Definition 97.13.1. Let $S$ be a locally Noetherian scheme.

1. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We say $\mathcal{X}$ satisfies openness of versality if given a scheme $U$ locally of finite type over $S$, an object $x$ of $\mathcal{X}$ over $U$, and a finite type point $u_0 \in U$ such that $x$ is versal at $u_0$, then there exists an open neighbourhood $u_0 \in U' \subset U$ such that $x$ is versal at every finite type point of $U'$.

2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We say $f$ satisfies openness of versality if given a scheme $U$ locally of finite type over $S$, an object $y$ of $\mathcal{Y}$ over $U$, openness of versality holds for $(\mathit{Sch}/U)_{fppf} \times _\mathcal {Y} \mathcal{X}$.

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