Zero Morphism

Examples

• In the category of groups or modules a zero morphism is a homomorphism f : G → H that maps all of G to the identity element of 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 is a family of zero morphisms for C.

• 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 family of zero morphisms for C.
• The category Set (sets with functions as morphisms) does not have zero morphisms; nor does Top (topological spaces, with continuous functions).

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