Complete Category - Examples and Counterexamples

Examples and Counterexamples

• 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
• 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.