Complete category

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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.

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.

[edit] Theorems

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, 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 is complete if and only if it is cocomplete. A small complete category is necessarily thin.

[edit] Examples and counterexamples

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

[edit] References

• Adámek, Jiří; Horst Herrlich, and George E. Strecker (1990). Abstract and Concrete Categories. John Wiley & Sons. ISBN 0-471-60922-6. 
• Mac Lane, Saunders (1998). Categories for the Working Mathematician, (2nd ed.), Graduate Texts in Mathematics 5, Springer. ISBN 0-387-98403-8.