The Stacks project

Definition 94.10.1. 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

  1. is preserved under any base change, and

  2. is fppf local on the base, see Descent on Spaces, Definition 74.10.1.

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 (, has property $\mathcal{P}$.

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