Definition 77.5.1. Let $B \to S$ as in Section 77.3.

1. A group algebraic space over $B$ is a pair $(G, m)$, where $G$ is an algebraic space over $B$ and $m : G \times _ B G \to G$ is a morphism of algebraic spaces over $B$ with the following property: For every scheme $T$ over $B$ the pair $(G(T), m)$ is a group.

2. A morphism $\psi : (G, m) \to (G', m')$ of group algebraic spaces over $B$ is a morphism $\psi : G \to G'$ of algebraic spaces over $B$ such that for every $T/B$ the induced map $\psi : G(T) \to G'(T)$ is a homomorphism of groups.

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