Definition 93.12.1. Let $S$ be a base scheme contained in $\mathit{Sch}_{fppf}$. An *algebraic stack over $S$* is a category

over $(\mathit{Sch}/S)_{fppf}$ with the following properties:

The category $\mathcal{X}$ is a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$.

The diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces.

There exists a scheme $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a $1$-morphism $(\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ which is surjective and smooth

^{1}.

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