Germ (mathematics) - Relation With Sheaves

Relation With Sheaves

The idea of germ is behind the definition of sheaves and presheaves. A presheaf on a topological space X is an assignment of an Abelian group to each open set U in X. Typical examples of Abelian groups here are: real valued functions on U, differential forms on U, vector fields on U, holomorphic functions on U (when X is a complex space), constant functions on U and differential operators on U.

If then there is a restriction map, satisfying certain compatibility conditions. For a fixed x, one says that elements and are equivalent at x if there is a neighbourhood of x with res_WU(f) = res_WV(g) (both elements of ). The equivalence classes form the stalk at x of the presheaf . This equivalence relation is an abstraction of the germ equivalence described above.