mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a complete category is a category in which all small

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

s exist. That is, a category ''C'' is complete if every

diagram A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three- ...

''F'' : ''J'' → ''C'' (where ''J'' is

small Small may refer to: Science and technology * SMALL, an ALGOL-like programming language * Small (anatomy), the lumbar region of the back * ''Small'' (journal), a nano-science publication * <small>, an HTML element that defines smaller text ...

) has a limit in ''C''.

Dually Dually may refer to: *Dualla, County Tipperary, a village in Ireland *A pickup truck with dual wheels on the rear axle * DUALLy, s platform for architectural languages interoperability * Dual-processor See also * Dual (disambiguation) Dual or ...

, a cocomplete category is one in which all small colimits 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 Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...

) 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 In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions su ...

that a category is complete if and only if it has equalizers (of all pairs of morphisms) and all (small) products. Since equalizers may be constructed from pullbacks 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 coequalizers and all (small) coproducts, or, equivalently, pushouts 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 pullbacks and a terminal object. The dual statements are also equivalent. A small category ''C'' is complete if and only if it is cocomplete. A small complete category is necessarily thin. A

posetal category In mathematics, specifically category theory, a posetal category, or thin category, is a category whose homsets each contain at most one morphism. As such, a posetal category amounts to a preordered class (or a preordered set, if its objects for ...

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

*The following categories are bicomplete: **Set, the category of sets **Top, the

category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again contin ...

**Grp, the category of groups **Ab, the

category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab is ...

**Ring, the category of rings **''K''-Vect, the category of vector spaces over a field ''K'' **''R''-Mod, the

category of modules In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring o ...

over a

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

''R'' **CmptH, the category of all compact Hausdorff spaces **Cat, the category of all small categories **Whl, the category of wheels **sSet, the category of

simplicial sets In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

*The following categories are finitely complete and finitely cocomplete but neither complete nor cocomplete: **The category of finite sets **The category of

finite abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

s **The category of

finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...

vector spaces *Any (

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) 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 In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...

. *The

partially ordered class In mathematics, a preordered class is a class equipped with a preorder. Definition When dealing with a class ''C'', it is possible to define a class relation on ''C'' as a subclass of the power class ''C \times C'' . Then, it is convenient to u ...

of all

ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...

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. A nontrivial group has pullbacks and pushouts, but not products, coproducts, equalizers, coequalizers, terminal objects, or initial objects.

Theorems

Examples and nonexamples

References

Further reading