Kleisli dualities and Vietoris coalgebras - Nora, Pedro Miguel Teixeira Olhero Pessoa (2020)
In this thesis we aim for a systematic way of extending Stone-Halmos duality theorems to categories including all compact Hausdorff spaces. To achieve this goal, we combine duality theory and quantale-enriched category theory. Our main idea is that, when passing from the two-element discrete space to a cogenerator of the category of compact Hausdorff spaces, all other involved structures should be substituted by corresponding enriched versions. Accordingly, we work with the unit interval [0, 1] and present duality theory for ordered compact spaces and (suitably defined) finitely cocomplete categories enriched in [0, 1]. In the second part, we study limits in categories of coalgebras whose underlying functor is a Vietoris polynomial one — intuitively, the topological analogue of a Kripke polynomial functor.