Search results
Results from the WOW.Com Content Network
Wheat sheaves near King's Somborne.Here the individual sheaves have been put together into a stook ("stooked") to dry. A sheaf of grain on a plaque Sheafing machine. A sheaf (/ ʃ iː f /; pl.: sheaves) is a bunch of cereal-crop stems bound together after reaping, traditionally by sickle, later by scythe or, after its introduction in 1872, by a mechanical reaper-binder.
Especially in the context of sheaves on manifolds, sheaf cohomology can often be computed using resolutions by soft sheaves, fine sheaves, and flabby sheaves (also known as flasque sheaves from the French flasque meaning flabby). For example, a partition of unity argument shows that the sheaf of smooth functions on a manifold is soft.
In mathematics, a topos (US: / ˈ t ɒ p ɒ s /, UK: / ˈ t oʊ p oʊ s, ˈ t oʊ p ɒ s /; plural topoi / ˈ t ɒ p ɔɪ / or / ˈ t oʊ p ɔɪ /, or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site).
In more abstract language, the sheaves on form a reflective subcategory of the presheaves (Mac Lane–Moerdijk Sheaves in Geometry and Logic p. 86). In topos theory , for a Lawvere–Tierney topology and its sheaves, there is an analogous result (ibid. p. 227).
A morphism of sheaves is an isomorphism, epimorphism, or monomorphism, respectively, if and only if the induced morphisms on all stalks have the same property. (However it is not true that two sheaves, all of whose stalks are isomorphic, are isomorphic, too, because there may be no map between the sheaves in question.) In particular:
Sheaves is the plural of either of two nouns: Sheaf (disambiguation) Sheave This page was last edited on 1 October 2024, at 19:34 (UTC). Text is available under the ...
A morphism of presheaves or of sheaves is a natural transformation of functors. The category of all sheaves on C is the topos defined by the site (C, J). Using the Yoneda lemma, it is possible to show that a presheaf on the category O(X) is a sheaf on the topology defined above if and only if it is a sheaf in the classical sense.
Algebraic spaces are similar to schemes, and much of the theory of schemes extends to algebraic spaces. For example, most properties of morphisms of schemes also apply to algebraic spaces, one can define cohomology of quasicoherent sheaves, this has the usual finiteness properties for proper morphisms, and so on.