Grothendieck topology

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. 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 which do not come from 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 define. 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, 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 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

Motivation

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 subsets U of X and whose morphisms are the inclusion maps V → U of open sets U and V of X. We will call such maps open immersions, just as in the context of schemes. 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 $\cup$ _i U_i = U. In this definition, U_i is an open subset of X. Grothendieck topologies replace each U_i with an entire family of open subsets; in this example, U_i is replaced by the family of all open immersions V_ij → U_i. 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 {V_ij → U_i}_j as i varies is a covering family of U. Sieves and covering families can be axiomatized, and once this is done open sets and pointwise covering can be replaced by other notions which describe other properties of the space X.

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(−, c); (this is the Yoneda embedding applied to c). In the case of O(X), a sieve S on an open set U selects a collection of open subsets of U which is stable under inclusion. More precisely, consider that for any open subset V of U, S(V) will be a subset of Hom(V, U), which has only one element, the open immersion V → U. Then V will be considered "selected" by S if and only if S(V) is nonempty. If W is a subset of V, then there is a morphism S(V) → S(W) given by composition with the inclusion W → V. If S(V) is non-empty, it follows that S(W) 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(−, X)} Hom(−, Y) together with its natural embedding in Hom(−, Y). More concretely, for each object Z of C, fS(Z) = { g: Z → Y | fg $\in$ S(Z) }, and fS inherits its action on morphisms by being a subfunctor of Hom(−, Y). In the classical example, the pullback of a collection {V_i} of subsets of U along an inclusion W → U is the collection {V_i∩W}.

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(c) 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(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.

Axioms

The conditions we impose on a Grothendieck topology are:

(T 1) (Base change) If S is a covering sieve on X, and f: Y → X is a morphism, then the pullback fS is a covering sieve on Y.
(T 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 fT is a covering sieve on Y. Then T is a covering sieve on X.
(T 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 J_i} 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.

Alternative axioms

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:

(PT 0) (Existence of fibered products) For all objects X of C, and for all morphisms X₀ → X which appear in some covering family of X, and for all morphisms Y → X, the fibered product X₀ ×_X Y exists.
(PT 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.
(PT 2) (Local character) 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.
(PT 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.

For categories with fibered products, there is a converse. Given a collection of arrows {X_α → X}, we construct a sieve S by letting S(Y) be the set of all morphisms Y → X that factor through some arrow X_α → X. This is called the sieve generated by {X_α → X}. Now choose a topology. Say that {X_α → X} 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 1_X : X → X is the identity arrow, then {1_X} 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 Y → X is Hom(−, X). Consequently, if we restrict our attention to topologies, (PT 3) and (PT 3') are equivalent.

Sites and sheaves

Examples of sites

The discrete and indiscrete topologies

Let C be any category. To define the discrete topology, we declare all sieves to be covering sieves. If C has all fibered products, this is equivalent to declaring all families to be covering families. To define the indiscrete topology, we declare only the sieves of the form Hom(−, X) to be covering sieves. The indiscrete topology is also known as the biggest or chaotic topology, and it is generated by the pretopology which has only isomorphisms for covering families. A sheaf on the indiscrete site is the same thing as a presheaf.

The canonical topology

Let C be any category. The Yoneda embedding gives a functor Hom(−, X) for each object X of C. The canonical topology is the biggest topology such that every representable presheaf Hom(−, X) is a sheaf. A covering sieve or covering family for this site is said to be strictly universally epimorphic. A topology which is less fine than the canonical topology, that is, for which every covering sieve is strictly universally epimorphic, is called subcanonical. Subcanonical sites are exactly the sites for which every presheaf of the form Hom(−, X) is a sheaf. Most sites encountered in practice are subcanonical.

Small site associated to a topological space

We repeat the example which we began with above. Let X be a topological space. We defined O(X) to be the category whose objects are the open sets of X and whose morphisms are inclusions of open sets. The covering sieves on an object U of O(X) were those sieves S satisfying the following condition:

If W is the union of all the sets V such that S(V) is non-empty, then W = U.

This topology can also naturally be expressed as a pretopology. We say that a family of inclusions {V_α $\sube$ U} is a covering family if and only if the union $\cup$ V_α equals U. This site is called the small site associated to a topological space X.

