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 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.

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Examples

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).