Category (mathematics) - Types of Morphisms

Types of Morphisms

A morphism f : a → b is called

• a monomorphism (or monic) if fg₁ = fg₂ implies g₁ = g₂ for all morphisms g₁, g₂ : x → a.
• an epimorphism (or epic) if g₁f = g₂f implies g₁ = g₂ for all morphisms g₁, g₂ : b → x.
• a bimorphism if it is both a monomorphism and an epimorphism.
• a retraction if it has a right inverse, i.e. if there exists a morphism g : b → a with fg = 1_b.
• a section if it has a left inverse, i.e. if there exists a morphism g : b → a with gf = 1_a.
• an isomorphism if it has an inverse, i.e. if there exists a morphism g : b → a with fg = 1_b and gf = 1_a.
• an endomorphism if a = b. The class of endomorphisms of a is denoted end(a).
• an automorphism if f is both an endomorphism and an isomorphism. The class of automorphisms of a is denoted aut(a).

Every retraction is an epimorphism. Every section is a monomorphism. The following three statements are equivalent:

• f is a monomorphism and a retraction;
• f is an epimorphism and a section;
• f is an isomorphism.

Relations among morphisms (such as fg = h) can most conveniently be represented with commutative diagrams, where the objects are represented as points and the morphisms as arrows.