Sheaves are defined on open sets, but the underlying topological space
It is reasonable to attempt to isolate the behavior of a sheaf at a single fixed point
Conceptually speaking, we do this by looking at small neighborhoods of the point.
Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.
, is: Here the direct limit is indexed over all the open sets containing
, with order relation induced by reverse inclusion (
By definition (or universal property) of the direct limit, an element of the stalk is an equivalence class of elements
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.
Notice that the only open sets of the one point space
, however, we get: For some categories C the direct limit used to define the stalk may not exist.
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
This is because the germ records the function's power series expansion, and all analytic functions are by definition locally equal to their power series.
Using analytic continuation, we find that the germ at a point determines the function on any connected open set where the function can be everywhere defined.
(This does not imply that all the restriction maps of this sheaf are injective!)
In contrast, for the sheaf of smooth functions on a smooth manifold, germs contain some local information, but are not enough to reconstruct the function on any open neighborhood.
On any sufficiently small neighborhood containing the origin,
is identically one, so at the origin it has the same germ as the constant function with value 1.
From what the germ tells us, the bump could be infinitely wide, that is,
, because the latter function is not identically one on any neighborhood of the origin.
(This extra information is related to the fact that the stalk of the sheaf of smooth functions at the origin is a non-Noetherian ring.
The Krull intersection theorem says that this cannot happen for a Noetherian ring.)
On any topological space, the skyscraper sheaf associated to a closed point
This idea makes more sense if one adopts the common visualisation of functions mapping from some space above to a space below; with this visualisation, any function that maps
This feature is the basis of the construction of Godement resolutions, used for example in algebraic geometry to get functorial injective resolutions of sheaves.
As outlined in the introduction, stalks capture the local behaviour of a sheaf.
As a sheaf is supposed to be determined by its local restrictions (see gluing axiom), it can be expected that the stalks capture a fair amount of the information that the sheaf is encoding.
This is indeed true: In particular: Both statements are false for presheaves.
However, stalks of sheaves and presheaves are tightly linked: