The Stacks project

Definition 8.4.1. Let $\mathcal{C}$ be a site. A stack over $\mathcal{C}$ is a category $p : \mathcal{S} \to \mathcal{C}$ over $\mathcal{C}$ which satisfies the following conditions:

  1. $p : \mathcal{S} \to \mathcal{C}$ is a fibred category, see Categories, Definition 4.32.5,

  2. for any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and any $x, y \in \mathcal{S}_ U$ the presheaf $\mathit{Mor}(x, y)$ (see Definition 8.2.2) is a sheaf on the site $\mathcal{C}/U$, and

  3. for any covering $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ of the site $\mathcal{C}$, any descent datum in $\mathcal{S}$ relative to $\mathcal{U}$ is effective.


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