nLab
quotient stack

Contents

Context

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

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

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|PY is a G-bundle,f is G-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

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

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

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

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

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

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

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

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

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

𝒟(Y):={PpY,PfX|PY is a G-bundle,f is G-equivariant}. \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 GG-equivariant isomorphisms. We fix the notation [X/G][X/G] for 𝒟\mathcal{D} and call it the quotient stack.

Properties

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.

References

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Last revised on December 21, 2018 at 01:06:23. See the history of this page for a list of all contributions to it.