funktorze

Funktorze is the locative singular form of the Polish noun funktor, which corresponds to the English term “functor.” In mathematics and computer science, a funktor is a structure-preserving mapping between categories. The Polish form appears in sentences like “w funktorze,” meaning “in a functor.”

Definition and basic properties: A funktor F from a category C to a category D assigns to each object X in C an object F(X) in D, and to each morphism f: X → Y in C a morphism F(f): F(X) → F(Y) in D. It preserves identities and composition: F(id_X) = id_{F(X)} and F(g ∘ f) = F(g) ∘ F(f) for all composable morphisms f, g in C. This preservation guarantees a faithful translation of structure from one context to another.

Types and variants: Functors can be covariant, preserving the direction of arrows, or contravariant, reversing it. An endofunctor is a functor where C = D. Other related notions include bifunctors (two-argument functors) and functor categories, which collect all functors between given categories.

Examples: In set theory, the identity functor Id_C on any category C simply maps each object and morphism to itself. The power set functor P: Set → Set maps each set to its set of subsets and each function to its direct image map. The Hom-functor maps objects to hom-sets and morphisms to function composition. In programming, a Functor is a type constructor equipped with a map (fmap) operation that applies a function to values inside a computational context, as in functional programming languages.

History and usage: The concept was introduced by Eilenberg and Mac Lane in the 1940s and became foundational in category theory. Today, functors are used to model structure-preserving translations across mathematics, logic, and computer science. See also natural transformations and endofunctors.