Homomorphism - Relation To Category Theory

Relation To Category Theory

Since homomorphisms are morphisms, the following specific kinds of morphisms defined in any category are defined for homomorphisms as well. However, the definitions in category theory are somewhat technical. In the important special case of module homomorphisms, and for some other classes of homomorphisms, there are much simpler descriptions, as follows:

• An isomorphism is a bijective homomorphism.

• An epimorphism (sometimes called a cover) is a surjective homomorphism.

• A monomorphism (sometimes called an embedding or extension) is an injective homomorphism.

• An endomorphism is a homomorphism from an object to itself.

• An automorphism is an endomorphism which is also an isomorphism, i.e., an isomorphism from an object to itself.

These descriptions may be used in order to derive several interesting properties. For instance, since a function is bijective if and only if it is both injective and surjective, a module homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism.

For endomorphisms and automorphisms, the descriptions above coincide with the category theoretic definitions; the first three descriptions do not. For instance, the precise definition for a homomorphism f to be iso is not only that it is bijective, and thus has an inverse f-1, but also that this inverse is a homomorphism, too. This has the important consequence that two objects are completely indistinguishable as far as the structure in question is concerned, if there is an isomorphism between them. Two such objects are said to be isomorphic.

Actually, in the algebraic setting (at least within the context of universal algebra) this extra condition on isomorphisms is automatically satisfied. However, the same is not true for epimorphisms; for instance, the inclusion of Z as a (unitary) subring of Q is not surjective, but an epimorphic ring homomorphism. This inclusion thus also is an example of a ring homomorphism which is both mono and epi, but not iso.

Relationships between different kinds of module homomorphisms.
H = set of Homomorphisms, M = set of Monomorphisms,
P = set of Epimorphisms, S = set of Isomorphisms,
N = set of Endomorphism, A = set of Automorphisms.
Notice that: M ∩ P = S, S ∩ N = A,
(M ∩ N) \ A and (P ∩ N) \ A contain only homomorphisms from some infinite modules to themselves.