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Grothendieck topology

A Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C, and with that the definition of general cohomology theories. A category together with a Grothendieck topology on it is called a site. This tool is mainly used in algebraic geometry, for instance to define étale cohomology[?]. Note that a Grothendieck topology is not a topology in the classical sense.

The motivating example is the following: start with a topological space X and consider the sheaf of all continuous real-valued functions defined on X. This associates to every open set U in X the set F(U) of real-valued 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 pre-sheaf of sets on C. Our functor F has a special property: if you have an open covering (Vi) of the set U, and you are given mutually compatible elements of F(Vi), 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 {φi : Vi -> U}i in I, called covering families of U, such that the following axioms are satisfied:

  • if φ1 : U1 -> U is an isomorphism, then {φ1 : U1 -> U} is a covering family of U.
  • if {φi : Vi -> U}i in I is a covering family of U and f : U1 -> U is a morphism, then the pullback[?] Pi = U1 ×UVi exists for every i in I, and the induced family {πi : Pi -> U1}i in I is a covering family of U1.
  • if {φi : Vi -> U}i in I is a covering family of U, and if for every i in I, {φij : Vij -> Vi}j in Ji is a covering family of Vi, then {φi φij : Vij -> U}i in I and j in Ji is a covering family for U.

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 {φi : Vi -> U}i in I, the map F(U) -> Πi in I F(Vi) is the equalizer[?] of the two natural maps Πi in I F(Vi) -> Π(i, j) in I x I F(Vi ×U Vj).

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.

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