Editing Group representation (section)

===Representations in other categories===
Every group ''G'' can be viewed as a [[category (mathematics)|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 ''C'' is a [[functor]] from ''G'' to ''C''. Such a functor selects an object ''X'' in ''C'' and a group homomorphism from ''G'' to Aut(''X''), the [[automorphism group]] of ''X''.

In the case where ''C'' is '''Vect'''<sub>''K''</sub>, the [[category of vector spaces]] over a field ''K'', this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of ''G'' in the [[category of sets]].

When ''C'' is '''Ab''', the [[category of abelian groups]], the objects obtained are called [[G-module|''G''-modules]].

For another example consider the [[category of topological spaces]], '''Top'''. Representations in '''Top''' are homomorphisms from ''G'' to the [[homeomorphism]] group of a topological space ''X''.

Two types of representations closely related to linear representations are:
*[[projective representation]]s: in the category of [[projective space]]s. These can be described as "linear representations [[up to]] scalar transformations". 
*[[affine representation]]s: in the category of [[affine space]]s. For example, the [[Euclidean group]] acts affinely upon [[Euclidean space]].