The Vietoris functor and modal operators on rings of continuous functions

TL;DR

This paper establishes a duality between certain algebraic and coalgebraic structures related to the Vietoris functor, extending Gelfand duality and generalizing previous results on Stone spaces.

Contribution

It introduces new endofunctors on categories of bounded archimedean $ ext{l}$-algebras and their complete subcategories, linking algebraic and coalgebraic frameworks via duality.

Findings

Dual adjunction between algebra and coalgebra categories for the Vietoris functor.

Extension of Gelfand duality to a broader algebraic-coalgebraic setting.

Generalization of existing results on Stone spaces to more general compact Hausdorff spaces.

Abstract

We introduce an endofunctor $H$ on the category $bal$ of bounded archimedean $\ell$-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 also introduce an endofunctor $Hu$ on the reflective subcategory of $bal$ consisting of uniformly complete objects of $bal$ and show that Gelfand duality lifts to a dual equivalence between $Alg(Hu)$ and $Coalg(V)$. On the one hand, this generalizes a result of \cite{Abr88,KKV04} for the category of coalgebras of the Vietoris endofunctor on the category of Stone spaces. On the other hand, it yields an alternate proof of a recent result of \cite{BCM20a}.