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.
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.
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¶
- 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) — more general patterns this builds on
-
Differentiable stack is a kind of Representation Prime
The proposed strict upward parent is
prime:representation.
Hierarchy path (1) — routes to 1 parentless root
- Differentiable stack → Representation → Abstraction
Neighborhood in Abstraction Space¶
Differentiable stack sits in a moderately populated region (50th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Differential Topology & Geometric Structure (11 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 · 2026-09-08