Equivariant map

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.

The centroid of a triangle (where the three red segments meet) is equivariant under affine transformations : the centroid of a transformed triangle is the same point as the transformation of the centroid of the triangle.