Remarks
For some categories C the direct limit used to define the stalk may not exist. However, it exists for most categories which occur in practice, such as the category of sets or most categories of algebraic objects such as abelian groups or rings.
There is a natural morphism F(U) → Fx for any open set U containing x: it takes a section s in F(U) to its germ, that is, its equivalence class in the direct limit. This is a generalization of the usual concept of a germ, which can be recovered by looking at the stalks of the sheaf of continuous functions on X.
Read more about this topic: Stalk (sheaf)
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