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

(references for what the following paragraphs are getting at are listed below)

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

Supposing that $G$ acts freely and properly on $X$, the quotient stack $[X/G]$ will be the stack $\underline{X/G}$. This action yields a principal $G$-bundle of manifolds $X\rightarrow X/G$, which gives a morphism of stacks $\underline{X}\rightarrow \underline{X/G}$. We refer to this stack morphism $\underline{X}\rightarrow \underline{X/G}$ as a principal $G$-bundle of stacks.

More precisely, a stack morphism $\underline{M}\rightarrow \mathcal{D}$ is said to be representable if given a manifold $N$ and a stack morphism $\underline{N}\rightarrow \mathcal{D}$, the fiber product $\underline{M}\times_{\mathcal{D}}\underline{N}$ is a manifold. A representable morphism of stacks is said to be a principal $G$ bundle of stacks if the map $\underline{M}\times_{\mathcal{D}}\underline{N}\rightarrow N$ is a principal $G$-bundle of manifolds. The stack morphism $\underline{X}\rightarrow \underline{X/G}$ is a principal $G$-bundle of stacks, since the map $X\rightarrow X/G$ is a principal $G$-bundle of manifolds.

The property “$\underline{X}\rightarrow \underline{X/G}$ is a principal $G$-bundle” is the main ingredient in the definition of the quotient stack $[X/G]$. Irrespective of whether or not $G$ acts freely and properly on $X$, we still want to define a quotient stack as a stack $\mathcal{D}$ such that $\underline{X}\rightarrow \mathcal{D}$ is a principal $G$-bundle of stacks in a “minimal” way.

The quotient stack of the action of $G$ on $X$ is a stack $\mathcal{D}$ equipped with a principal $G$-bundle of stacks $\underline{X}\rightarrow \mathcal{D}$ such that any other principal $G$-bundle of stacks $\underline{X}\rightarrow \mathcal{C}$ factors through $\underline{X}\rightarrow \mathcal{D}$.

If $G$ acts freely and properly, then an obvious choice for $\mathcal{D}$ is the stack $\underline{X/G}$. By the universal property, $\mathcal{D}$ must be precisely the stack appearing in the definition of quotient stacks, i.e.

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

Morphisms of quotient stacks are isomorphisms of principal $G$-bundles that commute with $G$-equivariant morphisms. Fixing notation, we write $[X/G]$ for $\mathcal{D}$ and refer to this as 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

As stackification of action groupoids

The definition of quotient stacks as stackifications of (presheaves of) action groupoids is considered for instance in:

• Gérard Laumon, Laurent Moret-Bailly, Prop. 4.3.1 in: Champs algébriques, Ergebn. der Mathematik und ihrer Grenzgebiete 39, Springer (2000) [doi:10.1007/978-3-540-24899-6]

• Stacks Project, Definition 77.20.1 in: Quotient Stacks [tag:044Q]

As presheaves of equivariant maps out of principal bundles

The construction of quotient stacks $X\sslash G$ as prestacks of $G$-principal bundles equipped with $G$-equivariant maps to $X$ is considered for instance in:

• Jochen Heinloth, Exp. 1.5 in: Notes on differentiable stacks (2004) [pdf, pdf]

• Frank Neumann, p. 28 in: Algebraic Stacks and Moduli of Vector Bundles, impa (2011) [pdf, pdf]

• Michael Groechenig, Def. 5.23 in: Algebraic Stacks, Lecture notes (2014) [web, pdf, pdf]

• Daniel Halpern-Leistner, Section 7.1 of: Moduli theory, Lecture notes (2020) [pdf, pdf]

Discussion of sufficient conditions for this construction to really yield a stack (instead of just a prestack):

• Elena Caviglia, Generalized principal bundles and quotient stacks, Theory and Applications of Categories 39 20 (2023) 567-597 [arXiv:2302.07852, tac:39-20]

As fibrations over $\mathbf{B}G$

The characterization of quotient stacks $X \sslash G$ as fibrations over delooping stacks $\ast \sslash G \simeq \mathbf{B}G$ with homotopy fiber $X$ (cf. also the discussion at $\infty$-action):

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Exp. 4.4 in: Principal $\infty$-bundles – General theory, J. Hom. Rel. Struc. 10 4 (2015) 749-801 [arXiv:1207.0248, doi:10.1007/s40062-014-0083-6]

• Hisham Sati, Urs Schreiber, Prop. 2.79 in: Proper Orbifold Cohomology [arXiv:2008.01101]

• Hisham Sati, Urs Schreiber, Prop. 0.2.1 (ii) & Prop. 3.2.76 in: Equivariant principal $\infty$-bundles [arXiv:2112.13654]

• Halpern-Leistner (2020), Exc. 7.2

More

See also:

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