How, in intuitive terms, does sheafification work?

Question

In many books one can find results about the sheafification functor which constructs the "free" sheaf on a given presheaf.

What, in intuitive terms, does the sheafification functor explicitly do? How does it systematically equip any presheaf with a gluing mechanism?

Answer 1

Let $U=\bigcup_i U_i$ be an open cover. Whenever we have sections $f_i \in \mathcal F(U_i)$, that agree on the intersections, for any $x \in U$ the definition $$f_x := \text{ stalk of } f_i \text{ at } x \text{ for some }i \text{ with } x \in U_i$$ is independent of the choice of $i$, hence well-defined.

So we equip a presheaf with a glueing mechanism in the sense, that we just define such a collection $(f_x)_{x \in U}$ to be a section over $U$.

Maybe you can think of this like that: The sheafification defines a global section to be a collection of local data. This adds a glueing mechanism to the whole story.

You can also think about (pre)sheaves of functions. Functions on a (global) set are just collections of local data (the evaluation of the function at each point). This is why sheafification is not necessary for such presheaves (like sheaf of holomorphic functions etc.).