Duality for coalgebras for Vietoris and monadicity

TL;DR

This paper establishes monadicity results for coalgebras of Vietoris functors on compact spaces, extending duality theories and providing axiomatizations for related algebraic structures.

Contribution

It proves monadicity of the opposite category of Vietoris coalgebras over Set and extends duality results to stably compact spaces with axiomatizations.

Findings

Opposite of Vietoris coalgebras is monadic over Set

Extends duality to stably compact spaces

Provides axiomatizations for related algebraic varieties

Abstract

We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over Set. We deliver an analogous result for the upper, lower and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jonsson-Tarski duality for modal algebras beyond the 0-dimensional setting.