Grothendieck topology

History and idea

At a time when cohomology for sheaves on topological spaces was well established, Alexander Grothendieck wanted to define cohomology theories for other structures, his schemes. He thought of a sheaf on a topological space as a "measuring rod" for that space, and the cohomology of such a measuring rod as a rough measure for the underlying space. His goal was thus to produce a structure which would allow the definition of more general sheaves or "measuring rods"; once that was done, the model of topological cohomology theories could be followed almost verbatim.

Motivating example

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 (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.

Formal definition

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:

• if φ₁ : U₁ -> U is an isomorphism, then {φ₁ : U₁ -> U} is a covering family of U.
• if {φ_i : V_i -> U}_{i in I} is a covering family of U and f : U₁ -> U is a morphism, then the pullback P_i = U₁ ×_UV_i exists for every i in I, and the induced family {π_i : P_i -> U₁}_{i in I} is a covering family of U₁.
• if {φ_i : V_i -> U}_{i in I} is a covering family of U, and if for every i in I, {φⁱ_j : Vⁱ_j -> V_i}_{j in Ji} is a covering family of V_i, then {φ_iφⁱ_j : Vⁱ_j -> U}_{i in I and j in Ji} is a covering family for U.

A presheaf of sets 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 induced 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).

In analogy, one can also define presheaves and sheaves of abelian groups, by considering contravariant functors F : C -> Ab.

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 of abelian groups 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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