Incategory theory, a branch ofmathematics, aGrothendieck topology is a structure on a categoryC that makes the objects ofC act like theopen sets of atopological space. A category together with a choice of Grothendieck topology is called asite.
Grothendieck topologies axiomatize the notion of anopen cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to definesheaves on a category and theircohomology. This was first done inalgebraic geometry andalgebraic number theory byAlexander Grothendieck to define theétale cohomology of ascheme. It has been used to define other cohomology theories since then, such asℓ-adic cohomology,flat cohomology, andcrystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as toJohn Tate's theory ofrigid analytic geometry.
There is a natural way to associate a site to an ordinarytopological space, and Grothendieck's theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namelysobriety, this is completely accurate—it is possible to recover a sober space from its associated site. However simple examples such as theindiscrete 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. InArtin (1962) it meant what is now called a Grothendieck pretopology, and some authors still use this old meaning.Giraud (1964) modified the definition to usesieves 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.
André Weil's famousWeil conjectures proposed that certain properties ofequations withintegral coefficients should be understood as geometric properties of thealgebraic variety that they define. His conjectures postulated that there should be acohomology 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 introducedétale maps into algebraic geometry as algebraic analogues of local analytic isomorphisms inanalytic geometry. He used étale coverings to define an algebraic analogue of thefundamental group of a topological space. SoonJean-Pierre Serre noticed that some properties of étale coverings mimicked those ofopen immersions, and that consequently it was possible to make constructions that imitated thecohomology 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.
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 mapsopen immersions, just as in the context ofschemes. Then a presheaf on is acontravariant functor from to the category of sets, and a sheaf is a presheaf that satisfies thegluing axiom (here including the separation axiom). The gluing axiom is phrased in terms ofpointwise 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 asieve. Pointwise covering is replaced by the notion of acovering 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.
In a Grothendieck topology, the notion of a collection of open subsets ofU stable under inclusion is replaced by the notion of asieve. Ifc is any given object inC, asieve onc is asubfunctor of the functor Hom(−,c); (this is theYoneda embedding applied toc). In the case ofO(X), a sieveS on an open setU selects a collection of open subsets ofU that is stable under inclusion. More precisely, consider that for any open subsetV ofU,S(V) will be a subset of Hom(V,U), which has only one element, the open immersionV →U. ThenV will be considered "selected" byS if and only ifS(V) is nonempty. IfW is a subset ofV, then there is a morphismS(V) →S(W) given by composition with the inclusionW →V. IfS(V) is non-empty, it follows thatS(W) is also non-empty.
IfS is a sieve onX, andf:Y →X is a morphism, then left composition byf gives a sieve onY called thepullback ofSalongf, denoted byfS. It is defined as thefibered productS ×Hom(−,X) Hom(−,Y) together with its natural embedding in Hom(−,Y). More concretely, for each objectZ ofC,fS(Z) = {g:Z →Y |fgS(Z) }, andfS inherits its action on morphisms by being a subfunctor of Hom(−,Y). In the classical example, the pullback of a collection {Vi} of subsets ofU along an inclusionW →U is the collection {Vi∩W}.
AGrothendieck topologyJ on a categoryC is a collection,for each object c of C, of distinguished sieves onc, denoted byJ(c) and calledcovering sieves ofc. This selection will be subject to certain axioms, stated below. Continuing the previous example, a sieveS on an open setU inO(X) will be a covering sieve if and only if the union of all the open setsV for whichS(V) is nonempty equalsU; in other words, if and only ifS gives us a collection of open sets thatcoverU in the classical sense.
The conditions we impose on a Grothendieck topology are:
The base change axiom corresponds to the idea that if {Ui} coversU, then {Ui ∩V} should coverU ∩V. The local character axiom corresponds to the idea that if {Ui} coversU and {Vij}jJi coversUi for eachi, then the collection {Vij} for alli andj should coverU. Lastly, the identity axiom corresponds to the idea that any set is covered by itself via the identity map.
