Lemma 19.9.2. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C} = (\mathcal{A}, \text{Cov})$ be the site defined in Lemma 19.9.1. Then $X \mapsto h_ X$ defines a fully faithful, exact functor

$\mathcal{A} \longrightarrow \textit{Ab}(\mathcal{C}).$

Moreover, the site $\mathcal{C}$ has enough points.

Proof. Suppose that $f : V \to U$ is a surjective morphism of $\mathcal{A}$. Let $K = \mathop{\mathrm{Ker}}(f)$. Recall that $V \times _ U V = \mathop{\mathrm{Ker}}((f, -f) : V \oplus V \to U)$, see Homology, Example 12.5.6. In particular there exists an injection $K \oplus K \to V \times _ U V$. Let $p, q : V \times _ U V \to V$ be the two projection morphisms. Note that $p - q : V \times _ U V \to V$ is a morphism such that $f \circ (p - q) = 0$. Hence $p - q$ factors through $K \to V$. Let us denote this morphism by $c : V \times _ U V \to K$. And since the composition $K \oplus K \to V \times _ U V \to K$ is surjective, we conclude that $c$ is surjective. It follows that

$V \times _ U V \xrightarrow {p - q} V \to U \to 0$

is an exact sequence of $\mathcal{A}$. Hence for an object $X$ of $\mathcal{A}$ the sequence

$0 \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(U, X) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(V, X) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(V \times _ U V, X)$

is an exact sequence of abelian groups, see Homology, Lemma 12.5.8. This means that $h_ X$ satisfies the sheaf condition on $\mathcal{C}$.

The functor is fully faithful by Categories, Lemma 4.3.5. The functor is a left exact functor between abelian categories by Homology, Lemma 12.5.8. To show that it is right exact, let $X \to Y$ be a surjective morphism of $\mathcal{A}$. Let $U$ be an object of $\mathcal{A}$, and let $s \in h_ Y(U) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(U, Y)$ be a section of $h_ Y$ over $U$. By Homology, Lemma 12.5.14 the projection $U \times _ Y X \to U$ is surjective. Hence $\{ V = U \times _ Y X \to U\}$ is a covering of $U$ such that $s|_ V$ lifts to a section of $h_ X$. This proves that $h_ X \to h_ Y$ is a surjection of abelian sheaves, see Sites, Lemma 7.11.2.

The site $\mathcal{C}$ has enough points by Sites, Proposition 7.39.3. $\square$

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