Editing Complete category

{{Short description|Category with all limits of (small) diagrams}}In [[mathematics]], a '''complete category''' is a [[category (mathematics)|category]] in which all small [[limit (category theory)|limit]]s exist. That is, a category ''C'' is complete if every [[diagram (category theory)|diagram]] ''F'' : ''J'' → ''C'' (where ''J'' is [[small category|small]]) has a limit in ''C''. [[Duality (category theory)|Dually]], a '''cocomplete category''' is one in which all small [[colimit]]s exist. A '''bicomplete category''' is a category which is both complete and cocomplete.

The existence of ''all'' limits (even when ''J'' is a [[proper class]]) is too strong to be practically relevant. Any category with this property is necessarily a [[thin category]]: for any two objects there can be at most one morphism from one object to the other.

A weaker form of completeness is that of finite completeness. A category is '''finitely complete''' if all finite limits exists (i.e. limits of diagrams indexed by a finite category ''J''). Dually, a category is '''finitely cocomplete''' if all finite colimits exist.

==Theorems==

It follows from the [[existence theorem for limits]] that a category is complete [[if and only if]] it has [[Equaliser (mathematics)|equalizers]] (of all pairs of morphisms) and all (small) [[product (category theory)|product]]s. Since equalizers may be constructed from [[pullback (category theory)|pullback]]s and binary products (consider the pullback of (''f'', ''g'') along the diagonal Δ), a category is complete if and only if it has pullbacks and products.

Dually, a category is cocomplete if and only if it has [[coequalizer]]s and all (small) [[coproduct]]s, or, equivalently, [[pushout (category theory)|pushout]]s and coproducts.

Finite completeness can be characterized in several ways. For a category ''C'', the following are all equivalent:
*''C'' is finitely complete,
*''C'' has equalizers and all finite products,
*''C'' has equalizers, binary products, and a [[terminal object]],
*''C'' has [[pullback (category theory)|pullback]]s and a terminal object.
The dual statements are also equivalent.

A [[small category]] ''C'' is complete if and only if it is cocomplete.<ref>Abstract and Concrete Categories, Jiří Adámek, Horst Herrlich, and George E. Strecker, theorem 12.7, page 213</ref> A small complete category is necessarily thin.

A [[posetal category]] vacuously has all equalizers and coequalizers, whence it is (finitely) complete if and only if it has all (finite) products, and dually for cocompleteness.  Without the finiteness restriction a posetal category with all products is automatically cocomplete, and dually, by a theorem about complete lattices.

==Examples and nonexamples==
{{unreferenced section|date=August 2012}}

*The following categories are bicomplete:
**'''Set''', the [[category of sets]]
**'''Top''', the [[category of topological spaces]]
**'''Grp''', the [[category of groups]]
**'''Ab''', the [[category of abelian groups]]
**'''Ring''', the [[category of rings]]
**'''''K''-Vect''', the [[category of vector spaces]] over a [[field (mathematics)|field]] ''K''
**'''''R''-Mod''', the [[category of modules]] over a [[commutative ring]] ''R''
**'''CmptH''', the category of all [[compact Hausdorff space]]s
**'''Cat''', the [[category of all small categories]]
**'''Whl''', the category of [[wheel theory|wheels]]
**'''sSet''', the category of [[Simplicial set|simplicial sets]]<ref>{{Cite book|title=Categorical Homotopy Theory.|last=Riehl|first=Emily|author-link=Emily Riehl|date=2014|publisher=Cambridge University Press|isbn=9781139960083|location=New York|pages=32|oclc=881162803}}</ref>
*The following categories are finitely complete and finitely cocomplete but neither complete nor cocomplete:
**The category of [[finite set]]s
**The category of [[finite abelian group]]s
**The category of [[finite-dimensional]] vector spaces
*Any ([[pre-abelian category|pre]])[[abelian category]] is finitely complete and finitely cocomplete.
*The category of [[complete lattices]] is complete but not cocomplete.
*The [[category of metric spaces]], '''Met''', is finitely complete but has neither binary coproducts nor infinite products.
*The [[category of fields]], '''Field''', is neither finitely complete nor finitely cocomplete.
*A [[poset]], considered as a small category, is complete (and cocomplete) if and only if it is a [[complete lattice]].
*The [[partially ordered class]] of all [[ordinal number]]s is cocomplete but not complete (since it has no terminal object).
*A group, considered as a category with a single object, is complete if and only if it is [[trivial group|trivial]]. A nontrivial group has pullbacks and pushouts, but not products, coproducts, equalizers, coequalizers, terminal objects, or initial objects.

== References ==

<references />

== Further reading ==

*{{cite book | last = Adámek | first = Jiří |author2=Horst Herrlich |author3=George E. Strecker  | year = 1990 | url = http://katmat.math.uni-bremen.de/acc/acc.pdf | title = Abstract and Concrete Categories | publisher = John Wiley & Sons | isbn = 0-471-60922-6}}
*{{cite book | first = Saunders | last = Mac Lane | authorlink = Saunders Mac Lane | year = 1998 | title = Categories for the Working Mathematician | title-link = Categories for the Working Mathematician | series = Graduate Texts in Mathematics '''5''' | edition = (2nd ed.) | publisher = Springer | isbn = 0-387-98403-8}}

[[Category:Limits (category theory)]]