\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*{Čech groupoid}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Codescent}{Codescent}\dotfill \pageref*{Codescent} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $\{U_i \to X\}$ a [[cover]] of a [[space]] $X$, the corresponding \textbf{ech groupoid} is the [[internal groupoid]]

\begin{displaymath}
C(\{U_i\}) = (\coprod_{i j} U_i \cap U_j \rightrightarrows \coprod_i U_i)
\end{displaymath}
whose set of objects is the [[disjoint union]] $\coprod_i U_i$ of the covering patches, and the set of morphisms is the disjoint union of the [[intersections]] $U_i \cap U_j$ of these patches.

This is the $2$-[[coskeleton]] of the full [[Čech nerve]]. See there for more details.

If we speak about [[generalized element|generalized points]] of the $U_i$ (which are often just ordinary points, in applications), then

\begin{itemize}%
\item an [[object]] of $C(\{U_i\})$ is a pair $(x,i)$ where $x$ is a point in $U_i$;


\item there is a unique [[morphism]] $(x,i) \to (x,j)$ for all pairs of objects labeled by the same $x$ such that $x \in U_i \cap U_j$;


\item hence the composition of morphism is of the form

\begin{displaymath}
\itexarray{
     && (x,j)
     \\
     & \nearrow &=& \searrow
     \\
    (x,i) &&\to&& (x,k)
  }
  \,.
\end{displaymath}


\end{itemize}
\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\begin{defn}
\label{CechGroupoid}\hypertarget{CechGroupoid}{}
\textbf{(Cech groupoid)}

Let $\mathcal{C}$ be a [[site]], and $X \in \mathcal{C}$ an [[object]] of that site. For each [[covering]] family $\{ U_i \overset{\iota_i}{\to} X\}$ of $X$ in the given [[coverage]], the \emph{[[Cech groupoid]]} is the [[presheaf of groupoids]]

\begin{displaymath}
C(\{U_i\}) 
    \;\in\; 
  [\mathcal{C}^{op}, Grpd]
  \;\simeq\;
  Grpd\left( [\mathcal{C}^{op}, Set]  \right)
\end{displaymath}
which, regarded as an [[internal category]] in the [[category of presheaves]] over $\mathcal{C}$, has as [[presheaf]] of [[objects]] the [[coproduct]]

\begin{displaymath}
Obj_{C(\{U_i\})} \;\coloneqq\; \underset{i}{\coprod}  y(U_i)
\end{displaymath}
of the [[representable presheaf|presheaves represented]] (under the [[Yoneda embedding]]) by the [[covering]] objects $U_i$, and as [[presheaf]] of [[morphisms]] the [[coproduct]] over all [[fiber products]] of these:

\begin{displaymath}
Mor_{C(\{U_i\})} 
    \;\coloneqq\; 
  \underset{i,j}{\coprod}  y(U_i) \times_{y(X)} y(U_j)
  \,.
\end{displaymath}
This means that for any $V \in \mathcal{C}$ the [[groupoid]] assigned by $C(\{U_i\})$ has as set of objects [[pairs]] consisting of an index $i$ and a morphism $V \overset{\kappa_i}{\to} U_i$ in $\mathcal{C}$, and there is a unique morphism between two such objects

\begin{displaymath}
\kappa_i \longrightarrow \kappa_j
\end{displaymath}
precisely if

\begin{equation}
\iota_i \circ \kappa_i
  \;=\;
  \iota_j \circ \kappa_j
  \phantom{AAAAAAAA}
  \itexarray{
    && V
    \\
    & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}}
    \\
    U_i && && U_j
    \\
    & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}}
    \\
    && X
  }
\label{CechMatchingCondition}\end{equation}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{Codescent}{}\subsubsection*{{Codescent}}\label{Codescent}

We discuss (Prop. \ref{CechGroupoidCoRepresents} below) how the Cech groupoid co-represents \emph{[[matching families]]} as they appear in the definition of [[sheaves]].

For reference, we first recall that definition:

\begin{defn}
\label{CompatibleElements}\hypertarget{CompatibleElements}{}
\textbf{([[matching family]] -- [[descent object]])}

Let $\mathcal{C}$ be a [[small category]] equipped with a [[coverage]], hence a [[site]] and consider a [[presheaf]] $\mathbf{Y} \in [\mathcal{C}^{op}, Set]$ (Example \ref{CategoryOfPresheaves}) over $\mathcal{C}$.

