Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology.
Under meager point-set hypotheses, namely sobriety, this is completely accurate—it is possible to recover a sober space from its associated site.
His conjectures postulated that there should be a cohomology theory of algebraic varieties that gives number-theoretic information about their defining equations.
He used étale coverings to define an algebraic analogue of the fundamental group of a topological space.
Soon Jean-Pierre Serre noticed that some properties of étale coverings mimicked those of open immersions, and that consequently it was possible to make constructions that imitated the cohomology functor
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 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.
If the collection of all covering families satisfies certain axioms, then we say that they form a Grothendieck pretopology.
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.
Sheaves on a pretopology have a particularly simple description: For each covering family {Xα → X}, the diagram must be an equalizer.
Similarly, one can define presheaves and sheaves of abelian groups, 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.
The indiscrete topology is generated by the pretopology that has only isomorphisms for covering families.
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.
Consider the comma category Spc/X of topological spaces with a fixed continuous map to X.
Notice that Spc is the big site associated to the one point space.
Mfd is a subcategory of Spc, and open immersions are continuous (or smooth, or analytic, etc.
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.
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.
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.
The big site is formed by taking the entire category of schemes and their morphisms, together with the covering sieves specified by the topology.
To be a scheme-theoretic open immersion it must also induce an isomorphism on structure sheaves, which this map does not do.
fppf stands for fidèlement plate de présentation finie, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat, of finite presentation, and is quasi-finite.
[2] In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover.
In the crystalline topology, which is the basis of this theory, the underlying category has objects given by infinitesimal thickenings together with divided power structures.
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.
Since the original map on topological spaces is said to send X to Y, the morphism of sites is said to as well.