Zero morphism

In mathematics, a group homomorphism or module homomorphism f : G → H that maps all of G to the identity element of H is called a zero morphism. In category theory, the concept of zero morphism is defined more generally. Suppose C is a category, and for any two objects X and Y in C we are given a morphism 0_XY : X → Y with the following property: for any two morphism f : R → S and g : U → V we obtain a commutative diagram

                    f

             R -----------> S

             |              |

             |              |

             |0_RU           |0_SV

             |              |

             V       g      V

             U -----------> V

i.e. we have 0_SV f = g 0_RU. Then the morphisms 0_XY are called a family of zero morphisms in C.

By taking f or g to be the identity morphism in the diagram above, we see that the composition of any morphism with a zero morphism results in a zero morphism. Furthermore, if a category has a family of zero morphisms, then this family is unique.

If a category has zero morphisms, then one can define the notions of kernel and cokernel in that category.

Examples

The zero morphisms in the category of groups or modules as introduced above are zero morphisms in the new general sense.

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.
If C has a zero object Z, then from X there is a unique morphism to Z, and from Z there is a unique morphism to Y. Composing these two gives a morphism from X to Y. The family of all morphisms so constructed is a family of zero morphisms for C.
The category of all sets with functions as morphisms does not have zero morphisms; neither does the category of all topological spaces, with continuous maps as morphisms.