TITLE: Question on sheafification of a presheaf
QUESTION [17 upvotes]: In chapter 2 of GTM 52 by Robin Hartshone there are definition of presheaf and the associated sheaf of a given presheaf.
I found that the definition of the sheafification is rather less natural and too rigorous. Harthshone did not give any non trivial concrete presheaf and its sheafification.
My questions are :

From the definition of a presheaf $\mathcal{F}$(as Hartshone defined) how can one think about its sheafification $\mathcal{F}^{+}$ as a collection of map : $s: U\rightarrow \cup \mathcal{F}_{p}$ for each open subset $U$ and why is $\cup \mathcal{F}_{p}$ rather than other sets ?
Could you please show me a nontrivial, concrete example of a presheaf(that is not a sheaf itself) and its sheafification ?

Thanks !

REPLY [9 votes]: First I give an example of sheafification which I think it is important: Let $X=\mathbb{R}^n$, for each open subset $U$ we define $\mathcal{F}(U)$ be the set of all constant functions from $U$ to $\mathbb{R}$. Then easy to see $\mathcal{F}$ is a presheaf on $X$. Now given $f_1\in\mathcal{F}(U)$ and $f_2\in\mathcal{F}(V)$ where $U\cap V=\emptyset$, there is now constant function $f$ on $U\cup V$ such that $f_{|U}=f_1$ and $f_{|V}=f_2$. So $\mathcal{F}$ is not a sheaf!. Now we define the sheaf $\mathcal{F}^+$ which is associated to $\mathcal{F}$: for each $U$, $\mathcal{F}^+(U)=$ the set of all locally constant functions on from $U$ to $\mathbb{R}$. Now you can observe that the sections of $\mathcal{F}^+$ on $U$ are created by glueing the sections of $\mathcal{F}$ in smaller open subset. This is the key point of thinking about sheafification, because for presheaf we can not glue the sections together.
Now for your question (1): in general the sections of a presheaf are not functions, so you can not glue them together. In order to glue them together you must somehow transfer them to functions, the canonically way to do this is Identify each $s\in\mathcal{F}(U)$ with the function $\bar{s}: U\rightarrow \cup \mathcal{F}_{p}$. $\bar{s}$ capture all behaviors of $s$. And now you can glue together $\bar{s}$ to have the sheaf $\mathcal{F}+$. You can read chapter 2 in the very readable book "Sheaf theory" written by Tennison to get more details.