# Sheafification of a presheaf through the etale space

I have some problems to show that the following contruction defines a sheafification:

Let $\mathcal F$ be a presheaf on $X$, and let $Et(\mathcal F)$ be the etale space associated to $\mathcal F$, with $π:Et(\mathcal F)\rightarrow X$ that is the canonical map which sends a germ $s_x$ in $x$. If with $U$ we indicate a generic open set of $X$, then the set of sections of $π$ on $U$ is

$\mathcal {F}^+(U)=$ {$\widetilde{s}:U\rightarrow Et(\mathcal F)\;with\;\widetilde {s}(x)=s_x\; \forall s\in\mathcal {F}(U)$}

We give a certain topology on $Et(\mathcal F)$ and make $π$ and $\widetilde s$ continuous functions. In this way whe define the sheaf $\mathcal F^+$ of continuous sections of $π$, and the morphism (for all $U$)

$\phi(U):\mathcal F(U)\rightarrow\mathcal F^+(U)$ such that $s\mapsto\widetilde s$

Now if $\mathcal F^+$ satisfies the "universal property", it is the sheafification of $\mathcal F$. Suppose that $\psi$ is a morphism from $\mathcal F$ in a generic sheaf $\mathcal G$; how can I prove that exists a unique morphism $\theta:\mathcal F\rightarrow\mathcal G$ such that $\psi=\theta\circ\phi$?

The definition $\theta(\widetilde s)=\psi(s)$ doesn't work because $s_x=t_x$ for all $x$ doesn't imply $s=t$ in $\mathcal F(U)$ since $\mathcal F$ is a presheaf.

• This can be found in standard texts like Hartshorne (without mentioning etale space) or in Mac Lane-Moerdijk. – Benjamin Steinberg May 19 '12 at 16:01
• Yes, but this is another equivalent construction of a sheafification and I should prove that it is equivalent to that introduced in the Hartshorne's text. – Galoisfan May 19 '12 at 16:12
• This question also appeared on Math.SE: math.stackexchange.com/questions/146996/… – Emerton May 20 '12 at 5:35

Given some section $\tilde{s} : U \to \mathrm{Et}(\mathcal{F})$, we have for all $x \in U$ some element in $\mathcal{F}_x$ and therefore in $\mathcal{G}_x$. The continuity of the map $\tilde{s}$ ensures that we can actually lift these germs to local sections around $x$. Since $\mathcal{F} \to \mathcal{G}$ is a homomorphism of presheaves, it doesn't matter on which neighborhood we work, and since $\mathcal{G}$ is a sheaf, we can glue these local sections to some section in $\mathcal{G}(U)$. The rest is also easy, you should be able to do this on your own.
The construction of the sheafification you mention is nicely described e.g. in Tennison's book Sheaf Theory, in all details . The functor $\Gamma$ from the category of sheaf spaces to the category of presheaves, giving the sheaf of sections of a sheaf space, and the functor $L$ (your Et), from the category of presheaves to the category of sheaf spaces, are a pair of adjoint functors, and the composition $\Gamma L$ is the wanted reflection.