Exponentiable morphisms of domains

Abstract

Given a mapfin the category ω-Cpoof ω-complete posets, exponentiability offin ω-Cpoeasily implies exponentiability offin the categoryPosof posets, while the converse is not true. We investigate the extra conditions needed onfexponentiable inPosto be exponentiable in ω-Cpoby showing the existence of partial products of the two-point ordered setS={0<1} (Theorem 2.8). Using this characterisation and the embedding through the Scott topology of ω-Cpoin the categoryTopof topological spaces, we compare exponentiability in each setting and find that a morphism in ω-Cpothat is exponentiable in bothTopandPosis exponentiable in ω-Cpoalso. Furthermore, we show that the exponentiability inTopandPosare independent of each other.