nLab
quotient stack

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 quotient stack $V// G$ is the quotient of this action but formed not in $\mathcal{T}$ but under the inclusion

$$\mathcal{T} \hookrightarrow \mathbf{H}$$

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 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”.

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

$$[X/G](Y):=\{P\xrightarrow{p} Y, P\xrightarrow{f}X | P\rightarrow Y \text{ is a G-bundle,} f \text{ is } G\text{-equivariant}\}.$$

Morphisms of objects are $G$-equivariant isomorphisms. This definition is taken from Heinloth’s 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.

1. 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 $BG$.

2. 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 $\underline{X}$.

3. 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 $\underline{X/G}$.

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

• 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.

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 this page.

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

• If the action of the Lie group $G$ on the manifold $X$ is not necessarily free and proper, what we get is 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 $[X/G]$.

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 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

$$\mathcal{D}(Y):=\{P\xrightarrow{p} Y, P\xrightarrow{f}X | P\rightarrow Y \text{ is a G-bundle,} f \text{ is } G\text{-equivariant}\}.$$

Morphisms of objects are $G$-equivariant isomorphisms. We fix the notation $[X/G]$ for $\mathcal{D}$ and call it the quotient stack.

Properties

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 $G$-gerbe.

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

Related concepts

• quotient, quotient space, quotient type

• action groupoid, global quotient orbifold

• local quotient stack

• Borel construction

• (infinity,1)-colimit, infinity-action

• mapping stack

• orbifold, Deligne-Mumford stack

• geometric invariant theory

References

(…)

• Jack Morava, Theories of anything (arXiv:1202.0684)

• J. Heinloth?, Some notes on Differentiable stacks(stacks)