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 0XY : 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 -----------> Vi.e. we have 0SV f = g 0RU. Then the morphisms 0XY 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