Morphism. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map. 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 structure-preserving functions. In category theory, morphisms are sometimes also called arrows. Definition[edit] There are two operations which are defined on every morphism, the domain (or source) and the codomain (or target). If a morphism f has domain X and codomain Y, we write f : X → Y. Thus a morphism is represented by an arrow from its domain to its codomain. Morphisms satisfy two axioms: Identity Associativity Some special morphisms[edit] Monomorphism Epimorphism.
Isomorphism (types of mappings) Epimorphism (mapping types) Types of mappings.