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
a -> 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.
Associative & Identity Value
- Number under addition with ID value of 0
- Number under multiplication with ID value of 1
- Etc.