Complete category

Theorems

• 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.

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
• *sSet, the category of simplicial sets
• 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 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 if and only if it is a complete lattice.
• The partially ordered class of all ordinal numbers is cocomplete but not complete.
• 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.