Morphism

A generalisation of homomorphism.

Relations among morphisms (such as fg = h) are often depicted using commutative diagrams, with "points" (corners) representing objects and "arrows" representing morphisms.

Morphisms can have any of the following properties. A morphism f : a → b is a:

monomorphism (or monic) if f ∘ g1 = f ∘ g2 implies g1 = g2 for all morphisms g1, g2 : x → a.
epimorphism (or epic) if g1 ∘ f = g2 ∘ f implies g1 = g2 for all morphisms g1, g2 : b → x.
bimorphism if f is both epic and monic.
isomorphism if there exists a morphism g : b → a such that f ∘ g = 1b and g ∘ f = 1a.[b]
endomorphism if a = b. end(a) denotes the class of endomorphisms of a.
automorphism if f is both an endomorphism and an isomorphism. aut(a) denotes the class of automorphisms of a.
retraction if a right inverse of f exists, i.e. if there exists a morphism g : b → a with f ∘ g = 1b.
section if a left inverse of f exists, i.e. if there exists a morphism g : b → a with g ∘ f = 1a.
Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent:

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

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