Morphisms
A morphism
is a mapping between sets. The different types of morphism are
Homomorphism: preserves the structure (e.g.
) where
and
are the operations on the domain and codomain respectively. For example exp is a morphism from
to
where
is the set of positive real numbers.Epimorphism: a homomorphism that is surjective or onto.
is a homomorphism from
onto the set consisting of the single element 0. (with the operation of either ordinary addition or ordinary multiplication.)Monomorphism: a homomorphism that is injective or one to one. The homomorphism
illustrated above is a monomorphism.Isomorphism: a homomorphism that is bijective (one to one and onto); isomorphic objects are equivalent.
is an isomorphism from
to the set
with the operation of addition on both domain and codomain.Endomorphism: a homomorphism from a set to a subset of itself. Eg
is an endomorphism from
to 
Automorphism: a bijective endomorphism (an isomorphism from an set onto itself, essentially just a re - labeling of elements):
where z is a complex number of magnitude 1 is an isomorphism. The effect ofis to rotate
by
anticlockwise about the origin.
Every morphism send the identity in the domain to the identity in the codomain. This is easy to prove:
If
is the identity element in the domain with operation
then
so
is the identity element in the codomain.
The relationship between the different types of morphism may be written
automorphisms
isomorphismsmonomorphismshomomorphisms.
Endomorphisms do not fit easiliy into this relationship since they are not isomorphisms and must be from a set into the same set.