Complete category

# Complete category

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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a complete category is a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

in which all small limit
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....

s exist. That is, a category C is complete if every diagram
Diagram (category theory)
In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function...

F : JC where J is small 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) is too strong to be practically relevant. Any category with this property is necessarily a thin category
Preordered 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 use the language of relations on a set.A preordered class is a...

: 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
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

it has equalizers (of all pairs of morphisms) and all (small) product
Product (category theory)
In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...

s. Since equalizers may be constructed from pullback
Pullback (category theory)
In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan X \rightarrow Z \leftarrow Y...

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
Coequalizer
In category theory, a coequalizer is a generalization of a quotient by an equivalence relation to objects in an arbitrary category...

s and all (small) coproduct
Coproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...

s, or, equivalently, pushout
Pushout (category theory)
In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain: it is the colimit of the span X \leftarrow Z \rightarrow Y.The pushout is the...

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
Pullback (category theory)
In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan X \rightarrow Z \leftarrow Y...

s 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
Posetal category
In mathematics, a posetal category is a category whose homsets each contain at most one morphism. As such a posetal category amounts to a preordered set...

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 counterexamples

• The following categories are bicomplete:
• Set, the category of sets
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...

• Top, the category of topological spaces
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 continuous...

• Grp, the category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...

• Ab, the category of abelian groups
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....

• Ring, the category of rings
Category of rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings and whose morphisms are ring homomorphisms...

• K-Vect, the category of vector spaces
Category of vector spaces
In mathematics, especially category theory, the category K-Vect has all vector spaces over a fixed field K as objects and K-linear transformations as morphisms...

over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

K
• R-Mod, the category of modules over a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

R
• CmptH, the category of all compact Hausdorff spaces
• Cat, the category of all small categories
• The following categories are finitely complete and finitely cocomplete but neither complete nor cocomplete:
• The category of finite sets
• The category of finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

s
• The category of finite-dimensional vector spaces
• Any (pre
Pre-Abelian category
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.Spelled out in more detail, this means that a category C is pre-abelian if:...

)abelian category
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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

is finitely complete and finitely cocomplete.
• The category of complete lattices is complete but not cocomplete.
• The category of metric spaces
Category of metric spaces
In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map...

, 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
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...

.
• The partially ordered class of all ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...

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
Trivial group
In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...

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