The Stacks project

Definition 7.3.1. Let $\mathcal{C}$ be a category, and let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of presheaves of sets.

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

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

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View Definition 7.3.1 as pdf