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Grothendieck topology

In mathematics, 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 used in algebraic number theory and algebraic geometry, mainly to define étale cohomology of schemess, but also for flat cohomology and crystalline cohomology. Note that a Grothendieck topology is not a topology in the classical sense.

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 schemess. 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 (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.

Formal definition

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:

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 {φi : Vi -> U}i in I, the induced 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).

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.