Grothendieck topology

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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, then the elements of F(U) all arise from piecing together mutually compatible elements of F(Vi). This turns F into a sheaf, and a Grothendieck topology on C is an attempt to capture the essense of what is needed to define sheaves on C.

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).