funktorielle

Funktorielle is a term used in mathematical discourse, particularly in category theory and related areas, to describe constructions or properties that are compatible with functors. In this sense, something funktorielle behaves in a way that can be organized into a functor, mapping objects to objects and morphisms to morphisms in a structure-preserving way.

Definition and meaning. A construction is called funktorielle when it can be functorially extended along the arrows of a given category. Concretely, if a process assigns to each object X an object F(X) and to each morphism f: X → Y a morphism F(f): F(X) → F(Y) such that identities and compositions are preserved, then the construction is funktorielle. This formalizes the idea that the construction commutes with the structure of the category.

Examples. Classic examples of funktorielle constructions include the power set functor P: Set → Set, which sends a function f: X → Y to f[P(X)(U)] = f[U] for subsets U ⊆ X, and various hom-functors, such as Hom(A, −) or Hom(−, A), which assign to each morphism a corresponding map of hom-objects. In topology, singular homology H_n(−) is funktorielle: a continuous map f: X → Y induces a homomorphism H_n(f): H_n(X) → H_n(Y).

Importance. Recognizing a construction as funktorielle clarifies its behavior under maps, enables naturality arguments, and supports the use of functorial language across areas such as algebra, topology, and geometry. See also: functor, natural transformation, functoriality, covariant and contravariant functors.