Equivariant Map

In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. Specifically, let G be a group and let X and Y be two associated G-sets. A function f : X → Y is said to be equivariant if

f(g·x) = g·f(x)

for all g ∈ G and all x in X. Note that if one or both of the actions are right actions the equivariance condition must be suitably modified:

f(x·g) = f(x)·g ; (right-right)

f(x·g) = g−1·f(x) ; (right-left)

f(g·x) = f(x)·g−1 ; (left-right)

Equivariant maps are homomorphisms in the category of G-sets (for a fixed G). Hence they are also known as G-maps or G-homomorphisms. Isomorphisms of G-sets are simply bijective equivariant maps.

The equivariance condition can also be understood as the following commutative diagram. Note that denotes the map that takes an element and returns .