\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\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{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{fundamental groupoid of a cubical set and the cubical nerve of a groupoid}

[[!redirects fundamental groupoid of a cubical set]] [[!redirects cubical nerve of a groupoid]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{introduction}{Introduction}\dotfill \pageref*{introduction} \linebreak
\noindent\hyperlink{preliminaries}{Preliminaries}\dotfill \pageref*{preliminaries} \linebreak
\noindent\hyperlink{fundamental_groupoid_for_2truncated_cubical_sets}{Fundamental groupoid for 2-truncated cubical sets}\dotfill \pageref*{fundamental_groupoid_for_2truncated_cubical_sets} \linebreak
\noindent\hyperlink{2truncated_nerve_functor}{2-truncated nerve functor}\dotfill \pageref*{2truncated_nerve_functor} \linebreak
\noindent\hyperlink{adjunction_between_the_fundamental_groupoid_functor_for_2truncated_cubical_sets_and_the_2truncated_cubical_nerve_functor}{Adjunction between the fundamental groupoid functor for 2-truncated cubical sets and the 2-truncated cubical nerve functor}\dotfill \pageref*{adjunction_between_the_fundamental_groupoid_functor_for_2truncated_cubical_sets_and_the_2truncated_cubical_nerve_functor} \linebreak
\noindent\hyperlink{fundamental_groupoid_functor}{Fundamental groupoid functor}\dotfill \pageref*{fundamental_groupoid_functor} \linebreak
\noindent\hyperlink{nerve_functor}{Nerve functor}\dotfill \pageref*{nerve_functor} \linebreak
\noindent\hyperlink{adjunction_between_the_fundamental_groupoid_functor_and_the_nerve_functor}{Adjunction between the fundamental groupoid functor and the nerve functor}\dotfill \pageref*{adjunction_between_the_fundamental_groupoid_functor_and_the_nerve_functor} \linebreak


\hypertarget{introduction}{}\subsection*{{Introduction}}\label{introduction}

We construct an adjunction between the category of [[groupoid | groupoids]] and the category of [[cubical set | cubical sets]], the left adjoint of which is the \emph{fundamental groupoid} of a cubical set, and the right adjoint of which is the (cubical) \emph{nerve} of a groupoid.

\hypertarget{preliminaries}{}\subsection*{{Preliminaries}}\label{preliminaries}

\begin{defn}
\label{}\hypertarget{}{}
We denote the category of [[groupoid | groupoids]] by $\mathsf{Grpd}$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
We make use throughout of the notation of [[category of cubes]], [[cubical set]], and [[cubical truncation, skeleton, and co-skeleton]].

\end{defn}
\hypertarget{fundamental_groupoid_for_2truncated_cubical_sets}{}\subsection*{{Fundamental groupoid for 2-truncated cubical sets}}\label{fundamental_groupoid_for_2truncated_cubical_sets}

\begin{defn}
\label{NotationFundamentalGroupoid}\hypertarget{NotationFundamentalGroupoid}{}
We denote by $\Pi^{\leq 2}_{1} : \mathsf{Set}^{\square_{\leq 2}^{op}} \rightarrow \mathsf{Grpd}$ the functor defined as follows.

1) To a 2-truncated cubical set $X$, we associate the groupoid $\Pi^{\leq 2}_{1}(X)$ defined as follows.

a) The objects of $\Pi^{\leq 2}_{1}(X)$ are the 0-cubes of $X$.

b) The arrows of $\Pi^{\leq 2}_{1}(X)$ are zig-zags of 1-cubes of $X$ up to the notion of equivalence defined below, where by a zig-zag of 1-cubes of $X$ we mean, for some integer $n \geq 0$, a set of 1-cubes of $X$ whose faces match up as follows.

\begin{displaymath}
\itexarray{
      x_{0} & \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n}
   }
\end{displaymath}
We identify a pair of zig-zags if one can be obtained from the other by a sequence of the following manipulations.

