Sheaf (mathematics)

Tool to track locally defined data attached to the open sets of a topological space

Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or sheaves of rings, for example, depending on the type of data assigned to the open sets.

The restriction morphisms are required to satisfy two additional (functorial) properties:

any collection of pairwise compatible sections can be uniquely glued together

In addition to the constant presheaf mentioned above, which is usually not a sheaf, there are further examples of presheaves that are not sheaves:

Morphisms of sheaves are, roughly speaking, analogous to functions between them. In contrast to a function between sets, which have no additional structure, morphisms of sheaves are those functions which preserve the structure inherent in the sheaves. This idea is made precise in the following definition.

The definition of sheaves by étalé spaces is older than the definition given earlier in the article. It is still common in some areas of mathematics such as mathematical analysis.

Sheaf cohomology captures this phenomenon. More precisely, for an exact sequence of sheaves of abelian groups

This isomorphism is an example of a base change theorem. There is another adjunction