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\section*{quotient stack}

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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{motivation_for_definition_of_quotient_stack}{Motivation for definition of quotient stack.}\dotfill \pageref*{motivation_for_definition_of_quotient_stack} \linebreak
\noindent\hyperlink{universal_property__for_quotient_stack}{Universal property (??) for Quotient stack}\dotfill \pageref*{universal_property__for_quotient_stack} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_principal_and_associated_bundles}{Relation to principal and associated bundles}\dotfill \pageref*{relation_to_principal_and_associated_bundles} \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 $\mathcal{T}$ a [[sheaf topos]], $G \in Grp(\mathcal{T})$ a [[group object]] and $V \in \mathcal{T}$ any object, and for $\rho \colon V \times G \to V$ an [[action]] of $G$ on $V$ , the \textbf{quotient stack} $V// G$ is the [[quotient]] of this action but formed not in $\mathcal{T}$ but under the inclusion

\begin{displaymath}
\mathcal{T} \hookrightarrow \mathbf{H}
\end{displaymath}
into the [[(2,1)-topos]] over the given [[site]] of definition: it is the quotient after regarding the action as an [[infinity-action]] in $\mathbf{H}$.

This is the geometric version of the notion of \emph{[[action groupoid]]}. A wider notion of the quotient stack may be defined using more general internal groupoids. Indeed, the [[small fibration]] obtained by the [[externalization]] of an internal groupoid in a site with pullbacks will be a [[fibered category]] which is a candidate for a quotient stack in this context. For many interesting sites, sometimes under additional conditions on the internal groupoid, the resulting small fibration is indeed a stack.

If the [[stabilizer subgroups]] of the [[action]] are [[finite groups]], then the quotient stack is an [[orbifold]]/[[Deligne-Mumford stack]] --the ``quotient orbifold''.

\hypertarget{motivation_for_definition_of_quotient_stack}{}\subsection*{{Motivation for definition of quotient stack.}}\label{motivation_for_definition_of_quotient_stack}

Let $G$ be a Lie group action on a manifold $X$ (left action).

We define the quotient stack $[X/G]$ as

\begin{displaymath}
[X/G](Y):=\{P\xrightarrow{p} Y, P\xrightarrow{f}X | P\rightarrow Y \text{ is a G-bundle,} f  \text{ is } G\text{-equivariant}\}.
\end{displaymath}
Morphisms of objects are $G$-equivariant isomorphisms. This definition is taken from Heinloth's \href{https://www.uni-due.de/~hm0002/stacks.pdf}{Some notes on Differentiable stacks}.

Given a Lie group action of $G$ on $X$, if we want to associate a stack, we start with simpler cases which allows us to guess how to define $[X/G]$ in general.

\begin{enumerate}%
\item Suppose $X$ is trivial and $G$ acts trivially on $X=\{*\}$ then $[X/G]$ should only depend on $G$. We know what stack to associate for a Lie group $G$ i.e., $BG$. Thus, $[X/G]$ should just be $BG$.


\item Suppose $G$ is trivial and $G$ acts on $X$, $[X/G]$ should only depend on $X$. We know what stack to associate for a manifold $X$ i.e., $\underline{X}$. Thus, $[X/G]$ should just be $\underline{X}$.


\item Suppose $G$ is non trivial and $X$ is non trivial and that the action of $G$ on $X$ is free (and proper) so that $X/G$ is a manifold. We know what stack to associate for a manifold $X/G$ i.e., $\underline{X/G}$. Thus, $[X/G]$ should just be $\underline{X/G}$.



\end{enumerate}
For general case of $G$ acting on $X$, we get a Lie groupoid, called the Translation groupoid (or action groupoid) usually denoted by $G\ltimes X$.

\begin{itemize}%
\item Given a manifold $M$, we have a stack associated to it, namely $\underline{M}$. Given a Lie group $G$, we have a stack associated to it, namely $BG$. Given a Lie groupoid $\mathcal{G}$, we have a stack associated to it, namely $B\mathcal{G}$ i.e., the stack of principal groupoid $\mathcal{G}$ bundles.

\end{itemize}
For action groupoid $\mathcal{G}=G\ltimes X$, let $B\mathcal{G}$ be the corresponding stack of principal $\mathcal{G}$ bundles. It turns out that $B\mathcal{G}$ is same $[X/G]$ defined above. More details to be found in \href{https://mathoverflow.net/questions/319038/motivation-for-definition-of-quotient-stack}{this page}.

\begin{itemize}%
\item If action of the Lie group $G$ on the manifold $X$ is free and proper, what we get is \textbf{a manifold} $X/G$. Stack associated to this manifold is $\underline{X/G}$ which we call to be the quotient stack, denote by $[X/G]$.


