Grothendieck topology

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In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexandre Grothendieck to define the étale cohomology of a scheme. It has been used to define many other cohomology theories since then, such as l-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry.

Grothendieck topologies are not comparable to the classical notion of a topology on a space. While it is possible to interpret sober spaces in terms of Grothendieck topologies, more pathological spaces have no such representation. Conversely, not all Grothendieck topologies correspond to topological spaces.

Introduction

Main article: Background and genesis of topos theory

André Weil's famous Weil conjectures proposed that certain properties of equations with integral coefficients should be understood as geometric properties of the algebraic variety that they defined. His conjectures postulated that there should be a cohomology theory of algebraic varieties which gave number-theoretic information about their defining equations. This cohomology theory was known as the "Weil cohomology", but using the tools he had available, Weil was unable to construct it.

In the early 1960s, Alexandre Grothendieck introduced étale maps into algebraic geometry as algebraic analogues of local analytic isomorphisms in analytic geometry. He used étale coverings to define an algebraic analogue of the fundamental group of a topological space. Soon Jean-Pierre Serre noticed that some properties of étale coverings mimicked those of open immersions, and that consequently it was possible to make constructions which imitated the cohomology functor H¹. Grothendieck saw that it would be possible to use Serre's idea to define a cohomology theory which he suspected would be the Weil cohomology. To define this cohomology theory, Grothendieck needed to replace the usual, topological notion of an open covering with one that would use étale coverings instead. Grothendieck also saw how to phrase the definition of covering abstractly; this is where the definition of a Grothendieck topology comes from.

Definition

The classical definition of a sheaf begins with a topological space X. A sheaf associates information to the open sets of X. This information can be phrased abstractly by letting O(X) be the category whose objects are the open sets of X and whose morphisms are open immersions. Then a presheaf on X is a contravariant functor from O(X) to the category of sets, and a sheaf is a presheaf which satisfies the gluing axiom. The gluing axiom is phrased in terms of pointwise covering, i.e., {U_i} covers U if and only if ∪_i U_i = U.

A Grothendieck topology encodes the information about covering without any reference to the space itself. The notion of covering is replaced by the notion of a sieve. A sieve is a subfunctor of a functor of the form Hom(−, c) for some object c; in other words, if S is a sieve, then S(c′) ⊆ Hom(c′, c) for some object c, and for any morphism f, S(f) is the restriction of Hom(f, c), the pullback by f. In the case of O(X), a sieve represents a collection of open sets contained in a larger open set. Each sieve corresponds to a possible way of covering U. For example, if S is a sieve on an open set U, then S(V) is a subset of Hom(V, U), which has only one element, the open immersion V → U. S(V) will contain this map if and only if V is one of the sets that S uses to cover U.

A Grothendieck topology J on a category C is defined by giving, for each object c of C, a collection J(c) of sieves on c, subject to certain conditions. These sieves are called covering sieves. Continuing the previous example, a sieve S on an open set U in O(X) will be a covering sieve if and only if the union of all the open sets V for which S(V) is nonempty equals U; in other words, if and only if S gives us a collection of open sets which cover U in the classical sense.

The conditions we impose on a Grothendieck topology are:

1. (Base change) Let S be a covering sieve on X, and let f: Y → X. Let f^∗S be the fibered product S ×_{Hom(−, X)} Hom(−, Y) together with its natural embedding in Hom(−, Y); equivalently, for each object Z of C, f^∗S(Z) = { g: Z → Y | fg ∈ S(Z) }. Then f^∗S, the pullback of S along f, is a covering sieve.
2. (Local character) Let S be a covering sieve on X, and let T be any sieve on X. Suppose that for each object Y of C and each arrow f: Y → X in S(Y), the pullback sieve f^∗T is a covering sieve on Y. Then T is a covering sieve on X.
3. (Identity) Hom(−, X) is a covering sieve on X for any object X in C.

The base change axiom corresponds to the idea that if {U_i} covers U, then {U_i ∩ V} should cover U ∩ V. The local character axiom corresponds to the idea that if {U_i} covers U and {V_ij}_{j ∈ Ji} covers U_i for each i, then the collection {V_ij} for all i and j should cover U. Lastly, the identity axiom corresponds to the idea that any set is covered by all its possible subsets.

In fact, it is possible to put these axioms in another form where their geometric character is more apparent. Assume that the underlying category C has fibered products. Instead of specifying sieves, we can specify that certain collections of maps with a common codomain should cover their codomain. These collections are called covering families. If the collection of all covering families satisfies certain axioms, then we say that they form a Grothendieck pretopology. These axioms are:

1. (Stability under base change) For all objects X of C, all morphisms Y → X, and all covering families {X_α → X}, the family {X_α ×_X Y → Y} is a covering family.
2. (Stability under composition) If {X_α → X} is a covering family, and if for all α, {X_βα → X_α} is a covering family, then the family of composites {X_βα → X_α → X} is a covering family.
3. (Isomorphisms) If f: Y → X is an isomorphism, then {f} is a covering family.

For any pretopology, the collection of all sieves that contain a covering family from the pretopology is always a Grothendieck topology.

Sites and sheaves

Let C be a category and let J be a Grothendieck topology on C. The pair (C, J) is called a site.

A presheaf on a category is a contravariant functor from C to the category of all sets. Note that for this definition C is not required to have a topology. A sheaf on a site, however, should allow gluing, just like sheaves in classical topology. Consequently, we define a sheaf on a site to be a presheaf F such that for all objects X and all covering sieves S on X, the natural map Hom(Hom(−, X), F) → Hom(S, F) induced by the inclusion of S into Hom(−, X) is a bijection. Halfway in between a presheaf and a sheaf is the notion of a separated presheaf, where the natural map above is required to be only an injection, not a bijection, for all sieves S.

Using the Yoneda lemma, it is possible to show that a presheaf on the category O(X) is a sheaf on the topology defined above if and only if it is a sheaf in the classical sense.

Sheaves on a pretopology have a particularly simple description: For each covering family {X_α → X}, the diagram

<math>F(X)\to \prod_{\alpha\in A} F(X_\alpha) {\to\atop\to} \prod_{\alpha,\beta \in A} F(X_\alpha\times_X X_\beta)</math>

must be an equalizer. For a separated presheaf, the first arrow need only be injective.

Similarly, one can define presheaves and sheaves of abelian groups, rings, modules, and so on. One can require either that a presheaf F is a contravariant functor to the category of abelian groups (or rings, or modules, etc.), or that F be an abelian group (ring, module, etc.) object in the category of all contravariant functors from C to the category of sets. These two definitions are equivalent.es:Topología de Grothendieck ko:그로텐디크 위상