Lemma 7.10.4 (00W5)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 7: Sites and Sheaves
Section 7.10: Sheafification
Lemma 7.10.4 (cite)

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Lemma 7.10.4. The association $\mathcal{F} \mapsto (\mathcal{F} \to \mathcal{F}^+)$ is a functor.

Proof. Instead of proving this we state exactly what needs to be proven. Let $\mathcal{F} \to \mathcal{G}$ be a map of presheaves. Prove the commutativity of:

\[ \xymatrix{ \mathcal{F} \ar[r] \ar[d] & \mathcal{F}^{+} \ar[d] \\ \mathcal{G} \ar[r] & \mathcal{G}^{+} } \]

$\square$

Comments (2)

Comment #8569 by Alejandro González Nevado on July 16, 2023 at 18:39

SS: The association sending a presheaf to the map of presheaves sending this presheaf to its zeroth Čech cohomology is a functor.

Comment #9148 by Stacks project on May 13, 2024 at 11:23

Not pithy enough

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