Given an [[object]] $X \in \mathcal{C}$ and a [[covering]] $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of it we say that a \emph{[[matching family]]} (of probes of $\mathbf{Y}$) is a [[tuple]] $(\phi_i \in \mathbf{Y}(U_i))_{i \in I}$ such that for all $i,j \in I$ and [[pairs]] of [[morphisms]] $U_i \overset{\kappa_i}{\leftarrow} V \overset{\kappa_j}{\to} U_j$ satisfying

\begin{equation}
\iota_i \circ \kappa_i
  \;=\;
  \iota_j \circ \kappa_j
  \phantom{AAAAAAAA}
  \itexarray{
    && V
    \\
    & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}}
    \\
    U_i && && U_j
    \\
    & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}}
    \\
    && X
  }
\label{MatchingCondition}\end{equation}
we have

\begin{equation}
\mathbf{Y}(\kappa_i)(\phi_i)
  \;=\;
  \mathbf{Y}(\kappa_j)(\phi_j)
  \,.
\label{GluingCondition}\end{equation}
We write

\begin{equation}
Match\big( 
    \{U_i\}_{i \in I}
    \,,\,
    \mathbf{Y}
  \big)
  \subset
  \underset{i}{\prod} \mathbf{Y}(U_i)
  \;\in\;
  Set
\label{SetOfMatching}\end{equation}
for the set of [[matching families]] for the given presheaf and covering.

This is also called the \emph{[[descent object]]} of $\mathbf{Y}$ for \emph{[[descent]]} along the [[covering]] $\{U_i \overset{\iota_i}{\to}X\}$.

\end{defn}
\begin{example}
\label{CechGroupoidCoRepresents}\hypertarget{CechGroupoidCoRepresents}{}
\textbf{([[Cech groupoid]] co-represents [[matching families]] -- [[codescent]])}

For [[Grpd]] regarded as a [[cosmos]] for [[enriched category theory]], via its [[cartesian closed category]]-structire, and $\mathcal{C}$ a [[site]], let

\begin{displaymath}
\mathbf{Y} \in [\mathcal{C}^{op}, Set] \hookrightarrow [\mathcal{C}^{op}, Grpd]
\end{displaymath}
be a [[presheaf]] on $\mathcal{C}$, regarded as a [[Grpd]]-[[enriched presheaf]], let $X \in \mathcal{C}$ be any [[object]] and $\{U_i \overset{\iota_i}{\to} X\}_i$ a [[covering]] family with induced [[Cech groupoid]] $C(\{U_i\}_i)$ (Def.\ref{CechGroupoid}).

Then there is an [[isomorphism]]

\begin{displaymath}
[\mathcal{C}^{op},Grpd]
  \left(
    C\left(\{U_i\}_i\right), \, \mathbf{Y}
  \right)
  \;\simeq\;
  Match\left( \{U_i\}_i, \, \mathbf{Y} \right)
\end{displaymath}
between the [[hom-object|hom-groupoid]] of [[Grpd]]-[[enriched presheaves]] and the set of [[matching families]] (Def. \ref{CompatibleElements}).

Since therefor the Cech-groupoid co-represents the [[descent object]], it is sometimes called the \emph{[[codescent object]]} along the given covering.

Moreover, under this identification the canonical morphism $C\left( \{U_i\}_i \right) \overset{p_{\{U_i\}_i}}{\longrightarrow} X$ induces the comparison morphism

\begin{displaymath}
\itexarray{    
    [\mathcal{C}^{op}, Grpd]\left( X, \, \mathbf{Y} \right)
    & \simeq &
    \mathbf{Y}(X) 
    \\
    {}^{
      \mathllap{
        [\mathcal{C}^{op}, Grpd](p_{\{U_i\}_i}, \mathbf{Y})
      }
    }\downarrow && \downarrow
    \\
    [\mathcal{C}^{op},Grpd]
    \left(
      C\left(\{U_i\}_i\right), \, \mathbf{Y}
    \right)
    &\simeq&
    Match\left( \{U_i\}_i, \, \mathbf{Y} \right)
  }
  \,.
\end{displaymath}
In conclusion, this means that the [[presheaf]] $\mathbf{Y}$ is a [[sheaf]] (Def. \ref{Sheaf}) precisely if homming Cech groupoid projections into it produces an isomorphism.

\end{example}
\begin{proof}
The hom-groupoid is computed as the [[end]]

\begin{displaymath}
[\mathcal{C}^{op},Grpd]
  \left(
    C\left(\{U_i\}_i\right), \, \mathbf{Y}
  \right)
  \;=\;
  \int_{V \in \mathcal{C}}
  \left[
    C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
  \right]
  \,,
\end{displaymath}
where the ``integrand'' is the [[functor category]] (here: a [[groupoid]]) from the [[Cech groupoid]] at a given $V$ to the set (regarded as a groupoid) assigned by $\mathbf{Y}$ to $V$.

