Group representation theory MOC

Category of representations

The category of representations 𝖱𝖾𝗉𝕂(𝐺) of a group 𝐺 over a field 𝕂 has representations carried by 𝕂-vector spaces as its objects, and equivariant linear maps as morphisms between them. If 𝐺 is viewed as a category, and a representations as a functor Γ :𝐺 →𝖵𝖾𝖼𝗍𝕂, 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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