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Definition 39.13.1. Let $S$ be a scheme.

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

    \[ (U(T), R(T), s, t, c) \]

    is a groupoid category in the sense described above.

  2. A morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoid schemes over $S$ is given by morphisms of schemes $f : U \to U'$ and $f : R \to R'$ with the following property: For any scheme $T$ over $S$ 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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