Duality for powerset coalgebras. (English) Zbl 07471716
Summary: 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 \(\mathcal{H}\) on CABA such that the category Alg\((\mathcal{H})\) of algebras for \(\mathcal{H}\) is dually equivalent to the category Coalg(\(\mathcal{P}\)) of coalgebras for the powerset endofunctor \(\mathcal{P}\) on Set. As a consequence, we derive Thomason duality from Tarski duality, thus paralleling how Jónsson-Tarski duality is derived from Stone duality.
MSC:
| 18B35 | Preorders, orders, domains and lattices (viewed as categories) |
| 18F70 | Frames and locales, pointfree topology, Stone duality |
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