Definition 58.56.1. Let $G$ be a topological group.

1. A $G$-module, sometimes called a discrete $G$-module, is an abelian group $M$ endowed with a left action $a : G \times M \to M$ by group homomorphisms such that $a$ is continuous when $M$ is given the discrete topology.

2. A morphism of $G$-modules $f : M \to N$ is a $G$-equivariant homomorphism from $M$ to $N$.

3. The category of $G$-modules is denoted $\text{Mod}_ G$.

Let $R$ be a ring.

1. An $R\text{-}G$-module is an $R$-module $M$ endowed with a left action $a : G \times M \to M$ by $R$-linear maps such that $a$ is continuous when $M$ is given the discrete topology.

2. A morphism of $R\text{-}G$-modules $f : M \to N$ is a $G$-equivariant $R$-module map from $M$ to $N$.

3. The category of $R\text{-}G$-modules is denoted $\text{Mod}_{R, G}$.

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