Generalized powerlocales via relation lifting

This paper introduces an endofunctor VT on the category of frames, parametrized by an endofunctor T on the category Set that satisfies certain constraints. This generalizes Johnstone’s construction of the Vietoris powerlocale, in the sense that his construction is obtained by taking for T the finite covariant power set functor. Our construction of the T -powerlocale VTL out of a frame L is based on ideas from coalgebraic logic and makes explicit the connection between the Vietoris construction and Moss’s coalgebraic cover modality. We show how to extend certain natural transformations between set functors to natural transformations between T -powerlocale functors. Finally, we prove that the operation VT preserves some properties of frames, such as regularity, zero-dimensionality, and the combination of zerodimensionality and compactness.