The motivating example is the following: start with a topological space X and consider the sheaf of all continuous realvalued functions defined on X. This associates to every open set U in X the set F(U) of realvalued continuous functions defined on U. Whenver U is a subset of V, we have a "restriction map" from F(V) to F(U). If we interpret the topological space X as a category, with the open sets being the objects and a morphism from U to V if and only if U is a subset of V, then F is revealed as a contravariant functor from this category into the category of sets. In general, every contravariant functor from a category C to the category of sets is therefore called a presheaf of sets on C. Our functor F has a special property: if you have an open covering (V_{i}) of the set U, and you are given mutually compatible elements of F(V_{i}), then there exists precisely one element of F(U) which restricts to all the given ones. This is the defining property of a sheaf, and a Grothendieck topology on C is an attempt to capture the essence of what is needed to define sheaves on C.
Formally, a Grothendieck topology on C is given by specifying for each object U of C families of morphisms {φ_{i} : V_{i} > U}_{i in I}, called covering families of U, such that the following axioms are satisfied:
A presheaf on the category C is a contravariant functor F : C > Set. If C is equipped with a Grothendieck topology, then a presheaf is called a sheaf on C if, for every covering family {φ_{i} : V_{i} > U}_{i in I}, the map F(U) > Π_{i in I} F(V_{i}) is the equalizer[?] of the two natural maps Π_{i in I} F(V_{i}) > Π_{(i, j) in I x I } F(V_{i} ×_{U} V_{j}).
Once a site (a category C with a Grothendieck topology) is given, one can consider the category of all sheaves on this site. This is a topos, and in fact the notion of topos originated here. The category of sheaves is also a Grothendieck category[?], which essentially means that one can define cohomology theories for these sheaves — the reason for the whole construction.
Search Encyclopedia

Featured Article
