nLab quotient stack



(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos



For 𝒯\mathcal{T} a sheaf topos, GGrp(𝒯)G \in Grp(\mathcal{T}) a group object and V𝒯V \in \mathcal{T} any object, and for ρ:V×GV\rho \colon V \times G \to V an action of GG on VV , the quotient stack V//GV// G is the quotient of this action but formed not in 𝒯\mathcal{T} but under the inclusion

𝒯H \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 H\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 GG be a Lie group action on a manifold XX (left action).

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

[X/G](Y):={PpY,PfX|PYis a G-bundle,fisG-equivariant}. [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 GG-equivariant isomorphisms. This definition is taken from Heinloth’s Some notes on Differentiable stacks.

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

  1. Suppose XX is trivial and GG acts trivially on X={*}X=\{*\} then [X/G][X/G] should only depend on GG. We know what stack to associate for a Lie group GG i.e., BGBG. Thus, [X/G][X/G] should just be BGBG.

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

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

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

  • Given a manifold MM, we have a stack associated to it, namely M̲\underline{M}. Given a Lie group GG, we have a stack associated to it, namely BGBG. Given a Lie groupoid 𝒢\mathcal{G}, we have a stack associated to it, namely B𝒢B\mathcal{G} i.e., the stack of principal groupoid 𝒢\mathcal{G} bundles.

For action groupoid 𝒢=GX\mathcal{G}=G\ltimes X, let B𝒢B\mathcal{G} be the corresponding stack of principal 𝒢\mathcal{G} bundles. It turns out that B𝒢B\mathcal{G} is same [X/G][X/G] defined above. More details to be found in this page.

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

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

Universal property (??) for Quotient stack

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

Let GG be a Lie group and XX be a manifold with a GG-action.

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

More precisely, a stack morphism M̲𝒟\underline{M}\rightarrow \mathcal{D} is said to be representable if given a manifold NN and a stack morphism N̲𝒟\underline{N}\rightarrow \mathcal{D}, the fiber product M̲× 𝒟N̲\underline{M}\times_{\mathcal{D}}\underline{N} is a manifold. A representable morphism of stacks is said to be a principal GG bundle of stacks if the map M̲× 𝒟N̲N\underline{M}\times_{\mathcal{D}}\underline{N}\rightarrow N is a principal GG-bundle of manifolds. The stack morphism X̲X/G̲\underline{X}\rightarrow \underline{X/G} is a principal GG-bundle of stacks, since the map XX/GX\rightarrow X/G is a principal GG-bundle of manifolds.

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

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

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

𝒟(Y):={PpY,PfX|PYis aG-bundle,fisG-equivariant}. \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 GG-bundles that commute with GG-equivariant morphisms. Fixing notation, we write [X/G][X/G] for 𝒟\mathcal{D} and refer to this as the quotient stack.


Relation to principal and associated bundles

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

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


As stackification of action groupoids

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

As presheaves of equivariant maps out of principal bundles

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

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

As fibrations over BG\mathbf{B}G

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


See also:

Last revised on June 1, 2023 at 19:18:07. See the history of this page for a list of all contributions to it.