Pfeilzuordnung

Pfeilzuordnung is a term used in mathematics and logic to describe the assignment or mapping of arrows (Pfeile) between objects within a diagram, category, or other structured setting. The core idea is to specify how morphisms or arrows are associated with the objects they connect, in a way that respects the surrounding structure.

In category theory, a prototypical form of Pfeilzuordnung is a functor. A Funktor F between categories assigns to each object X in a category C an object F(X) in a category D, and to each arrow f: X → Y in C an arrow F(f): F(X) → F(Y) in D. This assignment preserves identities (F(idX) = idF(X)) and composition (F(g ∘ f) = F(g) ∘ F(f)). In this sense, a Pfeilzuordnung that preserves these two operations is precisely a functor. Conversely, natural transformations describe higher-level Pfeilzuordnungen between functors.

Outside category theory, Pfeilzuordnung may refer more loosely to the orientation of arrows in a graph or to the interpretation of implications (arrows) in logic, where each relationship is treated as an arrow between entities or propositions.

Examples often cited include the forgetful functor from groups to sets, which maps each group to its underlying set and each group homomorphism to the corresponding function between sets; this is a canonical Pfeilzuordnung in the categorical sense.

Terminology varies: German texts may prefer speaking of Abbildungen der Pfeile, Pfeilzuordnung, or simply Funktor. The concept is fundamental for describing how structures relate and how structure-preserving maps are organized. See also category theory, Funktor, Morphismus, Diagramm.