Geodesic
A Gelfand duality for continuous lattices
Theory and applications of categories, Tome 41 (2024), pp. 1-20

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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.

Publié le :
Classification : 06B35, 06D10, 18F70, 18A35
Keywords: continuous lattice, completely distributive lattice, duality, free cocompletion 
@article{TAC_2024_41_a0,
     author = {Ruiyuan Chen},
     title = {A {Gelfand} duality for continuous lattices},
     journal = {Theory and applications of categories},
     pages = {1--20},
     publisher = {mathdoc},
     volume = {41},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2024_41_a0/}
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Ruiyuan Chen. A Gelfand duality for continuous lattices. Theory and applications of categories, Tome 41 (2024), pp. 1-20. http://geodesic.mathdoc.fr/item/TAC_2024_41_a0/