The Stacks project

Definition 97.8.1. Let $S$ be a scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. We say that $F$ is algebraic if for every scheme $T$ and every object $\xi $ of $\mathcal{Y}$ over $T$ the $2$-fibre product

\[ (\mathit{Sch}/T)_{fppf} \times _{\xi , \mathcal{Y}} \mathcal{X} \]

is an algebraic stack over $S$.

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View Definition 97.8.1 as pdf