## Morphisms

A morphismis a mapping between sets. The different types of morphism are

• Homomorphism: preserves the structure (e.g.) whereandare the operations on the domain and codomain respectively. For example exp is a morphism fromtowhereis the set of positive real numbers.

• Epimorphism: a homomorphism that is surjective  or onto.is a homomorphism fromonto 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 homomorphismillustrated above is a monomorphism.

• Isomorphism: a homomorphism that is bijective (one to one and onto); isomorphic objects are equivalent.is an isomorphism fromto the setwith the operation of addition on both domain and codomain.

• Endomorphism: a homomorphism from a set to a subset of itself. Egis an endomorphism fromto

• 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 rotateby anticlockwise about the origin.

Every morphism send the identity in the domain to the identity in the codomain. This is easy to prove:

Ifis the identity element in the domain with operationthen

sois the identity element in the codomain.

The relationship between the different types of morphism may be written

automorphismsisomorphismsmonomorphismshomomorphisms.

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