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Zero morphism

In mathematics, a group homomorphism or module homomorphism f : GH 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 : XY with the following property: for any two morphism f : RS and g : UV we obtain a commutative diagram
             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.