Big site associated to a topological space

Let Spc be the category of all topological spaces. Given any family of functions {u_α : V_α → X}, we say that it is a surjective family or that the morphisms u_α are jointly surjective if $\cup$ u_α(V_α) equals X. We define a pretopology on Spc by taking the covering families to be surjective families all of whose members are open immersions. Let S be a sieve on Spc. S is a covering sieve for this topology if and only if:

For all Y and every morphism f : Y → X in S(Y), there exists a V and a g : V → X such that g is an open immersion, g is in S(V), and f factors through g.
If W is the union of all the sets f(Y), where f : Y → X is in S(Y), then W = X.

Fix a topological space X. Consider the comma category Spc/X of topological spaces with a fixed continuous map to X. The topology on Spc induces a topology on Spc/X. The covering sieves and covering families are almost exactly the same; the only difference is that now all the maps involved commute with the fixed maps to X. This is the big site associated to a topological space X . Notice that Spc is the big site associated to the one point space. This site was first considered by Jean Giraud.

The big and small sites of a manifold

Let M be a manifold. M has a category of open sets O(M) because it is a topological space, and it gets a topology as in the above example. For two open sets U and V of M, the fiber product U ×_M V is the open set U ∩ V, which is still in O(M). This means that the topology on O(M) is defined by a pretopology, the same pretopology as before.

Let Mfd be the category of all manifolds and continuous maps. (Or smooth manifolds and smooth maps, or real analytic manifolds and analytic maps, etc.) Mfd is a subcategory of Spc, and open immersions are continuous (or smooth, or analytic, etc.), so Mfd inherits a topology from Spc. This lets us construct the big site of the manifold M as the site Mfd/M. We can also define this topology using the same pretopology we used above. Notice that to satisfy (PT 0), we need to check that for any continuous map of manifolds X → Y and any open subset U of Y, the fibered product U ×_Y X is in Mfd/M. This is just the statement that the preimage of an open set is open. Notice, however, that not all fibered products exist in Mfd because the preimage of a smooth map at a critical value need not be a manifold.

Topologies on the category of schemes

The category of schemes, denoted Sch, has a tremendous number of useful topologies. A complete understanding of some questions may require examining a scheme using several different topologies. All of these topologies have associated small and big sites. The big site is formed by taking the entire category of schemes and their morphisms, together with the covering sieves specified by the topology. The small site over a given scheme is formed by only taking the objects and morphisms which are part of a cover of the given scheme.

The most elementary of these is the Zariski topology. Let X be a scheme. X has an underlying topological space, and this topological space determines a Grothendieck topology. The Zariski topology on Sch is generated by the pretopology whose covering families are jointly surjective families of scheme-theoretic open immersions. The covering sieves S for Zar are characterized by the following two properties:

For all Y and every morphism f : Y → X in S(Y), there exists a V and a g : V → X such that g is an open immersion, g is in S(V), and f factors through g.
If W is the union of all the sets f(Y), where f : Y → X is in S(Y), then W = X.

Despite their outward similarities, the topology on Zar is not the restriction of the topology on Spc! This is because there are morphisms of schemes which are topologically open immersions but which are not scheme-theoretic open immersions. For example, let A be a non-reduced ring and let N be its ideal of nilpotents. The quotient map A → A/N induces a map Spec A/N → Spec A which is the identity on underlying topological spaces. To be a scheme-theoretic open immersion it must also induce an isomorphism on structure sheaves, which this map does not do. In fact, this map is a closed immersion.

The étale topology is finer than the Zariski topology. It was the first Grothendieck topology to be closely studied. Its covering families are jointly surjective families of étale morphisms. It is finer than the Nisnevich topology, but neither finer nor coarser than the cdh and l′ topologies.

There are two flat topologies, the fppf topology and the fpqc topology. fppf stands for fidèlement plate de présentation finie, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat, of finite presentation, and is quasi-finite. fpqc stands for fidèlement plate et quasi-compacte, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat. In both categories, a covering family is defined be a family which is a cover on Zariski open subsets.^[1] In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover.^[2] These topologies are closely related to descent. The fpqc topology is finer than all the topologies mentioned above, and it is very close to the canonical topology.

Grothendieck introduced crystalline cohomology to study the p-torsion part of the cohomology of characteristic p varieties. In the crystalline topology which is the basis of this theory, covering maps are given by infinitesimal thickenings together with divided power structures. The crystalline covers of a fixed scheme form a category with no final object.

Continuous and cocontinuous functors

There are two natural types of functors between sites. They are given by functors which are compatible with the topology in a certain sense.