\item If the action of the Lie group $G$ on the manifold $X$ is not necessarily free and proper, what we get is \textbf{a Lie groupoid} denoted (among other symbols) by $X//G$. Stack associated to this Lie groupoid $X//G$ is $B(X//G)$ which we call to be the quotient stack, denote by $[X/G]$.



\end{itemize}
\hypertarget{universal_property__for_quotient_stack}{}\subsection*{{Universal property (??) for Quotient stack}}\label{universal_property__for_quotient_stack}

Let $G$ be a Lie group and $X$ be a manifold with a $G$ action on it.

Suppose $G$ acts freely, properly on $X$ then, we have mentioned that the quotient stack $[X/G]$ has to be the stack $\underline{X/G}$. The proper, free action of $G$ on $X$ gives a principal $G$ bundle $X\rightarrow X/G$. This $X\rightarrow X/G$ gives a map of stacks $\underline{X}\rightarrow \underline{X/G}$. We call the map of stacks $\underline{X}\rightarrow \underline{X/G}$ to be a principal $G$ bundle.

A map of stacks $\underline{M}\rightarrow \mathcal{D}$ is said to be representable morphism if given a manifold $N$ and a map of stacks $\underline{N}\rightarrow \mathcal{D}$, the fiber product $\underline{M}\times_{\mathcal{D}}\underline{N}$ is a manifold.

A map of stacks $\underline{M}\rightarrow \mathcal{D}$ is said to be a principal $G$ bundle if it is a representable morphism and the map of manifolds $\underline{M}\times_{\mathcal{D}}\underline{N}\rightarrow N$ is a principal $G$ bundle.

It is easy to see that the map of stacks $\underline{X}\rightarrow \underline{X/G}$ is a principal $G$ bundle as the map of manifolds $X\rightarrow X/G$ is a principal $G$ bundle.

We see the property ``$\underline{X}\rightarrow \underline{X/G}$ is a principal $G$ bundle'' as main ingredient to define the quotient stack $[X/G]$. Irrespective of $G$ acting freely and properly on $X$, we want to define quotient stack as a stack $\mathcal{D}$ such that $\underline{X}\rightarrow \mathcal{D}$ is a principal $G$ bundle in minimal terms.

More precisely, by quotient stack of the action of $G$ on $X$, we mean a stack $\mathcal{D}$ that \textbf{comes with a map of stacks $\underline{X}\rightarrow \mathcal{D}$} that is a principal $G$ bundle (in the sense defined above) any map of stacks $\underline{X}\rightarrow \mathcal{C}$ that is a principal $G$ bundle factors through this map $\underline{X}\rightarrow \mathcal{D}$.

If $G$ acts freely and properly, then obvious choice for $\mathcal{D}$ is the stack $\underline{X/G}$.

Using the universal property, it turns out that $\mathcal{D}$ has to be the stack in the definition of quotient stack

\begin{displaymath}
\mathcal{D}(Y):=\{P\xrightarrow{p} Y, P\xrightarrow{f}X | P\rightarrow Y \text{ is a G-bundle,} f  \text{ is } G\text{-equivariant}\}.
\end{displaymath}
Morphisms of objects are $G$-equivariant isomorphisms. We fix the notation $[X/G]$ for $\mathcal{D}$ and call it the quotient stack.

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

\hypertarget{relation_to_principal_and_associated_bundles}{}\subsubsection*{{Relation to principal and associated bundles}}\label{relation_to_principal_and_associated_bundles}

For $V = *$ the [[terminal object]], one writes $\mathbf{B}G \coloneqq *// G$. This is the [[moduli stack]] for $G$-[[principal bundles]]. It is also the trivial \emph{$G$-[[gerbe]]}.

There is a canonical projection $\overline{\rho} \;\colon\; V// G \to \mathbf{B}G$. This is the [[universal associated infinity-bundle|universal rho-associated bundle]].

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

\begin{itemize}%
\item [[quotient]], [[quotient space]], [[quotient type]]


\item [[action groupoid]]


\item [[local quotient stack]]


\item [[Borel construction]]


\item [[(infinity,1)-colimit]], [[infinity-action]]


\item [[mapping stack]]


\item [[orbifold]], [[Deligne-Mumford stack]]


\item [[geometric invariant theory]]



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

(\ldots{})

\begin{itemize}%
\item [[Jack Morava]], \emph{Theories of anything} (\href{http://arxiv.org/abs/1202.0684}{arXiv:1202.0684})


\item [[J. Heinloth]], \emph{Some notes on Differentiable stacks}(\href{https://www.uni-due.de/~hm0002/stacks.pdf}{stacks})



\end{itemize}
[[!redirects quotient stacks]]

[[!redirects orbifold quotient]] [[!redirects orbifold quotients]]

[[!redirects quotient orbifold]] [[!redirects quotient orbifolds]]

[[!redirects stacky quotient]]



\end{document}