i) We may remove or add a pair of arrows (anywhere in the zig-zag) of the form

\begin{displaymath}
\itexarray{
      x_{0} & \overset{f}{\rightarrow} & x_{1} & \overset{f}{\leftarrow} & x_{0}
   }
\end{displaymath}
or of the following form.

\begin{displaymath}
\itexarray{
      x_{0} & \overset{f}{\leftarrow} & x_{1} & \overset{f}{\rightarrow} & x_{0}
   }
\end{displaymath}
ii) We may replace an entire zig-zag

\begin{displaymath}
\itexarray{
      x_{0} & \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n}
   }
\end{displaymath}
with a zig-zag

\begin{displaymath}
\itexarray{
      x'_{0} & \overset{f'_{1}}{\rightarrow} & x'_{1} \overset{g'_{1}}{\leftarrow} & x'_{2} & \cdots & x'_{2n-2} & \overset{f'_{n}}{\rightarrow} & x'_{2n-1} \overset{g'_{n}}{\leftarrow} & x'_{2n}
   }
\end{displaymath}
if there is, for every $1 \leq i \leq n$, a 2-cube of $X$ whose horizontal 1-cubes are as follows

\begin{displaymath}
\itexarray{
      x_{2i-2}   & \overset{f_{i}}{\to}   & x'_{2i-1} \\
      \downarrow &                        & \downarrow \\
      x'_{2i-2}  & \underset{f'_{i}}{\to} & x'_{2i-1}
   }
\end{displaymath}
and there is, for every $1 \leq i \leq n$, a 2-cube of $X$ whose horizontal 1-cubes are as follows.

\begin{displaymath}
\itexarray{
      x_{2i-1}   & \overset{g_{i}}{\leftarrow}   & x'_{2i} \\
      \downarrow &                        & \downarrow \\
      x'_{2i-1}  & \underset{g'_{i}}{\leftarrow} & x'_{2i}
   }
\end{displaymath}
c) The source of a zig-zag as at the beginning of b) is $x_{0}$, and the target is $x_{2n}$.

d) Composition of arrows is given by concatenation of zig-zags (it is immediately verified that this is well-defined with respect to the equivalence relation of b)).

e) The identity arrow on an object $x$ of $\Pi^{\leq 2}_{1}(X)$ is the zig-zag with $n = 0$ consisting simply of $x$.

f) The inverse of an arrow

\begin{displaymath}
\itexarray{
      x_{0} & \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n}
   }
\end{displaymath}
of $\Pi_{1}(X)$ is the following arrow (it is immediately verified that this is well-defined with respect to the equivalence relation of b)).

\begin{displaymath}
\itexarray{
      x_{2n} & \overset{g_{n}}{\rightarrow} & x_{2n-1} \overset{f_{n}}{\leftarrow} & x_{2n-2} & \cdots & x_{2} & \overset{g_{1}}{\rightarrow} & x_{1} \overset{f_{1}}{\leftarrow} & x_{0}
   }
\end{displaymath}
2) To a morphism of 2-truncated cubical sets $F : X \rightarrow Y$, we associate the functor $\Pi^{\leq 2}_{1}(F) : \Pi^{\leq 2}_{1}(X) \rightarrow \Pi^{\leq 2}_{1}(Y)$ defined as follows.

a) On objects, $\Pi^{\leq 2}_{1}(F)$ is the same as $F$.

b) To a zig-zag as follows

\begin{displaymath}
\itexarray{
      x_{0}  &  \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n}
   }
\end{displaymath}
we associate the following zig-zag.

\begin{displaymath}
\itexarray{
      F(x_{0}) &  \overset{F(f_{1})}{\rightarrow} & F(x_{1}) \overset{F(g_{1})}{\leftarrow} & F(x_{2}) & \cdots & F(x_{2n-2}) & \overset{F(f_{n})}{\rightarrow} & F(x_{2n-1}) \overset{F(g_{n})}{\leftarrow} & F(x_{2n})
   }
\end{displaymath}
It is immediately verified that this is well-defined with respect to the equivalence relation of 1) b).

