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
             |              |
             |              |
             |0RU           |0SV
             |              |
             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.