Equivariant Map - Categorical Description

Categorical Description

Equivariant maps can be generalized to arbitrary categories in a straightforward manner. Every group G can be viewed as a category with a single object (morphisms in this category are just the elements of G). Given an arbitrary category C, a representation of G in the category C is a functor from G to C. Such a functor selects an object of C and a subgroup of automorphisms of that object. 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, Vect_K.

Given two representations, ρ and σ, of G in C, an equivariant map between those representations is simply a natural transformation from ρ to σ. Using natural transformations as morphisms, one can form the category of all representations of G in C. This is just the functor category CG.

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. An equivariant map is then a continuous map f : X → Y between representations which commutes with the action of G.