Page 212 - Textos de Matemática Vol. 46

202 D. HOFMANN AND P. NORA
However, the main ingredients to identify this category as the full subcate- gory of DLat?,_ defined by all co-Heyting algebras were already provided by McKinsey and Tarski in 1946.
In order to present this argumentation, we carefully recall in Section 1 vari- ous aspects of spectral spaces and Stone spaces which are the spaces occurring on the topological side of the famous duality theorems of Stone for distributive lattices and Boolean algebras. Special emphasis is given to the larger class of stably compact spaces and their relationship with ordered compact Hausdor↵ spaces. We also briefly present the extension of Stone’s result to categories of continuous relations, an idea attributed to Halmos. These continuous rela- tions and, more generally, spectral distributors, are best understood using the Vietoris monad which is the topic of Section 2. In particular, we identify ad- junctions in the Kleisli category of the lower Vietors monad on the category of stably compact spaces and spectral maps. Based on this description, we present Esakia spaces as the idempotent split completion of Stone spaces, and in Section 3 we use this fact to deduce Esakia dualities using the technique of idempotent split completion. Moreover, we extend the notion of Esakia space to all stably compact spaces and show in Section 4 that the category of (general- ised) Esakia spaces and spectral distributors is the idempotent split completion of the category of compact Hausdor↵ spaces and continuous relations. Finally, the idempotent split completion of Kleisli categories is ultimately linked to the notion of split algebra for a monad, which is the topic of Section 5.
1. Stone and Halmos dualities
The aim of this section is to collect some well-known facts about duality theory for Boolean algebras and distributive lattices and about the topological spaces which occur as their duals. As much as possible we try to indicate original sources.
Naturally, we begin with the classical Stone dualities stating (in modern lan- guage) that the category Stone of Stone spaces (= zero-dimensional compact Hausdor↵ topological spaces) and continuous maps is dually equivalent to the category Boole of Boolean algebras and homomorphisms (see [42])
Stoneop ' Boole;
and that the category Spec of spectral spaces and spectral maps is dually equivalent to the category DLat of distributive lattices1 and homomorphisms
(see [43])
Specop ' DLat.
1We note that for us a lattice is an ordered set with finite suprema and finite infima, hence every lattice has a largest element > and a smallest element ?.