Categorically-algebraic frameworks for Priestley duality.

*(English)*Zbl 1248.08003
Chajda, I. (ed.) et al., Proceedings of the 79th workshop on general algebra “79. Arbeitstagung Allgemeine Algebra”, 25th conference of young algebraists, Palacký University Olomouc, Olomouc, Czech Republic, February 12–14, 2010. Klagenfurt: Verlag Johannes Heyn (ISBN 978-3-7084-0407-3/pbk). Contributions to General Algebra 19, 187-208 (2010).

In this paper the author presents a generalisation of the theory of natural dualities [D. M. Clark and B. A. Davey, Natural dualities for the working algebraist. Cambridge: Cambridge University Press (1998; Zbl 0910.08001)] in the framework of categorically algebraic topology [S. A. Solovyov, Acta Univ. M. Belii, Ser. Math. 17, 57–100 (2010; Zbl 1223.08004)].

After introducing the basic concepts of categorically algebraic topology, the author presents conditions that are sufficient to obtain an adjunction between the opposite category of a variety of algebras and topological spaces with relational structure. The notions of spatiality and sobriety are introduced to describe the algebras and topological spaces, respectively, for which the previous adjunction restricts to an equivalence. Finally, the author observes how the classical Stone dualities for Boolean algebras and bounded distributive lattices, and Priestley duality for bounded distributive lattices fit into this framework.

For the entire collection see [Zbl 1201.08001].

After introducing the basic concepts of categorically algebraic topology, the author presents conditions that are sufficient to obtain an adjunction between the opposite category of a variety of algebras and topological spaces with relational structure. The notions of spatiality and sobriety are introduced to describe the algebras and topological spaces, respectively, for which the previous adjunction restricts to an equivalence. Finally, the author observes how the classical Stone dualities for Boolean algebras and bounded distributive lattices, and Priestley duality for bounded distributive lattices fit into this framework.

For the entire collection see [Zbl 1201.08001].

Reviewer: Leonardo Manuel Cabrer (Bern)