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
We refer to $\Pi^{\leq 2}_{1} : \mathsf{Set}^{\square_{\leq 2}^{op}} \rightarrow \mathsf{Grpd}$ as the \emph{fundamental groupoid} functor for 2-truncated cubical sets.

\end{defn}
\hypertarget{2truncated_nerve_functor}{}\subsection*{{2-truncated nerve functor}}\label{2truncated_nerve_functor}

\begin{defn}
\label{}\hypertarget{}{}
We denote by $N^{\leq 2} : \mathsf{Grpd} \rightarrow \mathsf{Set}^{\square_{\leq 2}^{op}}$ the functor defined as follows.

1) To a groupoid $\mathcal{A}$, we associate the 2-truncated cubical set $N^{\leq 2}(\mathcal{A})$ defined as follows.

a) The 0-cubes of $N^{\leq 2}(\mathcal{A})$ are the objects of $\mathcal{A}$.

b) The 1-cubes $f : a_{0} \rightarrow a_{1}$ of $N^{\leq 2}(\mathcal{A})$ are the arrows $f$ of $\mathcal{A}$ with source $a_{0}$ and target $a_{1}$.

c) The 2-cubes

\begin{displaymath}
\itexarray{
      a_{0}            & \overset{f_{0}}{\to}  & a_{1} \\
      f_{2} \downarrow & \sigma                & \downarrow f_{1} \\
      a_{2}            & \underset{f_{3}}{\to} & a_{3}
   }
\end{displaymath}
of $N^{\leq 2}(\mathcal{A})$ are the commutative squares $\sigma$ of $\mathcal{A}$ whose boundary looks the same as this.

d) The degenerate 1-cubes of $N^{\leq 2}(\mathcal{A})$ are the identity arrows of $\mathcal{A}$.

e) The degenerate 2-cubes of $N^{\leq 2}(\mathcal{A})$ are the commutative squares of $\Pi^{\leq 2}_{1}(\mathcal{A})$ which look as follows

\begin{displaymath}
\itexarray{
      x_{0}         & \overset{f}{\to}  & x_{1} \\
      id \downarrow & \sigma            & \downarrow id \\
      x_{0}         & \underset{f}{\to} & x_{1}
   }
\end{displaymath}
or as follows.

\begin{displaymath}
\itexarray{
      x_{0}        & \overset{id}{\to}  & x_{0} \\
      f \downarrow & \sigma             & \downarrow f \\
      x_{1}        & \underset{id}{\to} & x_{1}
   }
\end{displaymath}
2) To a functor $F : \mathcal{A} \rightarrow \mathcal{B}$, we associate the morphism of 2-truncated cubical sets $N^{\leq 2}(F) : N^{\leq 2}(\mathcal{A}) \rightarrow N^{\leq 2}(\mathcal{B})$ defined as follows.

a) On 0-cubes, $N^{\leq 2}(\mathcal{A})$ is the same as $F$.

b) On 1-cubes, $N^{\leq 2}(F)$ is the same as $F$.

c) On 2-cubes, $N^{\leq 2}(F)$ sends a commutative square

\begin{displaymath}
\itexarray{
      a_{0}            & \overset{f_{0}}{\to}  & a_{1} \\
      f_{2} \downarrow & \sigma                & \downarrow f_{1} \\
      a_{2}            & \underset{f_{3}}{\to} & a_{3}
   }
\end{displaymath}
of $\mathcal{A}$ to the commutative square

\begin{displaymath}
\itexarray{
      F(a_{0})            & \overset{F(f_{0})}{\to}  & F(a_{1}) \\
      F(f_{2}) \downarrow & F(\sigma)                & \downarrow F(f_{1}) \\
      F(a_{2})            & \underset{F(f_{3})}{\to} & F(a_{3})
   }
\end{displaymath}
of $\mathcal{A}$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
We refer to $N^{\leq 2} : \mathsf{Grpd} \rightarrow \mathsf{Set}^{\square_{\leq 2}^{op}}$ as the 2-truncated \emph{nerve} functor.

