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\begin{document}

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\section*{geometric stack}

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

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

[[!include higher geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{relation_to_groupoid_objects}{Relation to groupoid objects}\dotfill \pageref*{relation_to_groupoid_objects} \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 [[stack]] $X$ on a [[site]] $C$ is \textbf{geometric} if, roughly, it is [[representable functor|represented]] by a suitably well-behaved [[groupoid]] object $\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0)$ [[internal category|internal]] to $C$, i.e. if to an object $U \in C$ the stack assigns the (ordinary) [[groupoid]]

\begin{displaymath}
X : U \mapsto (C(U,\mathcal{G}_1) \stackrel{\longrightarrow}{\longrightarrow} C(U,\mathcal{G}_0))
  \,.
\end{displaymath}
A crucial difference between the groupoid object $\mathcal{G}$ in $C$ and the geometric stack $X$ is that the [[equivalence class]] of the stack in general contains \emph{more} (geometric) stacks than there are groupoid objects internally equivalent to $\mathcal{G}$: two groupoid objects with equivalent geometric stacks are called \textbf{Morita equivalent} groupoid objects.

\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

Geometric stacks for the following choices of sites $C$ are called

\begin{itemize}%
\item for $C =$ [[Top]] -- [[topological stack]];


\item for $C =$ [[Diff]] -- [[differentiable stack]];


\item for $C =$ [[CRing]]${}^{op}$ -- [[algebraic stack]];


\item for $C =$ [[complex manifold|CplxMfd]] -- [[complex analytic stack]];



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

There are slight variations in the literature on what precisely is required of a [[stack]] $X$ on a [[site]] $C$ with [[subcanonical coverage|subcanonical topology]] in order that it qualifies as \textbf{geometric}.

A general requirement is that

\begin{enumerate}%
\item the [[diagonal]] morphism $\Delta : X \to X \times X$ is a [[representable morphism of stacks]]


\item there exists an \emph{atlas} for the stack, in that there is a [[representable functor|representable]] $U \in C$ and a surjective morphism



\end{enumerate}
\begin{displaymath}
p : U \to X
    \,.
\end{displaymath}
This is necessarily itself representable, precisely if $\Delta_X$ is.

Further conditions are the following

\begin{itemize}%
\item for $C = Sch_{et}$ the [[site]] of [[scheme]]s with the [[etale topology]]

\begin{itemize}%
\item $\Delta_X$ is required to be [[quasicompact]] and [[separated]]


\item for [[Deligne-Mumford stack]]s $p$ is moreover required to be etale


\item for [[Artin stack]]s $p$ is required to be [[smooth morphism of schemes|smooth]].



\end{itemize}


\end{itemize}
\hypertarget{relation_to_groupoid_objects}{}\subsection*{{Relation to groupoid objects}}\label{relation_to_groupoid_objects}

The \emph{groupoid object} associated to a geometric stack $X$ with atlas $p : U \to X$ is the [[Cech groupoid]] of $p$ (this is simply the Cech groupoid of $p$ seen as a singleton cover) defined by $\mathcal{G}_0 := U$ and $\mathcal{G}_1 = U \times_X U$, where the latter is the 2-categorical [[pullback]]

\begin{displaymath}
\itexarray{
   \mathcal{G}_1 &\stackrel{s}{\to}& U
   \\
   \downarrow^{\mathrlap{t}} &{}^{\simeq}\swArrow& \downarrow^{\mathrlap{p}}
   \\
   U &\to& X
  }
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \textbf{geometric stack}


\item [[geometric ∞-stack]]



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

A good discussion of topological and differentiable stacks is around definition 2.3 in

\begin{itemize}%
\item [[Jochen Heinloth]], \emph{Some notes on differentiable stacks} (\href{http://www.uni-due.de/~hm0002/stacks.pdf}{pdf})

\end{itemize}
Differentiable stacks are discussed in

\begin{itemize}%
\item [[Kai Behrend]], [[Ping Xu]], \emph{Differentiable Stacks and Gerbes} (\href{http://front.math.ucdavis.edu/0605.5694}{arXiv}).

\end{itemize}
Specifically for the relation to groupoid objects see

3.1 and 3.3 in

\begin{itemize}%
\item Metzler, \emph{Topological and smooth stacks} (\href{http://arxiv.org/abs/math/0306176}{arXiv:math/0306176})

\end{itemize}
paragraphs 2.4.3, 3.4.3, 3.8, 4.3 in

\begin{itemize}%
\item G. Laumon, L. Moret-Bailly, \emph{Champs alg\'e{}briques} , Ergebn. der Mathematik und ihrer Grenzgebiete 39 , Springer-Verlag, Berlin, 2000

\end{itemize}
paragraph 4.4 in

\begin{itemize}%
\item [[Eugene Lerman]], \emph{Orbifolds as stacks?} (\href{http://arxiv.org/abs/0806.4160}{arXiv:0806.4160})

\end{itemize}
See also

\begin{itemize}%
\item [[Bertrand Toen]], \emph{[[Master course on algebraic stacks]]}.


\item [[Aise Johan de Jong]], \emph{[[The Stacks Project]]} (\href{http://www.math.columbia.edu/algebraic_geometry/stacks-git/book.pdf}{pdf}) (\href{http://www.math.columbia.edu/algebraic_geometry/stacks-git/}{project website})



\end{itemize}
Geometric stacks over the site of schemes modeled on [[smooth loci]] is in section 8 of

\begin{itemize}%
\item [[Dominic Joyce]], \emph{Algebraic geometry over $C^\infty$-rings} (\href{http://arxiv.org/abs/1001.0023}{arXiv:1001.0023})

\end{itemize}
[[!redirects geometric stack]] [[!redirects geometric stacks]] [[!redirects presentable stack]] [[!redirects presentable stacks]]



\end{document}