equivariants

Equivariants (or equivariant objects) are mathematical constructs equipped with a symmetry compatible with a group action. In the context of a group G acting on spaces X and Y, a map f: X → Y is called G-equivariant if it commutes with the action: f(g·x) = g·f(x) for all g in G and x in X. Intuitively, applying the group action before or after f yields the same result.

Equivariant maps are the morphisms in the category of G-spaces or G-sets; they preserve the symmetry. If Y has a trivial G-action, a G-equivariant map must send G-orbits into fixed points, so it is constant on orbits. In linear contexts, when X and Y are G-modules (vector spaces with linear G-action), a linear map f is G-equivariant if f(g·v) = g·f(v) for all g and v; these are precisely the morphisms in the category of G-modules.

Extensions include equivariant vector bundles and equivariant morphisms between G-schemes. In topology, one studies equivariant maps between G-spaces, leading to theories such as equivariant cohomology or Borel equivariant cohomology, where one forms the homotopy quotient X_G = EG ×_G X to incorporate symmetry into cohomology calculations. In representation theory and algebraic geometry, these notions unify symmetry, actions on fibers, and quotient constructions.

Equivariant techniques appear across mathematics and physics, providing tools to analyze fixed points, symmetry breaking, and localization phenomena. They formalize the idea that symmetry should be respected by the maps and structures one studies.