Morphism is the starting point of category theory

(written by lawrence krubner, however indented passages are often quotes). You can contact lawrence at:

I guess I knew this, but still it is interesting to read:

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the ancient Greek language: ὁμός (homos) meaning “same” and μορφή (morphe) meaning “form” or “shape”.

Homomorphisms of vector spaces are also called linear maps, and their study is the object of linear algebra.

The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory.

Being an isomorphism, an automorphism, or an endomorphism is a property of some homomorphisms, which may be defined in a way that may be generalized to any class of morphisms.