In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The study of morphisms and of the structures (called objects) over which they are defined, is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are functions preserving this structure. Nevertheless, morphisms are not necessarily functions, and objects over which morphisms are defined are not necessarily sets. Instead, a morphism is often thought of as an arrow linking an object called the domain to another object called the codomain. Hence morphisms do not so much map sets into sets, as embody a relationship between some posited domain and codomain. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions; in universal algebra, homomorphisms.

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