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 structure preserving functions; which articulate relations among various category domains. Because it is a functor which articulates the structure-preserving map between the two categories (algebraic structures), morphisms are not necessarily functions; and objects over which morphisms are defined, are not necessarily sets (see homomorphism). Instead, a morphism is often thought of as a relation, which is visualized as an arrow linking a category domain to another, related codomain. Hence morphisms do not so as much map sets into sets, as embody the relationship between a particular category domain and another category 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.
A category C consists of two classes, one of objects and the other of morphisms.
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. The collection of all morphisms from X to Y is denoted homC(X,Y) or simply hom(X, Y) and called the hom-set between X and Y. Some authors write MorC(X,Y) or Mor(X, Y). Note that the term hom-set is a bit of a misnomer as the collection of morphisms is not required to be a set.
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