## 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 of is 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 isomorphisms monomorphisms homomorphisms.

Endomorphisms do not fit easiliy into this relationship since they are not isomorphisms and must be from a set into the same set. 