Grothendieck topology

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.
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 Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define 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.
There is a natural way to associate a site to an ordinary topological space, and Grothendieck's theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namely sobriety, this is completely accurate—it is possible to recover a sober space from its associated site. However simple examples such as the indiscrete topological space show that not all topological spaces can be expressed using Grothendieck topologies. Conversely, there are Grothendieck topologies that do not come from topological spaces.
The term "Grothendieck topology" has changed in meaning. In it meant what is now called a Grothendieck pretopology, and some authors still use this old meaning. modified the definition to use sieves rather than covers. Much of the time this does not make much difference, as each Grothendieck pretopology determines a unique Grothendieck topology, though quite different pretopologies can give the same topology.

Overview

'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 define. His conjectures postulated that there should be a cohomology theory of algebraic varieties that gives 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, Alexander 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 that imitated the cohomology functor. Grothendieck saw that it would be possible to use Serre's idea to define a cohomology theory that 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

Motivation

The classical definition of a sheaf begins with a topological space. A sheaf associates information to the open sets of. This information can be phrased abstractly by letting be the category whose objects are the open subsets of and whose morphisms are the inclusion maps of open sets and of. We will call such maps open immersions, just as in the context of schemes. Then a presheaf on is a contravariant functor from to the category of sets, and a sheaf is a presheaf that satisfies the gluing axiom. The gluing axiom is phrased in terms of pointwise covering, i.e., covers if and only if. In this definition, is an open subset of. Grothendieck topologies replace each with an entire family of open subsets; in this example, is replaced by the family of all open immersions. Such a collection is called a sieve. Pointwise covering is replaced by the notion of a covering family; in the above example, the set of all as varies is a covering family of. Sieves and covering families can be axiomatized, and once this is done open sets and pointwise covering can be replaced by other notions that describe other properties of the space.

Sieves

In a Grothendieck topology, the notion of a collection of open subsets of U stable under inclusion is replaced by the notion of a sieve. If c is any given object in C, a sieve on c is a subfunctor of the functor Hom;. In the case of O, a sieve S on an open set U selects a collection of open subsets of U that is stable under inclusion. More precisely, consider that for any open subset V of U, S will be a subset of Hom, which has only one element, the open immersion V → U. Then V will be considered "selected" by S if and only if S is nonempty. If W is a subset of V, then there is a morphism S → S given by composition with the inclusion W → V. If S is non-empty, it follows that S is also non-empty.
If S is a sieve on X, and f: Y → X is a morphism, then left composition by f gives a sieve on Y called the pullback of S along f, denoted by fS. It is defined as the fibered product S ×_Hom Hom together with its natural embedding in Hom. More concretely, for each object Z of C, fS =, and fS inherits its action on morphisms by being a subfunctor of Hom. In the classical example, the pullback of a collection of subsets of U along an inclusion W → U is the collection.

Grothendieck topology

A Grothendieck topology J on a category C is a collection, for each object c of C, of distinguished sieves on c, denoted by J and called covering sieves of c. This selection will be subject to certain axioms, stated below. Continuing the previous example, a sieve S on an open set U in O will be a covering sieve if and only if the union of all the open sets V for which S is nonempty equals U; in other words, if and only if S gives us a collection of open sets that cover U in the classical sense.

Axioms

The conditions we impose on a Grothendieck topology are:

If is a covering sieve on, and is a morphism, then the pullback is a covering sieve on.
Let be a covering sieve on, and let be any sieve on. Suppose that for each object of and each arrow in, the pullback sieve is a covering sieve on. Then is a covering sieve on.
is a covering sieve on for any object in.

The base change axiom corresponds to the idea that if covers, then covers. The local character axiom corresponds to the idea that if covers and covers for each, then the collection covers. The identity axiom corresponds to the idea that, at least when is an open set, the set of open subsets of covers.

Grothendieck pretopologies

In fact, it is possible to put these axioms in another form where their geometric character is more apparent, assuming that the underlying category C contains certain fibered products. In this case, 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:

For all objects X of C, and for all morphisms X₀ → X that appear in some covering family of X, and for all morphisms Y → X, the fibered product X₀ ×_X Y exists.
For all objects X of C, all morphisms Y → X, and all covering families, the family is a covering family.
If is a covering family, and if for all α, is a covering family, then the family of composites is a covering family.
If f: Y → X is an isomorphism, then 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.
For categories with fibered products, there is a converse. Given a collection of arrows, we construct a sieve S by letting S be the set of all morphisms Y → X that factor through some arrow X_α → X. This is called the sieve generated by. Now choose a topology. Say that is a covering family if and only if the sieve that it generates is a covering sieve for the given topology. It is easy to check that this defines a pretopology.
is sometimes replaced by a weaker axiom:

If 1_X : X → X is the identity arrow, then is a covering family.

implies, but not conversely. However, suppose that we have a collection of covering families that satisfies through and, but not. These families generate a pretopology. The topology generated by the original collection of covering families is then the same as the topology generated by the pretopology, because the sieve generated by an isomorphism Y → X is Hom. Consequently, if we restrict our attention to topologies, and are equivalent.

Sites and sheaves

Let C be a category and let J be a Grothendieck topology on C. The pair 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, induced by the inclusion of S into Hom, 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. A morphism of presheaves or of sheaves is a natural transformation of functors. The category of all sheaves on C is the topos defined by the site.
Using the Yoneda lemma, it is possible to show that a presheaf on the category O 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, the diagram
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 that F be an abelian group object in the category of all contravariant functors from C to the category of sets. These two definitions are equivalent.