Definition 92.12.1. Let $S$ be a base scheme contained in $\mathit{Sch}_{fppf}$. An algebraic stack over $S$ is a category

$p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$

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

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

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

3. 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 smooth1.

 In future chapters we will denote this simply $U \to \mathcal{X}$ as is customary in the literature. Another good alternative would be to formulate this condition as the existence of a representable category fibred in groupoids $\mathcal{U}$ and a surjective smooth $1$-morphism $\mathcal{U} \to \mathcal{X}$.

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