Build sheaf from stalks If I have a topological space $T$ and for each $p \in T$ I have an object $A_p$ in some category $\mathscr{A}$, then how can I define a sheaf out of this? In other words can I build a sheaf with predetermined stalks (kind of like the quasi-coherent sheaf construction in Hartshorn)?

You can't recover a sheaf from its stalks alone. For instance, all vector bundles $\mathcal{E}$ (of fixed rank $r$) on a variety $X$ have the same stalk at any given point $x \in X$: $$\mathcal{E}_{X, x} \cong \mathcal{O}_{X, x}^{\oplus r}$$ Stalks provide local information, whereas a sheaf encompasses not only local information but also information about how that local information globalizes.

If you just want an arbitrary sheaf with prescribed stalks, then on a nice enough space $T$ you can take the direct sum of the corresponding skyscraper sheaves, but this is not going to recover a nice sheaf. A section of this thing over an open set $U$ will simply be the formal sum of finitely many $s_p \in A_p$, where $p$ ranges over a finite set of points in $U$.

(The second sense is a duplicate of When does a sheaf exist with prescribed stalks? which gives some valid conditions on $T$ in the question.)