\end{defn}
\hypertarget{adjunction_between_the_fundamental_groupoid_functor_for_2truncated_cubical_sets_and_the_2truncated_cubical_nerve_functor}{}\subsection*{{Adjunction between the fundamental groupoid functor for 2-truncated cubical sets and the 2-truncated cubical nerve functor}}\label{adjunction_between_the_fundamental_groupoid_functor_for_2truncated_cubical_sets_and_the_2truncated_cubical_nerve_functor}

\begin{defn}
\label{}\hypertarget{}{}
We denote by $\epsilon : \Pi^{\leq 2}_{1} \circ N^{\leq 2} \rightarrow id$ the natural transformation which to a groupoid $\mathcal{A}$ associates the functor $\epsilon(\mathcal{A}) : \Pi^{\leq 2}_{1} \circ N^{\leq 2}(\mathcal{A}) \rightarrow \mathcal{A}$ defined as follows.

1) On objects it is the identity.

2) To an arrow of $\Pi^{\leq 2}_{1} \circ N^{\leq 2}(\mathcal{A})$, given by a zig-zag of arrows

\begin{displaymath}
\itexarray{
      a_{0}  &  \overset{f_{1}}{\rightarrow} & a_{1} \overset{g_{1}}{\leftarrow} & a_{2} & \cdots & a_{2n-2} & \overset{f_{n}}{\rightarrow} & a_{2n-1} \overset{g_{n}}{\leftarrow} & a_{2n}
   }
\end{displaymath}
of $\mathcal{A}$, we associate the arrow of $\mathcal{A}$ given by the composition in $\mathcal{A}$ of the arrows

\begin{displaymath}
\itexarray{
      a_{0}  &  \overset{f_{1}}{\rightarrow} & a_{1} \overset{g_{1}^{-1}}{\rightarrow} & a_{2} & \cdots & a_{2n-2} & \overset{f_{n}}{\rightarrow} & a_{2n-1} \overset{g_{n}^{-1}}{\rightarrow} & a_{2n}
   }
\end{displaymath}
of $\mathcal{A}$.

It is straightforward to check that this is well-defined with respect to the equivalence relation of 1 b) of Notation \ref{NotationFundamentalGroupoid}, and that we indeed have a functor.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
We denote by $\eta : id \rightarrow N^{\leq 2} \circ \Pi^{\leq 2}_{1}$ the natural transformation which to a 2-truncated cubical set $X$ associates the morphism of 2-truncated cubical sets $\eta(X) : X \rightarrow N^{\leq 2} \circ \Pi^{\leq 2}_{1}(X)$ defined as follows.

1) On objects it is the identity.

2) To a 1-cube $f : x_{0} \rightarrow x_{1}$ of $X$ we associate the following zig-zag of 1-cubes of $X$, where the right arrow is the degeneracy on $x_{1}$.

\begin{displaymath}
\itexarray{
      x_{0}  &  \overset{f}{\rightarrow} & x_{1} \overset{id}{\leftarrow} & x_{1} }
\end{displaymath}
3) To a 2-cube

\begin{displaymath}
\itexarray{
      x_{0}            & \overset{f_{0}}{\to}  & x_{1} \\
      f_{2} \downarrow & \sigma                & \downarrow f_{1} \\
      x_{2}            & \underset{f_{3}}{\to} & x_{3}
   }
\end{displaymath}
of $X$ we associate the commutative square in $\Pi^{\leq 2}_{1}(X)$ given as follows.

\begin{displaymath}
\itexarray{
      x_{0}            & \overset{f_{0}}{\to}  & x_{1} & \overset{id}{\leftarrow}  & x_{1} \\
      f_{2} \downarrow &                       &       &                           & \downarrow f_{1} \\
      x_{2}            &                       &       &                           & x_{3} \\
      id    \downarrow &                       &       &                           & \downarrow id \\
      x_{2}            & \underset{f_{3}}{\to} & x_{2} & \underset{id}{\leftarrow} & x_{3} \\       
   }
\end{displaymath}
The following diagram of commutative squares in $\Pi^{\leq 2}_{1}(X)$ illustrates that the above square does indeed commute in $\Pi^{\leq 2}_{1}(X)$.

