Equivariant maps generalize the concept of invariants, functions whose value is unchanged by a symmetry transformation of their argument.
More generally, all triangle centers are also equivariant under similarity transformations (combinations of translation, rotation, reflection, and scaling),[1] and the centroid is equivariant under affine transformations.
In this way, the function mapping each triangle to its area or perimeter can be seen as equivariant for a multiplicative group action of the scaling transformations on the positive real numbers.
By contrast, the mean is not equivariant with respect to nonlinear transformations such as exponentials.
The median of a sample is equivariant for a much larger group of transformations, the (strictly) monotonic functions of the real numbers.
As a consequence, in important cases the construction of an intertwiner is enough to show the representations are effectively the same.
Equivariant maps can be generalized to arbitrary categories in a straightforward manner.
For example, a G-set is equivalent to a functor from G to the category of sets, Set, and a linear representation is equivalent to a functor to the category of vector spaces over a field, VectK.
For another example, take C = Top, the category of topological spaces.
A representation of G in Top is a topological space on which G acts continuously.