snd}}
\newcommand\Path{\varindex{Path}}
\newcommand\Category{\varindex{Category}}
\newcommand\TODO[1]{TODO: \emph{#1}}
\newcommand*{\QED}{\hfill\ensuremath{\square}}%
\newcommand\uexists{\exists!}
\newcommand\Arrow{\varindex{Arrow}}
\newcommand\embellish[1]{\widehat{#1}}
\newcommand\nattrans[1]{\embellish{#1}}
\newcommand\functor[1]{\embellish{#1}}
\newcommand\NTsym{\varindex{NT}}
\newcommand\NT[2]{\NTsym\ #1\ #2}
\newcommand\Endo[1]{\varindex{Endo}\ #1}
\newcommand\EndoR{\functor{\mathcal{R}}}
\newcommand\omapR{\mathcal{R}}
\newcommand\omapF{\mathcal{F}}
\newcommand\omapG{\mathcal{G}}
\newcommand\FunF{\functor{\omapF}}
\newcommand\FunG{\functor{\omapG}}
\newcommand\funExt{\varindex{funExt}}
\newcommand{\suc}[1]{\varindex{suc}\ #1}
\newcommand{\trans}{\varindex{trans}}
\newcommand{\toKleisli}{\varindex{toKleisli}}
\newcommand{\toMonoidal}{\varindex{toMonoidal}}
\newcommand\pairA{\mathcal{A}}
\newcommand\pairB{\mathcal{B}}
\newcommand{\joinNT}{\functor{\varindex{join}}}
\newcommand{\pureNT}{\functor{\varindex{pure}}}
\newcommand{\hrefsymb}[2]{\href{#1}{#2 \ExternalLink}}
\newcommand{\sourcebasepath}{http://web.student.chalmers.se/\textasciitilde hanghj/cat/doc/html/}
\newcommand{\docbasepath}{https://github.com/fredefox/cat/}
\newcommand{\sourcelink}[1]{\hrefsymb
    {\sourcebasepath#1.html}
    {\texttt{#1}}
  }
\newcommand{\gitlink}{\hrefsymb{\docbasepath}{\texttt{\docbasepath}}}
\newcommand{\doclink}{\hrefsymb{\sourcebasepath}{\texttt{\sourcebasepath}}}
\newcommand{\clll}{\mathrel{\bC.\mathord{\lll}}}
\newcommand{\dlll}{\mathrel{\bD.\mathord{\lll}}}
\newcommand\coe{\varindex{coe}}
\newcommand\Monoidal{\varindex{Monoidal}}
\newcommand\Kleisli{\varindex{Kleisli}}
