2020/10/30 by Guram Bezhanishvili, Luca Carai, Bezhanishvili, G. +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2010.16352
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 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 \citeAbr88,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 \citeBCM20a.