Editing Equivariant map (section)

==Generalization==
{{See also|Representation theory#Generalizations|Category of representations#Category-theoretic definition}}
{{unreferenced section|date=April 2016}}
Equivariant maps can be generalized to arbitrary [[category (mathematics)|categories]] in a straightforward manner. Every group ''G'' can be viewed as a category with a single object ([[morphism]]s 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 [[automorphism]]s 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'''<sub>''K''</sub>.

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]] ''C''<sup>''G''</sup>.

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 [[continuous function|continuously]]. An equivariant map is then a continuous map ''f'' : ''X'' → ''Y'' between representations which commutes with the action of ''G''.