Synonymer & Information om | Engelska ordet FUNCTOR

Exempel på hur man kan använda FUNCTOR i en mening

• The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively.
• Another way to state the above is to say G is a group object in a category C if for every object X in C, there is a group structure on the morphisms Hom(X, G) from X to G such that the association of X to Hom(X, G) is a (contravariant) functor from C to the category of groups.
• In the language of category theory, a homology theory is a type of functor from the category of the mathematical object being studied to the category of abelian groups and group homomorphisms, or more generally to the category corresponding to the associated chain complexes.
• There is a covariant functor from the category of abelian groups to the category of torsion groups that sends every group to its torsion subgroup and every homomorphism to its restriction to the torsion subgroup.
• The functor U is to be thought of as a forgetful functor, which assigns to every object of C its "underlying set", and to every morphism in C its "underlying function".
• Higher-order functions should not be confused with other uses of the word "functor" throughout mathematics, see Functor (disambiguation).
• Using category theory, this can be expressed by saying that localization is a functor that is left adjoint to a forgetful functor.
• The question of points was close to resolution by 1950; Alexander Grothendieck took a sweeping step (invoking the Yoneda lemma) that disposed of it—naturally at a cost, that every variety or more general scheme should become a functor.
• It follows that, if free objects exist in , the functor , called the free functor is a left adjoint to the faithful functor ; that is, there is a bijection.
• In concise terms, a monad is a monoid in the category of endofunctors of some fixed category (an endofunctor is a functor mapping a category to itself).
• It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below).
• This means that P is projective if and only if this functor preserves epimorphisms (surjective homomorphisms), or if it preserves finite colimits.
• In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.
• The key property that one has to prove here is that the counit of an adjunction is an isomorphism if and only if the right adjoint is a full and faithful functor.
• every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.
• These constructions can be applied to all topological spaces, and so singular homology is expressible as a functor from the category of topological spaces to the category of graded abelian groups.
• Suppose we are given a covariant left exact functor F : A → B between two abelian categories A and B.
• Both of these functors are, in fact, right inverses to U (meaning that UD and UI are equal to the identity functor on Set).
• The functor F yields an equivalence between A and a full subcategory of R-Mod in such a way that kernels and cokernels computed in A correspond to the ordinary kernels and cokernels computed in R-Mod.
• The covariant functor that associates to each sheaf F the group of global sections F(X) is left-exact.

Förberedelsen av sidan tog: 199,05 ms.