Zero Morphism - Examples

Examples

• In the category of groups (or of modules), a zero morphism is a homomorphism f : G → H that maps all of G to the identity element of H. The null object in the category of groups is the trivial group 1 = {1}, which is unique up to isomorphism. Every zero morphism can be factored through 1, i. e., f : G → 1 → H.

• More generally, suppose C is any category with a zero object 0. Then for all objects X and Y there is a unique sequence of morphisms

0_XY : X → 0 → Y

The family of all morphisms so constructed endows C with the structure of a category with zero morphisms.

• If C is a preadditive category, then every morphism set Mor(X,Y) is an abelian group and therefore has a zero element. These zero elements form a compatible family of zero morphisms for C making it into a category with zero morphisms.

• The category Set (sets with functions as morphisms) does not have a zero object, but it does have an initial object, the empty set ∅. The only right zero morphisms in Set are the functions ∅ → X for a set X.