mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a complete category is a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

in which all small limits exist. That is, a category ''C'' is complete if every

diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age o ...

''F'' : ''J'' → ''C'' (where ''J'' is

small Small means of insignificant size Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to three geometrical measures: length, area, or ...

) has a limit in ''C''. Dually, 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 f ...

) 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 In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

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)

coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...

s, 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 In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

, *''C'' has pullbacks and a terminal object. The dual statements are also equivalent. A

small category In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...

''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 (mathematics), category whose Category (mathematics)#Small and large categories, homsets each contain at most one morphism. As such, a posetal catego ...

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 In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...

**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 con ...

**Grp, the

category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories The ...

**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 o ...

**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 ...

over a

commutative ring In mathematics, a commutative ring is a Ring (mathematics), 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 prope ...

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

finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. Th ...

s **The category of finite abelian groups **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 d ...

vector spaces *Any ( pre)

abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category o ...

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 In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...

, 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 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 leas ...

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