Definition 77.8.1. Let $B \to S$ as in Section 77.3. Let $(G, m)$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$.

1. An action of $G$ on the algebraic space $X/B$ is a morphism $a : G \times _ B X \to X$ over $B$ such that for every scheme $T$ over $B$ the map $a : G(T) \times X(T) \to X(T)$ defines the structure of a $G(T)$-set on $X(T)$.

2. Suppose that $X$, $Y$ are algebraic spaces over $B$ each endowed with an action of $G$. An equivariant or more precisely a $G$-equivariant morphism $\psi : X \to Y$ is a morphism of algebraic spaces over $B$ such that for every $T$ over $B$ the map $\psi : X(T) \to Y(T)$ is a morphism of $G(T)$-sets.

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