funktorid

Funktorid is the Estonian (and related) term for functors in category theory, a mathematical concept used to describe structure-preserving mappings between categories. In a broader sense, funktori provide a formal way to translate objects and their relationships from one mathematical setting to another while respecting the underlying composition rules.

Formally, a funktor F from a category C to a category D consists of two components: a mapping that assigns to each object X in C an object F(X) in D, and a mapping that assigns to each morphism f: X → Y in C a morphism F(f): F(X) → F(Y) in D. These assignments preserve identities, meaning F(id_X) = id_{F(X)}, and preserve composition, so F(g ∘ f) = F(g) ∘ F(f) for all composable morphisms f and g in C.

There are covariant and contravariant funktori. Covariant funktori preserve the direction of arrows, while contravariant funktori reverse it, producing morphisms in the opposite direction. Funktori enable the transportation of structure, for example via forgetful functors from algebraic structures to sets, or funktori like the power set funktor P: Set → Set and Hom-funktori such as Hom(A,-).

In practice, funktori form the morphisms between categories, and their study interacts with natural transformations, adjunctions, and limits and colimits. They are central to applications across algebra, topology, logic, and computer science, providing a unified language for relating different mathematical contexts.