Generalized powerlocales via relation lifting

TL;DR

This paper generalizes the Vietoris powerlocale construction by introducing a functor-based approach that connects coalgebraic logic with frame theory, extending properties like regularity and zero-dimensionality.

Contribution

It develops a unified framework for powerlocales via relation lifting, generalizing Vietoris construction and linking it to coalgebraic modalities.

Findings

Generalizes Vietoris powerlocale using relation lifting.

Extends natural transformations between set functors to powerlocale functors.

Preserves properties like regularity, zero-dimensionality, and compactness.

Abstract

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 $\VT \bbL$ out of a frame $\bbL$ 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.