The Stacks project

6.19 Sheafification of presheaves of algebraic structures

Lemma 6.19.1. Let $X$ be a topological space. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$ on $X$. Then there exists a sheaf $\mathcal{F}^\# $ with values in $\mathcal{C}$ and a morphism $\mathcal{F} \to \mathcal{F}^\# $ of presheaves with values in $\mathcal{C}$ with the following properties:

  1. The map $\mathcal{F} \to \mathcal{F}^\# $ identifies the underlying sheaf of sets of $\mathcal{F}^\# $ with the sheafification of the underlying presheaf of sets of $\mathcal{F}$.

  2. For any morphism $\mathcal{F} \to \mathcal{G}$, where $\mathcal{G}$ is a sheaf with values in $\mathcal{C}$ there exists a unique factorization $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$.

Proof. The proof is the same as the proof of Lemma 6.18.2, with repeated application of Lemma 6.15.4 (see also Example 6.15.5). The main idea however, is to define $\mathcal{F}^\# (U)$ as the fibre product in $\mathcal{C}$ of the diagram

\[ \xymatrix{ & \Pi (\mathcal{F})(U) \ar[d] \\ \prod _{x \in U} \mathcal{F}_ x \ar[r] & \prod _{x \in U} \Pi (\mathcal{F})_ x } \]

compare Lemma 6.18.1. $\square$

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