The Stacks project

Definition 7.11.1. Let $\mathcal{C}$ be a site, and let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of sheaves of sets.

  1. We say that $\varphi $ is injective if for every object $U$ of $\mathcal{C}$ the map $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ is injective.

  2. We say that $\varphi $ is surjective if for every object $U$ of $\mathcal{C}$ and every section $s\in \mathcal{G}(U)$ there exists a covering $\{ U_ i \to U\} $ such that for all $i$ the restriction $s|_{U_ i}$ is in the image of $\varphi : \mathcal{F}(U_ i) \to \mathcal{G}(U_ i)$.

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View Definition 7.11.1 as pdf