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 topology, continuous functions; in universal algebra, homomorphisms; in group theory, group homomorphisms.
A category C consists of two classes, one of objects and the other of morphisms.
There are two operations 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 an arrow from its domain to its codomain. The set of all morphisms from X to Y is denoted hom_{C}(X,Y) or simply hom(X, Y) and called the hom-set between X and Y. (Some authors write Mor_{C}(X,Y) or Mor(X, Y)).
For every three objects X, Y, and Z, there exists a binary operation hom(X, Y) × hom(Y, Z) → hom(X, Z) called composition. The composite of is written g o f, gf, or even fg. The composition of morphisms is often represented by a commutative diagram. For example,
Morphisms satisfy two axioms:
When C is a concrete category, the identity morphism is just the identity function, and composition is just the ordinary composition of functions. Associativity then follows, because the composition of functions is associative.
Note that the domain and codomain are in fact part of the information determining a morphism. For example, in the category of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs (may have the same range), while having different codomains. The two functions are distinct from the viewpoint of category theory. Thus many authors require that the hom-classes hom(X, Y) be disjoint. In practice, this is not a problem because if this disjointness does not hold, it can be assured by appending the domain and codomain to the morphisms, (say, as the second and third components of an ordered triple).
It is also called a mono or a monic. The morphism f has a left inverse if there is a morphism g:Y → X such that g o f = id_{X}. The left inverse g is also called a retraction of f. Morphisms with left inverses are always monomorphisms, but the converse is not always true in every category; a monomorphism may fail to have a left-inverse.
A split monomorphism h : X → Y is a monomorphism having a left inverse g : Y → X, so that g o h = id_{X}. Thus h o g : Y → Y is idempotent, so that (h o g)^{2} = h o g.
In concrete categories, a function which has left inverse is injective. Thus in concrete categories, monomorphisms are often, but not always, injective. The condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism.
A split epimorphism is an epimorphism having a right inverse.
In concrete categories, a function which has a right inverse is surjective. Thus in concrete categories, epimorphisms are often, but not always, surjective. The condition of being a surjection is stronger than that of being an epimorphism, but weaker than that of being a split epimorphism. In the category of sets, every surjection has a section, a result equivalent to the axiom of choice.
Note that if a split monomorphism f has a left-inverse g, then g is a split epimorphism and has right-inverse f.
If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f. Inverse morphisms, if they exist, are unique. The inverse g is also an isomorphism with inverse f. Two objects with an isomorphism between them are said to be isomorphic or equivalent.
Note that while every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism. For example, in the category of commutative rings the inclusion Z → Q is a bimorphism which is not an isomorphism. However, any morphism that is both an epimorphism and a split monomorphism, or both a monomorphism and a split epimorphism, must be an isomorphism. A category, such as Set, in which every bimorphism is an isomorphism is known as a balanced category.
A split endomorphism is an idempotent endomorphism f if f admits a decomposition f = h o g with g o h = id. In particular, the Karoubi envelope of a category splits every idempotent morphism.
For more examples, see the entry category theory.