Definition 18.42.1. 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.

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 18.41.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.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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