# Complete category

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{{Short description|Category in which all small limits exist}}In [mathematics](/source/mathematics), a '''complete category''' is a [category](/source/category_(mathematics)) in which all small [limit](/source/limit_(category_theory))s exist. That is, a category ''C'' is complete if every [diagram](/source/diagram_(category_theory)) ''F'' : ''J'' → ''C'' (where ''J'' is [small](/source/small_category)) has a limit in ''C''. [Dually](/source/Duality_(category_theory)), a '''cocomplete category''' is one in which all small [colimit](/source/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](/source/proper_class)) is too strong to be practically relevant. Any category with this property is necessarily a [thin category](/source/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](/source/existence_theorem_for_limits) that a category is complete [if and only if](/source/if_and_only_if) it has [equalizers](/source/Equaliser_(mathematics)) (of all pairs of morphisms) and all (small) [product](/source/product_(category_theory))s. Since equalizers may be constructed from [pullback](/source/pullback_(category_theory))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](/source/coequalizer)s and all (small) [coproduct](/source/coproduct)s, or, equivalently, [pushout](/source/pushout_(category_theory))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](/source/terminal_object),
*''C'' has [pullback](/source/pullback_(category_theory))s and a terminal object.
The dual statements are also equivalent.

A [small category](/source/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](/source/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](/source/category_of_sets)<ref>{{Cite book |last=Mac Lane |first=Saunders |title=Categories for the working mathematician |date=2000 |publisher=Springer |isbn=978-0-387-98403-2 |edition=2. ed., [Nachdr.] |series=Graduate texts in mathematics |location=New York |pages=110-112}}</ref>
**'''Top''', the [category of topological spaces](/source/category_of_topological_spaces)
**'''Grp''', the [category of groups](/source/category_of_groups)
**'''Ab''', the [category of abelian groups](/source/category_of_abelian_groups)
**'''Ring''', the [category of rings](/source/category_of_rings)
**'''''K''-Vect''', the [category of vector spaces](/source/category_of_vector_spaces) over a [field](/source/field_(mathematics)) ''K''
**'''''R''-Mod''', the [category of modules](/source/category_of_modules) over a [commutative ring](/source/commutative_ring) ''R''
**'''CmptH''', the category of all [compact Hausdorff space](/source/compact_Hausdorff_space)s
**'''Cat''', the [category of all small categories](/source/category_of_all_small_categories)
**'''Whl''', the category of [wheels](/source/wheel_theory)
**'''sSet''', the category of [simplicial sets](/source/Simplicial_set)<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](/source/finite_set)s
**The category of [finite abelian group](/source/finite_abelian_group)s
**The category of [finite-dimensional](/source/finite-dimensional) vector spaces
*Any ([pre](/source/pre-abelian_category))[abelian category](/source/abelian_category) is finitely complete and finitely cocomplete.
*The category of [complete lattices](/source/complete_lattices) is complete but not cocomplete.
*The [category of metric spaces](/source/category_of_metric_spaces), '''Met''', is finitely complete but has neither binary coproducts nor infinite products.
*The [category of fields](/source/category_of_fields), '''Field''', is neither finitely complete nor finitely cocomplete.
*A [poset](/source/poset), considered as a small category, is complete (and cocomplete) if and only if it is a [complete lattice](/source/complete_lattice).
*The [partially ordered class](/source/partially_ordered_class) of all [ordinal number](/source/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](/source/trivial_group). 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 | publisher = Springer | isbn = 0-387-98403-8}}

{{Category theory}}

Category:Limits (category theory)

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