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# Zero morphism

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 0XY : XY with the following property: for any two morphism f : RS and g : UV we obtain a commutative diagram:

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

0XY : X → 0 → Y
The family of all morphisms so constructed is a family of zero morphisms for C.
03-10-2013 05:06:04