Exponentiable monomorphisms in categories of domains

We give a characterization of exponentiable monomorphisms in the categories ω-Cpo of ωω-complete posets, Dcpo of directed complete posets and ContD of continuous directed complete posets as those monotone maps ff that are convex and that lift an element (and then a queue) of any directed set (ωω-chain in the case of ω-Cpo) whose supremum is in the image of ff (Theorem 1.9). Using this characterization, we obtain that a monomorphism f:X→Bf:X→B in Dcpo (ω-Cpo, ContD) exponentiable in Top w.r.t. the Scott topology is exponentiable also in Dcpo (ω-Cpo, ContD). We prove that the converse is true in the category ContD, but neither in Dcpo, nor in ω-Cpo.