The Stacks project

Definition 34.9.12. Let $F$ be a contravariant functor on the category of schemes with values in sets.

  1. Let $\{ U_ i \to T\} _{i \in I}$ be a family of morphisms of schemes with fixed target. We say that $F$ satisfies the sheaf property for the given family if for any collection of elements $\xi _ i \in F(U_ i)$ such that $\xi _ i|_{U_ i \times _ T U_ j} = \xi _ j|_{U_ i \times _ T U_ j}$ there exists a unique element $\xi \in F(T)$ such that $\xi _ i = \xi |_{U_ i}$ in $F(U_ i)$.

  2. We say that $F$ satisfies the sheaf property for the fpqc topology if it satisfies the sheaf property for any fpqc covering.

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