Since $\mathbf{Y}(V)$ is just a set, that functor groupoid, too, is just a set, regarded as a groupoid. Its elements are the [[functors]] $C\left(\{U_i\}_i\right)(V) \longrightarrow \mathbf{Y}(V)$, which are equivalently those [[functions]] on sets of objects

\begin{displaymath}
\underset{i}{\coprod} y(U_i)(V)
  =
  Obj_{C\left(\{U_i\}_i\right)(V)} 
    \longrightarrow 
  Obj_{\mathbf{Y}(V)}
  =
  \mathbf{Y}(V)
\end{displaymath}
which respect the [[equivalence relation]] induced by the morphisms in the Cech groupoid at $V$.

Hence the hom-groupoid is a subset of the [[end]] of these [[function sets]]:

\begin{displaymath}
\begin{aligned}
    \int_{V \in \mathcal{C}}
    \left[
      C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
    \right]
    & \hookrightarrow
    \int_{V \in \mathcal{C}}
    \left[
      \underset{i}{\coprod} y(U_i)(V), \, \mathbf{Y}(V)
    \right]
    \\
    & \simeq
    \int_{V \in \mathcal{C}}
    \underset{i}{\prod}
    \left[
       y(U_i)(V), \, \mathbf{Y}(V)
    \right]
    \\
    & \simeq
    \underset{i}{\prod}
    \int_{V \in \mathcal{C}}
    \left[
       y(U_i)(V), \, \mathbf{Y}(V)
    \right]
    \\
    & \simeq
    \underset{i}{\prod}
    \mathbf{Y}(U_i)
  \end{aligned}
\end{displaymath}
Here we used: first that the [[internal hom]]-functor turns colimits in its first argument into limits (see at \emph{[[internal hom-functor preserves limits]]}), then that [[limits commute with limits]], hence that in particular [[ends]] commute with [[products]] , and finally the [[enriched Yoneda lemma]], which here comes down to just the plain [[Yoneda lemma]] . The end result is hence the same [[Cartesian product]] set that also the set of matching families is defined to be a subset of, in \eqref{SetOfMatching}.

This shows that an element in $\int_{V \in \mathcal{C}}
    \left[
      C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
    \right]$ is a [[tuple]] $(\phi_i \in \mathbf{Y}(U_i))_i$, subject to some condition. This condition is that for each $V \in \mathcal{C}$ the assignment

\begin{displaymath}
\itexarray{
    C\left(\{U_i\}_i\right)(V) 
      & \longrightarrow & 
    \mathbf{Y}(V)
    \\
    (V \overset{\kappa_i}{\to} U_i)
    &\mapsto&
    \kappa_i^\ast \phi_i
    =
    \mathbf{Y}(\kappa_i)(\phi_i)
  }
\end{displaymath}
constitutes a [[functor]] of [[groupoids]].

By definition of the [[Cech groupoid]], and since the [[codomain]] is a just [[set]] regarded as a [[groupoid]], this is the case precisely if

\begin{displaymath}
\mathbf{Y}(\kappa_i)(\phi_i)
  \;=\;
  \mathbf{Y}(\kappa_j)(\phi_j)
  \phantom{AAAA}
  \text{for all}\, i,j
  \,,
\end{displaymath}
which is exactly the condition \eqref{GluingCondition} that makes $(\phi_i)_i$ a matching family.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

For $X$ a [[smooth manifold]] and $\{U_i \to X\}$ an [[atlas]] by [[coordinate chart]]s, the ech groupoid is a [[Lie groupoid]] which is equivalent to $X$ as a Lie groupoid: $C(\{U_i\}) \stackrel{\simeq}{\to} X$

For $\mathbf{B}G$ the Lie groupoid with one object coming from a [[Lie group]] $G$ morphisms of Lie groupoids of the form

\begin{displaymath}
\itexarray{
     C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}G
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     X
  }
\end{displaymath}
are also called [[anafunctor]]s from $X$ to $\mathbf{B}G$. They correspond to smooth $G$-[[principal bundle]]s on $X$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Cech nerve]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

For instance

\begin{itemize}%
\item [[J. F. Jardine]], Example 5 in: \emph{Homotopy classification of gerbes} (\href{https://arxiv.org/abs/math/0605200}{arXiv:math/0605200})

\end{itemize}
[[!redirects Cech groupoid]] [[!redirects Cech groupoids]] [[!redirects Čech groupoids]]



\end{document}