The Stacks project

Definition 8.5.1. A stack in groupoids over a site $\mathcal{C}$ is a category $p : \mathcal{S} \to \mathcal{C}$ over $\mathcal{C}$ such that

  1. $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids over $\mathcal{C}$ (see Categories, Definition 4.35.1),

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

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

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View Definition 8.5.1 as pdf