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 -----------> V
i.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
- 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.
