sheafification of a sheaf

In the preceding subsection, we have not been explained what ``sheafification'' really means. Here is the definition.

LEMMA 1.28   Let $\mathcal G$ be a presheaf on a topological space $X$ . Then there exists a sheaf $\operatorname{sheaf}(\mathcal G)$ and a presheaf morphism

$$\iota_{\mathcal G}:\mathcal G \to \operatorname{sheaf}(\mathcal G)$$

such that the following property holds.

1. If there is another sheaf $\mathcal F$ with a presheaf morphism
   $$\alpha:\mathcal G \to \mathcal F,$$
   then there exists a unique sheaf homomorphism
   $$\tilde{\alpha}: \operatorname{sheaf}(\mathcal G) \to \mathcal F$$
   such that
   $$\alpha=\tilde{\alpha}\circ \iota_{\mathcal G}$$
   holds.

Furthermore, such $\operatorname{sheaf}(\mathcal G), \iota_{\mathcal G}$ is unique.

DEFINITION 1.29   The sheaf $\operatorname{sheaf}(\mathcal G)$ (together with $\iota_{\mathcal G}$ ) is called the sheafification of $\mathcal G$ .

The proof of Lemma 1.28 is divided in steps.

The first step is to know the uniqueness of such sheafification. It is most easily done by using universality arguments. ([1] has a short explanation on this topic.)

Then we divide the sheafification process in two steps.

LEMMA 1.30 (First step of sheafification)   Let $\mathcal G$ be a presheaf on a topological space $X$ . Then for each open set $U\subset X$ , we may define a equivalence relation on $\mathcal G(U)$ by

$$f \sim g \iff \left( \begin{aligned} &\text{there exists an o... ...ambda,U}g$ } \\ &\text{for any $\lambda$.} \end{aligned}\right)$$

Then we define

$$\mathcal G^{(1)}(U)=\mathcal G(U)/\sim.$$

Then $\mathcal G^{(1)}$ is a presheaf that satisfies the locality axiom of a sheaf. There is also a presheaf homomorphism from $\mathcal G$ to $\mathcal G^{(1)}$ . Furthermore, $\mathcal G^{(1)}$ is universal among such.

LEMMA 1.31 (Second step of sheafification)   Let $\mathcal G$ be a presheaf on a topological space $X$ which satisfies the locality axiom of a sheaf. Then we define a presheaf $\mathcal G^{(2)}$ in the following manner. First for any open covering $\{U_\lambda\}$ of an open set $U\subset X$ , we define

$$\mathcal G^{(2)}(U; \{U_\lambda\}) =\left\{ \{r_\lambda \} \in... ...&\text{for any $\lambda, \mu\in \Lambda$.} \end{aligned}\right\}$$

Then we define

$$\mathcal G^{(2)}(U)= \varinjlim_{\{U_\lambda\}} \mathcal G^{(2)}(U; \{U_\lambda\})$$

Then we may see that $G^{(2)}$ is a sheaf and that there exists a homomorphism from $G$ to $G^{(2)}$ . Furthermore, $G^{(2)}$ is universal among such.

Proofs of the above two lemma are routine work and are left to the reader.

Finish of the proof of Lemma 1.28: We put

$$\operatorname{sheaf}(\mathcal G)=((\mathcal G)^{(1)})^{(2)}$$

$\qedsymbol$

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