Editing Zero morphism (section)

== Examples ==
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|1= In the [[category of groups]] (or of [[module (mathematics)|modules]]), a zero morphism is a [[homomorphism]] ''f'' : ''G'' → ''H'' that maps all of ''G'' to the [[identity element]] of ''H''.  The zero object in the category of groups is the [[trivial group]] '''1''' = {1}, which is unique up to [[isomorphism]].  Every zero morphism can be factored through '''1''', i. e., ''f'' : ''G'' → '''1''' → ''H''.
|2= 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<sub>''XY''</sub> : ''X'' → '''0''' → ''Y''

The family of all morphisms so constructed endows '''C''' with the structure of a category with zero morphisms.
|3= If '''C''' is a [[preadditive category]], then every hom-set Hom(''X'',''Y'') is an [[abelian group]] and therefore has a zero element. These zero elements form a compatible family of zero morphisms for '''C''' making it into a category with zero morphisms.
|4= The [[category of sets]] does not have a zero object, but it does have an [[initial object]], the [[empty set]] ∅. The only right zero morphisms in '''Set''' are the functions ∅ → ''X'' for a set ''X''.
}}