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arXiv · 2010.16352

The Vietoris functor and modal operators on rings of continuous functions

G. Bezhanishvili·L. Carai·P. Morandi
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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}.

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G. Bezhanishvili, L. Carai, P. Morandi. 2020-10-30. The Vietoris functor and modal operators on rings of continuous functions. https://arxiv.org/abs/2010.16352

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