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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.

Proof. This is a special case of Lemma 85.19.1. \square

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