The Stacks project

Lemma 19.9.1. Let $\mathcal{A}$ be an abelian category. Let

\[ \text{Cov} = \{ \{ f : V \to U\} \mid f\text{ is surjective}\} . \]

Then $(\mathcal{A}, \text{Cov})$ is a site, see Sites, Definition 7.6.2.

Proof. Note that $\mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ is a set by our conventions about categories. An isomorphism is a surjective morphism. The composition of surjective morphisms is surjective. And the base change of a surjective morphism in $\mathcal{A}$ is surjective, see Homology, Lemma 12.5.14. $\square$

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