Editing Grothendieck topology (section)

== Sites and sheaves ==
{{See also|Topos}}

Let ''C'' be a category and let ''J'' be a Grothendieck topology on ''C''.  The pair (''C'', ''J'') is called a '''site'''.

A '''[[presheaf (category theory)|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(&minus;, ''X''), ''F'') → Hom(''S'', ''F''), induced by the inclusion of ''S'' into Hom(&minus;, ''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 {''X''<sub>''α''</sub> → ''X''}, the diagram

:<math>F(X) \rightarrow \prod_{\alpha\in A} F(X_\alpha) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod_{\alpha,\beta \in A} F(X_\alpha\times_X X_\beta)</math>

must be an [[equalizer (mathematics)|equalizer]].  For a separated presheaf, the first arrow need only be injective.

Similarly, one can define presheaves and sheaves of [[abelian group]]s, [[ring (mathematics)|ring]]s, [[module (mathematics)|module]]s, 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.