Presenting locale pullback via directed complete posets

This paper shows how to describe the pullbacks of directed complete posets (dcpos) along geometric morphisms. This extends Joyal and Tierney’s original results on the pullbacks of suplattices. It is then shown how to treat every frame as a dcpo and so locale pullback is described in this way. Applications are given describing triquotient assignments in terms of internal dcpo maps, leading to pullback stability results for triquotient maps. The main application here shows how dcpo maps between frames can be described in terms of certain external natural transformations.