The duality between the category Sets and the category CABool of complete atomic boolean algebras is an example of a general Stone-type duality. At the same time, Sets is a topos. Are there other known examples of Stone-type dualities involving toposes ? On the algebraic side of such a duality, I guess the "completeness" property is essential since its corresponds (on the topological side) to the property of being extremally-disconnected, something we would expect for the objects of a topos, i.e. behaving like "sets".
2026-03-28 03:00:30.1774666830
Toposes and Stone-type dualities
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An atom in an internally complete Heyting algebra in an elementary topos is an element (in the Kripke-Joyal semantics) for which the downward segment is isomorphic to the subobject classifier. The details on this and how to establish the characterization of P(X) as being a complete atomic Heyting algebra is (theorem 6) in the second chapter of “Lattice Theoretic and Logical Aspects of Elementary Topoi”, Christian Juul Mikkelsen, March 1976. Various Publications Series No. 25, Matematisk Institut, Aarhus Universitet.