NatHom

NatHom, short for natural homomorphism, is a term encountered in some areas of mathematics, particularly category theory and algebra, to describe a homomorphism that arises in a natural or functorial way from a construction. It is not a single fixed map but a family of maps that behaves uniformly with respect to the morphisms in the underlying category.

The key idea behind NatHom is naturality. A NatHom is compatible with morphisms in the source context, meaning that when objects are related by a morphism, the induced maps between the resulting objects commute with the original homomorphisms. This aligns with the broader notion of natural transformations, where a construction is functorial and respects structure-preserving maps.

In practice, NatHoms appear from universal properties or other functorial constructions. They often appear as canonical maps that accompany a construction, such as maps induced by a universal property, or as unit and counit maps associated with adjunctions. Because they are defined by a naturality condition rather than by ad hoc choice, NatHoms are often described in terms of functors, natural transformations, and universal properties rather than as separate, standalone morphisms.

Usage and terminology vary. The label NatHom is informal and not universally standardized; in formal treatments, the same concept is usually described using the language of natural transformations, Hom functors, or universal properties. Nevertheless, NatHom serves as a convenient shorthand for emphasizing the natural, structure-preserving character of certain canonical maps.

See also: natural transformation, Hom functor, universal property, adjunction.