Lemma 6.9.2 (0073)—The Stacks project

Lemma 6.9.2. Suppose the category $\mathcal{C}$ and the functor $F : \mathcal{C} \to \textit{Sets}$ have the following properties:

$F$ is faithful,
$\mathcal{C}$ has limits and $F$ commutes with them, and
the functor $F$ reflects isomorphisms.

Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$ . Then $\mathcal{F}$ is a sheaf if and only if the underlying presheaf of sets is a sheaf.

Proof. Assume that $\mathcal{F}$ is a sheaf. Then $\mathcal{F}(U)$ is the equalizer of the diagram above and by assumption we see $F(\mathcal{F}(U))$ is the equalizer of the corresponding diagram of sets. Hence $F(\mathcal{F})$ is a sheaf of sets.

Assume that $F(\mathcal{F})$ is a sheaf. Let $E \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be the equalizer of the two parallel arrows in Definition 6.9.1. We get a canonical morphism $\mathcal{F}(U) \to E$ , simply because $\mathcal{F}$ is a presheaf. By assumption, the induced map $F(\mathcal{F}(U)) \to F(E)$ is an isomorphism, because $F(E)$ is the equalizer of the corresponding diagram of sets. Hence we see $\mathcal{F}(U) \to E$ is an isomorphism by condition (3) of the lemma. $\square$

The Stacks project

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