Grothendieck topology

From Academic Kids

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 has been used in algebraic number theory and algebraic geometry, initially to define étale cohomology of schemes, but also for flat cohomology and crystalline cohomology, and in further ways.

Note that a Grothendieck topology is a true generalisation. It is not a topology in the classical sense, and may not be equivalent to giving one (although it can be used to faithfully model sober spaces).

Contents

History and idea

See main article Background and genesis of topos theory

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 (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 (see gluing axiom) and a Grothendieck topology on C is an attempt to capture the essence of what is needed to define sheaves on C.

Formal definition

A Grothendieck topology T on a category C consists of a collection of subfunctors (called covering sieves) R ⊂ hom(-, x) where x are objects of C satisfying the following axioms:

  • (identity) hom(-, x) is a covering sieve of x for any x in C
  • (base change) If R ⊂ hom(-, x) and φ: y → x in C then φ−1(R) ⊂ hom(-, y) is a covering sieve, where φ−1(R) = {z → y | (z → y → x) ∈ R}.
  • (local character) Let R, R′ ⊂ hom(-, x) be subfunctors of the covering sieve hom(-, x), and suppose R is a covering sieve. Then if φ−1(R′) ⊂ hom(-, y) is a covering sieve for all φ: y → x in R then R′ is also a covering sieve.

When the category C has pullbacks, a Grothendieck topology is usually given by specifying for each object U of C families of morphisms {φi : Vi -> U}i∈I, called covering families of U (analogous to the covering sieves in the general case), such that the following axioms are satisfied:

  • (identity) If φ1 : U1 -> U is an isomorphism, then {φ1 : U1 -> U} is a covering family of U.
  • (base change) If {φi : Vi -> U}i∈I is a covering family of U and f : U1 -> U is a morphism, then the pullback Pi = U1 ×UVi exists for every i∈I, and the induced family {πi : Pi -> U1}i∈I is a covering family of U1.
  • (local character) If {φi : Vi -> U}i∈I is a covering family of U, and if for every i∈I, {φij : Vij -> Vi}j∈Ji is a covering family of Vi, then {φiφij : Vij -> U}i∈I, j∈Ji is a covering family for U.

Presheaves and sheaves

A presheaf of sets on a category C is a contravariant functor

F : CSet.

If C has pullbacks and is equipped with a Grothendieck topology, then a presheaf on C is called a sheaf if, for every covering family {φi : ViU}i∈I, the induced map

F(U) → Πi

in I F(Vi) is the equalizer of the two natural maps

Πi ∈ I F(Vi) → Π(i, j)∈I x I F(Vi ×U Vj).

In analogy, one can also define presheaves and sheaves of abelian groups, by considering contravariant functors

F : CAb.

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.

Beyond cohomology

Other uses have been found for Grothendieck topologies, not limited to defining cohomology theory. One important area, from the point of view of number theory, is the definition of John Tate's rigid analytic geometry. Another application, increasingly prominent, is the Nisnevich topology.es:Topología de Grothendieck

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