The Stacks project

Definition 102.14.1. Let $\mathcal{X}$ be an algebraic stack.

  1. The lisse-étale site of $\mathcal{X}$ is the full subcategory $\mathcal{X}_{lisse,{\acute{e}tale}}$1 of $\mathcal{X}$ whose objects are those $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over a scheme $U$ such that $x : U \to \mathcal{X}$ is smooth. A covering of $\mathcal{X}_{lisse,{\acute{e}tale}}$ is a family of morphisms $\{ x_ i \to x\} _{i \in I}$ of $\mathcal{X}_{lisse,{\acute{e}tale}}$ which forms a covering of $\mathcal{X}_{\acute{e}tale}$.

  2. The flat-fppf site of $\mathcal{X}$ is the full subcategory $\mathcal{X}_{flat,fppf}$ of $\mathcal{X}$ whose objects are those $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over a scheme $U$ such that $x : U \to \mathcal{X}$ is flat. A covering of $\mathcal{X}_{flat,fppf}$ is a family of morphisms $\{ x_ i \to x\} _{i \in I}$ of $\mathcal{X}_{flat,fppf}$ which forms a covering of $\mathcal{X}_{fppf}$.

[1] In the literature the site is denoted $\text{Lis-ét}(\mathcal{X})$ or $\text{Lis-Et}(\mathcal{X})$ and the associated topos is denoted $\mathcal{X}_{\text{lis-é}t}$ or $\mathcal{X}_{\text{lis-et}}$. In the Stacks project our convention is to name the site and denote the corresponding topos by $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

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