Category of representations

The category of representations $𝖱 𝖾 𝗉 𝕂 (𝐺)$ 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 $Γ : 𝐺 \to 𝖵 𝖾 𝖼 𝗍 𝕂$ , then this becomes a Functor category. Namely,

𝖱 𝖾 𝗉 𝕂 (𝐺) ≃ 𝖵 𝖾 𝖼 𝗍 𝕂 𝐺

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

𝖱 𝖾 𝗉 𝕂 (𝐺) ≃ 𝕂 [𝑅] 𝖬 𝗈 𝖽

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Category of representations

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