In algebraic geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme S is (up to natural bijections) the set of all morphisms . The scheme X is then said to represent the functor and that classify geometric objects over S given by F.
The best known example is the Hilbert scheme of a scheme X (over some fixed base scheme), which, when it exists, represents a functor sending a scheme S to a flat family of closed subschemes of .
In some applications, it may not be possible to find a scheme that represents a given functor. This led to the notion of a stack, which is not quite a functor but can still be treated as if it were a geometric space. (A Hilbert scheme is a scheme, but not a stack because, very roughly speaking, deformation theory is simpler for closed schemes.)
Some moduli problems are solved by giving formal solutions (as opposed to polynomial algebraic solutions) and in that case, the resulting functor is represented by a formal scheme. Such a formal scheme is then said to be algebraizable if there is another scheme that can represent the same functor, up to some isomorphisms.
rational point#Definition
The notion is an analog of a classifying space in algebraic topology. In algebraic topology, the basic fact is that each principal G-bundle over a space S is (up to natural isomorphisms) the pullback of a universal bundle along some map from S to . In other words, to give a principal G-bundle over a space S is the same as to give a map (called a classifying map) from a space S to the classifying space of G.
A similar phenomenon in algebraic geometry is given by a linear system: to give a morphism from a projective variety to a projective space is (up to base loci) to give a linear system on the projective variety.
Yoneda's lemma says that a scheme X determines and is determined by its points.
Let X be a scheme. Its functor of points is the functor
Hom(−,X) : (Affine schemes)op ⟶ Sets
sending an affine scheme Y to the set of scheme maps .
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In mathematics, a topos (USˈtɒpɒs, UKˈtoʊpoʊs,_ˈtoʊpɒs; plural topoi ˈtɒpɔɪ or ˈtoʊpɔɪ, or toposes) is a that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory.
In mathematics, particularly , a representable functor is a certain functor from an arbitrary into the . Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. From another point of view, representable functors for a category C are the functors given with C. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory.
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne).
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