Abstract
Let CABA be the category of complete and atomic boolean algebras and complete boolean homomorphisms, and let CSL be the category of complete meet-semilattices and complete meet-homomorphisms. We show that the forgetful functor from CABA to CSL has a left adjoint. This allows us to describe an endofunctor H on CABA such that the category Alg(H) of algebras for H is dually equivalent to the category Coalg(P) of coalgebras for the powerset endofunctor P on Set. As a consequence, we derive Thomason duality from Tarski duality, thus paralleling how J\'onsson-Tarski duality is derived from Stone duality.
Highlights
It is a classic result in modal logic, known as Jonsson-Tarski duality, that the category MA of modal algebras is dually equivalent to the category DFr of descriptive frames
Let CABA be the category of complete atomic boolean algebras and complete boolean homomorphisms, and let CSL be the category of complete meet-semilattices and complete meet-homomorphisms
We show that the forgetful functor from CABA to CSL has a left adjoint
Summary
It is a classic result in modal logic, known as Jonsson-Tarski duality, that the category MA of modal algebras is dually equivalent to the category DFr of descriptive frames. Jonsson-Tarski duality is a generalization of the celebrated Stone duality between the category BA of boolean algebras and the category Stone of Stone spaces It was observed by Abramsky [Abr88] and Kupke, Kurz, and Venema [KKV04] that Jonsson-Tarski duality can be proved by lifting Stone duality using algebra/coalgebra methods. This can be done by utilizing the classic Vietoris construction (see, e.g., [Joh[82], Ch. III.4]). He worked with the subcategory of modal algebras consisting of closure algebras of McKinsey and Tarski [MT44]
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