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A Gelfand duality for continuous lattices

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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.

http://www.tac.mta.ca/tac/volumes/41/1/41-01.pdf

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