Grothendieck topology
Encyclopedia
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 set
s 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.
'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.
s. 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., {Ui} covers U if and only if
i Ui = U. In this definition, Ui is an open subset of X. Grothendieck topologies replace each Ui with an entire family of open subsets; in this example, Ui is replaced by the family of all open immersions Vij → Ui. 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 {Vij → Ui}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.
. 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 f
S. 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, f
S(Z) = { g: Z → Y | fg 
S(Z) }, and f
S inherits its action on morphisms by being a subfunctor of Hom(−, Y). In the classical example, the pullback of a collection {Vi} of subsets of U along an inclusion W → U is the collection {Vi∩W}.
U in the classical sense.
The base change axiom corresponds to the idea that if {
} covers U, then {Ui ∩ V} should cover U ∩ V. The local character axiom corresponds to the idea that if {Ui} covers U and {Vij}j 
Ji covers Ui for each i, then the collection {Vij} 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.
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 {Xα → X}, we construct a sieve S by letting S(Y) be the set of all morphisms Y → X that factor through some arrow Xα → X. This is called the sieve generated by {Xα → X}. Now choose a topology. Say that {Xα → 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) 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.
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. 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 (C, J).
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 {Xα → X}, 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 group
s, ring
s, module
s, 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.
This topology can also naturally be expressed as a pretopology. We say that a family of inclusions {Vα
U} is a covering family if and only if the union 
Vα equals U. This site is called the small site associated to a topological space X.
uα(Vα) 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:
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
.
. 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.
s, 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:
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 , 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 , 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. In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover. 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 structure
s. The crystalline covers of a fixed scheme form a category with no final object.
and 
denote the topoi associated to C and D, then the pushforward functor is 
.
us admits a left adjoint us called the pullback. us 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, 1.9.3).
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
, admits a right adjoint 
. Then v is cocontinuous if and only if 
sends sheaves to sheaves, that is, if and only if it restricts to a functor 
. In this case, the composite of 
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 
.
. 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, us is naturally isomorphic to v* and us is naturally isomorphic to v*. In particular, u is a morphism of sites.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. 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
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
on a category and their cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
. This was first done in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
by Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
to define the étale cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
of a scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
. It has been used to define other cohomology theories since then, such as l-adic cohomology, flat cohomology, and crystalline cohomology
Crystalline cohomology
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by and developed by . Its values are modules over rings of Witt vectors over the base field....
. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
's theory of rigid analytic geometry.
There is a natural way to associate a site to an ordinary topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, and Grothendieck's theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namely sobriety
Sober space
In mathematics, a sober space is a topological spacesuch that every irreducible closed subset of X is the closure of exactly one point of X: that is, has a unique generic point.-Properties and examples :...
, 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
André WeilAndré Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
's famous Weil conjectures
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
proposed that certain properties of equations with integral coefficients should be understood as geometric properties of the algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
that they define. His conjectures postulated that there should be a cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
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
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
. He used étale coverings to define an algebraic analogue of the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of a topological space. Soon Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
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
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
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.
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 schemeScheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s. 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
Gluing axiom
In mathematics, the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor...
. The gluing axiom is phrased in terms of pointwise covering, i.e., {Ui} covers U if and only if
i Ui = U. In this definition, Ui is an open subset of X. Grothendieck topologies replace each Ui with an entire family of open subsets; in this example, Ui is replaced by the family of all open immersions Vij → Ui. 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 {Vij → Ui}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 sieveSieve (category theory)
In category theory, a branch of mathematics, a sieve is a way of choosing arrows with a common codomain. It is a categorical analogue of a collection of open subsets of a fixed open set in topology. In a Grothendieck topology, certain sieves become categorical analogues of open covers in...
. If c is any given object in C, a sieve on c is a subfunctor
Subfunctor
In category theory, a branch of mathematics, a subfunctor is a special type of functor which is an analogue of a subset.-Definition:Let C be a category, and let F be a functor from C to the category of sets Set...
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
Pullback
Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback , and is frequently denoted by φ*...
of S along f, denoted by f
S. 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 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 {Vi} of subsets of U along an inclusion W → U is the collection {Vi∩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 coverCover (topology)
In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, ifC = \lbrace U_\alpha: \alpha \in A\rbrace...
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 f
S 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 f
T 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 {
} covers U, then {Ui ∩ V} should cover U ∩ V. The local character axiom corresponds to the idea that if {Ui} covers U and {Vij}j Ji covers Ui for each i, then the collection {Vij} 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 {Xα → X}, the family {Xα ×X Y → Y} is a covering family.
- (PT 2) (Local character) If {Xα → X} is a covering family, and if for all α, {Xβα → Xα} is a covering family, then the family of composites {Xβα → Xα → 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 {Xα → X}, we construct a sieve S by letting S(Y) be the set of all morphisms Y → X that factor through some arrow Xα → X. This is called the sieve generated by {Xα → X}. Now choose a topology. Say that {Xα → 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 1X : X → X is the identity arrow, then {1X} 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
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. 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 (C, J).
Using the Yoneda lemma
Yoneda lemma
In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
, 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 {Xα → X}, 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 group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s, ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
s, module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
s, 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.
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 {Vα
U} is a covering family if and only if the union Vα 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 {uα : Vα → X}, we say that it is a surjective family or that the morphisms uα are jointly surjective if uα(Vα) 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
Comma category
In mathematics, a comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W...
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
Jean Giraud (mathematician)
Jean Giraud was a French mathematician, a student of Alexander Grothendieck and the author of the book "Cohomologie non abélienne" ....
.
The big and small sites of a manifold
Let M be a manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the 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 schemeScheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s, 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
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
. 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
Étale topology
In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic...
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
Flat topology
In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also has played a fundamental role in the theory of descent...
, the fppf topology and the fpqc topology. fppf stands for , 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 , 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. In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover. These topologies are closely related to descent
Descent (category theory)
In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.A sophisticated...
. The fpqc topology is finer than all the topologies mentioned above, and it is very close to the canonical topology.
Grothendieck introduced crystalline cohomology
Crystalline cohomology
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by and developed by . Its values are modules over rings of Witt vectors over the base field....
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 structure
Divided power structure
In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form x^n / n! meaningful even when it is not possible to actually divide by n!.- Definition :Let A be a commutative ring with an ideal I...
s. 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 and denote the topoi associated to C and D, then the pushforward functor is .us admits a left adjoint us called the pullback. us 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, 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
, admits a right adjoint . Then v is cocontinuous if and only if sends sheaves to sheaves, that is, if and only if it restricts to a functor . In this case, the composite of 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 .Morphisms of sites
A continuous functor u : C → D is a morphism of sites D → C (not C → D) if us preserves finite limits. In this case, us and us determine a geometric morphism of topoi. 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, us is naturally isomorphic to v* and us is naturally isomorphic to v*. In particular, u is a morphism of sites.