Local vs. global in the definition of a sheaf

Solution 1:

Sheaves are given by local data, yes, but this local data is not only encoded in the stalks. As already indicated in the comment by Jared and the answer by Qiaochu, even in the question (that section about line bundles), stalks are not enough to understand a sheaf. Beyond examples, there are several ways to make the difference explicit:

One of the classical definitions of a sheaf is via étale spaces. If $X$ is a topological space, then an étale space over $X$ is just a continuous map $p : Y \to X$ which is a local homeomorphism. The corresponding sheaf in the usual sense is the sheaf of sections of $p$. The stalk at a point $x \in X$ is just the fiber $p^{-1}(x)$. The fibers of $p$ determine the set $Y$, but not its topology.

If $F,G$ are sheaves on a space $X$ (or equivalently étale spaces if you like), then a family of maps of stalks $F_x \to G_x$ ($x \in X$) comes from a (unique!) homomorphism of sheaves $F \to G$ iff the following continuity condition holds: For every section $s \in F(U)$ there is some open covering $U=\cup_i U_i$ such that on each $U_i$ there is some section $t \in G(U_i)$ such that $F_x \to G_x$ maps $s_x \mapsto t_x$ for all $x \in U_i$.

Solution 2:

I don't understand the question. Consider locally constant sheaves, which are sheaves of sections of covering spaces. With a nice path-connected base, these correspond to group actions of the fundamental group of the base. The stalks of this sheaf give you very little information: if the covering space is also path-connected, you have a transitive group action, and looking at stalks only tells you its cardinality. To get the rest of the data you need to look at monodromy, which isn't a local thing.