I just started studying complex geometry and in all the literature sheaf theory shows up. For me this is still a rather abstract object and I have a hard time understanding the difference between a (pre)Sheaf and a set.
As an example, in an article to explain Hodge Decomposition, we have $X$, a $m$-dimensional Riemannian manifold. Now the sheaf of smooth $n$-forms on $X$ is denoted by $\mathcal{A}_X^n$.
I don't understand why we cannot simply work with the set of $n$-forms. What is in this case the key difference between taking the sheaf of $n$-forms and just taking the set of all $n$-forms on X.