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\begin{document}

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\section*{effective Lie groupoid}

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

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{higher_lie_theory}{}\paragraph*{{Higher Lie theory}}\label{higher_lie_theory}

[[!include infinity-Lie 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{EquivalentCharacterizations}{Equivalent characterizations}\dotfill \pageref*{EquivalentCharacterizations} \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}

A [[Lie groupoid]] is said to be \emph{effective} if its morphisms act locally freely on [[germs]], in a sense.

(Beware that this use of the term is entirely independent of ``effective'' in the sense of [[Giraud theorem|Giraud's axioms]], as discussed are [[groupoid object in an (infinity,1)-category]].)

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $X_\bullet = (X_1 \stackrel{\longrightarrow}{\longrightarrow} Y)$ be a [[Lie groupoid]]. Equivalently, let $X$ be a [[differentiable stack]] equipped with an [[atlas]] $X_0 \to X$.

Then given any element $f$ in $X_1$, hence given a [[morphism]] $f \colon x \to y$, it induces a [[germ]] of a local [[diffeomorphism]] $\tilde f \colon (X_0,x) \to (X_0,y)$ as follows:

choose $U_x \subset X_0$ to be any [[neighbourhood]] of $x$ small enough such that the restricted [[source]] and [[target]] maps

\begin{displaymath}
\itexarray{
    X_1 \times_{X_0} U_x &\to& X_1
    \\
    {}^{\mathllap{s|_U, t|_U}}\downarrow && \downarrow^{\mathrlap{s,t}}
    \\
    U_x &\stackrel{}{\hookrightarrow} & X_0
  }
\end{displaymath}
are [[diffeomorphisms]]. Then define $\tilde f$ to be the germ $t \circ (s|_U)^{-1}$.

\begin{defn}
\label{EffectiveLieGroupoid}\hypertarget{EffectiveLieGroupoid}{}
The Lie groupoid $X_\bullet$ is called \emph{effective} if this assignment of morphisms to germs of local diffeomorphisms is [[injective]].

Similarly a [[differentiable stack]] is called an \emph{effective \'e{}tale stack} if it is represented by an effective \'e{}tale Lie groupoid.

\end{defn}
This means that the [[action]] of the [[automorphism group]] at any point $x$ on the germ at $x$ is [[faithful action|faithful]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{EquivalentCharacterizations}{}\subsubsection*{{Equivalent characterizations}}\label{EquivalentCharacterizations}

\begin{prop}
\label{}\hypertarget{}{}
The following are equivalent

\begin{enumerate}%
\item $X_\bullet$ is an effective Lie groupoid, def. \ref{EffectiveLieGroupoid};


\item the canonical smooth [[functor]] $X_\bullet \to \mathbb{H}(X_0)$ (see \href{&#233;tale+groupoid#RelationToHaefligerGroupoids}{here}) to the [[Haefliger groupoid]] of the manifold of objects is [[faithful functor|faithful]], i.e. gives a [[representable morphism of stacks]];


\item under the equivalence (\href{&#233;tale+groupoid#CharacterizationBySiteOfManifolds}{here}) between smooth \'e{}tale stacks and stacks on the [[site]] $SmthMfd^{et}$ of smooth manifolds with [[local diffeomorphisms]] between them, $X$ corresponds to a [[sheaf]] (i.e. to a [[0-truncated]] stack) on $SmthMfd^{et}$.



\end{enumerate}
\end{prop}
(\hyperlink{Carchedi12}{Carchedi 12, theorem 4.1}).

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

\begin{itemize}%
\item [[proper Lie groupoid]], [[étale groupoid]]

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

A standard textbook aacount is in

\begin{itemize}%
\item [[Ieke Moerdijk]], Janez Mrun \emph{Introduction to Foliations and Lie Groupoids} ,Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003)

\end{itemize}
Brief survey is in

\begin{itemize}%
\item [[David Carchedi]], \emph{A quick note on \'e{}tale stacks} (\href{http://people.mpim-bonn.mpg.de/carchedi/Etale_Stacks.pdf}{pdf})

\end{itemize}
Discussion in a more general context of [[étale stacks]] is in

\begin{itemize}%
\item [[David Carchedi]], \emph{\'E{}tale Stacks as Prolongations} (\href{http://arxiv.org/abs/1212.2282}{arXiv:1212.2282})

\end{itemize}


\end{document}