morphisme

In mathematics, a morphism (French: morphisme) is a structure-preserving map between objects of a category. The concept is central to category theory, where objects and morphisms (arrows) describe mathematical structures and their relationships. A morphism f: X → Y is understood as a rule that associates to each instance of the structure on X a corresponding instance on Y in a way that respects the relevant operations. Examples include a group homomorphism f: G → H preserving the group operation (f(ab) = f(a)f(b)), a ring homomorphism preserving addition and multiplication, a linear map between vector spaces, and a continuous map between topological spaces.

In a category, morphisms have a domain and a codomain, and they can be composed: if f: X → Y and g: Y → Z, then g ∘ f: X → Z is a morphism. Each object X has an identity morphism id_X: X → X that acts as a neutral element for composition. An isomorphism is a morphism with an inverse; monomorphisms and epimorphisms generalize injective and surjective maps, respectively, in many familiar categories.

Morphisms appear in many mathematical contexts. In topology they are continuous maps; in algebraic geometry, morphisms of schemes respect both the underlying topological space and the structure sheaves. In the category of sets, functions are morphisms; in the category of modules or vector spaces, linear maps are morphisms. The notion of morphism provides a unifying language for comparing and relating different objects through their structure-preserving maps.