In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category that makes the objects of 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
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 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 (here including the separation 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 stable under inclusion is replaced by the notion of a sieve. If is any given object in , a sieve on is a subfunctor of the functor (this is the Yoneda embedding applied to ). In the case of a sieve on an open set selects a collection of open subsets of that is stable under inclusion. More precisely, consider that for any open subset of will be a subset of which has only one element, the open immersion Then will be considered "selected" by if and only if is nonempty. If is a subset of then there is a morphism given by composition with the inclusion If is non-empty, it follows that is also non-empty.
If is a sieve on and is a morphism, then left composition by gives a sieve on called the pullback of along denoted by It is defined as the fibered product together with its natural embedding in More concretely, for each object of and inherits its action on morphisms by being a subfunctor of In the classical example, the pullback of a collection of subsets of along an inclusion is the collection
Grothendieck topology A Grothendieck topology on a category is a collection, for each object of , of distinguished sieves on denoted by and called covering sieves of This selection will be subject to certain axioms, stated below. Continuing the previous example, a sieve on an open set in will be a covering sieve if and only if the union of all the open sets for which is nonempty equals in other words, if and only if gives us a collection of open sets that cover in the classical sense.
Axioms The conditions we impose on a Grothendieck topology are:
(T 1) (Base change) If is a covering sieve on , and is a morphism, then the pullback is a covering sieve on
(T 2) (Local character) 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
(T 3) (Identity) 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 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:
(PT 0) (Existence of fibered products) For all objects of , and for all morphisms that appear in some covering family of and for all morphisms the fibered product exists. (PT 1) (Stability under base change) For all objects of , all morphisms and all covering families the family is a covering family. (PT 2) (Local character) If is a covering family, and if for all is a covering family, then the family of composites is a covering family. (PT 3) (Isomorphisms) If 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 by letting be the set of all morphisms that factor through some arrow 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.
(PT 3) is sometimes replaced by a weaker axiom:
(PT 3') (Identity) If is the identity arrow, then is a covering family.
(PT 3) implies (PT 3'), but not conversely. However, suppose that we have a collection of covering families that satisfies (PT 0) through (PT 2) and (PT 3'), but not (PT 3). 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 is Consequently, if we restrict our attention to topologies, (PT 3) and (PT 3') are equivalent.
Sites and sheaves
Let be a category and let be a Grothendieck topology on The pair is called a site.
A presheaf on a category is a contravariant functor from to the category of all sets. Note that for this definition 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 such that for all objects and all covering sieves on the natural map
induced by the inclusion of into 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 A morphism of presheaves or of sheaves is a natural transformation of functors. The category of all sheaves on is the topos defined by the site
Using the Yoneda lemma, it is possible to show that a presheaf on the category 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 is a contravariant functor to the category of abelian groups (or rings, or modules, etc.), or that be an abelian group (ring, module, etc.) object in the category of all contravariant functors from to the category of sets. These two definitions are equivalent.
Examples of sites
The discrete and indiscrete topologies
Let be any category. To define the discrete topology, we declare all sieves to be covering sieves. If has all fibered products, this is equivalent to declaring all families to be covering families. To define the indiscrete topology, also known as the coarse or chaotic topology,SGA IV, II 1.1.4. we declare only the sieves of the form to be covering sieves. The indiscrete topology is generated by the pretopology that has only isomorphisms for covering families. A sheaf on the indiscrete site is the same thing as a presheaf.
The canonical topology