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. 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 JOnsson-Tarski duality was derived from Stone duality in [Abr88, KKV04].
Duality for powerset coalgebras / G. Bezhanishvili, L. Carai, P. Morandi. - In: LOGICAL METHODS IN COMPUTER SCIENCE. - ISSN 1860-5974. - 18:1(2022 Feb 03), pp. 1-17. [10.46298/lmcs-18(1:27)2022]
Duality for powerset coalgebras
L. Carai;
2022
Abstract
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. 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 JOnsson-Tarski duality was derived from Stone duality in [Abr88, KKV04].
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