Definition 103.14.1. Let $\mathcal{X}$ be an algebraic stack.

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 $\mathcal{X}_{\acute{e}tale}$.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 $\mathcal{X}_{fppf}$.

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