Group representation theory MOC

Category of representations

The category of representations of a group over a field has representations carried by -vector spaces as its objects, and equivariant linear maps as morphisms between them. If is viewed as a category, and a representations as a functor , then this becomes a Functor category. Namely,

where an equivariant map is a Natural transformation. Equivalent representations are thereby naturally equivalent. An alternate viewpoint is to consider a representation as a module over a group, so

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