\begin{displaymath}
\itexarray{
      x_{0}         & \overset{f_{0}}{\rightarrow}  & x_{1}        & \overset{id}{\leftarrow} & x_{1}        & \overset{f_{1}}{\rightarrow}   & x_{2}         & \overset{id}{\leftarrow} & x_{2} \\
      id \downarrow &                               & \downarrow A &                          & \downarrow A &                            & \downarrow id &                          & \downarrow id \\
      x_{0}         & \underset{f_{2}}{\rightarrow} & x_{2}        & \underset{id}{\leftarrow} & x_{2}        & \underset{f_{3}}{\rightarrow} & x_{3}         & \underset{id}{\leftarrow} & x_{3}
   }
\end{displaymath}
Here A is $f_{2} \circ f_{0}^{-1}$. That the square

\begin{displaymath}
\itexarray{
      x_{1}        & \overset{f_{1}}{\to}     & x_{3} \\
      A \downarrow &                       & \downarrow id \\
      x_{2}        & \underset{f_{3}}{\to} & x_{3}
   }
\end{displaymath}
commutes follows easily from the commutativity of the square arising from the 2-cube $\sigma$ of $X$ above.

\end{defn}
\begin{defn}
\label{PropositionAdjunctionBetweenPi1TwoTruncAndNTwoTrunc}\hypertarget{PropositionAdjunctionBetweenPi1TwoTruncAndNTwoTrunc}{}
The natural transformations $\eta$ and $\epsilon$ define an adjunction between $\Pi^{\leq 2}_{1}$ and $N^{\leq 2}$.

\end{defn}
\begin{proof}
Straightforward verification that the triangle identities hold.

\end{proof}
\hypertarget{fundamental_groupoid_functor}{}\subsection*{{Fundamental groupoid functor}}\label{fundamental_groupoid_functor}

\begin{defn}
\label{}\hypertarget{}{}
Adopting the notation of [[cubical truncation, skeleton, and co-skeleton]], we denote by $\Pi_{1} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Grpd}$ the functor

\begin{displaymath}
\itexarray{
      \mathsf{Set}^{\square^{op}} & \overset{tr_{2}}{\rightarrow}  & \mathsf{Set}^{\square_{2}^{op}} & \overset{\Pi_{1}^{\leq 2}}{\rightarrow} & \mathsf{Grpd}.
   }
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
We refer to the functor $\Pi_{1} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Grpd}$ as the \emph{fundamental groupoid} functor.

\end{defn}
\hypertarget{nerve_functor}{}\subsection*{{Nerve functor}}\label{nerve_functor}

\begin{defn}
\label{}\hypertarget{}{}
Adopting the notation of [[cubical truncation, skeleton, and co-skeleton]], we denote by $N : \mathsf{Grpd} \rightarrow \mathsf{Set}^{\square^{op}}$ the functor

\begin{displaymath}
\itexarray{
      \mathsf{Grpd} & \overset{N^{\leq 2}}{\rightarrow} & \mathsf{Set}^{\square_{\leq 2} ^{op}} & \overset{cosk_{2}}{\rightarrow}  & \mathsf{Set}^{\square^{op}}.
   }
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
We refer to the functor $N : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Grpd}$ as the \emph{nerve} functor.

\end{defn}
\hypertarget{adjunction_between_the_fundamental_groupoid_functor_and_the_nerve_functor}{}\subsection*{{Adjunction between the fundamental groupoid functor and the nerve functor}}\label{adjunction_between_the_fundamental_groupoid_functor_and_the_nerve_functor}

Since $\Pi^{\leq 2}_{1}$ is left adjoint to $N^{\leq 2}$ by Proposition $\backslash$ref above, and since $tr_{2}$ is left adjoint to $cosk_{2}$, we have that $\Pi_{1}$ is left adjoint to $N$.



\end{document}