Morphism - Definition

Definition

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 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). Note that the term hom-set is a bit of a misnomer as the collection of morphisms is not required to be a set.

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 f : X → Y and g : Y → Z is written g ∘ f or gf. The composition of morphisms is often represented by a commutative diagram. For example,

Morphisms satisfy two axioms:

• Identity: for every object X, there exists a morphism id_X : X → X called the identity morphism on X, such that for every morphism f : A → B we have id_B ∘ f = f = f ∘ id_A.
• Associativity: h ∘ (g ∘ f) = (h ∘ g) ∘ f whenever the operations are defined.

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).