Differentiable stack

Represent quotient-like smooth geometry as a stack on manifolds admitting a smooth atlas, equivalently through a Lie-groupoid presentation considered up to Morita equivalence.

Version

v3 · 2026-09-19 · History

Domain-specific #

1667

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Differential Geometry, Differentiable Stacks and Lie Groupoids → Mathematics

Core Idea

A differentiable stack is a stack over smooth manifolds that admits a suitable smooth representable surjective atlas by a manifold; choosing an atlas yields a Lie groupoid, and equivalent presentations are related by Morita equivalence. Descent glues local objects and arrows, an atlas covers the stack by ordinary smooth geometry, and the atlas fiber product records overlaps as a Lie groupoid whose presentation-dependent details are quotiented by Morita equivalence.

Its autonomous residual is the smooth-site stack plus atlas or Morita-equivalent Lie-groupoid identity, not a generic stack, one chosen groupoid, or an arbitrary singular topological quotient.

How would you explain it like I'm…

 

No faithful explanation at this level. All three generators agree that a five-year-old picture could only be a glued, folded or squished shape, which is the coarse quotient space the concept exists to avoid, losing the isotropy and gluing data a differentiable stack keeps.

 

No faithful explanation at this level. Two of three generators judge that without groupoids and descent the story reduces to a manifold glued from patches or a folded-up shape, i.e. the coarse quotient that discards the isotropy the stack retains.

Symmetry-Remembering Smooth Spaces

A differentiable stack is a generalized smooth space used when an ordinary quotient, like a space divided by a symmetry group, would lose information or fail to be smooth. Instead of forming the quotient directly, you cover the stack with an ordinary smooth manifold, called an atlas. The overlaps of the atlas with itself form a Lie groupoid: a manifold of points, a manifold of arrows between them, and smooth rules for composing arrows. The arrows record which points are identified and how, including symmetries that fix a point. Different atlases give different groupoids, but these are considered the same stack when they are Morita equivalent, so no single presentation is the official one. Examples include orbifolds and quotients by group actions.

 

A differentiable stack is a stack over the site of smooth manifolds that admits a representable, surjective, suitably smooth atlas X → 𝒳 from an ordinary manifold. Descent data glue local objects and arrows, and the fiber product X ×_𝒳 X records the overlaps; together with X it forms a Lie groupoid presenting the stack. The groupoid depends on the atlas chosen, and different presentations of the same stack are related by Morita equivalence, so only Morita-invariant properties belong to the stack. This is not a generic stack, nor a single chosen groupoid, nor a coarse topological quotient: the construction retains isotropy and descent information that an orbit space discards. Verifying it requires stating the site and covering convention, checking descent, and confirming the atlas is representable, smooth, and surjective. Differentiable stacks give a framework for orbifolds, quotient stacks of group actions, foliation leaf spaces, and gerbes, but those applications are not the definition. Algebraic stacks live on a different site and are not interchangeable with them.

Scope of Application

Differentiable stack applies when the analyst can specify a stack in groupoids on a chosen site of smooth manifolds together with a representable smooth surjective atlas, or a Lie groupoid presentation and establish that the stack satisfies descent on the declared smooth site and possesses an atlas meeting the representability and smooth-surjective conditions of the selected convention, with geometric claims invariant under presentation equivalence. The entry locks ordinary differentiable stacks on smooth manifolds; derived, higher, analytic, algebraic, and diffeological stacks need separately typed foundations.

Clarity

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because differentiable stack can name either the stack object or an equivalence class of Lie groupoid presentations, and definition variants differ on built-in separation or diagonal conditions. The disciplined statement is that the object counts as Differentiable stack exactly when the stack satisfies descent on the declared smooth site and possesses an atlas meeting the representability and smooth-surjective conditions of the selected convention, with geometric claims invariant under presentation equivalence

Manages Complexity

The abstraction compresses quotient stacks, orbifolds, gerbes, foliation stacks, proper or étale stacks, separated variants, and equivalent atlas or principal-bundle formulations into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares site, cover topology, representability, atlas class, diagonal, source and target maps, isotropy, properness, étaleness, presentation, Morita equivalence, and coarse space and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning

1. Type the carrier. Establish a stack in groupoids on a chosen site of smooth manifolds together with a representable smooth surjective atlas, or a Lie groupoid presentation and reject examples from a different problem.
2. Lock the rule. Express that the stack satisfies descent on the declared smooth site and possesses an atlas meeting the representability and smooth-surjective conditions of the selected convention, with geometric claims invariant under presentation equivalence independently of one notation or implementation.

Knowledge Transfer

Transfer within differential geometry is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from For a Lie group \(G\) acting smoothly on a manifold \(M\), the quotient stack \([M/G]\) is presented by the action groupoid \(G\times M\rightrightarrows M\). to An orbifold can be represented by a proper étale Lie groupoid and therefore determines a differentiable stack with discrete finite isotropy under the standard hypotheses. demonstrates that continuity.

Relationships to Other Abstractions

Current abstraction Differentiable stack Domain-specific

Parents (1) — outgoing typed relations

• Differentiable stack is a kind of Representation Prime

  A differentiable stack literally represents quotient-like smooth geometry by compatible local manifolds and a Lie groupoid presentation while preserving equivalence under changes of presentation.

Hierarchy path (1) — routes to 1 parentless root

• Differentiable stack → is a kind of Representation → presupposes Abstraction

Neighborhood in Abstraction Space

Differentiable stack sits in a crowded region of the domain-specific corpus (31^st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Manifolds, Stacks & Classifying Spaces (18 abstractions)

Nearest neighbors

• Quotient stack — 0.89
• Quotient space of an algebraic stack — 0.89
• Gerbe — 0.88
• Algebraic space — 0.88
• Weakly symmetric space — 0.88

Computed from structural-signature embeddings · 2026-10-25