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 resWU(f) = resWV(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.
Read more about this topic: Germ (mathematics)
Famous quotes containing the words relation with, relation and/or sheaves:
“[Man’s] life consists in a relation with all things: stone, earth, trees, flowers, water, insects, fishes, birds, creatures, sun, rainbow, children, women, other men. But his greatest and final relation is with the sun.”
—D.H. (David Herbert)
“Hesitation increases in relation to risk in equal proportion to age.”
—Ernest Hemingway (1899–1961)
“Being young you have not known
The fool’s triumph, nor yet
Love lost as soon as won,
Nor the best labourer dead
And all the sheaves to bind.”
—William Butler Yeats (1865–1939)