A Gelfand duality for continuous lattices

Ruiyuan Chen

We prove that the category of continuous lattices and meet- and directed join-preserving maps is dually equivalent, via the hom functor to [0,1], to the category of complete Archimedean meet-semilattices equipped with a finite meet-preserving action of the monoid of continuous monotone maps of [0,1] fixing 1. We also prove an analogous duality for completely distributive lattices. Moreover, we prove that these are essentially the only well-behaved "sound classes of joins Φ, dual to a class of meets" for which "Φ-continuous lattice'' and "Φ-algebraic lattice" are different notions, thus for which a 2-valued duality does not suffice.

Keywords: continuous lattice, completely distributive lattice, duality, free cocompletion

2020 MSC: 06B35, 06D10, 18F70, 18A35

Theory and Applications of Categories, Vol. 41, 2024, No. 1, pp 1-20.

Published 2024-01-16.


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