Flasque or Flabby Sheaves
A flasque sheaf (also called a flabby sheaf) is a sheaf with the following property: if is the base topological space on which the sheaf is defined and
are open subsets, then the restriction map
is surjective, as a map of groups (rings, modules, etc.).
Flasque sheaves are useful because (by definition) sections of them extend. This means that they are some of the simplest sheaves to handle in terms of homological algebra. Any sheaf has a canonical embedding into the flasque sheaf of all possibly discontinuous sections of the étalé space, and by repeating this we can find a canonical flasque resolution for any sheaf. Flasque resolutions, that is, resolutions by means of flasque sheaves, are one approach to defining sheaf cohomology.
Flasque is a French word, that has sometimes been translated into English as flabby.
Flasque sheaves are soft and acyclic.
Read more about this topic: Injective Sheaf
Famous quotes containing the words flabby and/or sheaves:
“Happy is the novelist who manages to preserve an actual love letter that he received when he was young within a work of fiction, embedded in it like a clean bullet in flabby flesh and quite secure there, among spurious lives.”
—Vladimir Nabokov (1899–1977)
“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)