Editing Equivariant map (section)

{{short description|Maps whose domain and codomain are acted on by the same group, and the map commutes}}
In [[mathematics]], '''equivariance''' is a form of [[symmetry]] for [[function (mathematics)|function]]s from one space with symmetry to another (such as [[symmetric space]]s). A function is said to be an '''equivariant map''' when its domain and codomain are [[Group action (mathematics)|acted on]] by the same [[symmetry group]], and when the function [[commutative property|commutes]] with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation.

Equivariant maps generalize the concept of [[Invariant (mathematics)|invariants]], functions whose value is unchanged by a symmetry transformation of their argument. The value of an equivariant map is often (imprecisely) called an invariant.

In [[statistical inference]], equivariance under statistical transformations of data is an important property of various estimation methods; see [[invariant estimator]] for details. In pure mathematics, equivariance is a central object of study in [[equivariant topology]] and its subtopics [[equivariant cohomology]] and [[equivariant stable homotopy theory]].