Definition 78.4.1. Let $S$ be a scheme. Let $a : F \to G$ be a map of presheaves on $(\mathit{Sch}/S)_{fppf}$ which is representable by algebraic spaces. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which

is preserved under any base change, and

is fppf local on the base, see Descent on Spaces, Definition 72.9.1.

In this case we say that $a$ has *property $\mathcal{P}$* if for every scheme $U$ and $\xi : U \to G$ the resulting morphism of algebraic spaces $U \times _ G F \to U$ has property $\mathcal{P}$.

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