Morphism

In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not to be maps, but they can be composed in a way that is similar to function composition.

Morphisms and objects are constituents of a category. Morphisms, also called maps or arrows, relate two objects called the source and the target of the morphism. There is a partial operation, called composition, on the morphisms of a category that is defined if the target of the first object equals the source of the second object. The composition of morphisms behave like function composition (associativity of composition when it is defined, and existence of an identity morphism for every object).

Morphisms and categories recur in much of contemporary mathematics. Originally, they were introduced for homological algebra and algebraic topology. They belong to the foundational tools of Grothendieck's scheme theory, a generalization of algebraic geometry that applies also to algebraic number theory.