Knowra Hom functor Hom functor A functor that assigns morphism sets or groups to objects and maps induced by composition to morphisms. Its two basic forms are covariant Hom(A, −) and contravariant Hom(−, A).
Covariant functor : A functor that preserves the direction of morphisms. Hom(A, −) is covariant because postcomposition maps Hom(A, X) to Hom(A, Y).
Category (mathematics) : A structure consisting of objects, morphisms, identities, and associative composition. Hom functors are defined on categories and use their morphisms as inputs.
Yoneda lemma : A theorem identifying natural transformations from a representable functor with morphisms to its representing object. It shows that an object is determined by how other objects map to it.
Endofunctor : A functor whose source and target are the same category. Unlike a general Hom functor, an endofunctor need not arise from morphisms to or from a fixed object.
Contravariant functor : A functor that reverses the direction of morphisms. Hom(−, A) is contravariant because precomposition reverses morphism direction.
Morphism : A structure-preserving arrow between objects in a category. The elements of each Hom-set are morphisms between its two specified objects.
Universal property : A characterization of an object by a unique pattern of morphisms to or from other objects. Hom functors express universal properties as bijections or natural isomorphisms of morphism sets.
Endomorphism : A morphism from an object to itself. An endomorphism is one element of Hom(A, A), not the functor assigning Hom-values.
Functor : A structure-preserving map between categories that sends objects to objects and morphisms to morphisms. A Hom functor is a specific construction obeying the general laws of functors.
Hom-set : The set of morphisms between two fixed objects in a category. A set-valued Hom functor assigns such sets as its values.
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