Sites and Sheaves
See also: ToposLet C be a category and let J be a Grothendieck topology on C. The pair (C, J) is called a site.
A presheaf on a category is a contravariant functor from C to the category of all sets. Note that for this definition C is not required to have a topology. A sheaf on a site, however, should allow gluing, just like sheaves in classical topology. Consequently, we define a sheaf on a site to be a presheaf F such that for all objects X and all covering sieves S on X, the natural map Hom(Hom(−, X), F) → Hom(S, F), induced by the inclusion of S into Hom(−, X), is a bijection. Halfway in between a presheaf and a sheaf is the notion of a separated presheaf, where the natural map above is required to be only an injection, not a bijection, for all sieves S. 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.
Sheaves on a pretopology have a particularly simple description: For each covering family {Xα → X}, the diagram
must be an equalizer. For a separated presheaf, the first arrow need only be injective.
Similarly, one can define presheaves and sheaves of abelian groups, rings, modules, and so on. One can require either that a presheaf F is a contravariant functor to the category of abelian groups (or rings, or modules, etc.), or that F be an abelian group (ring, module, etc.) object in the category of all contravariant functors from C to the category of sets. These two definitions are equivalent.
Read more about this topic: Grothendieck Topology
Famous quotes containing the word sheaves:
“A thousand golden sheaves were lying there,
Shining and still, but not for long to stay
As if a thousand girls with golden hair
Might rise from where they slept and go away.”
—Edwin Arlington Robinson (18691935)