morphismi

Morphism is a structure-preserving map between two objects in a mathematical setting. The notion depends on the category or algebraic structure in question: a morphism f: A → B respects the defining operations and relations that characterize A and B.

In category theory, a category consists of objects and morphisms (arrows) with composition and identity. If f: A → B and g: B → C are morphisms, their composition g ∘ f: A → C is a morphism. Every object has an identity morphism id_A: A → A. Morphisms encode the ways objects relate while preserving structure, and they form the basic arrows of the theory.

Examples illustrate the scope of the concept. In Set, morphisms are functions between sets. In Groups, morphisms are group homomorphisms that preserve the group operation. In Vector Spaces, morphisms are linear maps. In Topology, morphisms are continuous maps. Other contexts include rings, modules, posets, and graphs, each with its respective preserving maps.

Special kinds of morphisms include isomorphisms, which have inverses and thus express structural equivalence; monomorphisms (left-cancellable) generalize injective maps; and epimorphisms (right-cancellable) generalize surjective maps. In categories with kernels, the kernel of a morphism measures the obstruction to being injective; cokernels play a dual role. Morphisms can be composed to form complex mappings, and universal properties often characterize morphisms by optimality or uniqueness.

In higher-level language, morphisms are the arrows of a category; functors map between categories by sending objects and morphisms, and natural transformations relate functors. The standard plural is morphisms; the form “morphismi” is not standard in English.