Morphism

A morphism is a concept of category theory that generalizes structure-preserving maps, such as homomorphisms between algebraic structures, functions between sets, and continuous functions between topological spaces. Morphisms and objects are the constituents of a category, where morphisms are also called maps or arrows and relate a source object to a target object; a partial operation called composition combines morphisms whenever the target of the first equals the source of the second, behaving like function composition with associativity and an identity morphism for every object. Morphisms and categories recur throughout contemporary mathematics; originally introduced for homological algebra and algebraic topology, they now belong to the foundational tools of Grothendieck's scheme theory, a generalization of algebraic geometry that also applies to algebraic number theory.