Continuous functors

If (C, J) and (D, K) are sites and u : C → D is a functor, then u is continuous if for every sheaf F on D with respect to the topology K, the presheaf Fu is a sheaf with respect to the topology J. Continuous functors induce functors between the corresponding topoi by sending a sheaf F to Fu. These functors are called pushforwards. If $\tilde C$ and $\tilde D$ denote the topoi associated to C and D, then the pushforward functor is $u_s�: \tilde D \to \tilde C$ .

u_s admits a left adjoint u^s called the pullback. u^s need not preserve limits, even finite limits.

In the same way, u sends a sieve on an object X of C to a sieve on the object uX of D. A continuous functor sends covering sieves to covering sieves. If J is the topology defined by a pretopology, and if u commutes with fibered products, then u is continuous if and only if it sends covering sieves to covering sieves and if and only if it sends covering families to covering families. In general, it is not sufficient for u to send covering sieves to covering sieves (see SGA IV 3, Exemple 1.9.3).

Cocontinuous functors

Again, let (C, J) and (D, K) be sites and v : C → D be a functor. If X is an object of C and R is a sieve on vX, then R can be pulled back to a sieve S as follows: A morphism f : Z → X is in S if and only if v(f) : vZ → vX is in R. This defines a sieve. v is cocontinuous if and only if for every object X of C and every covering sieve R of vX, the pullback S of R is a covering sieve on X.

Composition with v sends a presheaf F on D to a presheaf Fv on C, but if v is cocontinuous, this need not send sheaves to sheaves. However, this functor on presheaf categories, usually denoted $\hat v^*$ , admits a right adjoint $\hat v_*$ . Then v is cocontinuous if and only if $\hat v_*$ sends sheaves to sheaves, that is, if and only if it restricts to a functor $v_*�: \tilde C \to \tilde D$ . In this case, the composite of $\hat v^*$ with the associated sheaf functor is a left adjoint of v_* denoted v^*. Furthermore, v^* preserves finite limits, so the adjoint functors v_* and v^* determine a geometric morphism of topoi $\tilde C \to \tilde D$ .

Morphisms of sites

A continuous functor u : C → D is a morphism of sites D → C (not C → D) if u^s preserves finite limits. In this case, u^s and u_s determine a geometric morphism of topoi $\tilde C \to \tilde D$ . The reasoning behind the convention that a continuous functor C → D is said to determine a morphism of sites in the opposite direction is that this agrees with the intuition coming from the case of topological spaces. A continuous map of topological spaces X → Y determines a continuous functor O(Y) → O(X). Since the original map on topological spaces is said to send X to Y, the morphism of sites is said to as well.

A particular case of this happens when a continuous functor admits a left adjoint. Suppose that u : C → D and v : D → C are functors with u right adjoint to v. Then u is continuous if and only if v is cocontinuous, and when this happens, u^s is naturally isomorphic to v^* and u_s is naturally isomorphic to v_*. In particular, u is a morphism of sites.

References

^ SGA III₁, IV 6.3.
^ SGA III₁, IV 6.3, Proposition 6.3.1(v).

Bibliography

Artin, Michael (1962), Grothendieck topologies, Harvard University, Dept. of Mathematics
Demazure, Michel; Alexandre Grothendieck, eds. (1970) (in French). Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 1 (Lecture notes in mathematics 151). Berlin; New York: Springer-Verlag. pp. xv+564.
Artin, Michael; Alexandre Grothendieck, Jean-Louis Verdier, eds. (1972) (in French). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269). Berlin; New York: Springer-Verlag. xix+525.
Nisnevich, Yevsey A. (1989). "The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory". In J. F. Jardine and V. P. Snaith. Algebraic K-theory: connections with geometry and topology. Proceedings of the NATO Advanced Study Institute held in Lake Louise, Alberta, December 7--11, 1987. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 279. Dordrecht: Kluwer Academic Publishers Group. pp. 241–342. , available at Nisnevich's website

Grothendieck topology

Contents

Introduction

Definition

Motivation

Sieves

Grothendieck topology

Axioms

Alternative axioms

Sites and sheaves

Examples of sites

The discrete and indiscrete topologies

The canonical topology

Small site associated to a topological space

Big site associated to a topological space

The big and small sites of a manifold

Topologies on the category of schemes

Continuous and cocontinuous functors

Continuous functors

Cocontinuous functors

Morphisms of sites

References

Bibliography