The Stacks project

Definition 6.16.2. Let $X$ be a topological space.

1. A presheaf $\mathcal{F}$ is called a subpresheaf of a presheaf $\mathcal{G}$ if $\mathcal{F}(U) \subset \mathcal{G}(U)$ for all open $U \subset X$ such that the restriction maps of $\mathcal{G}$ induce the restriction maps of $\mathcal{F}$. If $\mathcal{F}$ and $\mathcal{G}$ are sheaves, then $\mathcal{F}$ is called a subsheaf of $\mathcal{G}$. We sometimes indicate this by the notation $\mathcal{F} \subset \mathcal{G}$.

2. A morphism of presheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called injective if and only if $\mathcal{F}(U) \to \mathcal{G}(U)$ is injective for all $U$ open in $X$.

3. A morphism of presheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called surjective if and only if $\mathcal{F}(U) \to \mathcal{G}(U)$ is surjective for all $U$ open in $X$.

4. A morphism of sheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called injective if and only if $\mathcal{F}(U) \to \mathcal{G}(U)$ is injective for all $U$ open in $X$.

5. A morphism of sheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called surjective if and only if for every open $U$ of $X$ and every section $s$ of $\mathcal{G}(U)$ there exists an open covering $U = \bigcup U_ i$ such that $s|_{U_ i}$ is in the image of $\mathcal{F}(U_ i) \to \mathcal{G}(U_ i)$ for all $i$.