In
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) 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: 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 suc ...
that a category is complete
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
it has
equalizers (of all pairs of morphisms) and all (small)
products. Since equalizers may be constructed from
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of 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
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
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of 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. 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 fo ...
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
**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 ''K''
**''R''-Mod, the
category of modules over a
commutative ring ''R''
**CmptH, the category of all
compact Hausdorff spaces
**Cat, the
category of all small categories
**Whl, the category of
wheels
A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the key components of the wheel and axle which is one of the six simple machines. Wheels, in conjunction with axles, allow heavy objects to b ...
**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 dimensi ...
*The following categories are finitely complete and finitely cocomplete but neither complete nor cocomplete:
**The category of
finite sets
**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 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
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
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
. A nontrivial group has pullbacks and pushouts, but not products, coproducts, equalizers, coequalizers, terminal objects, or initial objects.
References
Further reading
*
*{{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
Limits (category theory)