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Concept
Stalk (sheaf)
Summary
The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.
Sheaves are defined on open sets, but the underlying topological space consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a single fixed point of . Conceptually speaking, we do this by looking at small neighborhoods of the point. If we look at a sufficiently small neighborhood of , the behavior of the sheaf on that small neighborhood should be the same as the behavior of at that point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.
The precise definition is as follows: the stalk of at , usually denoted , is:
Here the direct limit is indexed over all the open sets containing , with order relation induced by reverse inclusion (, if ). By definition (or universal property) of the direct limit, an element of the stalk is an equivalence class of elements , where two such sections and are considered equivalent if the restrictions of the two sections coincide on some neighborhood of .
There is another approach to defining a stalk that is useful in some contexts. Choose a point of , and let be the inclusion of the one point space into . Then the stalk is the same as the sheaf . Notice that the only open sets of the one point space are and , and there is no data over the empty set. Over , however, we get:
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 or most categories of algebraic objects such as abelian groups or rings, which are namely cocomplete.
There is a natural morphism for any open set containing : it takes a section in 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 .
The constant sheaf associated to some set (or group, ring, etc).
Official source
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