In fact, it is possible to put these axioms in another form where their geometric character is more apparent, assuming that the underlying categoryC 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 calledcovering families. If the collection of all covering families satisfies certain axioms, then we say that they form aGrothendieck pretopology. These axioms are:
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 sieveS by lettingS(Y) be the set of all morphismsY →X that factor through some arrowXα →X. This is called the sievegenerated 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 isomorphismY →X is Hom(−,X). Consequently, if we restrict our attention to topologies, (PT 3) and (PT 3') are equivalent.
LetC be a category and letJ be a Grothendieck topology onC. The pair (C,J) is called asite.
Apresheaf on a category is a contravariant functor fromC to the category of all sets. Note that for this definitionC 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 asheaf on a site to be a presheafF such that for all objectsX and all covering sievesS onX, the natural map Hom(Hom(−,X),F) → Hom(S,F), induced by the inclusion ofS into Hom(−,X), is a bijection. Halfway in between a presheaf and a sheaf is the notion of aseparated presheaf, where the natural map above is required to be only an injection, not a bijection, for all sievesS. Amorphism of presheaves or of sheaves is a natural transformation of functors. The category of all sheaves onC is thetopos defined by the site (C,J).
Using theYoneda lemma, it is possible to show that a presheaf on the categoryO(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 anequalizer. For a separated presheaf, the first arrow need only be injective.
Similarly, one can define presheaves and sheaves ofabelian groups,rings,modules, and so on. One can require either that a presheafF is a contravariant functor to the category of abelian groups (or rings, or modules, etc.), or thatF be an abelian group (ring, module, etc.) object in the category of all contravariant functors fromC to the category of sets. These two definitions are equivalent.
LetC be any category. To define thediscrete topology, we declare all sieves to be covering sieves. IfC has all fibered products, this is equivalent to declaring all families to be covering families. To define theindiscrete topology, also known as thecoarse orchaotic topology,[1] we declare only the sieves of the form Hom(−,X) 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.
LetC be any category. The Yoneda embedding gives a functor Hom(−,X) for each objectX ofC. Thecanonical topology is the biggest (finest) topology such that every representable presheaf, i.e. presheaf of the form Hom(−,X), is a sheaf. A covering sieve or covering family for this site is said to bestrictly universally epimorphic because it consists of the legs of a colimit cone (under the full diagram on the domains of its constituent morphisms) and these colimits are stable under pullbacks along morphisms inC. A topology that is less fine than the canonical topology, that is, for which every covering sieve is strictly universally epimorphic, is calledsubcanonical. 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.
We repeat the example that we began with above. LetX be a topological space. We definedO(X) to be the category whose objects are the open sets ofX and whose morphisms are inclusions of open sets. Note that for an open setU and a sieveS onU, the setS(V) contains either zero or one element for every open setV. The covering sieves on an objectU ofO(X) are those sievesS satisfying the following condition:
This notion of cover matches the usual notion in point-set topology.
This topology can also naturally be expressed as a pretopology. We say that a family of inclusions {VαU} is a covering family if and only if the unionVα equalsU. This site is called thesmall site associated to a topological spaceX.
LetSpc be the category of all topological spaces. Given any family of functions {uα :Vα →X}, we say that it is asurjective family or that the morphismsuα arejointly surjective ifuα(Vα) equalsX. We define a pretopology onSpc by taking the covering families to be surjective families all of whose members are open immersions. LetS be a sieve onSpc.S is a covering sieve for this topology if and only if:
Fix a topological spaceX. Consider thecomma categorySpc/X of topological spaces with a fixed continuous map toX. The topology onSpc induces a topology onSpc/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 toX. This is thebig site associated to a topological spaceX. Notice thatSpc is the big site associated to the one point space. This site was first considered byJean Giraud.
LetM be amanifold.M has a category of open setsO(M) because it is a topological space, and it gets a topology as in the above example. For two open setsU andV ofM, the fiber productU ×MV is the open setU ∩V, which is still inO(M). This means that the topology onO(M) is defined by a pretopology, the same pretopology as before.
