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.