The Stacks project

Definition 18.43.1. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\Lambda $, etc.

  1. We say $\mathcal{F}$ is a constant sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\Lambda $, etc if it is isomorphic as a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\Lambda $, etc to a constant sheaf $\underline{E}$ as in Section 18.42.

  2. We say $\mathcal{F}$ is locally constant if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ such that $\mathcal{F}|_{U_ i}$ is a constant sheaf.

  3. If $\mathcal{F}$ is a sheaf of sets or groups, then we say $\mathcal{F}$ is finite locally constant if the constant values are finite sets or finite groups.

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