The Stacks project

94.11 Stacks in groupoids

Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Recall that a category $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ over $(\mathit{Sch}/S)_{fppf}$ is said to be a stack in groupoids (see Stacks, Definition 8.5.1) if and only if

1. $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ is fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$,

2. for all $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, for all $x, y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U)$ the presheaf $\mathit{Isom}(x, y)$ is a sheaf on the site $(\mathit{Sch}/U)_{fppf}$, and

3. for all coverings $\mathcal{U} = \{ U_ i \to U\} $ in $(\mathit{Sch}/S)_{fppf}$, all descent data $(x_ i, \phi _{ij})$ for $\mathcal{U}$ are effective.

For examples see Examples of Stacks, Section 95.9 ff.