GSets

GSets, usually written as G-sets, denotes the category of sets equipped with a fixed group action by a group G. An object is a pair (X, ·) where X is a set and ·: G × X → X is a left group action. A morphism between G-sets is a map f: X → Y that is equivariant, meaning f(g · x) = g · f(x) for all g ∈ G and x ∈ X. The category is often called G-Set or GSets.

Examples include the action of G on itself by left multiplication, giving the regular G-set G, and any set X with a permutation representation of G. Coset spaces G/H form transitive G-sets, where G acts by left multiplication; stabilizers of points are conjugates of H, and transverse orbits correspond to subgroups up to conjugacy.

A key feature of G-sets is their orbit decomposition: every G-set decomposes into a disjoint union of orbits, and each orbit is transitive and isomorphic to a coset space G/H for some subgroup H ≤ G. For finite G-sets, this leads to a decomposition into finitely many finite transitive G-sets, and the Orbit-Stabilizer Theorem describes the size of each orbit.

Constructions and functors are natural in this setting. Products carry the diagonal action: g · (x, y) = (g · x, g · y). Coproducts are disjoint unions with the diagonal action. Quotients by G-equivalence relations yield coequalizers. Restrictions to subgroups yield H-sets, and induction from H-sets to G-sets is given by the construction G ×_H X. Finite G-sets form a basis for the Burnside ring, with addition given by disjoint union and multiplication by cartesian product.

GSets provide a discrete framework for studying symmetry and permutation representations, with applications in combinatorics, representation theory, and related areas.