Definition 77.11.1. Let $B \to S$ as in Section 77.3.

1. A groupoid in algebraic spaces over $B$ is a quintuple $(U, R, s, t, c)$ where $U$ and $R$ are algebraic spaces over $B$, and $s, t : R \to U$ and $c : R \times _{s, U, t} R \to R$ are morphisms of algebraic spaces over $B$ with the following property: For any scheme $T$ over $B$ the quintuple

$(U(T), R(T), s, t, c)$

is a groupoid category.

2. A morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoids in algebraic spaces over $B$ is given by morphisms of algebraic spaces $f : U \to U'$ and $f : R \to R'$ over $B$ with the following property: For any scheme $T$ over $B$ the maps $f$ define a functor from the groupoid category $(U(T), R(T), s, t, c)$ to the groupoid category $(U'(T), R'(T), s', t', c')$.

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