The Stacks project

Lemma 6.16.1. Let $X$ be a topological space. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of sheaves of sets on $X$.

  1. The map $\varphi $ is a monomorphism in the category of sheaves if and only if for all $x \in X$ the map $\varphi _ x : \mathcal{F}_ x \to \mathcal{G}_ x$ is injective.

  2. The map $\varphi $ is an epimorphism in the category of sheaves if and only if for all $x \in X$ the map $\varphi _ x : \mathcal{F}_ x \to \mathcal{G}_ x$ is surjective.

  3. The map $\varphi $ is an isomorphism in the category of sheaves if and only if for all $x \in X$ the map $\varphi _ x : \mathcal{F}_ x \to \mathcal{G}_ x$ is bijective.

Proof. Omitted. $\square$

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