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The Stacks project

Theorem 59.13.2. Let \mathcal{C} be a site and \mathcal{F} a presheaf on \mathcal{C}.

  1. The rule

    U \mapsto \mathcal{F}^+(U) := \mathop{\mathrm{colim}}\nolimits _{\mathcal{U} \text{ covering of }U} \check H^0(\mathcal{U}, \mathcal{F})

    is a presheaf. And the colimit is a directed one.

  2. There is a canonical map of presheaves \mathcal{F} \to \mathcal{F}^+.

  3. If \mathcal{F} is a separated presheaf then \mathcal{F}^+ is a sheaf and the map in (2) is injective.

  4. \mathcal{F}^+ is a separated presheaf.

  5. \mathcal{F}^\# = (\mathcal{F}^+)^+ is a sheaf, and the canonical map induces a functorial isomorphism

    \mathop{\mathrm{Hom}}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}^\# , \mathcal{G})

    for any \mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C}).

Proof. See Sites, Theorem 7.10.10. \square


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