Enriched Stone-type dualities

• Autor(a) principal: Hofmann, Dirk
• Data de Publicação: 2018
• Outros Autores: Nora, Pedro
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10773/28975
• topic: Dual equivalence; Quantale-enriched category; Kleisli construction; Vietoris functor; Ordered compact Hausdorff space; Metric compact Hausdorff space

A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the two-element set with an appropriate structure. A prime example of such a situation is Stone's duality theorem for Boolean algebras and Boolean spaces, the latter being precisely those compact Hausdorff spaces which are cogenerated by the two-element discrete space. In this paper we aim for a systematic way of extending this duality theorem 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 and metric compact Hausdorff spaces and (suitably defined) finitely cocomplete categories enriched in [0, 1].