The Stacks project

Lemma 85.21.1. Let $\mathcal{C}$ be a site with fibre product and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above. Then

1. $a^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X) \to \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/U)_{total})$ is fully faithful with essential image the cartesian sheaves of sets,

2. $a^{-1} : \textit{Ab}(\mathcal{C}/X) \to \textit{Ab}((\mathcal{C}/U)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups.

In both cases $a_*$ provides the quasi-inverse functor.