The Vietoris functor and modal operators on rings of continuous functions

Abstract

We introduce an endofunctor H on the category baℓ of bounded archimedean ℓ-algebras and show that there is a dual adjunction between the category Alg(H) of algebras for H and the category Coalg(V) of coalgebras for the Vietoris endofunctor V on the category of compact Hausdorff spaces. We prove that Gelfand duality lifts to a dual equivalence between Coalg(V) and the full reflective subcategory Algu(H) of Alg(H). We provide an alternate view of Algu(H) by introducing an endofunctor Hu on the full reflective subcategory of baℓ consisting of uniformly complete objects of baℓ and showing that Alg(Hu) is isomorphic to Algu(H). On the one hand, these results generalize those of [1,19] for the category of coalgebras of the Vietoris endofunctor on the category of Stone spaces. On the other hand, they provide an alternate, more categorical proof of a recent result of [6].