funktorin

In category theory, a funktor (Finnish: funktori) is a map between categories that preserves the mathematical structure of the categories involved. It assigns to every object of a source category C an object of a target category D, and to every morphism in C a morphism in D, in a way that respects identities and composition. The genitive form of the noun is funktorin.

Formally, a funktor F from a category C to a category D consists of two functions: an object mapping F on objects, and a morphism mapping F on arrows. For every morphism f: X → Y in C, F assigns a morphism F(f): F(X) → F(Y) in D. These mappings satisfy F(id_X) = id_{F(X)} for every object X, and F(g ∘ f) = F(g) ∘ F(f) for all composable morphisms f: X → Y and g: Y → Z. This encapsulates the idea that a funktor preserves the categorical structure.

There are two common types of funktori: covariant and contravariant. A covariant funktor preserves the direction of arrows, mapping f: X → Y to F(f): F(X) → F(Y). A contravariant funktor reverses the direction, sending f: X → Y to F(f): F(Y) → F(X); in this case the composition rule becomes F(g ∘ f) = F(f) ∘ F(g).

Examples include the identity funktor id_C: C → C, which maps each object and morphism to itself; the forgetful funktor from groups to sets, which assigns to each group its underlying set and to each group homomorphism the same function between sets; and the power-set funktor P: Set → Set, sending a set to its set of subsets and a function to the image of subsets. Funktori are fundamental tools for translating structures across different mathematical contexts.