# Morphisms and homomorphisms

In section 5.1.1 where category is defined the author says

B. for every pair x, y ∈ Ob(C), a set HomC(x,y)∈Set; it is called the hom-set from x to y; its elements are called morphisms from x to y;2

Footnote 2 says

The reason for the notation Hom and the word hom-set is that morphisms are often called homomorphisms, e.g., in group theory.

But morphisms and homomorphisms are different, right ? Morphism can be anything, homomorphisms are more restrictive. Is that correct ? For eg: there can be morphisms between 2 groups that are not homomorphisms, right ?

## 2 thoughts on “Morphisms and homomorphisms”

1. True. He is just pointing out the origin of the word – morphism. Homomorphism is used in Group theory etc. Since Categories are higher abstractions than the groups a “homomorphism” like term morphism is being used.
Homomorphism of the groups preserve group operation for a mapping between groups. In similar fashion, morphisms preserve operations between objects of a category for mappings between objects of a given category.

2. Arun Mathews says:

For groups all morphisms need not be homomorphisms, right ? For two groups we could send all objects of one group to one object of another group and all morphisms of one group to the identity morphism of the object of the other group. All morphisms are homomorphisms but not the other way around.