Editing Zero morphism

{{Short description|Bi-universal property in category theory}}In [[category theory]], a branch of [[mathematics]], a '''zero morphism''' is a special kind of [[morphism]] exhibiting properties like the morphisms to and from a [[zero object]].

==Definitions==
Suppose '''C''' is a [[Category (mathematics)|category]], and ''f'' : ''X'' → ''Y'' is a morphism in '''C'''. The morphism ''f'' is called a '''constant morphism''' (or sometimes '''left zero morphism''') if for any [[Object (category theory)|object]] ''W'' in '''C''' and any {{nowrap|''g'', ''h'' : ''W'' → ''X''}}, ''fg'' = ''fh''. Dually, ''f'' is called a '''coconstant morphism''' (or sometimes '''right zero morphism''') if for any object ''Z'' in '''C''' and any ''g'', ''h'' : ''Y'' → ''Z'', ''gf'' = ''hf''. A '''zero morphism''' is one that is both a constant morphism and a coconstant morphism.

A '''category with zero morphisms''' is one where, for every two objects ''A'' and ''B'' in '''C''', there is a fixed morphism 0<sub>''AB''</sub> : ''A'' → ''B'', and this collection of morphisms is such that for all objects ''X'', ''Y'', ''Z'' in '''C''' and all morphisms ''f'' : ''Y'' → ''Z'', ''g'' : ''X'' → ''Y'', the following diagram commutes:

[[Image:ZeroMorphism.png|center|160px]]

The morphisms 0<sub>''XY''</sub> necessarily are zero morphisms and form a compatible system of zero morphisms.

If '''C''' is a category with zero morphisms, then the collection of 0<sub>''XY''</sub> is unique.<ref>{{cite web|url=https://math.stackexchange.com/q/189818 |title=Category with zero morphisms - Mathematics Stack Exchange |website=Math.stackexchange.com |date=2015-01-17 |access-date=2016-03-30}}</ref>

This way of defining a "zero morphism" and the phrase "a category with zero morphisms" separately is unfortunate, but if each [[hom-set]] has a unique "zero morphism", then the category "has zero morphisms".

== Examples ==
{{unordered list
|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''.
}}

==Related concepts==
If '''C''' has a zero object '''0''', given two objects ''X'' and ''Y'' in '''C''', there are canonical morphisms ''f'' : ''X'' → '''0''' and ''g'' : '''0''' → ''Y''. Then, ''gf'' is a zero morphism in Mor<sub>'''C'''</sub>(''X'', ''Y'').  Thus, every category with a zero object is a category with zero morphisms given by the composition 0<sub>''XY''</sub> : ''X'' → '''0''' → ''Y''.

If a category has zero morphisms, then one can define the notions of [[kernel (category theory)|kernel]] and [[cokernel]] for any morphism in that category.

==References==
*Section 1.7 of {{Citation
| last=Pareigis
| first=Bodo
| title=Categories and functors
| year=1970
| isbn=978-0-12-545150-5
| publisher=[[Academic Press]]
| series=Pure and applied mathematics
| volume=39
}}
* {{Citation| last1= Herrlich |first1= Horst |last2=Strecker |first2=George E. |year=2007 |title=Category Theory |publisher= Heldermann Verlag}}.

==Notes==
{{Reflist}}

[[Category:Morphisms]]
[[Category:0 (number)]]