|
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. This tool is mainly used in algebraic geometry, for instance to define [étale cohomology]?. |
|
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. |
|
Formally, to be continued... |
|
Formally, a Grothendieck category on C is given by specifying for each object U of C families of morphisms {φi : Vi -> U}i in I, called coverging 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). |
![[Home]](wikipedia.png)