Duality for powerset coalgebras

TL;DR

This paper establishes a duality between powerset coalgebras and certain algebraic structures by leveraging category theory, connecting well-known dualities like Thomason and Tarski dualities through a novel functorial approach.

Contribution

It introduces a new duality framework for powerset coalgebras using categorical adjunctions and constructs an endofunctor that links coalgebras to algebraic structures, extending classical dualities.

Findings

Derived Thomason duality from Tarski duality.

Established a dual equivalence between coalgebras for powerset functor and algebraic structures.

Connected categorical adjunctions to duality theories in logic and computer science.

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.