[[Group representation theory MOC]]
# Category of representations

The **category of representations** $\Rep_{\mathbb{K}}(G)$ of a [[group]] $G$ over a [[field]] $\mathbb{K}$ has representations carried by $\mathbb{K}$-[[Vector space|vector spaces]] as its objects,
and [[Category of G-spaces|equivariant]] [[Linear map|linear maps]] as morphisms between them.
If $G$ is viewed as a [[category]], and a representations as a [[functor]] $\Gamma : G \to \Vect_{\mathbb{K}}$,
then this becomes a [[Functor category]].
Namely,
$$
\begin{align*}
\Rep_{\mathbb{K}}(G) \simeq {\Vect_{\mathbb{K}}}^G
\end{align*}
$$
where an equivariant map is a [[Natural transformation]].
Equivalent representations are thereby [[Natural isomorphism|naturally equivalent]].
An alternate viewpoint is to consider a representation as a [[module over a group]], so
$$
\begin{align*}
\Rep_{\mathbb{K}}(G) \simeq \lMod{\mathbb{K}[R]}
\end{align*}
$$

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