Definition 88.10.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $f$ 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 68.9.1.

In this case we say that $f$ has *property $\mathcal{P}$* if for every $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and any $y \in \mathcal{Y}_ U$ the resulting morphism of algebraic spaces $f_ y : F_ y \to U$, see diagram (88.9.1.1), has property $\mathcal{P}$.

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