Representation theory
Representations in other categories

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 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, the category of vector spaces over a field F, this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of G in the category of sets.

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.

Three types of representations closely related to linear representations are:
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