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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.
== 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).
