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category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
s of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...
. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
. This was first done in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ...
by Alexander Grothendieck to define the
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 conject ...
of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and
crystalline cohomology In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H'n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed b ...
. 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 In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. Such spaces were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing ''p''-adic elliptic curves with bad re ...
. There is a natural way to associate a site to an ordinary
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, and Grothendieck's theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namely
sobriety Sobriety is the condition of not having any measurable levels or effects from alcohol or drugs. Sobriety is also considered to be the natural state of a human being at birth. A person in a state of sobriety is considered sober. Organizations of ...
, 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
sieve A sieve, fine mesh strainer, or sift, is a device for separating wanted elements from unwanted material or for controlling the particle size distribution of a sample, using a screen such as a woven mesh or net or perforated sheet material ...
s 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 André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. ...
's famous
Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. T ...
proposed that certain properties of
equation In mathematics, an equation is a formula that expresses the equality (mathematics), equality of two Expression (mathematics), expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may h ...
s with
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
coefficients should be understood as geometric properties of the
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. M ...
that they define. His conjectures postulated that there should be a
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
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 map In mathematics, more specifically in algebra, the adjective étale refers to several closely related concepts: * Étale morphism ** Formally étale morphism * Étale cohomology * Étale topology * Étale fundamental group * Étale group scheme * Ét ...
s into algebraic geometry as algebraic analogues of local analytic isomorphisms in
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engine ...
. He used étale coverings to define an algebraic analogue of the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of a topological space. Soon
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the i ...
noticed that some properties of étale coverings mimicked those of
open immersion Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (Y ...
s, and that consequently it was possible to make constructions that imitated the cohomology functor ''H''1. 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 ''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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
from ''O''(''X'') 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 ''U'' if and only if \bigcup_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_ \to 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 \_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 that 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 A sieve, fine mesh strainer, or sift, is a device for separating wanted elements from unwanted material or for controlling the particle size distribution of a sample, using a screen such as a woven mesh or net or perforated sheet material ...
. 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'' that 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 In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
of ''S'' along ''f'', denoted by ''f''^\ast''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''^\ast''S''(''Z'') = , and ''f''^\ast''S'' inherits its action on morphisms by being a subfunctor of Hom(−, ''Y''). In the classical example, the pullback of a collection of subsets of ''U'' along an inclusion ''W'' → ''U'' is the collection .


Grothendieck topology

A Grothendieck topology ''J'' on a category ''C'' is a collection, ''for each object c of C'', of distinguished sieves on ''c'', denoted by ''J''(''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 ''U'' is the union of all the open sets ''V''; in other words, if and only if ''S'' gives us a collection of open sets that cover ''U'' in the classical sense.


Axioms

The conditions we impose on a Grothendieck topology are: * (T 1) (Base change) If ''S'' is a covering sieve on ''X'', and ''f'': ''Y'' → ''X'' is a morphism, then the pullback ''f''\ast''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''\ast''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 should cover ''U'' ∩ ''V''. The local character axiom corresponds to the idea that if covers ''U'' and ''j \inJi'' covers ''Ui'' for each ''i'', then the collection for all ''i'' and ''j'' should cover ''U''. Lastly, the identity axiom corresponds to the idea that any set is covered by itself via the identity map.


Grothendieck pretopologies

In fact, it is possible to put these axioms in another form where their geometric character is more apparent, assuming that the underlying category ''C'' contains certain fibered products. In this case, instead of specifying sieves, we can specify that certain collections of maps with a common codomain should cover their codomain. These collections are called covering families. If the collection of all covering families satisfies certain axioms, then we say that they form a Grothendieck pretopology. These axioms are: * (PT 0) (Existence of fibered products) For all objects ''X'' of ''C'', and for all morphisms ''X''0 → ''X'' that appear in some covering family of ''X'', and for all morphisms ''Y'' → ''X'', the fibered product ''X''0 ×''X'' ''Y'' exists. * (PT 1) (Stability under base change) For all objects ''X'' of ''C'', all morphisms ''Y'' → ''X'', and all covering families , the family is a covering family. * (PT 2) (Local character) If is a covering family, and if for all α, is a covering family, then the family of composites is a covering family. * (PT 3) (Isomorphisms) If ''f'': ''Y'' → ''X'' is an isomorphism, then is a covering family. For any pretopology, the collection of all sieves that contain a covering family from the pretopology is always a Grothendieck topology. For categories with fibered products, there is a converse. Given a collection of arrows , we construct a sieve ''S'' by letting ''S''(''Y'') be the set of all morphisms ''Y'' → ''X'' that factor through some arrow ''X''''α'' → ''X''. This is called the sieve generated by . Now choose a topology. Say that is a covering family if and only if the sieve that it generates is a covering sieve for the given topology. It is easy to check that this defines a pretopology. (PT 3) is sometimes replaced by a weaker axiom: * (PT 3') (Identity) If 1''X'' : ''X'' → ''X'' 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 ''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 In mathematics, the Yoneda lemma is arguably the most important result in category theory. It 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 (viewi ...
, 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 , the diagram :F(X) \rightarrow \prod_ F(X_\alpha) \prod_ F(X_\alpha\times_X X_\beta) 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 In mathematics, 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 the order in which they are written. That is, the group operation is comm ...
s, rings, modules, 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.


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, also known as the coarse or chaotic topology, 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.


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 (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 be ''strictly 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 in C. A topology that 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 that 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. Note that for an open set ''U'' and a sieve ''S'' on ''U'', the set ''S''(''V'') contains either zero or one element for every open set ''V''. The covering sieves on an object ''U'' of ''O''(''X'') are 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 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 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 , 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 In mathematics, a comma category (a special case being a slice 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 objec ...
''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 Henri Gaston Giraud (; 8 May 1938 – 10 March 2012) was a French artist, cartoonist, and writer who worked in the Bandes dessinées, Franco-Belgian ''bandes dessinées'' (BD) tradition. Giraud garnered worldwide acclaim under the pseu ...
.


The big and small sites of a manifold

Let ''M'' be a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
. ''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 that are part of a cover of the given scheme. The most elementary of these is the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
. 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 that are topologically open immersions but that 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 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. The étale t ...
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 to be a family that is a cover on Zariski open subsets. In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover.SGA III1, IV 6.3, Proposition 6.3.1(v). 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 In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H'n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed b ...
to study the ''p''-torsion part of the cohomology of characteristic ''p'' varieties. In the ''crystalline topology'', which is the basis of this theory, the underlying category has objects given by infinitesimal thickenings together with
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 ...
s. Crystalline sites are examples of sites with no final object.


Continuous and cocontinuous functors

There are two natural types of functors between sites. They are given by functors that 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, 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.


See also

*
Fibered category Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which ''inverse images'' (or ''pull-backs'') of ...
* Lawvere–Tierney topology


Notes


References

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External links


The birthday of Grothendieck topologies

The birthday of Grothendieck topologies (non-archived version)
{{DEFAULTSORT:Grothendieck Topology Topos theory Sheaf theory