Lemma 7.49.1 (00ZH)—The Stacks project

Lemma 7.49.1. In the situation above.

The assignment $U \mapsto L\mathcal{F}(U)$ combined with the restriction mappings defined above is a presheaf.
The maps $\ell $ glue to give a morphism of presheaves $\ell : \mathcal{F} \to L\mathcal{F}$.
The rule $\mathcal{F} \mapsto (\mathcal{F} \xrightarrow {\ell } L\mathcal{F})$ is a functor.
If $\mathcal{F}$ is a subpresheaf of $\mathcal{G}$, then $L\mathcal{F}$ is a subpresheaf of $L\mathcal{G}$.
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 $S$.

Comments (3)

Comment #11628 by Zhenhua Wu on July 13, 2026 at 23:32

（3） It does not make sense to say $\mathcal{F}\mapsto (l:\mathcal{F}\to L\mathcal{F})$ is a functor, because the left one lies in $\mathrm{Fun}(\mathcal{C}^{op},\mathrm{Sets})$ while the right one lies in $\mathrm{Nat}(\mathcal{F},L\mathcal{F})$ which depends on $\mathcal{F}$ .

I think what you want to say is that it defines a natural transformation $\phi$ from $\mathrm{id}:PSh(\mathcal{C})\to PSh(\mathcal{C})$ to $L:PSh(\mathcal{C})\to PSh(\mathcal{C})$ via $\phi_{\mathcal{F}}:=(l: \mathcal{F}\to L\mathcal{F})$ .

Comment #11629 by Zhenhua Wu on July 14, 2026 at 00:00

（5）Here I know by an element of $\varphi \in \mathrm{Mor}_{PSh(\mathcal{C})}(S,\mathcal{F})$ you mean a set of compatitble elements $({s_i\in \mathcal{F}(U_i)})_{ \\{ U_i\to U \in S \\}}$ (compatible under restriction maps). And $l(\varphi)$ means $(l_{U_i}(s_i)\in L\mathcal{F}(U_i))$ . Though this identification is used before, it's not written down in any lemma so I suggest either you replace $\varphi$ by a set of compatitble elements, or you state a lemma regarding this identification beforehand.

Comment #11644 by Stacks Project on July 14, 2026 at 11:10

There is a functor from the category of abelian groups to the category of arrows of abelian groups that sends $M$ to $M \to M/2M$ . This is how one should read assertion (2). Yes, the language in part (5) could be improved on, but I am going to leave it as is for now.

The sections on topologies using sieves perhaps should be moved to the obsolete chapter (see the bold text in Section 7.47) or need to be revised significantly. If you are interested, don't just start editing/revising but email me first with a plan.

There are also:

2 comment(s) on Section 7.49: Sheafification in a topology

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