Morphisms

A morphism is an abstract arrow that represents a structure-preserving map between objects in a category. In category theory, a category consists of objects and morphisms with composition and identity laws: for any objects A, B, C there is a morphism f: A→B and a morphism g: B→C, whose composite g∘f: A→C is again a morphism, and every object A has an identity morphism id_A: A→A serving as a neutral element for composition.

In many concrete mathematical contexts, morphisms are familiar structure-preserving maps. For example, in the category of sets, morphisms are functions between sets. In groups, morphisms are group homomorphisms that preserve the group operation. In rings, ring homomorphisms preserve addition and multiplication (and often carry 1 to 1 in the common convention). In linear algebra, linear maps between vector spaces are morphisms in the category of vector spaces, equivalently module homomorphisms. In topology, morphisms are continuous maps between topological spaces. In algebraic geometry, morphisms include maps between schemes that respect their underlying geometric and algebraic structure.

Notation commonly uses Hom(A,B) to denote the set of all morphisms from A to B. A morphism f:A→B is an isomorphism if it has an inverse, indicating that A and B are, in the categorical sense, the same. Special morphisms such as monomorphisms and epimorphisms generalize injectivity and surjectivity: mono means left-cancellable, epi means right-cancellable in the categorical sense. Endomorphisms are morphisms from an object to itself, and automorphisms are isomorphisms from an object to itself.

Beyond basic structure, morphisms enable the study of universal properties, functors, and constructions that preserve or reflect the essential features of objects across contexts.