morphismes

Morphismes, the French term for morphisms, are the structure-preserving maps between objects in a category. In category theory, a category consists of objects and morphisms between them, together with a composition operation that assigns to f: A → B and g: B → C a composite g ∘ f: A → C, and identity morphisms id_A: A → A for every object A. Composition is associative, and identity morphisms act as identities for composition.

In concrete settings, morphisms take familiar forms: in Set they are functions; in Grp they are group homomorphisms; in Top they are continuous maps; in Vect_K they are linear maps; in Ring they are ring homomorphisms. These examples show how the abstract concept unifies diverse notions of structure-preserving maps.

Key properties include monomorphisms (left-cancellable), epimorphisms (right-cancellable), and isomorphisms (having inverses). Endomorphisms map an object to itself; automorphisms are isomorphisms from an object to itself. In many contexts, kernels, cokernels, and factorization systems refine the study of morphisms, leading to abelian categories and beyond. Morphismes thus provide the central language for comparing and transporting structure across objects, and form the backbone of category-theoretic reasoning.