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 !