Definition. Colimits in the category \mathbf {Cpo}^\mathbf {R} [dt-005F]

A cpo A is the colimit of an \omega -chain of cpos A_i if is both an upper bound, i.e. there exists retraction pairs (f_i,g_i) : A_i \xtofrom []{} A, and for any other upper bound B, for which there will exist retraction pairs (m_i,n_i) : A_i \xtofrom []{} B, there exists a unique retraction pair A \xtofrom [k']{k} B such that (m_i,n_i) = (k,k') \circ (f_i,g_i) for all i \in \mathbb {N}, where \circ is composition of retraction pairs. Viewed diagramatically, it is the same as Definition [dt-004S] where continuous functions are replaced by retraction pairs.