Morphismus

Morphismus, in English often called morphism, is a central notion in category theory and in many areas of mathematics. In its general sense an morphismus is an arrow between objects that expresses a structure-preserving relation. A morphismus f has a domain (source) A and a codomain (target) B, written as f: A → B, and it can be composed with other morphismen. Identity morphisms id_A exist for every object A and act as units for composition; composition is associative. A morphismus is invertible precisely when there exists a morphismus g: B → A with g ∘ f = id_A and f ∘ g = id_B; such morphismen are called isomorphisms, and their inverses are automorphisms when the objects coincide.

In concrete categories, morphismen take familiar forms that preserve the relevant structure:

- Set: functions between sets

- Group: group homomorphisms

- Ring: ring homomorphisms

- Top: continuous maps

- Vect_K: linear maps between vector spaces over a field K

- Schemes (algebraic geometry): regular or scheme morphisms, depending on context

Special classes of morphismen include monomorphisms (left-cancellable arrows, often corresponding to injective maps in familiar categories) and epimorphisms (right-cancellable arrows, often corresponding to surjective maps). Isomorphisms are morphismen with inverses, indicating structural equivalence between objects.

Beyond individual arrows, morphismen are the building blocks of functors, which map objects and morphismen between categories, and of natural transformations, which relate functors. Universal properties, limits, and adjunctions are formulated in terms of unique or universal morphismen satisfying certain conditions.

History and usage: the concept of morphismus was formalized with category theory by Saunders Mac Lane and Samuel Eilenberg in the 1940s–1950s, and the term has since become a standard tool across many mathematical disciplines.

---