nLab cpo

Definition

The abbreviation cpo stands for “complete partial order”,

(beware that it does not generally refer to a partial order that is complete in the sense that the latter word is used in category theory: A partial order that is complete as a category is instead generally called a complete lattice.)

Instead, a “cpo” generally refers to either a poset that has suprema for directed subsets (a.k.a. a dcpo) or for chains indexed by the ordinal $\omega$ (a.k.a. an $\omega$-cpo — a weaker condition). Note also that suprema are colimits in a poset rather than limits, so this is actually a cocompleteness condition rather than a completeness one.

The morphisms between cpos are usually taken to be the the Scott-continuous functions (continuous functions with respect to the Scott topology). Cpos and Scott-continuous maps form a category $CPO$.

Applications

Cpos are important in domain theory.

References

• Jiri Adamek, Horst Herrlich, and George Strecker, Abstract and concrete categories: the joy of cats. free online