Zero morphism

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In category theory, a zero morphism is a special kind of "trivial" morphism. 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:

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.

A morphism is zero if and only if it is constant and coconstant.

[edit] 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).