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ESAKIA SPACES VIA IDEMPOTENT SPLIT COMPLETION
DIRK HOFMANN AND PEDRO NORA
Dedicated to Manuela Sobral
Abstract. Under Stone/Priestley duality for distributive lattices, Esakia spaces correspond to Heyting algebras which leads to the well- known dual equivalence between the category of Esakia spaces and mor- phisms on one side and the category of Heyting algebras and Heyting morphisms on the other. Based on the technique of idempotent split com- pletion, we give a simple proof of a more general result involving certain relations rather than functions as morphisms. We also extend the notion of Esakia space to all stably compact spaces and show that these spaces define the idempotent split completion of compact Hausdor↵ spaces. Fi- nally, we exhibit connections with split algebras for related monads.
Introduction
These notes evolve around the observation that Esakia duality for Heyting al- gebras arises more naturally when considering the larger category SpecDist with objects spectral spaces and with morphisms spectral distributors. In fact, as we observed already in [25], in this category Esakia spaces define the idempotent split completion of Stone spaces. Furthermore, it is well-known that SpecDist is dually equivalent to the category DLat?,_ of distributive lattices and maps preserving finite suprema and that, under this equivalence, Stone spaces corre- spond to Boolean algebras. This tells us that the category of Esakia spaces and spectral distributors is dually equivalent to the idempotent split completion of the category Boole?,_ of Boolean algebras and maps preserving finite suprema.
Received: 1 August 2014 / Accepted: 8 January 2015.
2010 Mathematics Subject Classification. 03G05, 03G10, 18A40, 18C15, 18C20, 54H10. Key words and phrases. Boolean algebra, distributive lattice, Heyting algebra, dual equiv-
alence, Stone space, spectral space, Esakia space, Vietoris functor, idempotent split comple- tion, split algebra.
Partial financial assistance by Portuguese funds through CIDMA (Center for Research and Development in Mathematics and Applications), and the Portuguese Foundation for Science and Technology (“FCT – Funda¸c˜ao para a Ciˆencia e a Tecnologia”), within the project PEst-OE/MAT/UI4106/2014, and by the project NASONI under the contract FCOMP-01- 0124-FEDER-028923 is gratefully acknowledged.
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