Generalised powerlocales via relation lifting

This paper introduces an endofunctor VT on the category of frames that is parametrised by an endofunctor T on the category Set that satisfies certain constraints. This generalises 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 VT out of a frame 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 zero-dimensionality and compactness.

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