LetMfd 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 ofSpc, and open immersions are continuous (or smooth, or analytic, etc.), soMfd inherits a topology fromSpc. This lets us construct the big site of the manifoldM as the siteMfd/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 manifoldsX →Y and any open subsetU ofY, the fibered productU ×YX is inMfd/M. This is just the statement that the preimage of an open set is open. Notice, however, that not all fibered products exist inMfd because the preimage of a smooth map at a critical value need not be a manifold.
The category ofschemes, denotedSch, 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 that are part of a cover of the given scheme.
The most elementary of these is theZariski topology. LetX be a scheme.X has an underlying topological space, and this topological space determines a Grothendieck topology. The Zariski topology onSch is generated by the pretopology whose covering families are jointly surjective families of scheme-theoretic open immersions. The covering sievesS forZar are characterized by the following two properties:
Despite their outward similarities, the topology onZar isnot the restriction of the topology onSpc! This is because there are morphisms of schemes that are topologically open immersions but that are not scheme-theoretic open immersions. For example, letA be a non-reduced ring and letN be its ideal of nilpotents. The quotient mapA →A/N induces a map SpecA/N → SpecA, 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.
Theétale 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 theNisnevich topology, but neither finer nor coarser than thecdh and l′ topologies.
There are twoflat topologies, thefppf topology and thefpqc topology.fppf stands forfidèlement 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 forfidèlement 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 to be a family that is a cover on Zariski open subsets.[2] In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover.[3] These topologies are closely related todescent. Thefpqc topology is finer than all the topologies mentioned above, and it is very close to the canonical topology.
Grothendieck introducedcrystalline cohomology to study thep-torsion part of the cohomology of characteristicp varieties. In thecrystalline topology, which is the basis of this theory, the underlying category has objects given by infinitesimal thickenings together withdivided power structures. Crystalline sites are examples of sites with nofinal object.
There are two natural types of functors between sites. They are given by functors that are compatible with the topology in a certain sense.
If (C,J) and (D,K) are sites andu :C →D is a functor, thenu iscontinuous if for every sheafF onD with respect to the topologyK, the presheafFu is a sheaf with respect to the topologyJ. Continuous functors induce functors between the corresponding topoi by sending a sheafF toFu. These functors are calledpushforwards. If and denote the topoi associated toC andD, then the pushforward functor is.
us admits a left adjointus called thepullback.us need not preserve limits, even finite limits.
In the same way,u sends a sieve on an objectX ofC to a sieve on the objectuX ofD. A continuous functor sends covering sieves to covering sieves. IfJ is the topology defined by a pretopology, and ifu commutes with fibered products, thenu 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 isnot sufficient foru to send covering sieves to covering sieves (see SGA IV 3,Exemple 1.9.3).
Again, let (C,J) and (D,K) be sites andv :C →D be a functor. IfX is an object ofC andR is a sieve onvX, thenR can be pulled back to a sieveS as follows: A morphismf :Z →X is inS if and only ifv(f) :vZ →vX is inR. This defines a sieve.v iscocontinuous if and only if for every objectX ofC and every covering sieveR ofvX, the pullbackS ofR is a covering sieve onX.
Composition withv sends a presheafF onD to a presheafFv onC, but ifv is cocontinuous, this need not send sheaves to sheaves. However, this functor on presheaf categories, usually denoted, admits a right adjoint. Thenv 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 ofv* denotedv*. Furthermore,v* preserves finite limits, so the adjoint functorsv* andv* determine ageometric morphism of topoi.
A continuous functoru :C →D is amorphism of sitesD →C (notC →D) ifus preserves finite limits. In this case,us andus determine a geometric morphism of topoi. The reasoning behind the convention that a continuous functorC →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 spacesX →Y determines a continuous functorO(Y) →O(X). Since the original map on topological spaces is said to sendX toY, the morphism of sites is said to as well.
A particular case of this happens when a continuous functor admits a left adjoint. Suppose thatu :C →D andv :D →C are functors withu right adjoint tov. Thenu is continuous if and only ifv is cocontinuous, and when this happens,us is naturally isomorphic tov* andus is naturally isomorphic tov*. In particular,u is a morphism of sites.