In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagramF : 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.
It follows from the existence theorem for limits 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:
A small categoryC is complete if and only if it is cocomplete.[1] A small complete category is necessarily thin.
A 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.
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.
Referencesedit
^Abstract and Concrete Categories, Jiří Adámek, Horst Herrlich, and George E. Strecker, theorem 12.7, page 213
complete, category, mathematics, complete, category, category, which, small, limits, exist, that, category, complete, every, diagram, where, small, limit, dually, cocomplete, category, which, small, colimits, exist, bicomplete, category, category, which, both,. In mathematics a complete category is a category in which all small limits exist That is a category C is complete if every diagram 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 Contents 1 Theorems 2 Examples and nonexamples 3 References 4 Further readingTheorems editIt follows from the existence theorem for limits 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 D 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 1 A small complete category is necessarily thin A 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 editThis section does not cite any sources Please help improve this section by adding citations to reliable sources Unsourced material may be challenged and removed August 2012 Learn how and when to remove this message 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 sSet the category of simplicial sets 2 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 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 numbers 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 References edit Abstract and Concrete Categories Jiri Adamek Horst Herrlich and George E Strecker theorem 12 7 page 213 Riehl Emily 2014 Categorical Homotopy Theory New York Cambridge University Press p 32 ISBN 9781139960083 OCLC 881162803 Further reading editAdamek Jiri Horst Herrlich George E Strecker 1990 Abstract and Concrete Categories PDF John Wiley amp Sons ISBN 0 471 60922 6 Mac Lane Saunders 1998 Categories for the Working Mathematician Graduate Texts in Mathematics 5 2nd ed ed Springer ISBN 0 387 98403 8 Retrieved from https en wikipedia org w index php title Complete category amp oldid 948255993, wikipedia, wiki, book, books, library,
