The Stacks project

Lemma 7.49.1. In the situation above.

  1. The assignment $U \mapsto L\mathcal{F}(U)$ combined with the restriction mappings defined above is a presheaf.

  2. The maps $\ell $ glue to give a morphism of presheaves $\ell : \mathcal{F} \to L\mathcal{F}$.

  3. The rule $\mathcal{F} \mapsto (\mathcal{F} \xrightarrow {\ell } L\mathcal{F})$ is a functor.

  4. If $\mathcal{F}$ is a subpresheaf of $\mathcal{G}$, then $L\mathcal{F}$ is a subpresheaf of $L\mathcal{G}$.

  5. The map $\ell : \mathcal{F} \to L\mathcal{F}$ has the following property: For every section $s \in L\mathcal{F}(U)$ there exists a covering sieve $S$ on $U$ and an element $\varphi \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F})$ such that $\ell (\varphi )$ equals the restriction of $s$ to $S$.

Proof. Omitted. $\square$

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View Lemma 7.49.1 as pdf