Equivariance

Equivariance is the property of a map to the stable under the action of some algebra - usually a group of symmetries.

For example, suppose we have a group G that acts on two sets, X and Y, through actions

\begin{aligned}
- \cdot - &: G \times X \to X \\
- \ast - &:  G \times Y \to Y
\end{aligned}

Then a set-theoretic function f : X \to Y is equivariant exactly when

f(g \cdot x) = g \ast f(x)

for all x in X and g \in G.

Uses

Equivariant functions are ubuquitous in nominal methods in CS.