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.
Read more about this topic: Complete Category
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (18961966)