Lemma 18.3.1. Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of abelian sheaves on $\mathcal{C}$.

The category $\textit{Ab}(\mathcal{C})$ is an abelian category.

The kernel $\mathop{\mathrm{Ker}}(\varphi )$ of $\varphi $ is the same as the kernel of $\varphi $ as a morphism of presheaves.

The morphism $\varphi $ is injective (Homology, Definition 12.5.3) if and only if $\varphi $ is injective as a map of presheaves (Sites, Definition 7.3.1), if and only if $\varphi $ is injective as a map of sheaves (Sites, Definition 7.11.1).

The cokernel $\mathop{\mathrm{Coker}}(\varphi )$ of $\varphi $ is the sheafification of the cokernel of $\varphi $ as a morphism of presheaves.

The morphism $\varphi $ is surjective (Homology, Definition 12.5.3) if and only if $\varphi $ is surjective as a map of sheaves (Sites, Definition 7.11.1).

A complex of abelian sheaves

\[ \mathcal{F} \to \mathcal{G} \to \mathcal{H} \]is exact at $\mathcal{G}$ if and only if for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and all $s \in \mathcal{G}(U)$ mapping to zero in $\mathcal{H}(U)$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ in $\mathcal{C}$ such that each $s|_{U_ i}$ is in the image of $\mathcal{F}(U_ i) \to \mathcal{G}(U_ i)$.

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