Group Actions & Quotient Geometry

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Abstractions about groupoids, group actions, bundles, quotients, homogeneous spaces, stacks, gerbes, and symmetry-based geometric constructions.

14 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Action groupoid — The groupoid whose objects are points acted on by a group and whose arrows record group elements carrying one point to another.
• Associated bundle — A fiber bundle obtained from a principal G-bundle and a G-space by quotienting their product under the diagonal group action.
• Burnside's lemma — The orbit-counting result that the number of orbits of a finite group action equals the average number of elements fixed by a group element.
• Categorical quotient — A universal invariant morphism from an object with group action through which every other invariant morphism factors uniquely.
• Fundamental domain — A representative region for a group action containing one representative from each orbit, whose translates reconstruct the acted-on space up to boundary identifications.
• Gerbe — A stack locally equivalent to the classifying stack of a group, serving as a degree-two geometric analogue of a principal bundle and encoding obstruction and twisting data.
• Lens space — A closed manifold obtained by a cyclic quotient of an odd-dimensional sphere, or in dimension three by gluing two solid tori with a slope specified by coprime integers p and q.
• Medial magma — A magma whose binary operation satisfies (ab)(cd)=(ac)(bd) for all four elements.
• Paradoxical set — A set that can be partitioned into finitely many pieces and moved by a group action into two disjoint reconstructions of the whole, exposing nonamenability and the failure of finitely additive invariant size on all subsets.
• Pointed set — A set equipped with one distinguished basepoint, with morphisms required to preserve that point, forming a category that adds a canonical zero-like reference to otherwise unstructured sets.
• Principal homogeneous space — A nonempty set or space with a free and transitive action of a group, also called a torsor.
• Quotient stack — An algebraic stack [X/G] that retains stabilizers and families of objects while representing a group action's quotient.
• Splitting principle — A technique that pulls a vector bundle to a space where it decomposes into line bundles, performs calculations there, and transfers valid identities back through an injective cohomology map.
• Torus action — An algebraic or smooth group action of a torus on a variety or manifold, organizing points into orbits and exposing weights, fixed points, quotients, and combinatorial structure.